add cond_kl

App.py

@@ -13,7 +13,8 @@ from DPMInteractive import fixed_point_init_change, fixed_point_apply_iterate
 from DPMInteractive import forward_plot_part, backward_plot_part, fit_plot_part, fixed_plot_part
 from RenderMarkdown import md_introduction_block, md_transform_block, md_likelihood_block, md_posterior_block
 from RenderMarkdown import md_forward_process_block, md_backward_process_block, md_fit_posterior_block
-from RenderMarkdown import md_posterior_transform_block, md_deconvolution_block, md_reference_block, md_about_block
+from RenderMarkdown import md_posterior_transform_block, md_deconvolution_block, md_cond_kl_block, md_proof_ctr_block
+from RenderMarkdown import md_reference_block, md_about_block
 from Misc import g_css, js_head, js_load
 
 
@@ -314,7 +315,7 @@ def run_app():
         md_introduction_block()
 
         md_transform_block()
-
+ 
         rets = transform_block()
         trans_param = rets
 
@@ -345,10 +346,16 @@ def run_app():
         
         md_deconvolution_block()
 
+        md_cond_kl_block()
+        
+        md_proof_ctr_block()
+        
         md_reference_block()
         
         md_about_block()
 
+        gr.Markdown("<div><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br></div>", visible=True)
+
         # running initiation consecutively because of the bug of multithreading rendering mathtext in matplotlib
         demo.load(trans_param["method"], trans_param["inputs"], trans_param["outputs"], show_progress="minimal").\
             then(cond_param["method"], cond_param["inputs"], cond_param["outputs"], show_progress="minimal"). \
@@ -387,7 +394,7 @@ def gtx():
         md_reference_block()
         
         md_about_block()
-    
+        
     demo.queue()
     demo.launch(allowed_paths=["/"])
     return

ExtraBlock.js

@@ -3,14 +3,14 @@
 
 async function write_markdown() {
     let names = ["introduction", "transform", "likelihood", "posterior", "forward_process", "backward_process",
-                 "fit_posterior", "posterior_transform", "deconvolution", "reference", "about"];
+                 "fit_posterior", "posterior_transform", "deconvolution", "cond_kl", "proof_ctr", "reference", "about"];
     // names = names.slice(-1)
     
     let data = await fetch("file/data.json").then(response => response.json());
     
     names.forEach((name, index) => {
         let elem_zh = document.getElementById("md_" + name + "_zh");
-        if (elem_zh != null) { data[name+"_zh"] = elem_zh.outerHTML; }
+        if (elem_zh != null) { data[name+"_zh"] = elem_zh.outerHTML; console.log(name);}
 
         const elem_en = document.getElementById("md_" + name + "_en");
         if (elem_en != null) { data[name+"_en"] = elem_en.outerHTML; }
@@ -25,7 +25,7 @@ async function write_markdown() {
 
 async function insert_markdown() {
     let names = ["introduction", "transform", "likelihood", "posterior", "forward_process", "backward_process", 
-                 "fit_posterior", "posterior_transform", "deconvolution", "reference", "about"];
+                 "fit_posterior", "posterior_transform", "deconvolution", "cond_kl", "proof_ctr", "reference", "about"];
 
     let data = await fetch("file/data.json").then(response => response.json());
     
@@ -54,7 +54,7 @@ async function insert_markdown() {
 
 function control_language() {
     const names = ["introduction", "transform", "likelihood", "posterior", "forward_process",
-        "backward_process", "fit_posterior", "posterior_transform", "deconvolution", "reference", "about"];
+        "backward_process", "fit_posterior", "posterior_transform", "deconvolution", "cond_kl", "proof_ctr", "reference", "about"];
 
     var is_zh = document.getElementById("switch_language").checked;
     for (let i = 0; i < names.length; i++) {
@@ -142,6 +142,7 @@ function katex_render(name) {
     if (elem == null) { return; }
     var text = elem.innerText.replaceAll("{underline}", "_");
     text = "\\begin{align}\n" + text + "\n\\end{align}";
+    
     katex.render(text, elem, {displayMode: true});
 }
 
@@ -150,7 +151,13 @@ function insert_special_formula() {
     katex_render("zh_fit_0");
     katex_render("zh_fit_1");
     katex_render("zh_fit_2");
+    katex_render("zh_cond_kl_1");
+    katex_render("zh_cond_kl_2");
+    katex_render("zh_cond_kl_3");
     katex_render("en_fit_0");
     katex_render("en_fit_1");
     katex_render("en_fit_2");
+    katex_render("en_cond_kl_1");
+    katex_render("en_cond_kl_2");
+    katex_render("en_cond_kl_3");
 }

README.md

@@ -19,7 +19,7 @@ license: apache-2.0
 
 This is a Web App which provides a special introduction about the Diffusion Probability Model with interactive demos.
 
-How to run
+### How to run
 <ol> 
 	<li> Open <a href="https://huggingface.co/spaces/blairzheng/DPMInteractive">HuggingFace space</a> in browser directly. </li>
 	<li> Running locally, recommended.

RenderMarkdown.py

@@ -3,11 +3,13 @@ import gradio as gr
 
 from RenderMarkdownZh import md_introduction_zh, md_transform_zh, md_likelihood_zh, md_posterior_zh
 from RenderMarkdownZh import md_forward_process_zh, md_backward_process_zh, md_fit_posterior_zh
-from RenderMarkdownZh import md_posterior_transform_zh, md_deconvolution_zh, md_reference_zh, md_about_zh
+from RenderMarkdownZh import md_posterior_transform_zh, md_deconvolution_zh, md_cond_kl_zh, md_proof_ctr_zh
+from RenderMarkdownZh import md_reference_zh, md_about_zh
 
 from RenderMarkdownEn import md_introduction_en, md_transform_en, md_likelihood_en, md_posterior_en
 from RenderMarkdownEn import md_forward_process_en, md_backward_process_en, md_fit_posterior_en
-from RenderMarkdownEn import md_posterior_transform_en, md_deconvolution_en, md_reference_en, md_about_en
+from RenderMarkdownEn import md_posterior_transform_en, md_deconvolution_en, md_cond_kl_en, md_proof_ctr_en
+from RenderMarkdownEn import md_reference_en, md_about_en
 
 
 def md_introduction_block(md_type="offline"):
@@ -122,6 +124,32 @@ def md_deconvolution_block(md_type="offline"):
     return
 
 
+def md_cond_kl_block(md_type="offline"):
+    if md_type == "offline":
+        title = "Appendix A Conditional KL Divergence"
+        gr.Accordion(label=title, elem_classes="first_md", elem_id="cond_kl")
+    elif md_type == "zh":
+        md_cond_kl_zh()
+    elif md_type == "en":
+        md_cond_kl_en()
+    else:
+        raise NotImplementedError
+    return
+
+
+def md_proof_ctr_block(md_type="offline"):
+    if md_type == "offline":
+        title = "Appendix B Proof of Contraction"
+        gr.Accordion(label=title, elem_classes="first_md", elem_id="proof_ctr")
+    elif md_type == "zh":
+        md_proof_ctr_zh()
+    elif md_type == "en":
+        md_proof_ctr_en()
+    else:
+        raise NotImplementedError
+    return
+
+
 def md_reference_block(md_type="offline"):
     if md_type == "offline":
         gr.Accordion(label="Reference", elem_classes="first_md", elem_id="reference")

RenderMarkdownEn.py

@@ -12,7 +12,7 @@ def md_introduction_en():
         
         gr.Markdown(
             r"""
-            The Diffusion Probability Model[\\[1\\]](#dpm)[\\[2\\]](#ddpm) is currently the main method used in image and video generation, but due to its abstruse theory, many engineers are unable to understand it well. This article will provide a very easy-to-understand method to help readers grasp the principles of the Diffusion Model. Specifically, it will illustrate the Diffusion Model using examples of one-dimensional random variables in an interactive way, explaining several interesting properties of the Diffusion Model in an intuitive manner.
+            The Diffusion Probability Model[\[1\]](#dpm)[\[2\]](#ddpm) is currently the main method used in image and video generation, but due to its abstruse theory, many engineers are unable to understand it well. This article will provide a very easy-to-understand method to help readers grasp the principles of the Diffusion Model. Specifically, it will illustrate the Diffusion Model using examples of one-dimensional random variables in an interactive way, explaining several interesting properties of the Diffusion Model in an intuitive manner.
             
             The diffusion model is a probabilistic model. Probabilistic models mainly offer two functions: calculating the probability of a given sample appearing; and generating new samples. The diffusion model focuses on the latter aspect, facilitating the production of new samples, thus realizing the task of **generation**.
             
@@ -43,11 +43,11 @@ def md_transform_en():
             
             The first sub-transformation performs a linear transformation ($\sqrt{\alpha}X$) on the random variable $X$. According to the conclusion of the literature[\[3\]](#linear_transform), the linear transformation makes the probability distribution of $X$ **narrower and taller**, and the extent of **narrowing and heightening** is directly proportional to the value of $\alpha$. 
             
-            This can be specifically seen in Demo 1, where the first figure depicts a randomly generated one-dimensional probability distribution, and the second figure represents the probability distribution after the linear transformation. It can be observed that the curve of the third figure has become **narrower and taller** compared to the first image. Readers can experiment with different $\alpha$ to gain a more intuitive understanding.
+            This can be specifically seen in <a href="#demo_1">Demo 1</a>, where the first figure depicts a randomly generated one-dimensional probability distribution, and the second figure represents the probability distribution after the linear transformation. It can be observed that the curve of the third figure has become **narrower and taller** compared to the first image. Readers can experiment with different $\alpha$ to gain a more intuitive understanding.
             
             The second sub-transformation is **adding independent random noise**($\sqrt{1-\alpha}\epsilon$). According to the conclusion of the literature[\[4\]](#sum_conv), **adding independent random variables** is equivalent to performing convolution on the two probability distributions. Since the probability distribution of random noise is Gaussian, it is equivalent to performing a **Gaussian Blur** operation. After blurring, the original probability distribution will become smoother and more similar to the standard normal distribution. The degree of blurring is directly proportional to the noise level ($\sqrt{1-\alpha}$).
             
-            For specifics, one can see Demo 1, where the first figure is a randomly generated one-dimensional probability distribution, and the third  figure is the result after the transformation. It can be seen that the transformed probability distribution curve is smoother and there are fewer corners. The readers can test different $\alpha$ values to feel how the noise level affect the shape of the probability distribution. The last figure is the result after applying all two sub-transformations.
+            For specifics, one can see <a href="#demo_1">Demo 1</a>, where the first figure is a randomly generated one-dimensional probability distribution, and the third  figure is the result after the transformation. It can be seen that the transformed probability distribution curve is smoother and there are fewer corners. The readers can test different $\alpha$ values to feel how the noise level affect the shape of the probability distribution. The last figure is the result after applying all two sub-transformations.
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_transform_en")
     return
 
@@ -63,7 +63,7 @@ def md_likelihood_en():
             \begin{align}
                 q(z|x) &= \mathcal{N}(\sqrt{\alpha}x,\ 1-\alpha)    \tag{2.1}
             \end{align}
-            It can be understood by concrete examples in Demo 2. The third figure depict the shape of $q(z|x)$. From the figure, a uniform slanting line can be observed. This implies that the mean of $q(z|x)$ is linearly related to x, and the variance is fixed. The magnitude of $\alpha$ will determine the width and incline of the slanting line.
+            It can be understood by concrete examples in <a href="#demo_2">Demo 2</a>. The third figure depict the shape of $q(z|x)$. From the figure, a uniform slanting line can be observed. This implies that the mean of $q(z|x)$ is linearly related to x, and the variance is fixed. The magnitude of $\alpha$ will determine the width and incline of the slanting line.
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_likelihood_en")
     return
 
@@ -103,13 +103,13 @@ def md_posterior_en():
             <li>When the variance of the Gaussian function is large (large noise), or when $q(x)$ changes drastically, the shape of $q(x|z)$ will be more complex, and greatly differ from a Gaussian function, which makes it difficult to model and learn.</li>
             </ul>
             
-            The specifics can be seen in Demo 2. The fourth figure present the shape of the posterior $q(x|z)$, which shows an irregular shape and resembles a curved and uneven line. As $\alpha$ increases (noise decreases), the curve tends to be uniform and straight. Readers can adjust different $\alpha$ values and observe the relationship between the shape of posterior and the level of noise. In the last figure, the $\textcolor{blue}{\text{blue dash line}}$ represents $q(x)$, the $\textcolor{green}{\text{green dash line}}$ represents <b>GaussFun</b> in the equation 3.4, and the $\textcolor{orange}{\text{orange curve}}$ represents the result of multiplying the two function and normalizing it, which is the posterior probability $q(x|z=fixed)$ under a fixed z condition. Readers can adjust different values of z to observe how the fluctuation of $q(x)$ affect the shape of the posterior probability $q(x|z)$.
+            The specifics can be seen in <a href="#demo_2">Demo 2</a>. The fourth figure present the shape of the posterior $q(x|z)$, which shows an irregular shape and resembles a curved and uneven line. As $\alpha$ increases (noise decreases), the curve tends to be uniform and straight. Readers can adjust different $\alpha$ values and observe the relationship between the shape of posterior and the level of noise. In the last figure, the $\textcolor{blue}{\text{blue dash line}}$ represents $q(x)$, the $\textcolor{green}{\text{green dash line}}$ represents <b>GaussFun</b> in the equation 3.4, and the $\textcolor{orange}{\text{orange curve}}$ represents the result of multiplying the two function and normalizing it, which is the posterior probability $q(x|z=fixed)$ under a fixed z condition. Readers can adjust different values of z to observe how the fluctuation of $q(x)$ affect the shape of the posterior probability $q(x|z)$.
             
             The posterior $q(x|z)$ under two special states are worth considering.
             <ul>
-            <li>As $\alpha \to 0$, the variance of <b>GaussFun</b> tends to <b>$\infty$</b>, and $q(x|z)$ for different $z$ almost become identical, and almost the same as $q(x)$. Readers can set $\alpha$ to 0.001 in Demo 2 to observe the specific results.</li>
+            <li>As $\alpha \to 0$, the variance of <b>GaussFun</b> tends to <b>$\infty$</b>, and $q(x|z)$ for different $z$ almost become identical, and almost the same as $q(x)$. Readers can set $\alpha$ to 0.001 in <a href="#demo_2">Demo 2</a> to observe the specific results.</li>
                 
-            <li>As $\alpha \to 1$, the variance of <b>GaussFun</b> tends to <b>$0$</b>, The $q(x|z)$ for different $z$ values contract into a series of <em>Dirac delta functions</em> with different offsets equalling to $z$. However, there are some exceptions. When there are regions where $q(x)$ is zero, the corresponding $q(x|z)$ will no longer be a Dirac <em>delta function</em>, but a zero function. Readers can set $\alpha$ to 0.999 in Demo 2 to observe the specific results.</li>
+            <li>As $\alpha \to 1$, the variance of <b>GaussFun</b> tends to <b>$0$</b>, The $q(x|z)$ for different $z$ values contract into a series of <em>Dirac delta functions</em> with different offsets equalling to $z$. However, there are some exceptions. When there are regions where $q(x)$ is zero, the corresponding $q(x|z)$ will no longer be a Dirac <em>delta function</em>, but a zero function. Readers can set $\alpha$ to 0.999 in <a href="#demo_2">Demo 2</a> to observe the specific results.</li>
             </ul>
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_en")
     return
@@ -125,11 +125,11 @@ def md_forward_process_en():
             r"""
             For any arbitrary data distribution $q(x)$, the transform(equation 2.1) in section 2 can be continuously applied(equation 4.1~4.4). As the number of transforms increases, the output probability distribution will become increasingly closer to the standard normal distribution. For more complex data distributions, more iterations or larger noise are needed.
             
-            Specific details can be observed in Demo3. The first figure illustrates a randomly generated one-dimensional probability distribution. After seven transforms, this distribution looks very similar to the standard normal distribution. The degree of similarity increases with the number of iterations and the level of the noise. Given the same degree of similarity, fewer transforms are needed if the noise added at each step is larger (smaller $\alpha$ value). Readers can try different $\alpha$ values and numbers of transforms to see how similar the final probability distribution is.
+            Specific details can be observed in <a href="#demo_3_1">Demo 3.1</a>. The first figure illustrates a randomly generated one-dimensional probability distribution. After seven transforms, this distribution looks very similar to the standard normal distribution. The degree of similarity increases with the number of iterations and the level of the noise. Given the same degree of similarity, fewer transforms are needed if the noise added at each step is larger (smaller $\alpha$ value). Readers can try different $\alpha$ values and numbers of transforms to see how similar the final probability distribution is.
             
             The complexity of the initial probability distribution tends to be high, but as the number of transforms increases, the complexity of the probability distribution $q(z_t)$ will decrease. As concluded in section 4, a more complex probability distribution corresponds to a more complex posterior probability distribution. Therefore, in order to ensure that the posterior probability distribution is more similar to the Conditional Gaussian function (easier to learn), a larger value of $\alpha$ (smaller noise) should be used in the initial phase, and a smaller value of $\alpha$ (larger noise) can be appropriately used in the later phase to accelerate the transition to the standard normal distribution.
             
-            In the example of Demo 3.1, it can be seen that as the number of transforms increases, the corners of $q(z_t)$ become fewer and fewer. Meanwhile, the slanting lines in the plot of the posterior probability distribution $q(z_{t-1}|z_t)$ become increasingly straight and uniform, resembling more and more the conditional Gaussian distribution. 
+            In the example of <a href="#demo_3_1">Demo 3.1</a>, it can be seen that as the number of transforms increases, the corners of $q(z_t)$ become fewer and fewer. Meanwhile, the slanting lines in the plot of the posterior probability distribution $q(z_{t-1}|z_t)$ become increasingly straight and uniform, resembling more and more the conditional Gaussian distribution. 
 
             \begin{align}
                 Z_1   &= \sqrt{\alpha_1} X + \sqrt{1-\alpha_1}\epsilon_1 			\tag{4.1}   \newline
@@ -160,7 +160,7 @@ def md_forward_process_en():
                 q(z_T|x) &= \mathcal{N}(0.00635\ x,\ 0.99998)    \tag{4.9}
             \end{align}
             
-            If considering only $q(z_T)$, a single transformation can also be used, which is as follows:
+            If considering only marginal distribution $q(z_T)$, a single transformation can also be used, which is as follows:
             \begin{align}
                 Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon 			 \tag{4.10}
             \end{align}
@@ -179,7 +179,7 @@ def md_backward_process_en():
             r"""
             If the final probability distribution $q(z_T)$ and the posterior probabilities of each transform $q(x|z),q(z_{t-1}|z_t)$ are known, the data distribution $q(x)$ can be recovered through the Bayes Theorem and the Law of Total Probability, as shown in equations 5.1~5.4. When the final probability distribution $q(z_T)$ is very similar to the standard normal distribution, the standard normal distribution can be used as a substitute.
             
-            Specifics can be seen in Demo 3.2. In the example, $q(z_T)$ substitutes $\mathcal{N}(0,1)$, and the error magnitude is given through JS Divergence. The restored probability distribution $q(z_t)$ and $q(x)$ are identified by the $\textcolor{green}{\text{green curve}}$, and the original probability distribution is identified by the $\textcolor{blue}{\text{blue curve}}$. It can be observed that the data distribution $q(x)$ can be well restored, and the error (JS Divergence) will be smaller than the error caused by the standard normal distribution replacing $q(z_T)$.
+            Specifics can be seen in <a href="#demo_3_2">Demo 3.2</a>. In the example, $q(z_T)$ substitutes $\mathcal{N}(0,1)$, and the error magnitude is given through JS Divergence. The restored probability distribution $q(z_t)$ and $q(x)$ are identified by the $\textcolor{green}{\text{green curve}}$, and the original probability distribution is identified by the $\textcolor{blue}{\text{blue curve}}$. It can be observed that the data distribution $q(x)$ can be well restored, and the error (JS Divergence) will be smaller than the error caused by the standard normal distribution replacing $q(z_T)$.
             \begin{align}
                 q(z_{T-1}) &= \int q(z_{T-1},z_T)dz_T = \int q(z_{T-1}|z_T)q(z_T)dz_T               	    \tag{5.1}   \newline
                            & \dots	                                                                        \notag      \newline
@@ -188,15 +188,15 @@ def md_backward_process_en():
                 q(z_1)     &= \int q(z_1,z_2) dz_1    = \int q(z_1|z_2)q(z_2)dz_2                           \tag{5.3}   \newline
                 q(x)       &= \int q(x,z_1) dz_1      = \int q(x|z_1)q(z_1)dz_1                             \tag{5.4}   \newline
             \end{align}
-            In this article, the aforementioned transform is referred to as the <b>Posterior Transform</b>. For example, in equation 5.4, the input of the transform is the probability distribution function $q(z_1)$, and the output is the probability distribution function $q(x)$.The entire transform is determined by the posterior $q(x|z_1)$. This transform can also be considered as the linear weighted sum of a set of basis functions, where the basis functions are $q(x|z_1)$ under different $z_1$, and the weights of each basis function are $q(z_1)$. Some interesting properties of this transform will be introduced in Section 7.
+            In this article, the aforementioned transform is referred to as the <b>Posterior Transform</b>. For example, in equation 5.4, the input of the transform is the probability distribution function $q(z_1)$, and the output is the probability distribution function $q(x)$.The entire transform is determined by the posterior $q(x|z_1)$. This transform can also be considered as the linear weighted sum of a set of basis functions, where the basis functions are $q(x|z_1)$ under different $z_1$, and the weights of each basis function are $q(z_1)$. Some interesting properties of this transform will be introduced in <a href="#posterior_transform">Section 7</a>.
             
-            In Section 3, we have considered two special posterior probability distributions. Next, we analyze their corresponding <em>posterior transforms</em>.
+            In <a href="#posterior">Section 3</a>, we have considered two special posterior probability distributions. Next, we analyze their corresponding <em>posterior transforms</em>.
             <ul>
-                <li> When $\alpha \to 0$, the $q(x|z)$ for different $z$ are almost the same as $q(x)$. In other words, the basis functions of linear weighted sum are almost the same. In this state, no matter how the input changes, the output of the transformation is always $q(x)$."</li>
+                <li> When $\alpha \to 0$, the $q(x|z)$ for different $z$ are almost the same as $q(x)$. In other words, the basis functions of linear weighted sum are almost the same. In this state, no matter how the input changes, the output of the transformation is always $q(x)$.</li>
                 <li> When $\alpha \to 1$, the $q(x|z)$ for different $z$ values becomes a series of Dirac delta functions and zero functions. In this state, as long as the <em>support set</em> of the input distribution is included in the <em>support set</em> of $q(x)$, the output of the transformation will remain the same with the input.</li>
             </ul>
             
-            In Section 5, it is mentioned that the 1000 transformations used in the DDPM[\[2\]](#ddpm) can be represented using a single transformation
+            In <a href="#forward_process">Section 4</a>, it is mentioned that the 1000 transformations used in the DDPM[\[2\]](#ddpm) can be represented using a single transformation
             \begin{align}
                 Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon 			 \tag{5.5}
             \end{align}
@@ -204,6 +204,8 @@ def md_backward_process_en():
             Since $\\alpha=0.0000403$ is very small, the corresponding standard deviation of GaussFun (Equation 3.4) reaches 157.52. However, the range of $X$ is limited within $[-1, 1]$, which is far smaller than the standard deviation of GaussFun. Within the range of $x \\in [-1, 1]$, GaussFun should be close to a constant, showing little variation. Therefore, the $q(x|z_T)$ corresponding to different $z_T$ are almost the same as $q(x)$. In this state, the posterior transform corresponding to $q(x|z_T)$ does not depend on the input distribution, the output distribution will always be $q(x)$.
             
             <b>Therefore, theoretically, in the DDPM model, it is not necessary to use the standard normal distribution to replace $q(z_T)$. Any other arbitrary distributions can also be used as a substitute.</b>
+            
+            Readers can conduct a similar experiment themselves. In <a href="#demo_3_1">Demo 3.1</a>, set <em>start_alpha</em> to 0.25, <em>end_alpha</em> to 0.25, and <em>step</em> to 7. At this point, $q(z_7)=\sqrt{0.000061}X + \sqrt{1-0.000061} \epsilon$, which is roughly equivalent to DDPM's $q(z_T)$. Click on <b>apply</b> to perform the forward transform (plotted using $\textcolor{blue}{\text{blue curves}}$), which prepares for the subsequent restoring process. In <a href="#demo_3_2">Demo 3.2</a>, set the <em>noise_ratio</em> to 1, introducing 100% noise into the <em>tail distribution</em> $q(z_7)$. Changing the value of <em>nose_random_seed</em> will change the distribution of noise. Deselect <em>backward_pdf</em> to reduce screen clutter. Click on <b>apply</b> to restore $q(x)$ through posterior transform. You will see that, no matter what the shape of input $q(z_7)$ may be, the restored $q(x)$ is always exactly the same as the original $q(x)$. The JS Divergence is zero. The restoration process is plotted using a $\textcolor{red}{\text{red curve}}$.
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_backward_process_en")
     return
 
@@ -216,13 +218,13 @@ def md_fit_posterior_en():
 
         gr.Markdown(
             r"""
-            From the front part of Section 4, it is known that the posterior probability distributions are unknown and related to $q(x)$. Therefore, in order to recover the data distribution or sample from it, it is necessary to learn and estimate each posterior probability distribution.
+            From the front part of <a href="#posterior">Section 3</a>, it is known that the posterior probability distributions are unknown and related to $q(x)$. Therefore, in order to recover the data distribution or sample from it, it is necessary to learn and estimate each posterior probability distribution.
             
-            From the latter part of Section 4, it can be understood that when certain conditions are met, each posterior probability distribution $q(x|z), q(z_{t-1}|z_t)$ approximates the Gaussian probability distribution. Therefore, by constructing a set of conditional Gaussian probability models $p(x|z), p(z_{t-1}|z_t)$, we can learn to fit the corresponding $q(x|z), q(z_{t-1}|z_t)$.
+            From the latter part of <a href="#posterior">Section 3</a>, it can be understood that when certain conditions are met, each posterior probability distribution $q(x|z), q(z_{t-1}|z_t)$ approximates the Gaussian probability distribution. Therefore, by constructing a set of conditional Gaussian probability models $p(x|z), p(z_{t-1}|z_t)$, we can learn to fit the corresponding $q(x|z), q(z_{t-1}|z_t)$.
             
-            Due to the limitations of the model's representative and learning capabilities, there will be certain errors in the fitting process, which will further impact the accuracy of restored $q(x)$. The size of the fitting error is related to the complexity of the posterior probability distribution. As can be seen from Section 4, when $q(x)$ is more complex or the added noise is large, the posterior probability distribution will be more complex, and it will differ greatly from the Gaussian distribution, thus leading to fitting errors and further affecting the restoration of $q(x)$.
+            Due to the limitations of the model's representative and learning capabilities, there will be certain errors in the fitting process, which will further impact the accuracy of restored $q(x)$. The size of the fitting error is related to the complexity of the posterior probability distribution. As can be seen from <a href="#posterior">Section 3</a>, when $q(x)$ is more complex or the added noise is large, the posterior probability distribution will be more complex, and it will differ greatly from the Gaussian distribution, thus leading to fitting errors and further affecting the restoration of $q(x)$.
             
-            Refer to Demo 3.3 for the specifics. The reader can test different $q(x)$ and $\alpha$, observe the fitting degree of the posterior probability distribution $q(z_{t-1}|z_t)$ and the accuracy of restored $q(x)$. The restored probability distribution is ploted with $\textcolor{orange}{\text{orange}}$, and the error is also measured by JS divergence.
+            Refer to <a href="#demo_3_3">Demo 3.3</a> for the specifics. The reader can test different $q(x)$ and $\alpha$, observe the fitting degree of the posterior probability distribution $q(z_{t-1}|z_t)$ and the accuracy of restored $q(x)$. The restored probability distribution is ploted with $\textcolor{orange}{\text{orange}}$, and the error is also measured by JS divergence.
              
             Regarding the objective function for fitting, similar to other probability models, the cross-entropy loss can be optimized to make $p(z_{t-1}|z_t)$ approaching $q(z_{t-1}|z_t)$. Since $(z_{t-1}|z_t)$ is a conditional probability, it is necessary to fully consider all conditions. This can be achieved by averaging the cross-entropy corresponding to each condition weighted by the probability of each condition happening. The final form of the loss function is as follows.
             \begin{align}
@@ -232,7 +234,7 @@ def md_fit_posterior_en():
             
             KL divergence can also be optimized as the objective function. KL divergence and cross-entropy are equivalent[\\[10\\]](#ce_kl)
             <span id="en_fit_0">
-                loss &= \int q(z_t) KL(q(z_{t-1}|z_t)\|\textcolor{blue}{p(z_{t-1}|z_t)})dz_t                                                             \tag{6.3}      \newline
+                loss &= \int q(z_t) KL(q(z_{t-1}|z_t) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dz_t                                                             \tag{6.3}      \newline
                      &= \int q(z_t) \int q(z_{t-1}|z_t) \frac{q(z_{t-1}|z_t)}{\textcolor{blue}{p(z_{t-1}|z_t)}} dz_{t-1} dz_t                            \tag{6.4}      \newline 
                      &= -\int q(z_t)\ \underbrace{\int q(z_{t-1}|z_t) \log \textcolor{blue}{p(z_{t-1}|z_t)}dz_{t-1}}{underline}{\text{Cross Entropy}}\ dz_t + \underbrace{\int q(z_t) \int q(z_{t-1}|z_t) \log q(z_{t-1}|z_t)}{underline}{\text{Is Constant}} dz  \tag{6.5}
             </span>
@@ -256,8 +258,8 @@ def md_fit_posterior_en():
                      &\quad - \iint \int q(x)q(z_{t-1}, z_t|x) \log \textcolor{blue}{p(z_{t-1}|z_t)}dxdz_{t-1}dz_t - \textcolor{orange}{C_1}                                                \tag{6.11}          \newline
                      &= \iint \int q(x)q(z_{t-1},z_t|x) \log \frac{q(z_{t-1}|z_t,x)}{\textcolor{blue}{p(z_{t-1}|z_t)}}dxdz_{t-1}dz_t - \textcolor{orange}{C_1}                              \tag{6.12}          \newline
                      &= \iint q(x)q(z_t|x)\int q(z_{t-1}|z_t,x) \log \frac{q(z_{t-1}|z_t,x)}{\textcolor{blue}{p(z_{t-1}|z_t)}}dz_{t-1}\ dz_xdz_t - \textcolor{orange}{C_1}                  \tag{6.13}          \newline
-                     &= \iint \ q(x)q(z_t|x) KL[q(z_{t-1}|z_t,x)\|\textcolor{blue}{p(z_{t-1}|z_t)}]dxdz_t - \textcolor{orange}{C_1}                                                         \tag{6.14}          \newline
-                     &\propto \iint \ q(x)q(z_t|x) KL[q(z_{t-1}|z_t,x)\|\textcolor{blue}{p(z_{t-1}|z_t)}]dxdz_t                                                                             \tag{6.15}          \newline
+                     &= \iint \ q(x)q(z_t|x) KL[q(z_{t-1}|z_t,x) \Vert \textcolor{blue}{p(z_{t-1}|z_t)}]dxdz_t - \textcolor{orange}{C_1}                                                         \tag{6.14}          \newline
+                     &\propto \iint \ q(x)q(z_t|x) KL(q(z_{t-1}|z_t,x) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dxdz_t                                                                             \tag{6.15}          \newline
             \end{align}
 
             In the above formula, the term $C_1$ is a fixed value, which does not contain parameters to be optimized. Here, $q(x)$ is a fixed probability distribution, and $q(z_{t-1}|z_t)$ is also a fixed probability distribution, whose specific form is determined by $q(x)$ and the coefficient $\alpha$.
@@ -274,15 +276,15 @@ def md_fit_posterior_en():
             
             Based on the conclusion of the Consistent Terms proof and the relationship between cross entropy and KL divergence, an interesting conclusion can be drawn:
             <span id="en_fit_1">
-                \mathop{\min}{underline}{\textcolor{blue}{p}} \int q(z_t) KL(q(z_{t-1}|z_t)\|\textcolor{blue}{p(z_{t-1}|z_t)})dz_t  \iff  \mathop{\min}{underline}{\textcolor{blue}{p}} \iint \ q(x)q(z_t|x) KL[q(z_{t-1}|z_t,x)\|\textcolor{blue}{p(z_{t-1}|z_t)}]dxdz_t       \tag{6.19}
+                \mathop{\min}{underline}{\textcolor{blue}{p}} \int q(z_t) KL(q(z_{t-1}|z_t) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dz_t  \iff  \mathop{\min}{underline}{\textcolor{blue}{p}} \iint \ q(x)q(z_t|x) KL(q(z_{t-1}|z_t,x) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dxdz_t       \tag{6.19}
             </span>
             By comparing the expressions on the left and right, it can be observed that the objective function on the right side includes an additional variable $X$ compared to the left side. At the same time, there is an additional integral with respect to $X$, with the occurrence probability of $X$, denoted as $q(x)$, serving as the weighting coefficient for the integral.
             
             Following a similar proof method, a more general relationship can be derived:
             <span id="en_fit_2">
-                \mathop{\min}{underline}{\textcolor{blue}{p}}  KL(q(z)\|\textcolor{blue}{p(z)})  \iff  \mathop{\min}_{\textcolor{blue}{p}} \int \ q(x) KL(q(z|x)\|\textcolor{blue}{p(z)})dx         \tag{6.20}
+                \mathop{\min}{underline}{\textcolor{blue}{p}}  KL(q(z) \Vert \textcolor{blue}{p(z)})  \iff  \mathop{\min}_{\textcolor{blue}{p}} \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p(z)})dx         \tag{6.20}
             </span>
-            
+            A detailed derivation of this conclusion can be found in <a href="#cond_kl">Appendix A</a>.
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_fit_posterior_en")
     return
 
@@ -299,12 +301,12 @@ def md_posterior_transform_en():
                 q(x) &= \int q(x,z) dz = \int q(x|z)q(z)dz      \tag{7.1}
             \end{align}
             
-            Through extensive experiments with one-dimensional random variables, it was found that the <b>Posterior Transform</b> exhibits the characteristics of <b>Contraction Mapping</b>. This means that, for any two probability distributions $q_{i1}(z) and $q_{i2}(z), after posterior transform, we get $q_{o1}(x)$ and $q_{o2}(x)$. The distance between $q_{o1}(x)$ and $q_{o2}(x)$ is always less than the distance between $q_{i1}(x)$ and $q_{i2}(x)$. Here, the distance can be measured using JS divergence or Total Variance. Furthermore, the contractive ratio of this contraction mapping is positively related to the size of the added noise.
+            Through extensive experiments with one-dimensional random variables, it was found that the <b>Posterior Transform</b> exhibits the characteristics of <b>Contraction Mapping</b>. This means that, for any two probability distributions $q_{i1}(z)$ and $q_{i2}(z)$, after posterior transform, we get $q_{o1}(x)$ and $q_{o2}(x)$. The distance between $q_{o1}(x)$ and $q_{o2}(x)$ is always less than the distance between $q_{i1}(x)$ and $q_{i2}(x)$. Here, the distance can be measured using JS divergence or Total Variance. Furthermore, the contractive ratio of this contraction mapping is positively related to the size of the added noise.
             \begin{align}
                 dist(q_{o1}(z),\ q_{o2}(z)) < dist(q_{i1}(x),\ q_{i2}(x))                   \tag{7.2}
             \end{align}
     
-            Readers can refer to Demo 4.1, where the first three figure present a transform process. The first figure is an arbitrary data distribution $q(x)$, the third figure is the transformed probability distribution, and second figure is the posterior probability distribution $q(x|z)$. You can change the random seed to generate a new data distribution$q(x)$, and adjust the value of $\alpha$ to introduce different degrees of noise.
+            Readers can refer to <a href="#demo_4_1">Demo 4.1</a>, where the first three figure present a transform process. The first figure is an arbitrary data distribution $q(x)$, the third figure is the transformed probability distribution, and second figure is the posterior probability distribution $q(x|z)$. You can change the random seed to generate a new data distribution$q(x)$, and adjust the value of $\alpha$ to introduce different degrees of noise.
 
             The last two figures show contraction of the transform. The fourth figure displays two randomly generated input distributions and their distance, $div_{in}$. The fifth figure displays the two output distributions after transform, with the distance denoted as $div_{out}$.
             
@@ -312,7 +314,7 @@ def md_posterior_transform_en():
             
             According to the Banach fixed-point theorem<a href="#fixed_point">[5]</a>, a contraction mapping has a unique fixed point (converged point). That is to say, for any input distribution, the <b>Posterior Transform</b> can be applied continuously through iterations, and as long as the number of iterations is sufficient, the final output would be the same distribution. After a large number of one-dimensional random variable experiments, it was found that the fixed point (converged point) is <b>located near $q(x)$</b>. Also, the location is related to the value of $\alpha$; the smaller $\alpha$ (larger noise), the closer it is. 
             
-            Readers can refer to Demo 4.2, which illustrates an example of applying posterior transform iteratively. Choose an appropriate number of iterations, and click on the button of <em>Apply</em>, and the iteration process will be draw step by step. Each subplot shows the transformed output distribution($\textcolor{green}{\text{green curve}}$) from each transform, with the reference distribution $q(x)$ expressed as a $\textcolor{blue}{\text{blue curve}}$, as well as the distance $div$ between the output distribution and $q(x)$. It can be seen that as the number of iterations increases, the output distribution becomes more and more similar to $q(x)$, and will eventually stabilize near $q(x)$. For more complicated distributions, more iterations or greater noise may be required. The maximum number of iterations can be set to tens of thousands, but it'll take longer.
+            Readers can refer to <a href="#demo_4_2">Demo 4.2</a>, which illustrates an example of applying posterior transform iteratively. Choose an appropriate number of iterations, and click on the button of <em>Apply</em>, and the iteration process will be draw step by step. Each subplot shows the transformed output distribution($\textcolor{green}{\text{green curve}}$) from each transform, with the reference distribution $q(x)$ expressed as a $\textcolor{blue}{\text{blue curve}}$, as well as the distance $div$ between the output distribution and $q(x)$. It can be seen that as the number of iterations increases, the output distribution becomes more and more similar to $q(x)$, and will eventually stabilize near $q(x)$. For more complicated distributions, more iterations or greater noise may be required. The maximum number of iterations can be set to tens of thousands, but it'll take longer.
             
             For the one-dimensional discrete case, $q(x|z)$ is discretized into a matrix (denoted as $Q_{x|z}$), $q(z)$ is discretized into a vector (denoted as $\boldsymbol{q_i}$). The integration operation $\int q(x|z)q(z)dz$ is discretized into a **matrix-vector** multiplication operation, thus the posterior transform can be written as
             \begin{align}
@@ -321,12 +323,147 @@ def md_posterior_transform_en():
                     & \dots &                                                                                             \notag          \newline
                 \boldsymbol{q_o} &= (Q_{x|z})^n\ \boldsymbol{q_i} &         \quad\quad        &\text{n iteration}         \tag{7.5}       \newline
             \end{align}
-            In order to better understand the property of the transform, the matrix $(Q_{x|z})^n$ is also plotted in Demo 4.2. From the demo we can see that, as the iterations converge, the row vectors of the matrix $(Q_{x|z})^n$ will become a constant vector, that is, all components of the vector will be the same, which will appear as a horizontal line in the denisty plot.
+            In order to better understand the property of the transform, the matrix $(Q_{x|z})^n$ is also plotted in <a href="#demo_4_2">Demo 4.2</a>. From the demo we can see that, as the iterations converge, the row vectors of the matrix $(Q_{x|z})^n$ will become a constant vector, that is, all components of the vector will be the same, which will appear as a horizontal line in the denisty plot.
+            
+            In the <a href="#proof_ctr">Appendix B</a>, a proof will be provided that, when $q(x)$ and $\alpha$ satisfy some conditions, the posterior transform is a strict Contraction Mapping.
+            
+            The relationship between the converged distribution and the input distribution q(x) cannot be rigorously proven at present. 
+            
+            <h3 style="font-size:18px"> Anti-noise Capacity In Restoring Data Distribution</h3>
+            From the above analysis, we know that when certain conditions are satisfied, the <em>posterior transform</em> is a contraction mapping. Therefore, the following relationship exists: 
+            \begin{align}
+                dist(q(x),\ q_o(x)) < dist(q(z),\ q_i(z))         \tag{7.12}
+            \end{align}
+            Wherein, $q(z)$ is the ideal input distribution, $q(x)$ is the ideal output distribution, $q_i(x)$ is any arbitrary input distribution, and $q_o(x)$ is the output distribution obtained after transforming $q_i(z)$. 
+            
+            The above equation indicates that the distance between the output distribution $q_o(x)$ and the ideal output distribution q(x) will always be <b>less than</b> the distance between the input distribution $q_i(z)$ and the ideal input distribution q(x). Hence, the <em>posterior transform</em> has certain resistance to noise. This means that during the process of restoring $q(x)$(<a href="#backward_process">Section 5</a>), even if the <em>tail distribution</em> $q(z_T)$ contains some error, the error of the outputed distribution $q(x)$ will be smaller than the error of input after undergoing a series of transform.
+            
+            Refer specifically to <a href="#demo_3_2">Demo 3.2</a>, where by increasing the value of the <b>noise ratio</b>, noise can be added to the <em>tail distribution</em> $q(z_T)$. Clicking the "apply" button will gradually draw out the restoring process, with the restored distribution represented by a $\textcolor{red}{\text{red curve}}$, and the error size will be computed by the JS divergence. You will see that the error of restored $q(x)$ is always less than the error of $q(z_T)$.
+            
+            From the above discussion, we know that the smaller the $\alpha$ (the larger the noise used in the transform process), the greater the contractive ratio of the contraction mapping, and thus, the stronger the ability to resist noise.
+            
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform_en")
+    return
+
+
+def md_deconvolution_en():
+    global g_latex_del
+
+    title = "8. Can the data distribution be restored by deconvolution?"
+    with gr.Accordion(label=title, elem_classes="first_md", elem_id="deconvolution"):
+        gr.Markdown(
+            r"""
+            As mentioned in the <a href="#introduction">Section 1</a>, the transform of Equation 2.1 can be divided into two sub-transforms, the first one being a linear transform and the second being adding independent Gaussian noise. The linear transform is equivalent to a scaling transform of the probability distribution, so it has an inverse transformation. Adding independent Gaussian noise is equivalent to the execution of a convolution operation on the probability distribution, which can be restored through <b>deconvolution</b>. Therefore, theoretically, the data distribution $q(x)$ can be recovered from the final probability distribution $q(z_T)$ through <b>inverse linear transform</b> and <b>deconvolution</b>.
+            
+            However, in actuality, some problems do exist. Due to the extreme sensitivity of deconvolution to errors, having high input sensitivity, even a small amount of input noise can lead to significant changes in output[\[11\]](#deconv_1)[\[12\]](#deconv_2). Meanwhile, in the diffusion model, the standard normal distribution is used as an approximation to replace $q(z_T)$, thus, noise is introduced at the initial stage of recovery. Although the noise is relatively small, because of the sensitivity of deconvolution, the noise will gradually amplify, affecting the recovery.
             
+            In addition, the infeasibility of <b>deconvolution restoring</b> can be understood from another perspective. Since the process of forward transform (equations 4.1 to 4.4) is fixed, the convolution kernel is fixed. Therefore, the corresponding deconvolution transform is also fixed. Since the initial data distribution $q(x)$ is arbitrary, any probability distribution can be transformed into an approximation of $\mathcal{N}(0,I)$ through a series of fixed linear transforms and convolutions. If <b>deconvolution restoring</b> is feasible, it means that a fixed deconvolution can be used to restore any data distribution $q(x)$ from the $\mathcal{N}(0,I)$ , this is clearly <b>paradoxical</b>. The same input, the same transform, cannot have multiple different outputs.
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution_en")
+    return
+
+
+def md_cond_kl_en():
+    global g_latex_del
+
+    title = "Appendix A Conditional KL Divergence"
+    with gr.Accordion(label=title, elem_classes="first_md", elem_id="cond_kl"):
+        gr.Markdown(
+            r"""
+            This section mainly introduces the relationship between <b>KL divergence</b> and <b>conditional KL divergence</b>. Before the formal introduction, we will briefly introduce the definitions of entropy and conditional entropy, as well as the inequality relationship between them, in preparation for the subsequent proof 
+
+            <h3 style="font-size:20px">Entropy and Conditional Entropy</h3>
+            For any two random variables $Z, X$, the <b>Entropy</b> is defined as follows<a href="#entropy">[16]</a>：
+            \begin{align}
+               \mathbf{H}(Z) = \int -p(z)\log{p(z)}dz    \tag{A.1}
+            \end{align}
+            The <b>Conditional Entropy</b> is defined as followed <a href="#cond_entropy">[17]</a>：
+            \begin{align}
+               \mathbf{H}(Z|X) = \int p(x) \overbrace{\int -p(z|x)\log{p(z|x)}dz}^{\text{Entropy}}\ dx    \tag{A.2}
+            \end{align}
+            The following inequality relationship exists between the two：
+            \begin{align}
+               \mathbf{H}(Z|X) \le \mathbf{H}(Z)         \tag{A.3}
+            \end{align}
+            It is to say that the Conditional Entropy is always less than or equal to the Entropy, and they are equal only when X and Z are independent. The proof of this relationship can be found in the literature <a href="#cond_entropy">[17]</a>.
+
+            <h3 style="font-size:20px">KL Divergence and Conditional KL Divergence</h3>
+            In the same manner as the definition of Conditional Entropy, we introduce a new definition, <b>Conditional KL Divergence</b>, denoted as $KL_{\\mathcal{C}}$. Since KL Divergence is non-symmetric, there exist two forms as follows. 
+            \begin{align}
+               KL_{\mathcal{C}}(q(z|x) \Vert \textcolor{blue}{p(z)}) = \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p(z)})dx                        \tag{A.4}     \newline
+               KL_{\mathcal{C}}(q(z) \Vert \textcolor{blue}{p(z|x)}) = \int \ \textcolor{blue}{p(x)} KL(q(z) \Vert \textcolor{blue}{p(z|x)})dx      \tag{A.5}
+            \end{align}
+
+            Similar to Conditional Entropy, there also exists a similar inequality relationship for Conditional KL Divergence:
+            \begin{align}
+               KL_{\mathcal{C}}(q(z|x) \Vert \textcolor{blue}{p(z)}) \ge KL(q(z) \Vert \textcolor{blue}{p(z)})        \tag{A.6}   \newline
+               KL_{\mathcal{C}}(q(z) \Vert \textcolor{blue}{p(z|x)}) \ge KL(q(z) \Vert \textcolor{blue}{p(z)})        \tag{A.7}
+            \end{align}
+            It is to say that the Conditional KL Divergence is always less than or equal to the KL Divergence, and they are equal only when X and Z are independent.
+
+            The following provides proofs for the conclusions on Equation A.5 and Equation A.6 respectively.
+
+            For equation A.6, the proof is as follows:
+            \begin{align}
+                KL_{\mathcal{C}}(q(z|x) \Vert \textcolor{blue}{p(z)}) &= \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p(z)})dx      \tag{A.8}    \newline
+                    &= \iint q(x) q(z|x) \log \frac{q(z|x)}{\textcolor{blue}{p(z)}}dzdx                                             \tag{A.9}    \newline
+                    &= -\overbrace{\iint - q(x)q(z|x) \log q(z|x) dzdx}^{\text{Condtional Entropy }\mathbf{H}_q(Z|X)} - \iint q(x) q(z|x) \log \textcolor{blue}{p(z)} dzdx          \tag{A.10}   \newline
+                    &= -\mathbf{H}_q(Z|X) - \int \left\lbrace \int q(x) q(z|x)dx \right\rbrace \log \textcolor{blue}{p(z)}dz                                                        \tag{A.11}   \newline
+                    &= -\mathbf{H}_q(Z|X) + \overbrace{\int - q(z) \log p(z)dz}^{\text{Cross Entropy}}                                                                              \tag{A.12}   \newline
+                    &= -\mathbf{H}_q(Z|X) + \int q(z)\left\lbrace \log\frac{q(z)}{\textcolor{blue}{p(z)}} -\log q(z)\right\rbrace dz                                                \tag{A.13}   \newline
+                    &= -\mathbf{H}_q(Z|X) + \int q(z)\log\frac{q(z)}{\textcolor{blue}{p(z)}}dz + \overbrace{\int - q(z)\log q(z)dz}^{\text{Entropy } \mathbf{H}_q(Z)}               \tag{A.14}   \newline
+                    &= KL(q(z) \Vert \textcolor{blue}{p(z)}) + \overbrace{\mathbf{H}_q(Z) - \mathbf{H}_q(Z|X)}^{\ge 0}              \tag{A.15}  \newline
+                    &\le KL(q(z) \Vert \textcolor{blue}{p(z)})      \tag{A.16}
+            \end{align}
+            In this context, equation A.15 applies the conclusion that <b>Conditional Entropy is always less than or equal to Entropy</b>. Thus, the relationship in equation A.6 is derived. 
+
+            For equation A.6, the proof is as follows:
+            \begin{align}
+                KL(\textcolor{blue}{q(z)} \Vert p(z)) &= \int \textcolor{blue}{q(z)}\log\frac{\textcolor{blue}{q(z)}}{p(z)}dx    \tag{A.15}          \newline
+                        &= \int q(z)\log\frac{q(z)}{\int p(z|x)p(z)dz}dz 						 	\tag{A.16}          \newline
+                        &= \textcolor{orange}{\int p(x)dx}\int q(z)\log q(z)dz - \int q(z)\textcolor{red}{\log\int p(z|x)p(x)dx}dz	\qquad \ \textcolor{orange}{\int p(x)dx=1}			    \tag{A.17}      \newline
+                        &\le \iint p(x) q(z)\log q(z)dzdx - \int q(z)\textcolor{red}{\int p(x)\log p(z|x)dx}dz \ \qquad 	 \textcolor{red}{\text{jensen\ inequality}}                     \tag{A.18}      \newline
+                        &= \iint p(x)q(z)\log q(z)dzdx - \iint p(z)q(z)\log p(z|x)dzdx			 	    \tag{A.19}          \newline
+                        &= \iint p(x)q(z)(\log q(z) - \log p(z|x))dzdx								    \tag{A.20}          \newline
+                        &= \iint p(x)q(z)\log \frac{q(z)}{p(z|x)}dzdx								    \tag{A.21}          \newline
+                        &= \int p(x)\left\lbrace \int q(z)\log \frac{q(z)}{p(z|x)}dz\right\rbrace dx    \tag{A.22}          \newline
+                        &= \int p(x)KL(\textcolor{blue}{q(z)} \Vert p(z|x))dx                           \tag{A.23}          \newline
+                        &= KL_{\mathcal{C}}(q(z) \Vert \textcolor{blue}{p(z|x)})                        \tag{A.24}
+            \end{align}
+            Thus, the relationship in equation A.7 is obtained.
+
+            Another <b>important conclusion</b> can be drawn from equation A.15.
+
+            The KL Divergence is often used to fit the distribution of data. In this scenario, the distribution of the data is denoted by $q(z)$ and the parameterized model distribution is denoted by $\textcolor{blue}{p_\theta(z)}$. During the optimization process, since both $q(z|x)$ and $q(x)$ remain constant, the term $\mathbf{H}(Z) - \mathbf{H}(Z|X)$ in Equation A.15 is a constant. Thus, the following relationship is obtained:
+            <span id="zh_cond_kl_2">
+                \mathop{\min}{underline}{\textcolor{blue}{p_\theta}}  KL(q(z) \Vert \textcolor{blue}{p_\theta(z)})  \iff  \mathop{\min}{underline}{\textcolor{blue}{p_\theta}} \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p_\theta(z)})dx   \tag{A.25}
+            </span>
+
+            Comparing the above relationship with <b>Denoised Score Matching</b> <a href="#dsm">[18]</a>(equation A.26), some similarities can be observed. Both introduce a new variable $X$, and substitute the targeted fitting distribution q(z) with q(z|x). After the substitution, since q(z|x) is a conditional probability distribution, both consider all conditions and perform a weighted sum using the probability of the conditions occurring, $q(x)$, as the weight coefficient.
+            <span id="zh_cond_kl_3">
+                \mathop{\min}{underline}{\textcolor{blue}{\psi_\theta}} \frac{1}{2} \int q(z) \left\lVert \textcolor{blue}{\psi_\theta(z)} - \frac{\partial q(z)}{\partial z} \right\rVert^2 dz \iff  \mathop{\min}{underline}{\textcolor{blue}{\psi_\theta}} \int q(x)\ \overbrace{\frac{1}{2} \int q(z|x) \left\lVert \textcolor{blue}{\psi_\theta(z)} - \frac{\partial q(z|x)}{\partial z} \right\rVert^2 dz}^{\text{Score Matching of }q(z|x)}\ dx      \tag{A.26}
+            </span>
+
+            The operation of the above weighted sum is somewhat similar to <em> Elimination by Total Probability Formula </b>.
+            \begin{align}
+                q(z) = \int q(z,x) dx = \int q(x) q(z|x) dx     \tag{A.27}
+            \end{align}
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_cond_kl_en")
+    return
+
+
+def md_proof_ctr_en():
+    global g_latex_del
+
+    title = "Appendix B Proof of Contraction"
+    with gr.Accordion(label=title, elem_classes="first_md", elem_id="proof_ctr"):
+        gr.Markdown(
+            r"""
             <center> <img id="en_fig2" src="file/fig2.png" width="960" style="margin-top:12px"/> </center>
             <center> Figure 2: Only one component in support </center>
             
-            The following will prove that with some conditions, the posterior transform is a contraction mapping, and there exists a unique point, which is also the converged point. The proof assumes that the random variable is discrete, so the posterior transform can be regarded as a single step transition of a <b>discrete Markov Chain</b>. The posterior $q(x|z)$ corresponds to the <b>transfer matrix</b>. Continuous variables can be considered as discrete variables with infinite states.
+            The following will prove that with some conditions, the posterior transform is a contraction mapping, and there exists a unique point, which is also the converged point.
+             
+            The proof will be divided into several cases, and assumes that the random variable is discrete, so the posterior transform can be regarded as a single step transition of a <b>discrete Markov Chain</b>. The posterior $q(x|z)$ corresponds to the <b>transfer matrix</b>. Continuous variables can be considered as discrete variables with infinite states.
             <ol style="list-style-type:decimal">
             <li> When $q(x)$ is greater than 0, the posterior transform matrix $q(x|z)$ will be greater than 0 too. Therefore, this matrix is the transition matrix of an $\textcolor{red}{\text{irreducible}}\ \textcolor{green}{\text{aperiodic}}$ Markov Chain. According to the conclusion of the literature <a href="#mc_basic_p6">[13]</a>, this transformation is a contraction mapping with respect to Total Variance metric. Therefore, according to the Banach fixed-point theorem, this transformation has a unique fixed point(converged point). </li>
              
@@ -366,51 +503,20 @@ def md_posterior_transform_en():
             
             Additionally, there exists a more generalized relation about the posterior transform that is independent of $q(x|z)$: the Total Variance distance between two output distributions will always be <b>less than or equal to</b> the Total Variance distance between their corresponding input distributions, that is
             \begin{align}
-                dist(q_{o1}(x),\ q_{o2}(x)) <= dist(q_{i1}(z),\ q_{i2}(z))    \notag
+                dist(q_{o1}(x),\ q_{o2}(x)) <= dist(q_{i1}(z),\ q_{i2}(z))    \tag{B.1}
             \end{align}
             The proof is given below in discrete form:
             \begin{align}
-                      \lVert q_{o1}-q_{o2}\rVert_{TV} &= \lVert Q_{x|z}q_{i1} - Q_{x|z}q_{i2}\rVert_{TV}                                                                                                                    \tag{7.6}         \newline
-                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)q_{i1}(n) - \sum_{n}Q_{x|z}(m,n)q_{i2}(n)\textcolor{red}{|}                                                        \tag{7.7}         \newline
-                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                          \tag{7.8}         \newline
-                                                      &\leq \sum_{m}\sum_{n}Q_{x|z}(m,n)\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}             \qquad \qquad \qquad \text{Absolute value inequality}       \tag{7.9}         \newline
-                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|} \sum_{m} Q_{x|z}(m,n)            \qquad \qquad \qquad \sum_{m} Q_{x|z}(m,n) = 1              \tag{7.10}        \newline
-                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                                              \tag{7.11}
-            \end{align}
-            In this context, $Q_{x|z}(m,n)$ represents the element at the m-th row and n-th column of the matrix $Q_{x|z}$, and $q_{i1}(n)$ represents the n-th element of the vector $q_{i1}$. 
-            
-            The relationship between the converged distribution and the input distribution q(x) cannot be rigorously proven at present. 
-            
-            <h3 style="font-size:18px"> Anti-noise Capacity In Restoring Data Distribution</h3>
-            From the above analysis, we know that when certain conditions are satisfied, the <em>posterior transform</em> is a contraction mapping. Therefore, the following relationship exists: 
-            \begin{align}
-                dist(q(x),\ q_o(x)) < dist(q(z),\ q_i(z))         \tag{7.12}
+                      \lVert q_{o1}-q_{o2}\rVert_{TV} &= \lVert Q_{x|z}q_{i1} - Q_{x|z}q_{i2}\rVert_{TV}                                                                                                                    \tag{B.2}         \newline
+                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)q_{i1}(n) - \sum_{n}Q_{x|z}(m,n)q_{i2}(n)\textcolor{red}{|}                                                        \tag{B.3}         \newline
+                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                          \tag{B.4}         \newline
+                                                      &\leq \sum_{m}\sum_{n}Q_{x|z}(m,n)\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}             \qquad \qquad \qquad \text{Absolute value inequality}       \tag{B.5}         \newline
+                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|} \sum_{m} Q_{x|z}(m,n)            \qquad \qquad \qquad \sum_{m} Q_{x|z}(m,n) = 1              \tag{B.6}         \newline
+                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                                              \tag{B.7}
             \end{align}
-            Wherein, $q(z)$ is the ideal input distribution, $q(x)$ is the ideal output distribution, $q_i(x)$ is any arbitrary input distribution, and $q_o(x)$ is the output distribution obtained after transforming $q_i(z)$. 
-            
-            The above equation indicates that the distance between the output distribution $q_o(x)$ and the ideal output distribution q(x) will always be <b>less than</b> the distance between the input distribution $q_i(z)$ and the ideal input distribution q(x). Hence, the <em>posterior transform</em> has certain resistance to noise. This means that during the process of restoring $q(x)$(Section 5), even if the <em>tail distribution</em> $q(z_T)$ contains some error, the error of the outputed distribution $q(x)$ will be smaller than the error of input after undergoing a series of transform.
-            
-            Refer specifically to Demo 3.2, where by increasing the value of the <b>noise ratio</b>, noise can be added to the <em>tail distribution</em> $q(z_T)$. Clicking the "apply" button will gradually draw out the restoring process, with the restored distribution represented by a $\textcolor{red}{\text{red curve}}$, and the error size will be computed by the JS divergence. You will see that the error of restored $q(x)$ is always less than the error of $q(z_T)$.
-            
-            From the above discussion, we know that the smaller the $\alpha$ (the larger the noise used in the transform process), the greater the contractive ratio of the contraction mapping, and thus, the stronger the ability to resist noise.
-            
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform_en")
-    return
-
-
-def md_deconvolution_en():
-    global g_latex_del
-
-    title = "8. Can the data distribution be restored by deconvolution?"
-    with gr.Accordion(label=title, elem_classes="first_md", elem_id="deconvolution"):
-        gr.Markdown(
-            r"""
-            As mentioned in the section 2, the transformation of Equation 2.1 can be divided into two sub-transformations, the first one being a linear transformation and the second being adding independent Gaussian noise. The linear transformation is equivalent to a scaling transform of the probability distribution, so it has an inverse transformation. Adding independent Gaussian noise is equivalent to the execution of a convolution operation on the probability distribution, which can be restored through <b>deconvolution</b>. Therefore, theoretically, the data distribution $q(x)$ can be recovered from the final probability distribution $q(z_T)$ through <b>inverse linear transform</b> and <b>deconvolution</b>.
-            
-            However, in actuality, some problems do exist. Due to the extreme sensitivity of deconvolution to errors, having high input sensitivity, even a small amount of input noise can lead to significant changes in output[\[11\]](#deconv_1)[\[12\]](#deconv_2). Meanwhile, in the diffusion model, the standard normal distribution is used as an approximation to replace $q(z_T)$, thus, noise is introduced at the initial stage of recovery. Although the noise is relatively small, because of the sensitivity of deconvolution, the noise will gradually amplify, affecting the recovery.
-            
-            In addition, the infeasibility of <b>deconvolution restoring</b> can be understood from another perspective. Since the process of forward transform (equations 4.1 to 4.4) is fixed, the convolution kernel is fixed. Therefore, the corresponding deconvolution transform is also fixed. Since the initial data distribution $q(x)$ is arbitrary, any probability distribution can be transformed into an approximation of $\mathcal{N}(0,I)$ through a series of fixed linear transforms and convolutions. If <b>deconvolution restoring</b> is feasible, it means that a fixed deconvolution can be used to restore any data distribution $q(x)$ from the $\mathcal{N}(0,I)$ , this is clearly <b>paradoxical</b>. The same input, the same transform, cannot have multiple different outputs.
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution_en")
+            In this context, $Q_{x|z}(m,n)$ represents the element at the m-th row and n-th column of the matrix $Q_{x|z}$, and $q_{i1}(n)$ represents the n-th element of the vector $q_{i1}$.
+                     
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_proof_ctr_en")
     return
 
 
@@ -448,6 +554,8 @@ def md_reference_en():
             <a id="mc_basic_t7" href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf"> [13] Markov Chain:Basic Theory - Theorem 7 </a>
             
             <a id="mc_basic_d4" href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf"> [14] Markov Chain:Basic Theory - Definition 4 </a>
+            
+            <a id="vdm" href="https://arxiv.org/pdf/2107.00630"> [15] Variational Diffusion Models </a>
              
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_reference_en")
 
@@ -461,7 +569,7 @@ def md_about_en():
 
         gr.Markdown(
             r"""
-            <b>APP</b>: This APP is developed using Gradio and deployed on HuggingFace. Due to limited resources (2 cores, 16G memory), the response may be slow. For a better experience, it is recommended to clone the source code from <a href="https://github.com/blairstar/The_Art_of_DPM">github</a> and run it locally. This program only relies on Gradio, SciPy, and Matplotlib.
+            <b>APP</b>: This Web APP is developed using Gradio and deployed on HuggingFace. Due to limited resources (2 cores, 16G memory), the response may be slow. For a better experience, it is recommended to clone the source code from <a href="https://github.com/blairstar/The_Art_of_DPM">github</a> and run it locally. This program only relies on Gradio, SciPy, and Matplotlib.
             
             <b>Author</b>: Zhenxin Zheng, Senior computer vision engineer with ten years of algorithm development experience, Formerly employed by Tencent and JD.com, currently focusing on image and video generation.
             
@@ -493,6 +601,10 @@ def run_app():
         
         md_deconvolution_en()
         
+        md_cond_kl_en()
+        
+        md_proof_ctr_en()
+        
         md_reference_en()
         
         md_about_en()

RenderMarkdownZh.py

@@ -41,9 +41,9 @@ def md_transform_zh():
 
             此变换可分为两个子变换。
 
-            第一个子变换是对随机变量$X$执行一个线性变换($\sqrt{\alpha}X$)，根据文献[\[3\]](#linear_transform)的结论，线性变换使$X$的概率分布“变窄变高”，并且"变窄变高"的程度与$\alpha$的值成正比；具体可看Demo 1，左1图为随机生成的一维的概率分布，左2图是经过线性变换后的概率分布，可以看出，与左1图相比，左2图的曲线“变窄变高”了。读者可亲自测试不同的$\alpha$值，获得更直观的理解。
+            第一个子变换是对随机变量$X$执行一个线性变换($\sqrt{\alpha}X$)，根据文献[\[3\]](#linear_transform)的结论，线性变换使$X$的概率分布“变窄变高”，并且"变窄变高"的程度与$\alpha$的值成正比。具体可看<a href="#demo_1">Demo 1</a>，左1图为随机生成的一维的概率分布，左2图是经过线性变换后的概率分布，可以看出，与左1图相比，左2图的曲线“变窄变高”了。读者可亲自测试不同的$\alpha$值，获得更直观的理解。
 
-            第二个子变换是“加上独立的随机噪声”($\sqrt{1-\alpha}\epsilon$)，根据文献[\[4\]](#sum_conv)的结论，“加上独立的随机变量”等效于对两个概率分布执行卷积，由于随机噪声的概率分布为高斯形状，所以相当于执行”高斯模糊“的操作。经过模糊后，原来的概率分布将变得更加平滑，与标准正态分布将更加相似。模糊的程度与噪声大小($1-\alpha$)正相关。具体可看Demo 1，左1图是随机生成的一维概率分布，左3图是经过变换后的结果，可以看出，变换后的曲线变光滑了，棱角变少了。读者可测试不同的$\alpha$值，感受噪声大小对概率分布曲线形状的影响。左4图是综合两个子变换后的结果。
+            第二个子变换是“加上独立的随机噪声”($\sqrt{1-\alpha}\epsilon$)，根据文献[\[4\]](#sum_conv)的结论，“加上独立的随机变量”等效于对两个概率分布执行卷积，由于随机噪声的概率分布为高斯形状，所以相当于执行”高斯模糊“的操作。经过模糊后，原来的概率分布将变得更加平滑，与标准正态分布将更加相似。模糊的程度与噪声大小($1-\alpha$)正相关。具体可看<a href="#demo_1">Demo 1</a>，左1图是随机生成的一维概率分布，左3图是经过变换后的结果，可以看出，变换后的曲线变光滑了，棱角变少了。读者可测试不同的$\alpha$值，感受噪声大小对概率分布曲线形状的影响。左4图是综合两个子变换后的结果。
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_transform_zh")
     return
 
@@ -59,7 +59,7 @@ def md_likelihood_zh():
             \begin{align}
                 q(z|x) &= \mathcal{N}(\sqrt{\alpha}x,\ 1-\alpha)    \tag{2.1}
             \end{align}
-            具体可看Demo 2，左3图展示了$q(z|x)$的形状，从图中可以看到一条均匀的斜线，这意味着$q(z|x)$的均值与x线性相关，方差固定不变。$\alpha$值的大小将决定斜线宽度和倾斜程度。
+            具体可看<a href="#demo_2">Demo 2</a>，左3图展示了$q(z|x)$的形状，从图中可以看到一条均匀的斜线，这意味着$q(z|x)$的均值与x线性相关，方差固定不变。$\alpha$值的大小将决定斜线宽度和倾斜程度。
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_likelihood_zh")
     return
 
@@ -96,12 +96,12 @@ def md_posterior_zh():
                 <li> 当高斯函数的方差较大(较大噪声)，或者$q(x)$剧烈变化时，$q(x|z)$的形状将较复杂，与高斯函数有较大的差别，难以建模学习。</li>
             </ul>
             
-            具体可看Demo 2，左4图给出后验概率分布$q(x|z)$的形态，可以看出，其形状较不规则，像一条弯曲且不均匀的曲线。当$\alpha$较大时(噪声较小)，曲线将趋向于均匀且笔直。读者可调整不同的$\alpha$值，观察后验概率分布与噪声大小的关系；左5图，$\textcolor{blue}{蓝色虚线}$给出$q(x)$，$\textcolor{green}{绿色虚线}$给出式3.4中的GaussFun，$\textcolor{orange}{黄色实线}$给出两者相乘并归一化的结果，即固定z条件下后验概率$q(x|z=fixed)$。读者可调整不同z值，观察$q(x)$的波动变化对后验概率$q(x|z)$形态的影响。
+            具体可看<a href="#demo_2">Demo 2</a>，左4图给出后验概率分布$q(x|z)$的形态，可以看出，其形状较不规则，像一条弯曲且不均匀的曲线。当$\alpha$较大时(噪声较小)，曲线将趋向于均匀且笔直。读者可调整不同的$\alpha$值，观察后验概率分布与噪声大小的关系；左5图，$\textcolor{blue}{蓝色虚线}$给出$q(x)$，$\textcolor{green}{绿色虚线}$给出式3.4中的GaussFun，$\textcolor{orange}{黄色实线}$给出两者相乘并归一化的结果，即固定z条件下后验概率$q(x|z=fixed)$。读者可调整不同z值，观察$q(x)$的波动变化对后验概率$q(x|z)$形态的影响。
             
             两个特殊状态下的后验概率分布$q(x|z)$值得考虑一下。
             <ul>
-                <li> 当$\alpha \to 0$时，GaussFun的方差趋向于<b>无穷大</b>，不同$z$值的$q(x|z)$几乎变成一致，并与$q(x)$几乎相同。读者可在Demo 2中，将$\alpha$设置为0.01，观察具体的结果。</li>
-                <li> 当$\alpha \to 1$时，GaussFun的方差趋向于<b>无穷小</b>，不同$z$值的$q(x|z)$收缩成一系列不同偏移量的Dirac delta函数, 偏移量等于$z$。但有一些例外，当q(x)存在为零的区域时，其对应的q(x|z)将不再为Dirac delta函数，而是零函数。可在Demo 2中，将$\alpha$设置为0.999，观察具体的结果。</li>
+                <li> 当$\alpha \to 0$时，GaussFun的方差趋向于<b>无穷大</b>，不同$z$值的$q(x|z)$几乎变成一致，并与$q(x)$几乎相同。读者可在<a href="#demo_2">Demo 2</a>中，将$\alpha$设置为0.01，观察具体的结果。</li>
+                <li> 当$\alpha \to 1$时，GaussFun的方差趋向于<b>无穷小</b>，不同$z$值的$q(x|z)$收缩成一系列不同偏移量的Dirac delta函数, 偏移量等于$z$。但有一些例外，当q(x)存在为零的区域时，其对应的q(x|z)将不再为Dirac delta函数，而是零函数。可在<a href="#demo_2">Demo 2</a>中，将$\alpha$设置为0.999，观察具体的结果。</li>
             </ul>
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_zh")
     return
@@ -117,11 +117,11 @@ def md_forward_process_zh():
             r"""
             对于任意的数据分布$q(x)$，均可连续应用上述的变换(如式4.1~4.4)，随着变换的次数的增多，输出的概率分布将变得越来越接近于标准正态分布。对于较复杂的数据分布，需要较多的次数或者较大的噪声。
 
-            具体可看Demo 3.1，第一子图是随机生成的一维概率分布，经过7次的变换后，最终的概率分布与标准正态分布非常相似。相似的程度与迭代的次数和噪声大小正相关。对于相同的相似程度，如果每次所加的噪声较大(较小的$\alpha$值)，那所需变换的次数将较少。读者可尝试不同的$\alpha$值和次数，观测最终概率分布的相似程度。
+            具体可看<a href="#demo_3_1">Demo 3.1</a>，第一子图是随机生成的一维概率分布，经过7次的变换后，最终的概率分布与标准正态分布非常相似。相似的程度与迭代的次数和噪声大小正相关。对于相同的相似程度，如果每次所加的噪声较大(较小的$\alpha$值)，那所需变换的次数将较少。读者可尝试不同的$\alpha$值和次数，观测最终概率分布的相似程度。
 
             起始概率分布的复杂度会比较高，随着变换的次数增多，概率分布$q(z_t)$的复杂度将会下降。根据第3节结论，更复杂的概率分布对应更复杂的后验概率分布，所以，为了保证后验概率分布与高斯函数较相似(较容易学习)，在起始阶段，需使用较大的$\alpha$(较小的噪声)，后期阶段可适当使用较小的$\alpha$(较大的噪声)，加快向标准正态分布转变。
 
-            在Demo3的例子可以看到，随着变换次数增多，$q(z_t)$的棱角变得越来越少，同时，后验概率分布$q(z_{t-1}|z_t)$图中的斜线变得越来越笔直匀称，越来越像条件高斯分布。
+            在<a href="#demo_3_1">Demo 3.1</a>的例子可以看到，随着变换次数增多，$q(z_t)$的棱角变得越来越少，同时，后验概率分布$q(z_{t-1}|z_t)$图中的斜线变得越来越笔直匀称，越来越像条件高斯分布。
 
             \begin{align}
                 Z_1   &= \sqrt{\alpha_1} X + \sqrt{1-\alpha_1}\epsilon_1 			\tag{4.1}   \newline
@@ -152,7 +152,7 @@ def md_forward_process_zh():
                 q(z_T|x) &= \mathcal{N}(0.00635\ x,\ 0.99998)    \tag{4.9}
             \end{align}
             
-            如果只考虑$q(z_T)$，也可使用一次变换代替，变换如下:
+            如果只考虑边际分布$q(z_T)$，也可使用一次变换代替，变换如下:
             \begin{align}
                 Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon 			 \tag{4.10}
             \end{align}
@@ -171,7 +171,7 @@ def md_backward_process_zh():
             r"""
             如果知道了最终的概率分布$q(z_T)$及各个转换过程的后验概率$q(x|z),q(z_{t-1}|z_t)$，则可通过“贝叶斯公式”和“全概率公式”恢复数据分布$q(x)$，见式5.1~5.4。当最终的概率分布$q(z_T)$与标准正态分布很相似时，可用标准正态分布代替。
             
-            具体可看Demo 3.2。示例中$q(z_T)$使用$\mathcal{N}(0,1)$代替，同时通过JS Div给���了误差大小。恢复的概率分布$q(z_t)$及$q(x)$使用$\textcolor{green}{绿色曲线}$标识，原始的概率分布使用$\textcolor{blue}{蓝色曲线}$标识。可以看出，数据分布$q(x)$能够被很好地恢复回来，并且误差(JS Divergence)会小于标准正态分布替换$q(z_T)$引起的误差。
+            具体可看<a href="#demo_3_2">Demo 3.2</a>。示例中$q(z_T)$使用$\mathcal{N}(0,1)$代替，同时通过JS Div给出了误差大小。恢复的概率分布$q(z_t)$及$q(x)$使用$\textcolor{green}{绿色曲线}$标识，原始的概率分布使用$\textcolor{blue}{蓝色曲线}$标识。可以看出，数据分布$q(x)$能够被很好地恢复回来，并且误差(JS Divergence)会小于标准正态分布替换$q(z_T)$引起的误差。
             \begin{align}
                 q(z_{T-1}) &= \int q(z_{T-1},z_T)dz_T = \int q(z_{T-1}|z_T)q(z_T)dz_T               	    \tag{5.1}   \newline
                            & \dots	                                                                        \notag      \newline
@@ -180,21 +180,23 @@ def md_backward_process_zh():
                 q(z_1)     &= \int q(z_1,z_2) dz_1    = \int q(z_1|z_2)q(z_2)dz_2                           \tag{5.3}   \newline
                 q(x)       &= \int q(x,z_1) dz_1      = \int q(x|z_1)q(z_1)dz_1                             \tag{5.4}   \newline
             \end{align}
-            在本文中，将上述恢复过程(式5.1~5.4)所使用的变换称之为“后验概率变换”。例如，在式5.4中，变换的输入为概率分布函数$q(z_1)$，输出为概率分布函数$q(x)$，整个变换由后验概率分布$q(x|z_1)$决定。此变换也可看作为一组基函数的线性加权和，基函数为不同条件下的$q(x|z_1)$，各个基函数的权重为$q(z_1)$。在第7节，将会进一步介绍此变换的一些有趣性质。
+            在本文中，将上述恢复过程(式5.1~5.4)所使用的变换称之为“后验概率变换”。例如，在式5.4中，变换的输入为概率分布函数$q(z_1)$，输出为概率分布函数$q(x)$，整个变换由后验概率分布$q(x|z_1)$决定。此变换也可看作为一组基函数的线性加权和，基函数为不同条件下的$q(x|z_1)$，各个基函数的权重为$q(z_1)$。在<a href="#posterior_transform">第7节</a>，将会进一步介绍此变换的一些有趣性质。
             
-            在第3节中，我们考虑了两个特殊的后验概率分布。接下来，分析其对应的”后验概率变换“。
+            在<a href="#posterior">第3节</a>中，我们考虑了两个特殊的后验概率分布。接下来，分析其对应的”后验概率变换“。
             <ul>
                 <li> 当$\alpha \to 0$时，不同$z$值的$q(x|z)$均与$q(x)$几乎相同，也就是说，线性加权和的基函数几乎相同。此状态下，不管输入如何变化，变换的输出总为$q(x)$。</li>
                 <li> 当$\alpha \to 1$时，不同$z$值的$q(x|z)$收缩成一系列不同偏移量的Dirac delta函数及零函数。此状态下，只要输入分布的支撑集(support set)包含于$q(x)$的支撑集，变换的输出与输入将保持一致。</li>
             </ul>
             
-            在第5节中提到，DDPM[\[2\]](#ddpm)论文所使用的1000次变换可使用一次变换表示：
+            在<a href="#forward_process">第4节</a>中提到，DDPM[\[2\]](#ddpm)论文所使用的1000次变换可使用一次变换表示：
             \begin{align}
                 Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon 			 \tag{5.5}
             \end{align}
             由于$\alpha=0.0000403$非常小，其对应的GaussFun(式3.4)的标准差达到157.52，而$X$的范围限制在$[-1, 1]$，远小于GaussFun的标准差。在$x \in [-1, 1]$范围内，GaussFun应该接近于常量，没有什么变化，所以不同的$z_T$对应的$q(x|z_T)$均与$q(x)$几乎相同。在这种状态下，对于$q(x|z_T)$相应的后验概率变换，不管输入分布是什么，输出分布都将是$q(x)$。
             
             <b>所以，理论上，在DDPM模型中，无需非得使用标准正态分布代替$q(z_T)$，也可使用其它任意的分布代替。</b>
+            
+            读者可亲自做一个类似的实验。在<a href="#demo_3_1">Demo 3.1</a>中，将start_alpha设置0.25，end_alpha也设置为0.25，step设置为7，此时$q(z_7)=\sqrt{0.000061}X + \sqrt{1-0.000061}\epsilon$，与DDPM的$q(z_T)$基本相似。点击<b>apply</b>执行前向变换($\textcolor{blue}{蓝色曲线}$)，为接下来的反向恢复做准备。在<a href="#demo_3_2">Demo 3.2</a>中，noise_ratio设置为1，为末端分布$q(z_7)$引入100%的噪声，切换nose_random_seed的值可改变噪声的分布，取消选择backward_pdf，减少画面的干扰。点击<b>apply</b>将通过后验概率变换恢复$q(x)$，将会看到，不管输入的$q(z_7)$的形状如何，恢复的$q(x)$均与原始的$q(x)$完全相同, JS Divergence为0，���复的过程使用$\textcolor{red}{红色曲线}$画出。
              
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_backward_process_zh")
     return
@@ -209,13 +211,13 @@ def md_fit_posterior_zh():
         # because of the render bug in gradio markdown, some formulas are render in ExtraBlock.js
         gr.Markdown(
             r"""
-            由第四节前半部分可知，各个后验概率分布是未知的，并且与$q(x)$有关。所以，为了恢复数据分布或者从数据分布中采样，需要对各个后验概率分布进行学习估计。
+            由<a href="#posterior">第3节</a>前半部分可知，各个后验概率分布是未知的，并且与$q(x)$有关。所以，为了恢复数据分布或者从数据分布中采样，需要对各个后验概率分布进行学习估计。
 
-            由第四节后半部分可知，当满足一定条件时，各个后验概率分布$q(x|z)、q(z_{t-1}|z_t)$近似于高斯概率分布，所以可通过构建一批条件高斯概率模型$p(x|z),p(z_{t-1}|z_t)$，学习拟合对应的$q(x|z),q(z_{t-1}|z_t)$。
+            由<a href="#posterior">第3节</a>后半部分可知，当满足一定条件时，各个后验概率分布$q(x|z)、q(z_{t-1}|z_t)$近似于高斯概率分布，所以可通过构建一批条件高斯概率模型$p(x|z),p(z_{t-1}|z_t)$，学习拟合对应的$q(x|z),q(z_{t-1}|z_t)$。
 
-            由于模型表示能力和学习能力的局限性，拟合过程会存在一定的误差，进一步会影响恢复$q(x)$的准确性。拟合误差大小与后验概率分布的形状有关。由第4节可知，当$q(x)$较复杂或者所加噪声较大时，后验概率分布会较复杂，与高斯分布差别较大，从而导致拟合误差，进一步影响恢复$q(x)$。
+            由于模型表示能力和学习能力的局限性，拟合过程会存在一定的误差，进一步会影响恢复$q(x)$的准确性。拟合误差大小与后验概率分布的形状有关。由<a href="#posterior">第3节</a>可知，当$q(x)$较复杂或者所加噪声较大时，后验概率分布会较复杂，与高斯分布差别较大，从而导致拟合误差，进一步影响恢复$q(x)$。
             
-            具体可看Demo 3.3，读者可测试不同复杂程度的$q(x)$和$\alpha$，观看后验概率分布$q(z_{t-1}|z_t)$的拟合程度，以及恢复$q(x)$的准确度。恢复的概率分布使用$\textcolor{orange}{橙色}$标识，同时也通过JS divergence给出误差。
+            具体可看<a href="#demo_3_3">Demo 3.3</a>，读者可测试不同复杂程度的$q(x)$和$\alpha$，观看后验概率分布$q(z_{t-1}|z_t)$的拟合程度，以及恢复$q(x)$的准确度。恢复的概率分布使用$\textcolor{orange}{橙色}$标识，同时也通过JS divergence给出误差。
 
             关于拟合的目标函数，与其它概率模型类似，可$\textcolor{red}{优化交叉熵损失}$，使$p(z_{t-1}|z_t)$逼近于$q(z_{t-1}|z_t)$。由于$(z_{t-1}|z_t)$是条件概率，所以需要综合考虑各个条件，以<b>各个条件发生的概率$q(z_t)$</b>加权平均<b>各个条件对应的交叉熵</b>。最终的损失函数形式如下:
             \begin{align}
@@ -224,7 +226,7 @@ def md_fit_posterior_zh():
             \end{align}
             也可以KL散度作为目标函数进行优化，KL散度与交叉熵是等价的[\[10\]](#ce_kl)。
             <span id="zh_fit_0">
-                loss &= \int q(z_t) KL(q(z_{t-1}|z_t)\|\textcolor{blue}{p(z_{t-1}|z_t)})dz_t                                                             \tag{6.3}      \newline
+                loss &= \int q(z_t) KL(q(z_{t-1}|z_t) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dz_t                                                             \tag{6.3}      \newline
                      &= \int q(z_t) \int q(z_{t-1}|z_t) \frac{q(z_{t-1}|z_t)}{\textcolor{blue}{p(z_{t-1}|z_t)}} dz_{t-1} dz_t                            \tag{6.4}      \newline 
                      &= -\int q(z_t)\ \underbrace{\int q(z_{t-1}|z_t) \log \textcolor{blue}{p(z_{t-1}|z_t)}dz_{t-1}}{underline}{\text{Cross Entropy}}\ dz_t + \underbrace{\int q(z_t) \int q(z_{t-1}|z_t) \log q(z_{t-1}|z_t)}{underline}{\text{Is Constant}} dz  \tag{6.5}
             </span>
@@ -248,8 +250,8 @@ def md_fit_posterior_zh():
                      &\quad - \iint \int q(x)q(z_{t-1}, z_t|x) \log \textcolor{blue}{p(z_{t-1}|z_t)}dxdz_{t-1}dz_t - \textcolor{orange}{C_1}                                                \tag{6.11}          \newline
                      &= \iint \int q(x)q(z_{t-1},z_t|x) \log \frac{q(z_{t-1}|z_t,x)}{\textcolor{blue}{p(z_{t-1}|z_t)}}dxdz_{t-1}dz_t - \textcolor{orange}{C_1}                              \tag{6.12}          \newline
                      &= \iint q(x)q(z_t|x)\int q(z_{t-1}|z_t,x) \log \frac{q(z_{t-1}|z_t,x)}{\textcolor{blue}{p(z_{t-1}|z_t)}}dz_{t-1}\ dz_xdz_t - \textcolor{orange}{C_1}                  \tag{6.13}          \newline
-                     &= \iint \ q(x)q(z_t|x) KL[q(z_{t-1}|z_t,x)\|\textcolor{blue}{p(z_{t-1}|z_t)}]dxdz_t - \textcolor{orange}{C_1}                                                         \tag{6.14}          \newline
-                     &\propto \iint \ q(x)q(z_t|x) KL[q(z_{t-1}|z_t,x)\|\textcolor{blue}{p(z_{t-1}|z_t)}]dxdz_t                                                                             \tag{6.15}          \newline
+                     &= \iint \ q(x)q(z_t|x) KL(q(z_{t-1}|z_t,x) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dxdz_t - \textcolor{orange}{C_1}                                                         \tag{6.14}          \newline
+                     &\propto \iint \ q(x)q(z_t|x) KL(q(z_{t-1}|z_t,x) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dxdz_t                                                                             \tag{6.15}          \newline
             \end{align}
 
             上式中的$C_1$项是一个固定值，不包含待优化的参数，其中，$q(x)$是固定的概率分布，$q(z_{t-1}|z_t)$也是固定概率分布，具体形式由$q(x)$及系数$\alpha$确定。
@@ -265,14 +267,15 @@ def md_fit_posterior_zh():
              
             根据一致项证明的结论，以及交叉熵与KL散度的关系，可得出一个有趣的结论：
             <span id="zh_fit_1">
-                \mathop{\min}{underline}{\textcolor{blue}{p}} \int q(z_t) KL(q(z_{t-1}|z_t)\|\textcolor{blue}{p(z_{t-1}|z_t)})dz_t  \iff  \mathop{\min}{underline}{\textcolor{blue}{p}} \iint \ q(x)q(z_t|x) KL[q(z_{t-1}|z_t,x)\|\textcolor{blue}{p(z_{t-1}|z_t)}]dxdz_t
+                \mathop{\min}{underline}{\textcolor{blue}{p}} \int q(z_t) KL(q(z_{t-1}|z_t) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dz_t  \iff  \mathop{\min}{underline}{\textcolor{blue}{p}} \iint \ q(x)q(z_t|x) KL(q(z_{t-1}|z_t,x) \Vert \textcolor{blue}{p(z_{t-1}|z_t)})dxdz_t
             </span>
             比较左右两边的式子，可以看出，右边的目标函数比左边的目标函数多了一个条件变量$X$，同时也多了一个关于$X$积分，并且以$X$的发生的概率$q(x)$作为积分的加权系数。
             
             依照类似的思路，可推导出一个更通用的关系：
             <span id="zh_fit_2">
-                \mathop{\min}{underline}{\textcolor{blue}{p}}  KL(q(z)\|\textcolor{blue}{p(z)})  \iff  \mathop{\min}_{\textcolor{blue}{p}} \int \ q(x) KL(q(z|x)\|\textcolor{blue}{p(z)})dx
+                \mathop{\min}{underline}{\textcolor{blue}{p}}  KL(q(z) \Vert \textcolor{blue}{p(z)})  \iff  \mathop{\min}_{\textcolor{blue}{p}} \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p(z)})dx
             </span>
+            关于此结论的详细推导，可见<a href="#cond_kl">Appendix A</a>。
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_fit_posterior_zh")
     return
 
@@ -284,7 +287,7 @@ def md_posterior_transform_zh():
 
         gr.Markdown(
             r"""
-            <h3 style="font-size:18px"> Contraction Mapping and Fixed Point </h3>
+            <h3 style="font-size:18px"> 压缩映射及收敛点 </h3>
             \begin{align}
                 q(x) &= \int q(x,z) dz = \int q(x|z)q(z)dz      \tag{7.1}
             \end{align}
@@ -294,25 +297,161 @@ def md_posterior_transform_zh():
                 dist(q_{o1}(z),\ q_{o2}(z)) < dist(q_{i1}(x),\ q_{i2}(x))                   \tag{7.2}
             \end{align}
 
-            读者可查看Demo 4.1，左侧三个图呈现一个变换的过程，左1图是任意的数据分布$q(x)$，左3图是变换后的概率分布，左2图是后验概率分布。可更改随机种子生成新的数据分布，调整$\alpha$值引入不同程度的噪声。左侧最后两个图展示变换的“压缩性质”，左4图展示随机生成的两个输入分布，同时给出其距离度量值$div_{in}$；左5图展示经过变换后的两个输出分布，输出分布之间的距离标识为$div_{out}$。读者可改变输入的随机种子，切换不同的输入。可在图中看到，对于任意的输入，$div_{in}$总是小于$div_{out}$。另外，也可改变$\alpha$的值，将会看到，$\alpha$越小(噪声越大)，$\frac{div_{out}}{div_{in}}$的比值也越小，即收缩率越大。
-
+            读者可查看<a href="#demo_4_1">Demo 4.1</a>，左侧三个图呈现一个变换的过程，左1图是任意的数据分布$q(x)$，左3图是变换后的概率分布，左2图是后验概率分布。可更改随机种子生成新的数据分布，调整$\alpha$值引入不同程度的噪声。左侧最后两个图展示变换的“压缩性质”，左4图展示随机生成的两个输入分布，同时给出其距离度量值$div_{in}$；左5图展示经过变换后的两个输出分布，输出分布之间的距离标识为$div_{out}$。读者可改变输入的随机种子，切换不同的输入。可在图中看到，对于任意的输入，$div_{in}$总是小于$div_{out}$。另外，也可改变$\alpha$的值，将会看到，$\alpha$越小(噪声越大)，$\frac{div_{out}}{div_{in}}$的比值也越小，即收缩率越大。
+            
             由Banach fixed-point theorem<a href="#fixed_point">[5]</a>可知，压缩映射存在惟一一个定点(收敛点)。也就是说，对于任意的输入分布，可以连续迭代应用“后验概率变换”，只要迭代次数足够多，最终都会输出同一个分布。经过大量一维随机变量实验发现，定点(收敛点)<b>位于$q(x)$附近</b>。并且，与$\alpha$的值有关，$\alpha$越小(噪声越大)，离得越近。
             
-            读者可看Demo 4.2，此部分展示迭代收敛的例子。选择合适的迭代次数，点中“apply iteration transform”，将逐步画出迭代的过程，每个子图均会展示各自变换后的输出分布($\textcolor{green}{绿色曲线}$)，收敛的参考点分布$q(x)$以$\textcolor{blue}{蓝色曲线}$画出，同时给出输出分布与$q(x)$之间的距离$dist$。可以看出，随着迭代的次数增加，输出分布与$q(x)$越来越相似，并最终会稳定在$q(x)$附近。对于较复杂的分布，可能需要较多迭代的次数或者较大的噪声。迭代次数可以设置为上万步，但会花费较长时间。
+            读者可看<a href="#demo_4_2">Demo 4.2</a>，此部分展示迭代收敛的例子。选择合适的迭代次数，点中“apply iteration transform”，将逐步画出迭代的过程，每个子图均会展示各自变换后的输出分布($\textcolor{green}{绿色曲线}$)，收敛的参考点分布$q(x)$以$\textcolor{blue}{蓝色曲线}$画出，同时给出输出分布与$q(x)$之间的距离$dist$。可以看出，随着迭代的次数增加，输出分布与$q(x)$越来越相似，并最终会稳定在$q(x)$附近。对于较复杂的分布，可能需要较多迭代的次数或者较大的噪声。迭代次数可以设置为上万步，但会花费较长时间。
 
-            对于一维离散的情况，$q(x|z)$将离散成一个矩阵(暂记为$Q_{x|z}$)，$q(z)$离散成一个向量(记为$\boldsymbol{q_i}$)，积分操作$\int q(x|z)q(z)dz$将离散成"矩阵-向量"乘法操作，所以后验概率变换可写成
+            对于一维离散的情况，$q(x|z)$将离散成一个矩阵(记为$Q_{x|z}$)，$q(z)$离散成一个向量(记为$\boldsymbol{q_i}$)，积分操作$\int q(x|z)q(z)dz$将离散成"矩阵-向量"乘法操作，所以后验概率变换可写成
             \begin{align}
                 \boldsymbol{q_o} &= Q_{x|z}\ \boldsymbol{q_i} &             \quad\quad        &\text{1 iteration}         \tag{7.3}       \newline
                 \boldsymbol{q_o} &= Q_{x|z}\ Q_{x|z}\ \boldsymbol{q_i} &    \quad\quad        &\text{2 iteration}         \tag{7.4}       \newline
                     & \dots &                                                                                             \notag          \newline
                 \boldsymbol{q_o} &= (Q_{x|z})^n\ \boldsymbol{q_i} &         \quad\quad        &\text{n iteration}         \tag{7.5}       \newline
             \end{align}
-            于是，为了更深入地理解变换的特点，Demo 4.2也画出矩阵$(Q_{x|z})^n$的结果。从图里可以看到，当迭代趋向收敛时，矩阵$(Q_{x|z})^n$的行向量将变成一个常数向量，即向量的各分量都相等。在二维密度图里将表现为一条横线。
+            于是，为了更深入地理解变换的特点，<a href="#demo_4_2">Demo 4.2</a>也画出矩阵$(Q_{x|z})^n$的结果。从图里可以看到，当迭代趋向收敛时，矩阵$(Q_{x|z})^n$的行向量将变成一个常数向量，即向量的各分量都相等。在二维密度图里将表现为一条横线。
+
+            在<a href="#proof_ctr">Appendix B</a>中，将会提供一个证明，当$q(x)$和$\alpha$满足一些条件时，后验概率变换是一个严格的压缩映射。
+
+            关于定点分布与输入分布q(x)之间距离的关系，目前尚不能严格证明。
+            
+            <h3 style="font-size:18px"> 恢复数据分布过程中的抗噪声能力 </h3>
+            由上面的分析可知，当满足一些条件时，"后验概率变换"是一个压缩映射，所以存在如下的关系：
+            \begin{align}
+                dist(q(x),\ q_o(x)) < dist(q(z),\ q_i(z))         \tag{7.12}
+            \end{align}
+            其中，$q(z)$是理想的输入分布，$q(x)$理想的输出分布，$q_i(x)$是任意的输入分布，$q_o(x)$是$q_i(z)$经过变换后的输出分布。
+            
+            上式表明，输出的分布$q_o(x)$与理想输出分布q(x)之间的距离总会</em>小于</em>输入分布$q_i(z)$与理想输入分布q(x)的距离。于是，"后验概率变换"具备一定的抵抗噪声能力。这意味着，在恢复$q(x)$的过程中(<a href="#backward_process">第5节</a>)，哪怕输入的“末尾分布$q(z_T)”$存在一定的误差，经过一系列变换后，输出的“数据分布$q(x)$“的误差也会比输入的误差更小。
+            
+            具体可看<a href="#demo_3_2">Demo 3.2</a>，通过增加“noise ratio”的值可以向“末尾分布$q(z_T)$”添加噪声，点击“apply”按钮将逐步画出恢复的过程，恢复的分布以$\textcolor{red}{红色曲线}$画出，同时也会通过JS散度标出误差的大小。将会看到，恢复的$q(x)$的误差总是小于$q(z_T)$的误差。
+            
+            由上面的讨论可知，$\alpha$越小(即变换过程中使用的噪声越大)，压缩映射的压缩率越大，于是，抗噪声的能力也越强。
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform_zh")
+    return
+
+
+def md_deconvolution_zh():
+    global g_latex_del
+    
+    title = "8. Can the data distribution be restored by deconvolution?"
+    with gr.Accordion(label=title, elem_classes="first_md", elem_id="deconvolution"):
+
+        gr.Markdown(
+            r"""
+            在<a href="#introduction">第1节</a>中提到，式2.1的变换可分为两个子变换，第一个子变换为”线性变换“，第二个为“加上独立高斯噪声”。线性变换相当于对概率分布进行拉伸变换，所以存在逆变换。"加上独立高斯噪声”相当于对概率分布执行卷积操作，卷积操作可通过逆卷积恢复。所以，理论上，可通过“逆线性变换”和“逆卷积”从最终的概率分布$q(z_T)$恢复数据分布$q(x)$。
+            
+            但实际上，会存在一些问题。由于逆卷积对误差极为敏感，具有很高的输入灵敏度，很小的输入噪声就会引起输出极大的变化[\[11\]](#deconv_1)[\[12\]](#deconv_2)。而在扩散模型中，会使用标准正态分布近似代替$q(z_T)$，因此，在恢复的起始阶段就会引入噪声。虽然噪声较小，但由于逆卷积的敏感性，噪声会逐步放大，影响恢复。
+            
+            另外，也可以从另一个角度理解“逆卷积恢复”的不可行性。由于前向变换的过程(式4.1~4.4)是确定的，所以卷积核是固定的，因此，相应的“逆卷积变换“也是固定的。由于起始的数据分布$q(x)$可以是任意的分布，所以，通过一系列固定的“卷积正变换”，可以将任意的概率分布转换成近似$\mathcal{N}(0,I)$的分布。如“逆卷积变换“可行，则意味着，可用一个固定的“逆卷积变换"，将$\mathcal{N}(0,I)$分布恢复成任意的数据分布$q(x)$，这明显是一个悖论。同一个输入，同一个变换，不可能会有多个输出。
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution_zh")
+    return
+
+
+def md_cond_kl_zh():
+    global g_latex_del
+
+    title = "Appendix A Conditional KL Divergence"
+    with gr.Accordion(label=title, elem_classes="first_md", elem_id="cond_kl"):
+        gr.Markdown(
+            r"""
+            本节主要介绍<b>KL散度</b>与<b>条件KL散度</b>之间的关系。在正式介绍之前，先简单介绍<b>熵</b>和<b>条件熵</b>的定义，以及两者之间存在的不等式关系，为后面的证明作准备。
+
+            <h3 style="font-size:18px">熵及条件熵</h3>
+            对于任意两个随机变量$Z,X$，<b>熵</b>(Entropy)定义如下<a href="#entropy">[16]</a>：
+            \begin{align}
+               \mathbf{H}(Z) = \int -p(z)\log{p(z)}dz    \tag{A.1}
+            \end{align}
+            <b>条件熵</b>(Conditional Entropy)的定义如下<a href="#cond_entropy">[17]</a>：
+            \begin{align}
+               \mathbf{H}(Z|X) = \int p(x) \overbrace{\int -p(z|x)\log{p(z|x)}dz}^{\text{Entropy}}\ dx    \tag{A.2}
+            \end{align}
+            两者存在如下的不等式关系：
+            \begin{align}
+               \mathbf{H}(Z|X) \le \mathbf{H}(Z)         \tag{A.3}
+            \end{align}
+            也就是说，条件熵总是小于或者等于熵，当且仅当X与Z相互独立时，两者相等。此关系的证明可看文献<a href="#cond_entropy">[17]</a>。
+            
+            <h3 style="font-size:18px"> KL散度及条件KL散度 </h3>
+            仿照条件熵定义的方式，引入一个新定义，<b>条件KL散度</b>，记为$KL_{\mathcal{C}}$。由于KL散度的定义是非对称的，所以存在两种形式，如下：
+            \begin{align}
+               KL_{\mathcal{C}}(q(z|x) \Vert \textcolor{blue}{p(z)}) = \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p(z)})dx                        \tag{A.4}     \newline
+               KL_{\mathcal{C}}(q(z) \Vert \textcolor{blue}{p(z|x)}) = \int \ \textcolor{blue}{p(x)} KL(q(z) \Vert \textcolor{blue}{p(z|x)})dx      \tag{A.5}
+            \end{align}
+            
+            与条件熵类似，条件KL散度也存在类似的不等式关系：
+            \begin{align}
+               KL_{\mathcal{C}}(q(z|x) \Vert \textcolor{blue}{p(z)}) \ge KL(q(z) \Vert \textcolor{blue}{p(z)})        \tag{A.6}   \newline
+               KL_{\mathcal{C}}(q(z) \Vert \textcolor{blue}{p(z|x)}) \ge KL(q(z) \Vert \textcolor{blue}{p(z)})        \tag{A.7}
+            \end{align}
+            也就是说，条件KL散度总是大于或者等于KL散度，当且仅当X与Z相互独立时，两者相等。
+             
+            下面对式A.5和式A.6的结论分别证明。
+            
+            对于式A.6，证明如下：
+            \begin{align}
+                KL_{\mathcal{C}}(q(z|x) \Vert \textcolor{blue}{p(z)}) &= \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p(z)})dx      \tag{A.8}    \newline
+                    &= \iint q(x) q(z|x) \log \frac{q(z|x)}{\textcolor{blue}{p(z)}}dzdx                                             \tag{A.9}    \newline
+                    &= -\overbrace{\iint - q(x)q(z|x) \log q(z|x) dzdx}^{\text{Condtional Entropy }\mathbf{H}_q(Z|X)} - \iint q(x) q(z|x) \log \textcolor{blue}{p(z)} dzdx          \tag{A.10}   \newline
+                    &= -\mathbf{H}_q(Z|X) - \int \left\lbrace \int q(x) q(z|x)dx \right\rbrace \log \textcolor{blue}{p(z)}dz                                                        \tag{A.11}   \newline
+                    &= -\mathbf{H}_q(Z|X) + \overbrace{\int - q(z) \log p(z)dz}^{\text{Cross Entropy}}                                                                              \tag{A.12}   \newline
+                    &= -\mathbf{H}_q(Z|X) + \int q(z)\left\lbrace \log\frac{q(z)}{\textcolor{blue}{p(z)}} -\log q(z)\right\rbrace dz                                                \tag{A.13}   \newline
+                    &= -\mathbf{H}_q(Z|X) + \int q(z)\log\frac{q(z)}{\textcolor{blue}{p(z)}}dz + \overbrace{\int - q(z)\log q(z)dz}^{\text{Entropy } \mathbf{H}_q(Z)}               \tag{A.14}   \newline
+                    &= KL(q(z) \Vert \textcolor{blue}{p(z)}) + \overbrace{\mathbf{H}_q(Z) - \mathbf{H}_q(Z|X)}^{\ge 0}              \tag{A.15}  \newline
+                    &\le KL(q(z) \Vert \textcolor{blue}{p(z)})      \tag{A.16}
+            \end{align}
+            其中式A.15应用了"条件熵总是小于或者等于熵"的结论。于是，得到式A.6的关系。
             
+            对于式A.7，证明如下：
+            \begin{align}
+                KL(\textcolor{blue}{q(z)} \Vert p(z)) &= \int \textcolor{blue}{q(z)}\log\frac{\textcolor{blue}{q(z)}}{p(z)}dx    \tag{A.15}          \newline
+                        &= \int q(z)\log\frac{q(z)}{\int p(z|x)p(z)dz}dz 						 	\tag{A.16}          \newline
+                        &= \textcolor{orange}{\int p(x)dx}\int q(z)\log q(z)dz - \int q(z)\textcolor{red}{\log\int p(z|x)p(x)dx}dz	\qquad \ \textcolor{orange}{\int p(x)dx=1}			    \tag{A.17}      \newline
+                        &\le \iint p(x) q(z)\log q(z)dzdx - \int q(z)\textcolor{red}{\int p(x)\log p(z|x)dx}dz \ \qquad 	 \textcolor{red}{\text{jensen\ inequality}}                     \tag{A.18}      \newline
+                        &= \iint p(x)q(z)\log q(z)dzdx - \iint p(z)q(z)\log p(z|x)dzdx			 	    \tag{A.19}          \newline
+                        &= \iint p(x)q(z)(\log q(z) - \log p(z|x))dzdx								    \tag{A.20}          \newline
+                        &= \iint p(x)q(z)\log \frac{q(z)}{p(z|x)}dzdx								    \tag{A.21}          \newline
+                        &= \int p(x)\left\lbrace \int q(z)\log \frac{q(z)}{p(z|x)}dz\right\rbrace dx    \tag{A.22}          \newline
+                        &= \int p(x)KL(\textcolor{blue}{q(z)} \Vert p(z|x))dx                           \tag{A.23}          \newline
+                        &= KL_{\mathcal{C}}(q(z) \Vert \textcolor{blue}{p(z|x)})                        \tag{A.24}
+            \end{align}
+            于是，得到式A.7的关系。
+           
+            从式A.15可得出另外一个<b>重要的结论</b>。
+            
+            KL散度常用于拟合数据的分布。在此场景中，数据潜在的分布用$q(z)$表示，参数化的模型分布用$\textcolor{blue}{p_\theta(z)}$表示。在优化的过程中，由于$q(z|x)$和$q(x)$均保持不变，所以式A.15中的$\mathbf{H}(Z) - \mathbf{H}(Z|X)$为一个常数项。于是，可得到如下的关系
+            <span id="zh_cond_kl_2">
+                \mathop{\min}{underline}{\textcolor{blue}{p_\theta}}  KL(q(z) \Vert \textcolor{blue}{p_\theta(z)})  \iff  \mathop{\min}{underline}{\textcolor{blue}{p_\theta}} \int \ q(x) KL(q(z|x) \Vert \textcolor{blue}{p_\theta(z)})dx   \tag{A.25}
+            </span>
+            
+            把上述的关系与Denoised Score Matching<a href="#dsm">[18]</a>作比较，可发现一些相似的地方。两者均引入一个新变量$X$，并且将拟合的目标分布q(z)代替为q(z|x)。代替后，由于q(z|x)是条件概率分布，所以，两者均考虑了所有的条件，并以条件发生的概率$q(x)$作为权重系数执行加权和。
+            <span id="zh_cond_kl_3">
+                \mathop{\min}{underline}{\textcolor{blue}{\psi_\theta}} \frac{1}{2} \int q(z) \left\lVert \textcolor{blue}{\psi_\theta(z)} - \frac{\partial q(z)}{\partial z} \right\rVert^2 dz \iff  \mathop{\min}{underline}{\textcolor{blue}{\psi_\theta}} \int q(x)\ \overbrace{\frac{1}{2} \int q(z|x) \left\lVert \textcolor{blue}{\psi_\theta(z)} - \frac{\partial q(z|x)}{\partial z} \right\rVert^2 dz}^{\text{Score Matching of }q(z|x)}\ dx      \tag{A.26}
+            </span>
+            
+            上述加权和的操作有点类似于"全概率公式消元"。
+            \begin{align}
+                q(z) = \int q(z,x) dx = \int q(x) q(z|x) dx     \tag{A.27}
+            \end{align}
+            
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_cond_kl_zh")
+    return
+
+
+def md_proof_ctr_zh():
+    global g_latex_del
+
+    title = "Appendix B Proof of Contraction"
+    with gr.Accordion(label=title, elem_classes="first_md", elem_id="proof_ctr"):
+        gr.Markdown(
+            r"""
             <center> <img src="file/fig2.png" width="960" style="margin-top:12px"/> </center>
             <center> Figure 2: Only one component in support </center>
-             
-            下面分几种情况证明，后验概率变换是一个压缩映射，并存在惟一收敛点。证明的过程假设随机变量是离散型的，因此，后验概率变换可看作是一个<b>离散Markov Chain</b>的一步转移，后验概率$q(x|z)$对应于<b>转移矩阵</b>(Transfer Matrix)。连续型的变量可认为是无限多状态的离散型变量。
+            
+            本节将证明，当$q(x)$及$\alpha$满足一些条件时，后验概率变换是一个压缩映射，并存在惟一收敛点。
+            
+            下面分四种情况进行证明。证明的过程假设随机变量是离散型的，因此，后验概率变换可看作是一个<b>离散Markov Chain</b>的一步转移，后验概率$q(x|z)$对应于<b>转移矩阵</b>(Transfer Matrix)。连续型的变量可认为是无限多状态的离散型变量。
             
             <ol style="list-style-type:decimal">
             <li> 当$q(x)$均大于0时，后验概率变换矩阵$q(x|z)$将大于0，于是此矩阵是一个$\textcolor{red}{不可约}\textcolor{green}{非周期}$的Markov Chain的转移矩阵，根据文献<a href="#mc_basic_p6">[13]</a>的结论，此变换是一个关于Total Variance度量的压缩映射，于是，根据Banach fixed-point theorem，此变换存在惟一定点(收敛点)。</li>
@@ -354,55 +493,23 @@ def md_posterior_transform_zh():
             
             另外，后验概率变换存在一个更通用的关系，与$q(x|z)$的具体值无关: 两个输出分布的之间的Total Variance距离总是会<b>小于等于</b>对应输入分布之间的Total Variance距离，即
             \begin{align}
-                dist(q_{o1}(x),\ q_{o2}(x)) <= dist(q_{i1}(z),\ q_{i2}(z))    \notag
+                dist(q_{o1}(x),\ q_{o2}(x)) \le dist(q_{i1}(z),\ q_{i2}(z))    \tag{B.1}
             \end{align}
             下面通过离散的形式给出证明：
             \begin{align}
-                      \lVert q_{o1}-q_{o2}\rVert_{TV} &= \lVert Q_{x|z}q_{i1} - Q_{x|z}q_{i2}\rVert_{TV}                                                                                                                    \tag{7.6}         \newline
-                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)q_{i1}(n) - \sum_{n}Q_{x|z}(m,n)q_{i2}(n)\textcolor{red}{|}                                                        \tag{7.7}         \newline
-                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                          \tag{7.8}         \newline
-                                                      &\leq \sum_{m}\sum_{n}Q_{x|z}(m,n)\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}             \qquad \qquad \qquad \text{Absolute value inequality}       \tag{7.9}         \newline
-                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|} \sum_{m} Q_{x|z}(m,n)            \qquad \qquad \qquad \sum_{m} Q_{x|z}(m,n) = 1              \tag{7.10}        \newline
-                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                                              \tag{7.11}
+                      \lVert q_{o1}-q_{o2}\rVert_{TV} &= \lVert Q_{x|z}q_{i1} - Q_{x|z}q_{i2}\rVert_{TV}                                                                                                                    \tag{B.2}         \newline
+                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)q_{i1}(n) - \sum_{n}Q_{x|z}(m,n)q_{i2}(n)\textcolor{red}{|}                                                        \tag{B.3}         \newline
+                                                      &=   \sum_{m}\textcolor{red}{|}\sum_{n}Q_{x|z}(m,n)(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                          \tag{B.4}         \newline
+                                                      &\leq \sum_{m}\sum_{n}Q_{x|z}(m,n)\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}             \qquad \qquad \qquad \text{Absolute value inequality}       \tag{B.5}         \newline
+                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|} \sum_{m} Q_{x|z}(m,n)            \qquad \qquad \qquad \sum_{m} Q_{x|z}(m,n) = 1              \tag{B.6}         \newline
+                                                      &=   \sum_{n}\textcolor{red}{|}(q_{i1}(n) - q_{i2}(n))\textcolor{red}{|}                                                                                              \tag{B.7}
             \end{align}
             其中，$Q_{x|z}(m,n)$表示矩阵$Q_{x|z}$的第m行第n列的元素，$q_{i1}(n)$表示向量$q_{i1}$的第n个元素。
             
-            关于定点分布与输入分布q(x)之间距离的关系，目前尚不能严格证明。
-            
-            <h3 style="font-size:18px"> 恢复数据分布过程中的抗噪声能力 </h3>
-            由上面的分析可知，当满足一些条件时，"后验概率变换"是一个压缩映射，所以存在如下的关系：
-            \begin{align}
-                dist(q(x),\ q_o(x)) < dist(q(z),\ q_i(z))         \tag{7.12}
-            \end{align}
-            其中，$q(z)$是理想的输入分布，$q(x)$理想的输出分布，$q_i(x)$是任意的输入分布，$q_o(x)$是$q_i(z)$经过变换后的输出分布。
-            
-            上式表明，输出的分布$q_o(x)$与理想输出分布q(x)之间的距离总会</em>小于</em>输入分布$q_i(z)$与理想输入分布q(x)的距离。于是，"后验概率变换"具备一定的抵抗噪声能力。这意味着，在恢复$q(x)$的过程中，哪怕输入的“末尾分布$q(z_T)”$存在一定的误差，经过一系列变换后，输出的“数据分布$q(x)$“的误差也会比输入的误差更小。
-            
-            具体可看Demo 3.2，通过增加“noise ratio”的值可以向“末尾分布$q(z_T)$”添加噪声，点击“apply”按钮将逐步画出恢复的过程，恢复的分布以$\textcolor{red}{红色曲线}$画出，同时也会通过JS散度标出误差的大小。将会看到，恢复的$q(x)$的误差总是小于$q(z_T)$的误差。
-            
-            由上面的讨论可知，$\alpha$越小(即变换过程中使用的噪声越大)，压缩映射的压缩率越大，于是，抗噪声的能力也越强。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform_zh")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_proof_ctr_zh")
     return
 
 
-def md_deconvolution_zh():
-    global g_latex_del
-    
-    title = "8. Can the data distribution be restored by deconvolution?"
-    with gr.Accordion(label=title, elem_classes="first_md", elem_id="deconvolution"):
-
-        gr.Markdown(
-            r"""
-            在第二节中提到，式2.1的变换可分为两个子变换，第一个子变换为”线性变换“，第二个为“加上独立高斯噪声”。线性变换相当于对概率分布进行拉伸变换，所以存在逆变换。"加上独立高斯噪声”相当于对概率分布执行卷积操作，卷积操作可通过逆卷积恢复。所以，理论上，可通过“逆线性变换”和“逆卷积”从最终的概率分布$q(z_T)$恢复数据分布$q(x)$。
-            
-            但实际上，会存在一些问题。由于逆卷积对误差极为敏感，具有很高的输入灵敏度，很小的输入噪声就会引起输出极大的变化[\[11\]](#deconv_1)[\[12\]](#deconv_2)。而在扩散模型中，会使用标准正态分布近似代替$q(z_T)$，因此，在恢复的起始阶段就会引入噪声。虽然噪声较小，但由于逆卷积的敏感性，噪声会逐步放大，影响恢复。
-            
-            另外，也可以从另一个角度理解“逆卷积恢复”的不可行性。由于前向变换的过程(式4.1~4.4)是确定的，所以卷积核是固定的，因此，相应的“逆卷积变换“也是固定的。由于起始的数据分布$q(x)$可以是任意的分布，所以，通过一系列固定的“卷积正变换”，可以将任意的概率分布转换成近似$\mathcal{N}(0,I)$的分布。如“逆卷积变换“可行，则意味着，可用一个固定的“逆卷积变换"，将$\mathcal{N}(0,I)$分布恢复成任意的数据分布$q(x)$，这明显是一个悖论。同一个输入，同一个变换，不可能会有多个输出。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution_zh")
-    return
-
-
-
 def md_reference_zh():
     global g_latex_del
 
@@ -438,6 +545,14 @@ def md_reference_zh():
             
             <a id="mc_basic_d4" href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf"> [14] Markov Chain:Basic Theory - Definition 4 </a>
             
+            <a id="vdm" href="https://arxiv.org/pdf/2107.00630"> [15] Variational Diffusion Models </a>
+            
+            <a id="entropy" href="https://en.wikipedia.org/wiki/Entropy"> [16] Entropy </a>
+            
+            <a id="cond_entropy" href="https://en.wikipedia.org/wiki/Conditional_entropy"> [17] Conditional Entropy </a>
+            
+            <a id="dsm" href="https://www.iro.umontreal.ca/~vincentp/Publications/smdae_techreport_1358_v1.pdf"> [18] A Connection Between Score Matching and Denoising autoencoders </a>
+            
             """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_reference_zh")
 
     return
@@ -450,7 +565,7 @@ def md_about_zh():
 
         gr.Markdown(
             r"""
-            <b>APP</b>: 本APP是使用Gradio开发，并部署在HuggingFace。由于资源有限(2核，16G内存)，所以可能会响应较慢。为了更好地体验，建议从<a href="https://github.com/blairstar/The_Art_of_DPM">github</a>复制源代码，在本地机器运行。本APP只依赖Gradio, SciPy, Matplotlib。
+            <b>APP</b>: 本Web APP是使用Gradio开发，并部署在HuggingFace。由于资源有限(2核，16G内存)，所以可能会响应较慢。为了更好地体验，建议从<a href="https://github.com/blairstar/The_Art_of_DPM">github</a>复制源代码，在本地机器运行。本APP只依赖Gradio, SciPy, Matplotlib。
             
             <b>Author</b>: 郑镇鑫，资深视觉算法工程师，十年算法开发经历，曾就职于腾讯京东等互联网公司，目前专注于视频生成(类似Sora)。
             
@@ -462,7 +577,7 @@ def md_about_zh():
 
 def run_app():
     
-    # with gr.Blocks(css=g_css, js="() => insert_special_formula(); ", head=js_head) as demo:
+    # with gr.Blocks(css=g_css, js="() => insert_special_formula() ", head=js_head) as demo:
     with gr.Blocks(css=g_css, js="() => {insert_special_formula(); write_markdown();}", head=js_head) as demo:
         md_introduction_zh()
 
@@ -481,7 +596,11 @@ def run_app():
         md_posterior_transform_zh()
         
         md_deconvolution_zh()
-         
+        
+        md_cond_kl_zh()
+        
+        md_proof_ctr_zh()
+
         md_reference_zh()
          
         md_about_zh()