Optimized Compressed Sensing via Incoherent Frames Designed by Convex Optimization

TL;DR

This paper introduces a new convex optimization-based method for designing highly incoherent frames, significantly improving upon previous approaches and enhancing compressed sensing performance.

Contribution

It provides a detailed analysis of the optimization problem and proposes a novel method that outperforms existing techniques for various frame types.

Findings

New method outperforms all existing frame design algorithms

Significant improvements in incoherence for low and high redundancy frames

Enhanced compressed sensing performance using the optimized frames

Abstract

The construction of highly incoherent frames, sequences of vectors placed on the unit hyper sphere of a finite dimensional Hilbert space with low correlation between them, has proven very difficult. Algorithms proposed in the past have focused in minimizing the absolute value off-diagonal entries of the Gram matrix of these structures. Recently, a method based on convex optimization that operates directly on the vectors of the frame has been shown to produce promising results. This paper gives a detailed analysis of the optimization problem at the heart of this approach and, based on these insights, proposes a new method that substantially outperforms the initial approach and all current methods in the literature for all types of frames, with low and high redundancy. We give extensive experimental results that show the effectiveness of the proposed method and its application to optimized compressed sensing.