Optimally conditioned sparse semi-orthonormal frames

Abstract

Frame theory has been extensively used for generating over-complete redundant representations of signals. When parts of the frame measurements of a signal are lost, iterative frame reconstruction methods are used for recovering the signal, where its convergence rate is associated with the tightness of the frame and the Second Largest Eigenvalue Modulus (SLEM) of its operator. Scaling is a popular noninvasive method employed for constructing optimally conditioned frames that are as tight as possible. This is possible by optimizing the condition number of the frame operator. In large vector spaces, frame sparsity is critical for constructing frames. Following these design factors, this paper defines particular types of frames, namely Semi-Orthonormal (SO) and Disjoint Semi-Orthonormal (DSO) frames that have a very sparse structure. Optimal conditioning of these frames, using scaling, has been addressed by optimizing the SLEM of the frame operator, which has been solved using its Semi-definite Programming (SDP) formulation. Based on the results derived from the SDP solution, an iterative algorithm has been proposed to determine the optimal scales and SLEM values for DSO frames of any size. These optimal results have been extended to the SO frames using a conjecture developed based on the dual variables in the SDP solution. Furthermore, the erasure robustness of SO and DSO frames has been addressed, where the maximal reconstruction error is minimized with respect to all possible erasure locations with a fixed cardinality. The erasure robustness and the optimal scaling have been examined by simulating the frame algorithm.