Model Selection: Two Fundamental Measures of Coherence and Their Algorithmic Significance

TL;DR

This paper introduces two fundamental measures of coherence for design matrices and analyzes a simple thresholding algorithm for model selection, showing its near-optimal performance under verifiable conditions without requiring prior knowledge or full rank assumptions.

Contribution

It generalizes the notion of incoherence, introduces worst-case and average coherence measures, and provides a nonasymptotic analysis of OST for model selection applicable to generic matrices.

Findings

OST is feasible under verifiable coherence conditions.

OST performs near-optimally in low SNR regimes for certain coherence bounds.

Analysis does not require prior model knowledge or full rank submatrices.

Abstract

The problem of model selection arises in a number of contexts, such as compressed sensing, subset selection in linear regression, estimation of structures in graphical models, and signal denoising. This paper generalizes the notion of \emph{incoherence} in the existing literature on model selection and introduces two fundamental measures of coherence---termed as the worst-case coherence and the average coherence---among the columns of a design matrix. In particular, it utilizes these two measures of coherence to provide an in-depth analysis of a simple one-step thresholding (OST) algorithm for model selection. One of the key insights offered by the ensuing analysis is that OST is feasible for model selection as long as the design matrix obeys an easily verifiable property. In addition, the paper also characterizes the model-selection performance of OST in terms of the worst-case coherence, \mu, and establishes that OST performs near-optimally in the low signal-to-noise ratio regime for N x C design matrices with \mu = O(N^{-1/2}). Finally, in contrast to some of the existing literature on model selection, the analysis in the paper is nonasymptotic in nature, it does not require knowledge of the true model order, it is applicable to generic (random or deterministic) design matrices, and it neither requires submatrices of the design matrix to have full rank, nor does it assume a statistical prior on the values of the nonzero entries of the data vector.