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

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 coherence 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 thresholding algorithm for model selection. One of the key insights offered by the ensuing analysis is that thresholding 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 thresholding in terms of the worst-case coherence and establishes that thresholding performs near-optimally in the low SNR regime for design matrices with O(N^{-1/2}) worst-case coherence. 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 arbitrary (random or deterministic) design matrices, and it does not assume a statistical prior on the magnitudes of the nonzero entries.