LOW RANK ESTIMATION OF SMOOTH KERNELS ON GRAPHS
成果类型：
Article
署名作者：
Koltchinskii, Vladimir; Rangel, Pedro
署名单位：
University System of Georgia; Georgia Institute of Technology
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/13-AOS1088
发表日期：
2013
页码：
604-640
关键词：
摘要：
Let (V, A) be a weighted graph with a finite vertex set V, with a symmetric matrix of nonnegative weights A and with Laplacian Delta. Let S-* : V x V bar right arrow R be a symmetric kernel defined on the vertex set V. Consider n i.i.d. observations (X-j, X'(j), Y-j), j = 1, ..., n, where X-j, X'(j) are independent random vertices sampled from the uniform distribution in V and Y-j is an element of R is a real valued response variable such that E(Y-j vertical bar X-j, X'(j)) = S-*(X-j,X'(j)), j = 1 , ..., n. The goal is to estimate the kernel S-* based on the data (X-1, X'(1), Y-1), ..., (X-n, X'(n), Y-n) and under the assumption that S-* is low rank and, at the same time, smooth on the graph (the smoothness being characterized by discrete Sobolev norms defined in terms of the graph Laplacian). We obtain several results for such problems including minimax lower bounds on the L-2-error and upper bounds for penalized least squares estimators both with nonconvex and with convex penalties.