Basis Pursuit (BP) Algorithm: Solving Underdetermined Systems via Linear Programming

In signal processing, the Basis Pursuit (BP) algorithm serves as a fundamental method for solving underdetermined systems of equations. The core implementation involves converting the original problem into a linear programming framework through mathematical reformulation. The BP algorithm employs iterative optimization procedures to progressively refine solutions, making it particularly valuable in applications such as image reconstruction and audio signal denoising. From a computational perspective, the algorithm typically utilizes optimization solvers (e.g., interior-point methods or simplex algorithms) to minimize the L1-norm of the solution vector. Practical implementations often incorporate prior knowledge or regularization constraints to enhance solution accuracy and numerical stability. The algorithm structure generally involves: - Problem formulation using constraint matrices and objective functions - Iterative convergence checks with tolerance thresholds - Sparsity-promoting mechanisms through norm minimization As a versatile computational tool, the BP algorithm significantly advances signal processing research by providing robust solutions for inverse problems and sparse representation challenges.