Binary, or one-bit, representations of data arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this work, we study the problem of data classification from binary data obtained from the sign pattern of low-dimensional projections and propose a framework with low computation and resource costs. We illustrate the utility of the proposed approach through stylized and realistic numerical experiments, and provide a theoretical analysis for a simple case. We hope that our framework and analysis will serve as a foundation for studying similar types of approaches.

Motivated by recent work on stochastic gradient descent methods, we develop two stochastic variants of greedy algorithms for possibly non-convex optimization problems with sparsity constraints. We prove linear convergence1in expectation to the solution within a specified tolerance. This generalized framework is specialized to the problems of sparse signal recovery in compressed sensing and low-rank matrix recovery, giving methods with provable convergence guarantees that often outperform their deterministic counterparts. We also analyze the settings, where gradients and projections can only be computed approximately, and prove the methods are robust to these approximations. We include many numerical experiments, which align with the theoretical analysis and demonstrate these improvements in several different settings.