In-network Sparsity-regularized Rank Minimization: Algorithms and Applications

TL;DR

This paper introduces distributed algorithms for recovering low-rank and sparse matrices from limited data, applicable to network traffic analysis, latency prediction, and RF mapping, with provable convergence to optimal solutions.

Contribution

It develops a novel distributed method for sparsity-regularized rank minimization that overcomes non-separability issues of nuclear norm minimization, ensuring convergence to the centralized solution.

Findings

Distributed algorithm converges to the global optimum.

Per-node tasks are computationally efficient.

Performance matches centralized methods in simulations.

Abstract

Given a limited number of entries from the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, recovery of the low-rank and sparse components is a fundamental task subsuming compressed sensing, matrix completion, and principal components pursuit. This paper develops algorithms for distributed sparsity-regularized rank minimization over networks, when the nuclear- and $\ell_1$-norm are used as surrogates to the rank and nonzero entry counts of the sought matrices, respectively. While nuclear-norm minimization has well-documented merits when centralized processing is viable, non-separability of the singular-value sum challenges its distributed minimization. To overcome this limitation, an alternative characterization of the nuclear norm is adopted which leads to a separable, yet non-convex cost minimized via the alternating-direction method of multipliers. The novel distributed iterations entail reduced-complexity per-node tasks, and affordable message passing among single-hop neighbors. Interestingly, upon convergence the distributed (non-convex) estimator provably attains the global optimum of its centralized counterpart, regardless of initialization. Several application domains are outlined to highlight the generality and impact of the proposed framework. These include unveiling traffic anomalies in backbone networks, predicting networkwide path latencies, and mapping the RF ambiance using wireless cognitive radios. Simulations with synthetic and real network data corroborate the convergence of the novel distributed algorithm, and its centralized performance guarantees.