Various Latent Variable Models

Various Latent Variable Models implementations in Pytorch, including VAE, VAE with AF Prior, VQ-VAE and VQ-VAE with Gated PixelCNN

Datasets

Datasets (231 MB) can be downloaded here, and contains CIFAR-10, MNIST and The Street View House Numbers (SVHN) dataset.

Full Covariance Gaussian	Full Covariance Gaussian 2	CIFAR-10	SVHN

Models

VAE on 2D Data:

Implemented is a VAE with the following characteristics:

• 2D latent variables $z$ with a standard normal prior, $p(z) = N(0, I)$
• An approximate posterior $q_\theta(z|x) = N(z; \mu_\theta(x), \Sigma_\theta(x))$, where $\mu_\theta(x)$ is the mean vector, and $\Sigma_\theta(x)$ is a diagonal covariance matrix
• A decoder $p(x|z) = N(x; \mu_\phi(z), \Sigma_\phi(z))$, where $\mu_\phi(z)$ is the mean vector, and $\Sigma_\phi(z)$ is a diagonal covariance matrix

VAE for images:

Implemented is a standard VAE with the following characteristics:

• 16-dim latent variables $z$ with standard normal prior $p(z) = N(0,I)$
• An approximate posterior $q_\theta(z|x) = N(z; \mu_\theta(x), \Sigma_\theta(x))$, where $\mu_\theta(x)$ is the mean vector, and $\Sigma_\theta(x)$ is a diagonal covariance matrix
• A decoder $p(x|z) = N(x; \mu_\phi(z), I)$, where $\mu_\phi(z)$ is the mean vector. (We are not learning the covariance of the decoder)

VAE with Autoregressive Flow prior - VLAE: implement a VAE with an Autoregressive Flow prior (Variational Lossy Autoencoder (VLAE)) with the following characteristics:

• 16-dim latent variables $z$ with a MADE prior, with $\epsilon \sim N(0, I)$
• An approximate posterior $q_\theta(z|x) = N(z; \mu_\theta(x), \Sigma_\theta(x))$, where $\mu_\theta(x)$ is the mean vector, and $\Sigma_\theta(x)$ is a diagonal covariance matrix
• A decoder $p(x|z) = N(x; \mu_\phi(z), I)$, where $\mu_\phi(z)$ is the mean vector. (We are not learning the covariance of the decoder)

For the MADE prior, it would suffice to use two hidden layers of size $512$. More explicitly, the MADE AF (mapping from $z\rightarrow \epsilon$) should output location $\mu_\theta(z)$ and scale parameters $\sigma_\theta(z)$ and do the following transformation on $z$: $$\epsilon = z \odot \sigma_\theta(z) + \mu_\theta(z)$$

where the $i$th element of $\sigma_\theta(z)$ is computed from $z_{1:i-1}$ (same for $\mu_\theta(z)$) and optimize the objective

$$-E_{z\sim q(z|x)}[\log{p(x|z)}] + E_{z\sim q(z|x)}[\log{q(z|x)} - \log{p(z)}]$$ where $$\log{p(z)} = \log{p(\epsilon)} + \log{\det\left|\frac{d\epsilon}{dz}\right|}$$

VQ-VAE:

Implemented is a VQ-VAE on the CIFAR-10 and SVHN. You may find Lilian Weng's blogpost to be useful, to understand this architecture.

Notes:

• Using a codebook with $K = 128$ latents each with a $D = 256$ dimensional embedding vector
• Each element in your $K\times D$ codebook should be initialized to be uniformly random in $[-1/K, 1/K]$
• Use batch size 128 with a learning rate of $10^{-3}$ and an Adam optimizer
• Center and scale the images to $[-1, 1]$
• Supposing that $z_e(x)$ is the encoder output, and $z_q(x)$ is the quantized output using the codebook, a straight-through estimator is implemented as follows (where below is fed into the decoder):
• (z_q(x) - z_e(x)).detach() + z_e(x)

In addition to training the VQ-VAE, we will also need to train a PixelCNN prior on the categorical latents in order to sample:

• Since the input is a 2D grid of discrete values, we should have an input (learned) embedding layer to map the discrete values to embeddings of length $64$
• Use a single Type A masked convolutions followed by 10-15 residual blocks, and $2$ $1\times 1$ convolutions of $512$ and $K$ channels respectively.

Results and samples

Model	Dataset	Samples	Samples with decoder noise
VAE 2D	Full Covariance Gaussian
VAE 2D	Full Covariance Gaussian 2

Model	Dataset	Samples	Interpolations	Reconstructions
VAE	CIFAR-10
VAE	SVHN
VLAE	CIFAR-10
VLAE	SVHN

Model	Dataset	Samples	Reconstructions
VQ-VAE	CIFAR-10
VQ-VAE	SVHN
VQ-VAE with Gated PixelCNN prior	CIFAR-10