Autoencoders Learn Generative Linear Models

TL;DR

This paper demonstrates that shallow, shared-weight autoencoders trained with gradient descent can effectively learn the parameters of common generative models like Gaussian mixtures, sparse coding, and non-negative sparsity.

Contribution

It provides the first rigorous analysis of gradient descent dynamics in weight-sharing autoencoders for multiple generative models, supporting their use in feature learning.

Findings

Autoencoders can recover model parameters under proper hyperparameters.

Gradient descent dynamics for weight-sharing autoencoders are rigorously characterized.

Autoencoders serve as effective feature learning modules for various data models.

Abstract

We provide a series of results for unsupervised learning with autoencoders. Specifically, we study shallow two-layer autoencoder architectures with shared weights. We focus on three generative models for data that are common in statistical machine learning: (i) the mixture-of-gaussians model, (ii) the sparse coding model, and (iii) the sparsity model with non-negative coefficients. For each of these models, we prove that under suitable choices of hyperparameters, architectures, and initialization, autoencoders learned by gradient descent can successfully recover the parameters of the corresponding model. To our knowledge, this is the first result that rigorously studies the dynamics of gradient descent for weight-sharing autoencoders. Our analysis can be viewed as theoretical evidence that shallow autoencoder modules indeed can be used as feature learning mechanisms for a variety of data models, and may shed insight on how to train larger stacked architectures with autoencoders as basic building blocks.