Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations

TL;DR

This paper introduces a group-theoretic framework for spectral estimation from single observations, unifying classical transforms and demonstrating applications across signal processing and machine learning.

Contribution

It establishes a novel algebraic approach to spectral estimation, linking group symmetry to variance reduction and unifying classical transforms as special cases.

Findings

Sample covariance variance governed by group order and snapshot count.

Classical transforms like DFT, DCT, KLT are unified as group-matched cases.

Monte Carlo experiments support the conjectured variance bounds.

Abstract

We establish that temporal averaging over multiple observations is the degenerate case of algebraic group action with the trivial group $G=\{e\}$. A General Replacement Theorem proves that a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation. The Trivial Group Embedding Theorem proves that the sample covariance is the accumulation of trivial-group estimates, with variance governed by a $(G,L)$ continuum as $1/(|G|\cdot L)$. The processing gain $10\log_{10}(M)$ dB equals the classical beamforming gain, establishing that this gain is a property of group order, not sensor count. The DFT, DCT, and KLT are unified as group-matched special cases. We conjecture a General Algebraic Averaging Theorem extending these results to arbitrary statistics, with variance governed by the effective group order $d_{\mathrm{eff}}$. Monte Carlo experiments on the first four sample moments across five group types confirm the conjecture to four-digit precision. The framework exploits the $structure$ of information (representation-theoretic symmetry of the data object) rather than the content, complementing Shannon's theory. Five applications are demonstrated: single-snapshot MUSIC, massive MIMO, single-pulse waveform classification, graph signal processing, and analysis of transformer LLMs. Techniques for blind group matching are described.