Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach

TL;DR

This paper introduces a convex programming method for robust 1-bit compressed sensing and sparse logistic regression, providing theoretical guarantees for accurate signal estimation under noise and broad model conditions.

Contribution

It presents the first unified convex approach with theoretical guarantees for noisy 1-bit compressed sensing and sparse logistic regression, applicable to general signal structures.

Findings

Accurate signal estimation from O(s log(n/s)) measurements even with high flip probability.

The convex program works for nearly all generalized linear models.

The approach accounts for adversarial noise and broad signal structures.

Abstract

This paper develops theoretical results regarding noisy 1-bit compressed sensing and sparse binomial regression. We show that a single convex program gives an accurate estimate of the signal, or coefficient vector, for both of these models. We demonstrate that an s-sparse signal in R^n can be accurately estimated from m = O(slog(n/s)) single-bit measurements using a simple convex program. This remains true even if each measurement bit is flipped with probability nearly 1/2. Worst-case (adversarial) noise can also be accounted for, and uniform results that hold for all sparse inputs are derived as well. In the terminology of sparse logistic regression, we show that O(slog(n/s)) Bernoulli trials are sufficient to estimate a coefficient vector in R^n which is approximately s-sparse. Moreover, the same convex program works for virtually all generalized linear models, in which the link function may be unknown. To our knowledge, these are the first results that tie together the theory of sparse logistic regression to 1-bit compressed sensing. Our results apply to general signal structures aside from sparsity; one only needs to know the size of the set K where signals reside. The size is given by the mean width of K, a computable quantity whose square serves as a robust extension of the dimension.