A new interpretation of the weighted pseudoinverse and its applications

TL;DR

This paper introduces a new interpretation of the weighted pseudoinverse through a linear operator framework, deriving generalized equations, a closed-form expression via GSVD, and an iterative algorithm for solving generalized least squares problems.

Contribution

It provides a novel linear operator perspective of the weighted pseudoinverse, leading to new characterization equations, a closed-form GSVD-based expression, and a gLSQR iterative solver.

Findings

The new interpretation links GLS, weighted pseudoinverse, GSVD, and gLSQR.

Derived generalized Moore-Penrose equations for the weighted pseudoinverse.

Numerical tests demonstrate the effectiveness of the proposed iterative algorithm.

Abstract

Consider the generalized linear least squares (GLS) problem $\min\|Lx\|_2 \ \mathrm{s.t.} \ \|M(Ax-b)\|_2=\min$. The weighted pseudoinverse $A_{ML}^{\dag}$ is the matrix that maps $b$ to the minimum 2-norm solution of this GLS problem. By introducing a linear operator induced by $\{A, M, L\}$ between two finite-dimensional Hilbert spaces, we show that the minimum 2-norm solution of the GLS problem is equivalent to the minimum norm solution of a linear least squares problem involving this linear operator, and $A_{ML}^{\dag}$ can be expressed as the composition of the Moore-Penrose pseudoinverse of this linear operator and an orthogonal projector. With this new interpretation, we establish the generalized Moore-Penrose equations that completely characterize the weighted pseudoinverse, give a closed-form expression of the weighted pseudoinverse using the generalized singular value decomposition (GSVD), and propose a generalized LSQR (gLSQR) algorithm for iteratively solving the GLS problem. We construct several numerical examples to test the proposed iterative algorithm for solving GLS problems. Our results highlight the close connections between GLS, weighted pseudoinverse, GSVD and gLSQR, providing new tools for both analysis and computations.