Summary
This work explains a mathematical tool called the Generalized Singular Value Decomposition (GSVD), which helps compare two processes acting on the same information to see which one dominates and in what way. The authors build the necessary math and statistics from scratch with examples. They explore the GSVD through different viewpoints, explain how it relates to robot teams working together, and share how to use it well in practice with verified computations. This helps understand and improve how groups of robots estimate and decide together based on multiple sources of data.
What this means in practice
- •For robotics engineers: Improve the design of collective estimation and decision algorithms in robot teams by rigorously comparing information sources.
- •For sensor fusion developers: Develop mathematically grounded methods to combine multimodal sensor data while quantifying relative contributions of each source.
A theory result. No direct application yet.
Abstract
The Generalized Singular Value Decomposition (GSVD) is a central tool for comparing two linear operators that act on the same unknown. Once the output metrics are fixed, it answers the basis-independent question "along which state directions does the first operator dominate the second, and by how much?". This foundational volume builds the GSVD and its surrounding estimation theory from the ground up. Part I re-derives every piece of linear algebra and statistical estimation machinery needed later -- inner products, the four subspaces, orthogonal matrices, the spectral theorem, quadratic forms, the ordinary SVD, Lie-group tangent perturbations, Jacobians, covariance, whitening, Fisher information, CCA, mutual information, discrimination criteria, and covariance intersection -- with self-contained derivations and worked numerical examples. Part II states the GSVD in both a transparent and a dimensionally general form, then proves it by two complementary routes. Part III works one small matrix pair through five different lenses, each computed by hand and verified numerically. Part IV separates information attribution, subspace geometry, statistical dependence, and decision-theoretic discrimination for collective systems. Part V evaluates the role and limits of the GSVD across the cognition loop. Part VI covers numerical practice, including a tested reference implementation. Every number printed in this document was verified in double precision. Where a statement is an assumption rather than a theorem, it is labelled as such.