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# 计算机代写|扩散模型代写Diffusion Model代考|COMP5318 Score-based Generative Models

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## 计算机代写|扩散模型代写Diffusion Model代考|Score-based Generative Models

We have shown that a Variational Diffusion Model can be learned simply by optimizing a neural network $\boldsymbol{s}_{\boldsymbol{\theta}}\left(\boldsymbol{x}_t, t\right)$ to predict the score function $\nabla \log p\left(\boldsymbol{x}_t\right)$. However, in our derivation, the score term arrived from an application of Tweedie’s Formula; this doesn’t necessarily provide us with great intuition or insight into what exactly the score function is or why it is worth modeling. Fortunately, we can look to another class of generative models, Score-based Generative Models [9, 10, 11], for exactly this intuition. As it turns out, we can show that the VDM formulation we have previously derived has an equivalent Score-based Generative Modeling formulation, allowing us to flexibly switch between these two interpretations at will.

To begin to understand why optimizing a score function makes sense, we take a detour and revisit energybased models [12, 13]. Arbitrarily flexible probability distributions can be written in the form:
$$p_{\boldsymbol{\theta}}(\boldsymbol{x})=\frac{1}{Z_{\boldsymbol{\theta}}} e^{-f_{\boldsymbol{\theta}}(\boldsymbol{x})}$$
where $f_{\boldsymbol{\theta}}(\boldsymbol{x})$ is an arbitrarily flexible, parameterizable function called the energy function, often modeled by a neural network, and $Z_{\boldsymbol{\theta}}$ is a normalizing constant to ensure that $\int p_{\boldsymbol{\theta}}(\boldsymbol{x}) d \boldsymbol{x}=1$. One way to learn such a distribution is maximum likelihood; however, this requires tractably computing the normalizing constant $Z_{\boldsymbol{\theta}}=\int e^{-f_{\boldsymbol{\theta}}(\boldsymbol{x})} d \boldsymbol{x}$, which may not be possible for complex $f_{\boldsymbol{\theta}}(\boldsymbol{x})$ functions.

## 计算机代写|扩散模型代写Diffusion Model代考|Guidance

So far, we have focused on modeling just the data distribution $p(\boldsymbol{x})$. However, we are often also interested in learning conditional distribution $p(\boldsymbol{x} \mid y)$, which would enable us to explicitly control the data we generate through conditioning information $y$. This forms the backbone of image super-resolution models such as Cascaded Diffusion Models [18], as well as state-of-the-art image-text models such as DALL-E 2 [19] and Imagen [7].

A natural way to add conditioning information is simply alongside the timestep information, at each iteration. Recall our joint distribution from Equation 32:
$$p\left(\boldsymbol{x}{0: T}\right)=p\left(\boldsymbol{x}_T\right) \prod{t=1}^T p_{\boldsymbol{\theta}}\left(\boldsymbol{x}{t-1} \mid \boldsymbol{x}_t\right)$$ Then, to turn this into a conditional diffusion model, we can simply add arbitrary conditioning information $y$ at each transition step as: $$p\left(\boldsymbol{x}{0: T} \mid y\right)=p\left(\boldsymbol{x}T\right) \prod{t=1}^T p_{\boldsymbol{\theta}}\left(\boldsymbol{x}{t-1} \mid \boldsymbol{x}_t, y\right)$$ For example, $y$ could be a text encoding in image-text generation, or a low-resolution image to perform super-resolution on. We are thus able to learn the core neural networks of a VDM as before, by predicting $\hat{\boldsymbol{x}}{\boldsymbol{\theta}}\left(\boldsymbol{x}t, t, y\right) \approx \boldsymbol{x}_0, \hat{\boldsymbol{\epsilon}}{\boldsymbol{\theta}}\left(\boldsymbol{x}t, t, y\right) \approx \boldsymbol{\epsilon}_0$, or $\boldsymbol{s}{\boldsymbol{\theta}}\left(\boldsymbol{x}_t, t, y\right) \approx \nabla \log p\left(\boldsymbol{x}_t \mid y\right)$ for each desired interpretation and implementation.

## 计算机代写|扩散模型代写Diffusion Model代考|Guidance

$$p(\boldsymbol{x} 0: T)=p\left(\boldsymbol{x}T\right) \prod t=1^T p{\boldsymbol{\theta}}\left(\boldsymbol{x} t-1 \mid \boldsymbol{x}t\right)$$ $$p(\boldsymbol{x} 0: T \mid y)=p(\boldsymbol{x} T) \prod t=1^T p{\boldsymbol{\theta}}\left(\boldsymbol{x} t-1 \mid \boldsymbol{x}_t, y\right)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。