Algebraic Diversity: Principles of a Group-Theoretic Approach to Signal Processing

TL;DR

This paper introduces algebraic diversity, a group-theoretic framework for signal processing that leverages symmetry to improve information extraction and reduce variance without additional observations.

Contribution

It develops a comprehensive algebraic approach to signal processing using group invariance, including methods for blind group matching and adaptive processing, extending classical techniques.

Findings

Matched group determines natural transform for analysis.

A polynomial-time eigenvalue problem identifies the matched group from data.

Blind equalization predictions match residual phase ambiguity within two degrees.

Abstract

We present principles of algebraic diversity (AD), a group-theoretic approach to signal processing exploiting signal symmetry to extract more information per observation, complementing classical methods that use temporal and spatial diversity. The transformations under which a signal's statistics are invariant form a matched group; this group determines the natural transform for analysis, and averaging an estimator over the group action reduces variance without requiring additional snapshots. The viewpoint is broadened in five directions beyond the single-observation measurement of a companion paper. Rank promotion admits AD on scalar data streams and identifies the law of large numbers as the trivial-group case of a $(G, L)$ continuum combining sample-count with group-orbit averaging. An eigentensor hierarchy handles signals with nested symmetry. A blind group-matching methodology identifies the matched group from data via a polynomial-time generalized eigenvalue problem on the unitary Lie algebra, placing the DFT, DCT, and Karhunen--Lo\`{e}ve transforms as distinguished points on a transform manifold. A cost-symmetry matching principle then extends AD from measurement to blind and adaptive signal processing generally; blind equalization is given as a detailed example, with the Constant Modulus Algorithm's residual phase ambiguity predicted analytically and matched within two degrees on 3GPP TDL multipath channels, and other blind problems in signal processing are mapped into the framework. Four theorems formalize a structural capacity $\kappa$, the R\'{e}nyi-2 analog of Shannon and von Neumann's R\'{e}nyi-1 entropies, quantifying how a signal's information is organized rather than how much information it contains. AD complements prior algebraic approaches including invariant estimation, minimax robust estimation, algebraic signal processing, and compressed sensing.