Sparse Recovery and Dictionary Learning from Nonlinear Compressive Measurements

TL;DR

This paper introduces a unified, convex optimization framework for sparse recovery and dictionary learning from nonlinear measurements like clipping, quantization, and 1-bit data, enabling direct learning from corrupted signals.

Contribution

It proposes a simple, convex, and differentiable cost function for nonlinear measurements and develops proximal algorithms for sparse coding and dictionary learning in this context.

Findings

Framework effectively handles various nonlinear measurement types

Algorithms successfully learn dictionaries from nonlinear corrupted data

Generalizes traditional linear sparse coding methods

Abstract

Sparse coding and dictionary learning are popular techniques for linear inverse problems such as denoising or inpainting. However in many cases, the measurement process is nonlinear, for example for clipped, quantized or 1-bit measurements. These problems have often been addressed by solving constrained sparse coding problems, which can be difficult to solve, and assuming that the sparsifying dictionary is known and fixed. Here we propose a simple and unified framework to deal with nonlinear measurements. We propose a cost function that minimizes the distance to a convex feasibility set, which models our knowledge about the nonlinear measurement. This provides an unconstrained, convex, and differentiable cost function that is simple to optimize, and generalizes the linear least squares cost commonly used in sparse coding. We then propose proximal based sparse coding and dictionary learning algorithms, that are able to learn directly from nonlinearly corrupted signals. We show how the proposed framework and algorithms can be applied to clipped, quantized and 1-bit data.