APPROXIMATE l0-PENALIZED ESTIMATION OF PIECEWISE-CONSTANT SIGNALS ON GRAPHS

• 成果类型: Article
• 署名作者: Fan, Zhou; Guan, Leying
• 署名单位: Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1656
• 发表日期: 2018
• 页码: 3217-3245
• 关键词: total variation minimization; energy minimization; change-points; fused lasso; regression; sparsification; algorithms; prediction; selection; cluster

摘要

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.