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

This paper develops theoretical results regarding noisy 1-bit compressed sensing and sparse binomial regression. We demonstrate that a single convex program gives an accurate estimate of the signal, or coefficient vector, for both of these models. We show that an s-sparse signal in Rn can be accurately estimated from m = O(s log(n/s)) single-bit measurements using a simple convex program. This remains true even if almost half of the measurements are randomly flipped. Worstcase (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(s log(n/s)) Bernoulli trials are sufficient to estimate a coefficient vector in Rn 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 lo...

• Type: Journal
• Year: 2012
• Where: CORR
• Authors: Yaniv Plan, Roman Vershynin
• Keywords: Bernoulli Trials, CORR 2012, Education, Generalized Linear Models, Logistic Regression