CVAE 回顾版

VAE回顾

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Why is the Reconstruction Term Often an L2 Distance?

First, let’s recap the two parts of the VAE loss (the Evidence Lower Bound, ELBO):

• KL Divergence Term: $DKL(q(z\mid x)\Vert p(z))$. This is the regularization term. It encourages your learned posterior distribution q(z∣x) (from the encoder) to be close to a simple prior distribution p(z) (e.g., a standard Gaussian). This helps ensure your latent space is well-behaved and continuous, allowing for smooth sampling.

• Reconstruction Term (Data Consistency): $Eq(z\mid x)[logp(x\mid z)]$. This is the term that makes sure your decoder can reconstruct the input data. It represents the expected log-likelihood of the data given the latent code, averaged over the possible latent codes provided by the encoder’s posterior.

The key to understanding this lies in the assumed likelihood distribution of the data, $p(x\mid z)$, which is modeled by the decoder.

Most commonly, for continuous data like images (e.g., pixel values), $p(x\mid z)$ is assumed to be a Gaussian (Normal) distribution.

Let’s assume $p(x\mid z)$ is a Gaussian distribution with a mean $\mu D(z)$ (output of the decoder) and some fixed variance $\sigma 2$ (often set to 1 for simplicity, or treated as a hyperparameter, or even learned).

The probability density function (PDF) for a single data point xi​ from a Gaussian is: …

When we put this into the VAE’s reconstruction loss, we are minimizing this is equivalent to minimizing $\sum i(xi-\mu D(z))2$.

This is precisely the Squared Euclidean Distance (or Squared L2 distance) between the original input $x$ and its reconstruction $\mu D(z)$ (the mean output of the decoder).

About CVAE

The “C” in CVAE stands for Conditional. A Conditional Variational Autoencoder (CVAE) extends the standard VAE by allowing you to control or specify what kind of data you want to generate. Instead of just generating a random sample from the learned data distribution, you can generate a sample that satisfies a specific condition.

Differences in Structure (Architecture)

• Concatenation for Input: Yes, this is very common and usually the most straightforward way to feed the condition c into both the encoder and decoder networks. It allows the networks to learn joint representations of x and c (for the encoder) or z and c (for the decoder). Other methods exist (like conditional batch normalization or attention mechanisms), but simple concatenation is widespread.

• Generated Output: Yes, the format of the generated output is the same as a VAE. If the VAE generates images, the CVAE also generates images. The key difference is that the CVAE’s output is controlled by the condition c.

• Components of Loss Function: Yes, the types of components (KL divergence and reconstruction loss) are fundamentally the same. The crucial distinction is that all probability distributions involved (q(z∣x), p(z), p(x∣z)) become conditional on c. So, while the components are the same, their precise mathematical definitions change to reflect the conditioning:

• Conditional Prior: A more sophisticated approach where a small “prior network” takes c as input and predicts the mean and variance for p(z∣c). This allows the latent space to be structured differently based on the condition, potentially leading to more flexible and powerful models, but also adding complexity.