Why Gabor Frames? Two Fundamental Measures of Coherence and Their Role in Model Selection

The problem of model selection arises in a number of contexts, such as subset selection in linear regression, estimation of structures in graphical models, and signal denoising. This paper generalizes the notion of "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. It utilizes these two measures of coherence to provide an in-depth analysis of two variants of a simple one-step thresholding (OST) algorithm for model selection and proves that OST is feasible for model selection as long as the design matrix obeys an easily verifiable property. One of the key insights offered by the ensuing analysis in this regard is that if the design matrix has reasonably small worst-case and average coherence then OST performs near-optimally when either (i) the energy of any nonzero entry of the signal is close to the average signal energy per nonzero entry or (ii) the signal-to-noise ratio in the measurement system is not too high. In contrast to some of the existing literature on model selection, this analysis in the paper is nonasymptotic in nature, it does not necessarily 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. Finally, two other key contributions of the paper are that (i) it provides bounds on the average coherence of Gaussian matrices and Gabor frames, and (ii) it extends the results on model selection using OST to low-complexity, model-order agnostic recovery of sparse signals.