IEEE Signal Processing - July 2018 - 14

Exploiting Structure in Big Data Analytics:
Sparse and Low-Rank Structures

Yudong Chen and Yuejie Chi

Harnessing Structures in Big Data
via Guaranteed Low-Rank Matrix Estimation
Recent theory and fast algorithms via convex and nonconvex optimization

L

©istockphoto.com/shuoshu

ow-rank modeling plays a pivotal role in signal processing
and machine learning, with applications ranging from collaborative filtering, video surveillance, and medical imaging
to dimensionality reduction and adaptive filtering. Many modern high-dimensional data and interactions thereof can be modeled as lying approximately in a low-dimensional subspace or
manifold, possibly with additional structures, and its proper
exploitations lead to significant cost reduction in sensing, computation, and storage. In recent years, there has been a plethora of
progress in understanding how to exploit low-rank structures
using computationally efficient procedures in a provable manner,
including both convex and nonconvex approaches. On one side,
convex relaxations such as nuclear norm minimization often lead
to statistically optimal procedures for estimating low-rank matrices, where first-order methods are developed to address the computational challenges; on the other side, there is emerging
evidence that properly designed nonconvex procedures, such as
projected gradient descent, often provide globally optimal solutions with a much lower computational cost in many problems.
This survey article provides a unified overview of these recent
advances in low-rank matrix estimation from incomplete measurements. Attention is paid to rigorous characterization of the
performance of these algorithms and to problems where the lowrank matrix has additional structural properties that require new
algorithmic designs and theoretical analysis.

Introduction

Digital Object Identifier 10.1109/MSP.2018.2821706
Date of publication: 27 June 2018

14

The ubiquity of advanced sensing and imaging technologies
produce vast amounts of data at an unprecedented rate. A
fundamental goal of signal processing is to extract, and possibly track the evolution of, the relevant structural information faithfully from such high-dimensional data, ideally with
a minimal amount of computation, storage, and human intervention. To overcome the curse of dimensionality, it is
important to exploit the fact that real-world data often possess some low-dimensional geometric structures. In particular, such structures allow for a succinct description of the
data by a number of parameters much smaller than the
IEEE Signal Processing Magazine

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July 2018

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Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018

Contents
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