Submodularity in action: From machine learning to signal processing applications

Tohidi, Ehsan; Amiri, Rouhollah; Coutino, Mario; Gesbert, David; Leus, Geert; Karbasi, Amin
IEEE Signal Processing Magazine, Vol.37, N°5, September 2020

Submodularity is a discrete domain functional property that can be interpreted as mimicking the role of the wellknown convexity/concavity properties in the continuous domain. Submodular functions exhibit strong structure that lead to efficient optimization algorithms with provable near-optimality guarantees. These characteristics, namely, efficiency and provable performance bounds, are of particular interest for signal processing (SP) and machine learning (ML) practitioners as a variety of discrete optimization problems are encountered in a wide range of applications. Conventionally, two general approaches exist to solve discrete problems: (i) relaxation into the continuous domain to obtain an approximate solution, or (ii) development of a tailored algorithm that applies directly in the discrete domain. In both approaches, worst-case performance guarantees are often hard to establish. Furthermore, they are often complex, thus not practical for large-scale problems. In this paper, we show how certain scenarios lend themselves to exploiting submodularity so as to construct scalable solutions with provable worst-case performance guarantees. We introduce a variety of submodularfriendly applications, and elucidate the relation of submodularity to convexity and concavity which enables efficient optimization. With a mixture of theory and practice, we present different flavors of submodularity accompanying illustrative real-world case studies from modern SP and ML. In all cases, optimization algorithms are presented, along with hints on how optimality guarantees can be established.

Systèmes de Communication
Eurecom Ref:
© 2020 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.