An efficient infinity norm minimization algorithm for under-determined inverse problems

The problem of solving under-determined systems of linear equations with minimum peak magnitude (${\ell }_{\infty }$ norm) has numerous applications in signal processing. These include Peak-to-Average Power Ratio (PAPR) reduction in MIMO-OFDM systems, vector quantization, approximate nearest neighbor search, optimal control in robotics, and power grid optimization. Several methods have been proposed to address this problem, but they often face limitations in computational speed or representation accuracy. Some methods also impose constraints on the frame matrix, such as restrictions on the type of its entries or its aspect ratio. In this paper, we present the Fast Iterative Peak Shrinkage Algorithm (FIPSA), which iterates over feasible solutions to consistently reduce peak magnitude and provably converge to near-optimal solutions. Our experimental results, conducted across various frame matrix types and aspect ratios, demonstrate that FIPSA consistently achieves near-minimal ${\ell }_{\infty }$ norm values. In addition, it operates 1.3 to 7.3 times faster than previous methods, while maintaining an average representation error of ${10}^{-15}$. Notably, these advancements are achieved without imposing any constraints on the frame matrix.