TELET: A Monotonic Algorithm to Design Large Dimensional Equiangular Tight Frames for Applications in Compressed Sensing

TL;DR

This paper introduces TELET, a monotonic iterative algorithm based on majorization minimization for constructing large-dimensional equiangular tight frames, which enhances compressed sensing performance with low mutual coherence.

Contribution

The paper presents TELET, a novel MM-based algorithm capable of efficiently constructing large-scale ETFs, overcoming previous size limitations and improving sensing matrix quality.

Findings

TELET generates ETFs with very low mutual coherence.

Constructed sensing matrices improve image reconstruction accuracy.

Algorithm demonstrates guaranteed convergence and monotonicity.

Abstract

An Equiangular tight frame (ETF) - also known as the Welch-bound-equality sequences - consists of a sequence of unit norm vectors whose absolute inner product is identical and minimal. Due to this unique property, these frames are preferred in different applications such as in constructing sensing matrices for compressed sensing systems, robust transmission, and quantum computing. Construction of ETFs involves solving a challenging non-convex minimax optimization problem, and only a few methods were successful in constructing them, albeit only for smaller dimensions. In this paper, we propose an iterative algorithm named TEchnique to devise Large dimensional Equiangular Tight-frames (TELET-frames) based on the majorization minimization (MM) procedure - in which we design and minimize a tight upper bound for the ETF cost function at every iteration. Since TELET is designed using the MM approach, it inherits useful properties of MM such as monotonicity and guaranteed convergence to a stationary point. Subsequently, we use the derived frames to construct optimized sensing matrix for compressed sensing systems. In the numerical simulations, we show that the proposed algorithm can generate complex and real frames (in the order of hundreds) with very low mutual coherence value when compared to the state-of-the-art algorithm, with a slight increase in computational cost. Experiments using synthetic data and real images reveal that the optimized sensing matrix obtained through the frames constructed by TELET performs better, in terms of image reconstruction accuracy, than the sensing matrix constructed using state-of-the-art methods.