General form of the Gauss-Seidel equation to linearly approximate the Moore-Penrose pseudoinverse in random non-square systems and high order tensors

TL;DR

The paper introduces a global analytical iterative framework that generalizes Gauss-Seidel to linearly approximate the Moore-Penrose pseudoinverse for random non-square systems and high-order tensors. It derives a smooth exponential-kernel objective guiding updates and develops a tensor-aware, beta-weighted iterative scheme that converges in cases where classical GS diverges, including non-square and noisy problems. The approach reproduces MP-pseudoinverse results with high fidelity across matrix and tensor contractions, and offers competitive complexity ($\mathcal{O}(nm^2)$) relative to MP methods, while avoiding gradient calculations. Extensive results across four contraction cases demonstrate near-perfect agreement with MP solutions, supporting practical utility for large, non-square, and noisy linear systems. A concluding annex provides an implementation blueprint in Python.

Abstract

The Gauss-Seidel method has been used for more than 100 years as the standard method for the solution of linear systems of equations under certain restrictions. This method, as well as Cramer and Jacobi, is widely used in education and engineering, but there is a theoretical gap when we want to solve less restricted systems, or even non-square or non-exact systems of equation. Here, the solution goes through the use of numerical systems, such as the minimization theories or the Moore-Penrose pseudoinverse. In this paper we fill this gap with a global analytical iterative formulation that is capable to reach the solutions obtained with the Moore-Penrose pseudoinverse and the minimization methodologies, but that analytically lies to the solutions of Gauss-Seidel, Jacobi, or Cramer when the system is simplified.

Figures (10)

• Figure 1: The kernel of a function as the exponential subspace of maxima. For a circle in a), a line in b) and the combination of them in c).
• Figure 2: Iterative convergence of the proposed model in 3 different systems.
• Figure 3: Convergence of the proposed formulation under different approaches.
• Figure 4: Maximum in red and the three lines that represent the equations of the system. Exponential kernel with $\gamma=2$ (a), linear function (b) and quadratic (c).
• Figure 5: Comparison of different linear and quadratic algorithms with the Moore-Penrose pseudoinverse.