The Annals of Statistics

Approximate $\ell_{0}$-penalized estimation of piecewise-constant signals on graphs

Zhou Fan and Leying Guan

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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.

Article information

Ann. Statist., Volume 46, Number 6B (2018), 3217-3245.

Received: April 2017
Revised: September 2017
First available in Project Euclid: 11 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation

Approximation algorithm graph cut effective resistance total-variation denoising


Fan, Zhou; Guan, Leying. Approximate $\ell_{0}$-penalized estimation of piecewise-constant signals on graphs. Ann. Statist. 46 (2018), no. 6B, 3217--3245. doi:10.1214/17-AOS1656.

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Supplemental materials

  • Supplementary Appendices. The supplementary appendices contain proofs of theoretical results.