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Submodularity is a discrete domain functional property that can be interpreted as mimicking the role of the well-known 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<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.09905v1">arXiv:2006.09905v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ksn2bqbdczechktpa6ivcpwcau">fatcat:ksn2bqbdczechktpa6ivcpwcau</a> </span>
more »... rete 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 submodular-friendly 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.
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