A Density Evolution framework for Preferential Recovery of Covariance and Causal Graphs from Compressed Measurements

TL;DR

This paper introduces a density evolution-based framework for designing sensing matrices that improve covariance and causal graph recovery from compressed measurements, especially emphasizing preferential sensing of important covariance parts.

Contribution

It presents a novel framework leveraging density evolution from coding theory for designing sensing matrices tailored for covariance and causal graph recovery from compressed data.

Findings

Matching state-of-the-art in regular sensing

Improved performance in preferential sensing

Feasibility demonstrated for causal graph recovery

Abstract

In this paper, we propose a general framework for designing sensing matrix $\boldsymbol{A} \in \mathbb{R}^{d\times p}$, for estimation of sparse covariance matrix from compressed measurements of the form $\boldsymbol{y} = \boldsymbol{A}\boldsymbol{x} + \boldsymbol{n}$, where $\boldsymbol{y}, \boldsymbol{n} \in \mathbb{R}^d$, and $\boldsymbol{x} \in \mathbb{R}^p$. By viewing covariance recovery as inference over factor graphs via message passing algorithm, ideas from coding theory, such as \textit{Density Evolution} (DE), are leveraged to construct a framework for the design of the sensing matrix. The proposed framework can handle both (1) regular sensing, i.e., equal importance is given to all entries of the covariance, and (2) preferential sensing, i.e., higher importance is given to a part of the covariance matrix. Through experiments, we show that the sensing matrix designed via density evolution can match the state-of-the-art for covariance recovery in the regular sensing paradigm and attain improved performance in the preferential sensing regime. Additionally, we study the feasibility of causal graph structure recovery using the estimated covariance matrix obtained from the compressed measurements.