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The bound that arises out of sparse recovery analysis in compressed sensinginvolves input signal sparsity and some property of the sensing matrix. Aneffort has therefore been made in the literature to optimize sensing matricesfor optimal recovery using this property. We discover, in the specific case ofoptimizing codes for the CACTI camera, that the popular method of mutualcoherence minimization does not produce optimal results: codes designed tooptimize effective dictionary coherence often perform worse than random codesin terms of mean squared reconstruction error.  This surprising phenomenon leads us to investigate the reliability of thecoherence bound for matrix optimization, in terms of its looseness. We examine,on simulated data, the looseness of the bound as it propagates across varioussteps of the inequalities in a derivation leading to the final bound. We thensimilarly examine an alternate bound derived by Tang, G. et al, based on the$\ell_1/\ell_{\infty}$ notion of sparsity, which is a compromise betweencoherence and the restricted isometry constant (RIC). Moreover, we also performa bound looseness analysis for the RIC as derived by Cai, T. et al. Theconclusion of these efforts is that coherence optimization is problematic notonly because of the coherence bound on the RIC, but also the RIC bound itself.These negative results imply that despite the success of previous work indesigning sensing matrices based on optimization of a matrix quality factor,one needs to exercise caution in using them for practical sensing matrixdesign.  We then introduce a paradigm for optimizing sensing matrices that overcomesthe looseness of compressed sensing upper bounds using an average case errorapproach. We show a proof-of-concept design using this paradigm that performsconvincingly better than coherence-based design in the CACTI case, and no worsefor general matrices.

Optimizing Matrices For Compressed Sensing Using Existing Goodness Measures: Negative Results, And An Alternative