Cheatsheet of Latex Code for Most Popular Machine Learning Equations

rockingdingo 2024-05-19 #GAN #VAE #KL-Divergence #Wasserstein #Mahalanobis


Cheatsheet of Latex Code for Most Popular Machine Learning Equations

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In this blog, we will summarize the latex code for most popular machine learning equations, including multiple distance measures, generative models, etc. There are various distance measurements of different data distribution, including KL-Divergence, JS-Divergence, Wasserstein Distance(Optimal Transport), Maximum Mean Discrepancy(MMD) and so on. We will provide the latex code for machine learning models in the following sections. We will also provide latex code of Generative Adversarial Networks(GAN), Variational AutoEncoder(VAE), Diffusion Models(DDPM) for generative models in the second section.

    Generative Models

  • Generative Adversarial Networks(GAN)

    Equation


    Latex Code
            \min_{G} \max_{D} V(D,G)=\mathbb{E}_{x \sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z \sim p_{z}(z)}[\log(1-D(G(z)))]
            
    Explanation

    GAN latex code is illustrated above. See paper for more details Generative Adversarial Networks

  • Variational AutoEncoder(VAE)

    Estimating the Log-likelihood and Posterior
    Equation


    Latex Code
            \log p_{\theta}(x)=\mathbb{E}_{q_{\phi}(z|x)}[\log p_{\theta}(x)] \\
            =\mathbb{E}_{q_{\phi}(z|x)}[\log \frac{p_{\theta}(x,z)}{p_{\theta}(z|x)}] \\
            =\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \times \frac{q_{\phi}(z|x)}{p_{\theta}(z|x)}]] \\
            =\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} ]] +D_{KL}(q_{\phi}(z|x) || p_{\theta}(z|x))\\
            
    Explanation

    Evidence Lower Bound
    Equation


    Latex Code
                \mathbb{L}_{\theta,\phi}(\mathbf{x})=\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}[\log p_{\theta}(\mathbf{x},\mathbf{z})-\log q_{\phi}(\mathbf{z}|\mathbf{x}) ]
            
    Explanation

    Reparameterization trick
    Equation

    Latex Code
                z = \mu + \epsilon \cdot \sigma
            
    Explanation

    VAE latex code is illustrated above. See paper for more details Auto-Encoding Variational Bayes

  • Diffusion Models(DDPM)

    Explanation

    See paper Denoising Diffusion Probabilistic Models for more details. See reference of the following blogpost https://lilianweng.github.io/posts/2021-07-11-diffusion-models/

    1.1 Forward Process
    Equation


    Latex Code
                q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I) \\q(x_{1:T}|x_{0})=\prod_{t=1}^{T}q(x_{t}|x_{t-1})
            
    1.2 Forward Process Reparameterization Trick
    Equation


    Latex Code
                x_{t}=\sqrt{\alpha_{t}}x_{t-1}+\sqrt{1-\alpha_{t}} \epsilon_{t-1}\\=\sqrt{\alpha_{t}\alpha_{t-1}}x_{t-2} + \sqrt{1-\alpha_{t}\alpha_{t-1}} \bar{\epsilon}_{t-2}\\=\text{...}\\=\sqrt{\bar{\alpha}_{t}}x_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon \\\alpha_{t}=1-\beta_{t}, \bar{\alpha}_{t}=\prod_{t=1}^{T}\alpha_{t}
            
    1.3 Reverse Process


    Latex Code
                p_\theta(\mathbf{x}_{0:T}) = p(\mathbf{x}_T) \prod^T_{t=1} p_\theta(\mathbf{x}_{t-1} \vert \mathbf{x}_t) \\
                p_\theta(\mathbf{x}_{t-1} \vert \mathbf{x}_t) = \mathcal{N}(\mathbf{x}_{t-1}; \boldsymbol{\mu}_\theta(\mathbf{x}_t, t), \boldsymbol{\Sigma}_\theta(\mathbf{x}_t, t))
            
    1.4 Reverse Process Variational Lower Bound


    Latex Code
                \begin{aligned}
                - \log p_\theta(\mathbf{x}_0) 
                &\leq - \log p_\theta(\mathbf{x}_0) + D_\text{KL}(q(\mathbf{x}_{1:T}\vert\mathbf{x}_0) \| p_\theta(\mathbf{x}_{1:T}\vert\mathbf{x}_0) ) \\
                &= -\log p_\theta(\mathbf{x}_0) + \mathbb{E}_{\mathbf{x}_{1:T}\sim q(\mathbf{x}_{1:T} \vert \mathbf{x}_0)} \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T}) / p_\theta(\mathbf{x}_0)} \Big] \\
                &= -\log p_\theta(\mathbf{x}_0) + \mathbb{E}_q \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} + \log p_\theta(\mathbf{x}_0) \Big] \\
                &= \mathbb{E}_q \Big[ \log \frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \\
                \text{Let }L_\text{VLB} 
                &= \mathbb{E}_{q(\mathbf{x}_{0:T})} \Big[ \log \frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \geq - \mathbb{E}_{q(\mathbf{x}_0)} \log p_\theta(\mathbf{x}_0)
                \end{aligned}
            
    1.5 Reverse Process Variational Lower Bound Decomposition Multiple KL-Divergence

    $$begin{aligned}L_\text{VLB} &= \mathbb{E}_{q(\mathbf{x}_{0:T})} \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \\&= \mathbb{E}_q \Big[ \log\frac{\prod_{t=1}^T q(\mathbf{x}_t\vert\mathbf{x}_{t-1})}{ p_\theta(\mathbf{x}_T) \prod_{t=1}^T p_\theta(\mathbf{x}_{t-1} \vert\mathbf{x}_t) } \Big] \\&= \mathbb{E}_q [\underbrace{D_\text{KL}(q(\mathbf{x}_T \vert \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_T))}_{L_T} + \sum_{t=2}^T \underbrace{D_\text{KL}(q(\mathbf{x}_{t-1} \vert \mathbf{x}_t, \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_{t-1} \vert\mathbf{x}_t))}_{L_{t-1}} \underbrace{- \log p_\theta(\mathbf{x}_0 \vert \mathbf{x}_1)}_{L_0} ]\end{aligned}$$
    Latex Code
                \begin{aligned}L_\text{VLB} &= \mathbb{E}_{q(\mathbf{x}_{0:T})} \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \\&= \mathbb{E}_q \Big[ \log\frac{\prod_{t=1}^T q(\mathbf{x}_t\vert\mathbf{x}_{t-1})}{ p_\theta(\mathbf{x}_T) \prod_{t=1}^T p_\theta(\mathbf{x}_{t-1} \vert\mathbf{x}_t) } \Big] \\&= \mathbb{E}_q [\underbrace{D_\text{KL}(q(\mathbf{x}_T \vert \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_T))}_{L_T} + \sum_{t=2}^T \underbrace{D_\text{KL}(q(\mathbf{x}_{t-1} \vert \mathbf{x}_t, \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_{t-1} \vert\mathbf{x}_t))}_{L_{t-1}} \underbrace{- \log p_\theta(\mathbf{x}_0 \vert \mathbf{x}_1)}_{L_0} ]\end{aligned}
            
    1.6 Reverse Process Variational Lower Bound Loss Function


    Latex Code
                \begin{aligned}
                L_\text{VLB} &= L_T + L_{T-1} + \dots + L_0 \\
                \text{where } L_T &= D_\text{KL}(q(\mathbf{x}_T \vert \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_T)) \\
                L_t &= D_\text{KL}(q(\mathbf{x}_t \vert \mathbf{x}_{t+1}, \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_t \vert\mathbf{x}_{t+1})) \text{ for }1 \leq t \leq T-1 \\
                L_0 &= - \log p_\theta(\mathbf{x}_0 \vert \mathbf{x}_1)
                \end{aligned}
            

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