Sharpness and well-conditioning of nonsmooth convex formulations in statistical signal recovery

Lijun Ding, Alex Wang

Abstract:

We study a sample complexity vs. conditioning tradeoff in modern signal recovery problems
where convex optimization problems are built from sampled observations. We begin by introducing
a set of condition numbers related to sharpness for a general class of convex optimization problems
covering sparse recovery, low-rank matrix sensing, and (abstract) phase retrieval. These condition
numbers can be used to control how noisy observations or numerical errors in the solution process
affect the accuracy of the recovered signal. We develop a first-order method for solving such
problems with a nearly dimension-independent linear convergence rate (even in the presence of
small or sparse noise). The condition numbers we introduce show up explicitly in the rate of
linear convergence in our new algorithm. In each of these settings, we show that the condition
numbers approach constants only a small factor above the statistical threshold.

Submission Type: Abstract