Learning and Inference in Imaginary Noise Models

TL;DR

This paper introduces the $\sigma$-VAE, a variational autoencoder with a decoder tailored for noisy data, demonstrating its robustness in inference on highly noisy unseen data and establishing theoretical equivalences among models with different noise levels.

Contribution

The paper develops the concept of imaginary noise models in VAEs, proving their equivalence via a $eta$-VAE expansion and empirically showing their effectiveness in noisy data inference.

Findings

Power law $ ext{KL} ext{ divergence} o \sigma^{- u}$ observed

Model performs well on extremely noisy MNIST samples unseen during training

Vanilla VAE fails under high noise conditions

Abstract

Inspired by recent developments in learning smoothed densities with empirical Bayes, we study variational autoencoders with a decoder that is tailored for the random variable $Y=X+N(0,\sigma^2 I_d)$. A notion of smoothed variational inference emerges where the smoothing is implicitly enforced by the noise model of the decoder; "implicit", since during training the encoder only sees clean samples. This is the concept of imaginary noise model, where the noise model dictates the functional form of the variational lower bound $\mathcal{L}(\sigma)$, but the noisy data are never seen during learning. The model is named $\sigma$-VAE. We prove that all $\sigma$-VAEs are equivalent to each other via a simple $\beta$-VAE expansion: $\mathcal{L}(\sigma_2) \equiv \mathcal{L}(\sigma_1,\beta)$, where $\beta=\sigma_2^2/\sigma_1^2$. We prove a similar result for the Laplace distribution in exponential families. Empirically, we report an intriguing power law $\mathcal{D}_{\rm KL} \sim \sigma^{-\nu}$ for the learned models and we study the inference in the $\sigma$-VAE for unseen noisy data. The experiments were performed on MNIST, where we show that quite remarkably the model can make reasonable inferences on extremely noisy samples even though it has not seen any during training. The vanilla VAE completely breaks down in this regime. We finish with a hypothesis (the XYZ hypothesis) on the findings here.