Principal Bit Analysis: Autoencoding with Schur-Concave Loss

TL;DR

This paper introduces Principal Bit Analysis, a method for autoencoding with Schur-concave constraints that optimally decomposes sources into principal components, leading to improved fixed-rate compression.

Contribution

It demonstrates that principal component decomposition is optimal under Schur-concave constraints and develops a practical fixed-rate compressor outperforming existing methods.

Findings

Principal component decomposition is optimal under Schur-concave constraints.

The proposed PBA method outperforms existing fixed-rate compression algorithms.

Autoencoders with Schur-concave constraints decompose sources into principal components.

Abstract

We consider a linear autoencoder in which the latent variables are quantized, or corrupted by noise, and the constraint is Schur-concave in the set of latent variances. Although finding the optimal encoder/decoder pair for this setup is a nonconvex optimization problem, we show that decomposing the source into its principal components is optimal. If the constraint is strictly Schur-concave and the empirical covariance matrix has only simple eigenvalues, then any optimal encoder/decoder must decompose the source in this way. As one application, we consider a strictly Schur-concave constraint that estimates the number of bits needed to represent the latent variables under fixed-rate encoding, a setup that we call \emph{Principal Bit Analysis (PBA)}. This yields a practical, general-purpose, fixed-rate compressor that outperforms existing algorithms. As a second application, we show that a prototypical autoencoder-based variable-rate compressor is guaranteed to decompose the source into its principal components.