Autoencoders and VAEs

What Is an Autoencoder?

Imagine compressing a photograph into a tiny summary – a few numbers that capture the essence of the image – then trying to reconstruct the full photograph from only that summary. An autoencoder does exactly this: an encoder network squeezes the input into a low-dimensional latent code $z$, and a decoder network expands $z$ back into the original input space. The bottleneck forces the network to learn which features truly matter.

A standard (deterministic) autoencoder minimizes reconstruction loss:

$\mathcal{L}_{\text{AE}} = \| x - \hat{x} \|^2$

where $x$ is the input and $\hat{x} = \text{Dec}(\text{Enc}(x))$ is the reconstruction.

A Variational Autoencoder (VAE), introduced by Kingma and Welling (2013), replaces the deterministic latent code with a probability distribution. The encoder outputs parameters $\mu$ and $\sigma$ of a Gaussian, and a latent sample is drawn via the reparameterization trick: $z = \mu + \sigma \odot \epsilon$, where $\epsilon \sim \mathcal{N}(0, I)$.

How It Works

Deterministic Autoencoders

The encoder $q_\phi(z|x)$ maps input $x$ to a fixed-length vector $z \in \mathbb{R}^d$. The decoder $p_\theta(x|z)$ maps $z$ back to the input space. Training minimizes pixel-wise MSE or binary cross-entropy between input and output. Common variants include:

• Undercomplete autoencoders: Bottleneck dimension $d$ is smaller than input dimension, forcing compression.
• Sparse autoencoders: Add an $L_1$ penalty on activations to encourage sparse codes.
• Denoising autoencoders: Corrupt input with noise and train to recover the clean signal (Vincent et al., 2008).

Variational Autoencoders

VAEs optimize the Evidence Lower Bound (ELBO):

$\mathcal{L}_{\text{VAE}} = \mathbb{E}_{q_\phi(z|x)}[\log p_\theta(x|z)] - D_{\text{KL}}(q_\phi(z|x) \| p(z))$

The first term is the reconstruction likelihood. The second term – the KL divergence – regularizes the encoder to produce latent distributions close to the prior $p(z) = \mathcal{N}(0, I)$.

For Gaussian encoder output $q_\phi(z|x) = \mathcal{N}(\mu, \text{diag}(\sigma^2))$, the KL term has a closed-form solution:

$D_{\text{KL}} = -\frac{1}{2}\sum_{j=1}^{d}(1 + \log \sigma_j^2 - \mu_j^2 - \sigma_j^2)$

The Reparameterization Trick

Sampling $z \sim q_\phi(z|x)$ is not differentiable. The trick rewrites $z = \mu + \sigma \odot \epsilon$ with $\epsilon \sim \mathcal{N}(0, I)$, pushing stochasticity outside the computational graph so gradients flow through $\mu$ and $\sigma$.

Beta-VAE and Disentanglement

Higgins et al. (2017) introduced $\beta$-VAE, which scales the KL term by $\beta > 1$:

$\mathcal{L}_{\beta\text{-VAE}} = \mathbb{E}[\log p_\theta(x|z)] - \beta \cdot D_{\text{KL}}(q_\phi(z|x) \| p(z))$

Higher $\beta$ encourages disentangled latent factors at the cost of reconstruction quality.

VQ-VAE

Van den Oord et al. (2017) proposed Vector Quantized VAE, replacing continuous latents with a discrete codebook of $K$ embedding vectors $e_k \in \mathbb{R}^d$. The encoder output $z_e(x)$ is quantized by mapping to the nearest codebook entry:

$z_q(x) = e_k \quad \text{where} \quad k = \arg\min_j \| z_e(x) - e_j \|_2$

The training loss combines reconstruction, codebook alignment, and commitment terms:

$\mathcal{L}_{\text{VQ}} = \| x - \hat{x} \|^2 + \| \text{sg}[z_e(x)] - e \|^2 + \beta \| z_e(x) - \text{sg}[e] \|^2$

where $\text{sg}[\cdot]$ is the stop-gradient operator and $\beta = 0.25$ is the commitment coefficient. Gradients for the encoder pass through the quantization via straight-through estimation. VQ-VAE achieves high-fidelity reconstruction and serves as a foundation for later models like DALL-E.

Why It Matters

1. Dimensionality reduction: Autoencoders learn nonlinear compressions superior to PCA for complex data like images.
2. Generative modeling: VAEs provide a principled framework to sample new images by drawing $z \sim \mathcal{N}(0, I)$ and decoding.
3. Latent space arithmetic: VAE latent spaces support interpolation – walking between two face images produces smooth transitions.
4. Downstream tasks: Pretrained encoders serve as feature extractors for classification, anomaly detection, and retrieval.
5. Foundation for modern architectures: VQ-VAE is a core component of latent diffusion models (Stable Diffusion) and discrete generative models.

Key Technical Details

• Typical latent dimensions: 64–512 for image autoencoders; VQ-VAE uses codebooks of 512–8192 vectors.
• VAEs trained on CelebA 64x64 typically achieve reconstruction FID around 40–60, significantly worse than GANs (~10) at the same resolution.
• The “posterior collapse” problem occurs when the decoder is too powerful and ignores $z$; mitigation strategies include KL annealing, free bits, and cyclical schedules.
• VQ-VAE-2 (Razavi et al., 2019) achieves FID 31 on 256x256 CelebA-HQ using a hierarchical discrete latent space.
• Training is stable compared to GANs – standard Adam optimizer with learning rate 1e-4 works reliably.

Common Misconceptions

• “VAEs generate blurry images because the model is bad.” The blurriness comes from optimizing pixel-wise reconstruction likelihood under a Gaussian assumption, which averages over modes. Using perceptual losses or adversarial training (VAE-GAN) substantially sharpens outputs.
• “The KL term is just a regularizer you can drop.” Without the KL term, the latent space has no structure and sampling from $\mathcal{N}(0, I)$ produces garbage. The KL term is what makes a VAE generative.
• “Autoencoders are the same as PCA.” A linear autoencoder with MSE loss recovers the PCA subspace, but nonlinear autoencoders learn much richer representations.
• “Larger latent dimensions are always better.” Beyond a task-dependent optimum, increasing latent dimension wastes capacity on noise and can degrade generalization. For CelebA faces, latent dimensions of 128–256 typically suffice.

Connections to Other Concepts

Further Reading