Learning and Inference in Imaginary Noise Models
This addresses the issue of robust inference in VAEs for noisy data, though it is incremental as it builds on existing smoothed density and VAE frameworks.
The paper tackles the problem of variational autoencoders (VAEs) failing on noisy data by introducing σ-VAE, a model that uses an imaginary noise model to learn smoothed densities without seeing noisy samples during training. The result shows that σ-VAE can make reasonable inferences on extremely noisy MNIST samples, unlike vanilla VAE, and reveals a power law relationship for KL divergence.
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 $Y=X+N(0,σ^2 I_d)$. 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 $\mathcal{L}(σ)$, but the noisy data are never seen during learning. The model is named $σ$-VAE. We prove that all $σ$-VAEs are equivalent to each other via a simple $β$-VAE expansion: $\mathcal{L}(σ_2) \equiv \mathcal{L}(σ_1,β)$, where $β=σ_2^2/σ_1^2$. We prove a similar result for the Laplace distribution in exponential families. Empirically, we report an intriguing power law $\mathcal{D}_{\rm KL} \sim σ^{-ν}$ for the learned models and we study the inference in the $σ$-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.