Coherence Optimization and Best Complex Antipodal Spherical Codes

Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases

Incoherent dictionary pair learning : application to a novel open-source database of chinese numbers

We enhance the efficacy of an existing dictionary pair learning algorithm by adding a dictionary incoherence penalty term. After presenting an alternating minimization solution, we apply the proposed incoherent dictionary pair learning (InDPL) method in classification of a novel open-source database of Chinese numbers. Benchmarking results confirm that the InDPL algorithm offers enhanced classification accuracy, especially when the number of training samples is limited

Algorithms for the Construction of Incoherent Frames Under Various Design Constraints

Unit norm finite frames are generalizations of orthonormal bases with many applications in signal processing. An important property of a frame is its coherence, a measure of how close any two vectors of the frame are to each other. Low coherence frames are useful in compressed sensing applications. When used as measurement matrices, they successfully recover highly sparse solutions to linear inverse problems. This paper describes algorithms for the design of various low coherence frame types: real, complex, unital (constant magnitude) complex, sparse real and complex, nonnegative real and complex, and harmonic (selection of rows from Fourier matrices). The proposed methods are based on solving a sequence of convex optimization problems that update each vector of the frame. This update reduces the coherence with the other frame vectors, while other constraints on its entries are also imposed. Numerical experiments show the effectiveness of the methods compared to the Welch bound, as well as other competing algorithms, in compressed sensing applications

Design of Incoherent Frames via Convex Optimization

This paper describes a new procedure for the design of incoherent frames used in the field of sparse representations. We present an efficient algorithm for the design of incoherent frames that works well even when applied to the construction of relatively large frames. The main advantage of the proposed method is that it uses a convex optimization formulation that operates directly on the frame, and not on its Gram matrix. Solving a sequence of convex optimization problems allows for the introduction of constraints on the frame that were previously considered impossible or very hard to include, such as non-negativity. Numerous experimental results validate the approach