Variational AutoEncoder VAE Equation Formula Latex Code and Summary

Variational AutoEncoder VAE

Tags: #machine learning #VAE

Equation

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

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))\\

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Introduction

Estimating the Log-likelihood and Posterior

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

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

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

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

$z = \mu + \epsilon \cdot \sigma$

Latex Code

            z = \mu + \epsilon \cdot \sigma

Explanation

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

Discussion

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Ruby Sanders

I'm burning the midnight oil to pass this test.

2023-02-04 00:00
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Ernest White reply to Ruby Sanders

You can make it...

2023-02-28 00:00:00.0
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Neil Richmond

If I could have one wish, it would be to pass this exam.

2023-09-01 00:00
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Kathy Bailey reply to Neil Richmond

Nice~

2023-09-13 00:00:00.0
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Ashley White

I have high hopes for passing this exam.

2023-10-31 00:00
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Dorothy Parker reply to Ashley White

You can make it...

2023-11-14 00:00:00.0
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