Open Access
December 2018 Approximate $\ell_{0}$-penalized estimation of piecewise-constant signals on graphs
Zhou Fan, Leying Guan
Ann. Statist. 46(6B): 3217-3245 (December 2018). DOI: 10.1214/17-AOS1656


We study recovery of piecewise-constant signals on graphs by the estimator minimizing an $l_{0}$-edge-penalized objective. Although exact minimization of this objective may be computationally intractable, we show that the same statistical risk guarantees are achieved by the $\alpha$-expansion algorithm which computes an approximate minimizer in polynomial time. We establish that for graphs with small average vertex degree, these guarantees are minimax rate-optimal over classes of edge-sparse signals. For spatially inhomogeneous graphs, we propose minimization of an edge-weighted objective where each edge is weighted by its effective resistance or another measure of its contribution to the graph’s connectivity. We establish minimax optimality of the resulting estimators over corresponding edge-weighted sparsity classes. We show theoretically that these risk guarantees are not always achieved by the estimator minimizing the $l_{1}$/total-variation relaxation, and empirically that the $l_{0}$-based estimates are more accurate in high signal-to-noise settings.


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Zhou Fan. Leying Guan. "Approximate $\ell_{0}$-penalized estimation of piecewise-constant signals on graphs." Ann. Statist. 46 (6B) 3217 - 3245, December 2018.


Received: 1 April 2017; Revised: 1 September 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 06965686
MathSciNet: MR3852650
Digital Object Identifier: 10.1214/17-AOS1656

Primary: 62G05

Keywords: Approximation algorithm , Effective resistance , graph cut , total-variation denoising

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 6B • December 2018
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