Autoencoders and VAEs

One-Line Summary: Autoencoders learn compressed latent representations by encoding inputs and reconstructing them, while Variational Autoencoders add a probabilistic structure that enables principled generation of new data.

Prerequisites: Convolutional neural networks, backpropagation, probability distributions, KL divergence basics

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:

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

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: $z=\mu +\sigma \odot ϵ$, where $ϵ\sim \mathcal{N}(0,I)$.

How It Works

Deterministic Autoencoders

The encoder $q_{\varphi }(z|x)$ 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):

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

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 $q_{\varphi }(z|x)=\mathcal{N}(\mu ,\text{diag}({\sigma }^2))$, 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 $z\sim q_{\varphi }(z|x)$ is not differentiable. The trick rewrites $z=\mu +\sigma \odot ϵ$ with $ϵ\sim \mathcal{N}(0,I)$, 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 $\beta >1$:

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

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:

$z_q(x)=e_k\text{where}k=\arg {\min }_j\Vert z_e(x)-e_j{\Vert }_2$

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

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

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

• diffusion-models.md: Use a pretrained VAE (or VQ-VAE) encoder/decoder to move diffusion into a compressed latent space.
• generative-adversarial-networks.md: VAE-GAN hybrids use adversarial losses to sharpen VAE reconstructions.
• image-super-resolution.md: Autoencoders provide the backbone architecture for many super-resolution networks.
• neural-style-transfer.md: Encoder features from autoencoders overlap conceptually with VGG features used in style transfer.

Further Reading

• Kingma and Welling, "Auto-Encoding Variational Bayes" (2013) -- The original VAE paper introducing the reparameterization trick and ELBO objective.
• Higgins et al., "beta-VAE: Learning Basic Visual Concepts with a Constrained Variational Framework" (2017) -- Disentangled representation learning via scaled KL.
• Van den Oord et al., "Neural Discrete Representation Learning" (2017) -- VQ-VAE with discrete codebooks.
• Razavi et al., "Generating Diverse High-Fidelity Images with VQ-VAE-2" (2019) -- Hierarchical VQ-VAE achieving near-GAN quality.
• Vincent et al., "Extracting and Composing Robust Features with Denoising Autoencoders" (2008) -- Denoising autoencoders and their connection to score matching.