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April 2013 Low rank estimation of smooth kernels on graphs
Vladimir Koltchinskii, Pedro Rangel
Ann. Statist. 41(2): 604-640 (April 2013). DOI: 10.1214/13-AOS1088


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 $S_{\ast}: V\times V\mapsto{\mathbb{R}}$ be a symmetric kernel defined on the vertex set $V$. Consider $n$ i.i.d. observations $(X_{j},X_{j}',Y_{j})$, $j=1,\ldots,n$, where $X_{j}$, $X_{j}'$ are independent random vertices sampled from the uniform distribution in $V$ and $Y_{j}\in{\mathbb{R}}$ is a real valued response variable such that ${\mathbb{E}}(Y_{j}|X_{j},X_{j}')=S_{\ast}(X_{j},X_{j}')$, $j=1,\ldots,n$. The goal is to estimate the kernel $S_{\ast}$ based on the data $(X_{1},X_{1}',Y_{1}),\ldots,(X_{n},X_{n}',Y_{n})$ and under the assumption that $S_{\ast}$ 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.


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Vladimir Koltchinskii. Pedro Rangel. "Low rank estimation of smooth kernels on graphs." Ann. Statist. 41 (2) 604 - 640, April 2013.


Published: April 2013
First available in Project Euclid: 26 April 2013

zbMATH: 1360.62272
MathSciNet: MR3099115
Digital Object Identifier: 10.1214/13-AOS1088

Primary: 60B20, 60G15, 62H12, 62J99

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 2 • April 2013
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