Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations
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 gr...
| Authors | Mitchell A. Thornton |
|---|---|
| Primary category | cs.LG |
| Published | 4 April 2026 |
| Last updated at source | 19 August 2026 |
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