Compressed sensing and optimal denoising of monotone signals

TL;DR

This paper investigates compressed sensing and denoising of monotone, sparsely varying signals, deriving measurement thresholds for successful recovery and optimal regularization parameters using the statistical dimension framework.

Contribution

It provides a closed-form expression for measurement requirements and optimal regularization for monotone signals, extending understanding of phase transitions in compressed sensing.

Findings

Measurement phase transition depends on change points and their locations.

Derived formula for the optimal regularizer weight for denoising.

Established high-probability success conditions for reconstruction.

Abstract

We consider the problems of compressed sensing and optimal denoising for signals $\mathbf{x_0}\in\mathbb{R}^N$ that are monotone, i.e., $\mathbf{x_0}(i+1) \geq \mathbf{x_0}(i)$, and sparsely varying, i.e., $\mathbf{x_0}(i+1) > \mathbf{x_0}(i)$ only for a small number $k$ of indices $i$. We approach the compressed sensing problem by minimizing the total variation norm restricted to the class of monotone signals subject to equality constraints obtained from a number of measurements $A\mathbf{x_0}$. For random Gaussian sensing matrices $A\in\mathbb{R}^{m\times N}$ we derive a closed form expression for the number of measurements $m$ required for successful reconstruction with high probability. We show that the probability undergoes a phase transition as $m$ varies, and depends not only on the number of change points, but also on their location. For denoising we regularize with the same norm and derive a formula for the optimal regularizer weight that depends only mildly on $\mathbf{x_0}$. We obtain our results using the statistical dimension tool.