Incoherent and Robust Projection Matrix Design Based on Equiangular Tight Frame

Meenakshi, Seshan Srirangarajan

Abstract: Designing a projection matrix to optimally select the informative samples from high-dimensional data is a challenging task. Several approaches have been proposed for this task, however conventional methods obtain the projection matrix from the corresponding Gram matrix without considering the underlying structure of the equiangular frame. The study propose a framework to optimize the projections based on the equivalent tight frame, which is in turn constructed from the target Gram matrix. The proposed work optimizes the projection matrix by restricting the eigenvalues of the corresponding Gram matrix to ensure reduced pairwise correlation and tightness of the frame. Additionally, an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2,1}$ </tex-math></inline-formula> -norm based regularization term and a projection matrix energy constraint are incorporated to reduce the effect of outliers and noisy data. This unified optimization problem results in an incoherent and robust projection matrix. Experiments are performed on synthetic data as well as real images. The performance evaluation is carried out in terms of mutual coherence, signal reconstruction accuracy, and peak signal-to-noise ratio (PSNR). The results show that the sensing error constraint enables the design of optimized projections especially when the signals are noisy and not exactly sparse which is the case in real-world scenarios.