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Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations

Mitchell A. Thornton

Abstract

We prove that temporal averaging over multiple observations can be replaced by algebraic group action on a single observation for second-order statistical estimation. A General Replacement Theorem establishes conditions under which a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation, and an Optimality Theorem proves that the symmetric group is universally optimal (yielding the KL transform). The framework unifies the DFT, DCT, and KLT as special cases of group-matched spectral transforms, with a closed-form double-commutator eigenvalue problem for polynomial-time optimal group selection. Five applications are demonstrated: MUSIC DOA estimation from a single snapshot, massive MIMO channel estimation with 64% throughput gain, single-pulse waveform classification at 90% accuracy, graph signal processing with non-Abelian groups, and a new algebraic analysis of transformer LLMs revealing that RoPE uses the wrong algebraic group for 70-80% of attention heads across five models (22,480 head observations), that the optimal group is content-dependent, and that spectral-concentration-based pruning improves perplexity at the 13B scale. All diagnostics require a single forward pass with no gradients or training.

Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations

Abstract

We prove that temporal averaging over multiple observations can be replaced by algebraic group action on a single observation for second-order statistical estimation. A General Replacement Theorem establishes conditions under which a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation, and an Optimality Theorem proves that the symmetric group is universally optimal (yielding the KL transform). The framework unifies the DFT, DCT, and KLT as special cases of group-matched spectral transforms, with a closed-form double-commutator eigenvalue problem for polynomial-time optimal group selection. Five applications are demonstrated: MUSIC DOA estimation from a single snapshot, massive MIMO channel estimation with 64% throughput gain, single-pulse waveform classification at 90% accuracy, graph signal processing with non-Abelian groups, and a new algebraic analysis of transformer LLMs revealing that RoPE uses the wrong algebraic group for 70-80% of attention heads across five models (22,480 head observations), that the optimal group is content-dependent, and that spectral-concentration-based pruning improves perplexity at the 13B scale. All diagnostics require a single forward pass with no gradients or training.

Paper Structure

This paper contains 96 sections, 14 theorems, 43 equations, 18 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Organization
  5. Related Work
  6. Mathematical Framework
  7. Signal Model and Classical Estimation
  8. Group Actions on Observation Vectors
  9. Conditions for Subspace Recovery
  10. The Cayley Graph Construction
  11. The General Replacement Theorem
  12. Optimality of the Symmetric Group
  13. The KL Optimality Chain
  14. Connection to Cayley Graph Spectra
  15. Commutativity and the KL Basis
  16. ...and 81 more sections

Key Result

Theorem 4

Let $\mathbf{x} = \mathbf{s} + \mathbf{n} \in \mathbb{C}^M$ be a single observation satisfying the signal model (eq:obs_model), and let $G$ be a finite group with unitary representation $\rho: G \to U(M)$ satisfying Conditions cond:equivariance and cond:ergodicity. Then the group-averaged estimator $\blacktriangleleft$$\blacktriangleleft$

Figures (18)

  • Figure 1: Eigenvalue SNR versus number of permutations $n$ drawn from $S_{10}$ using four ordering strategies. $M = 10$, single source at $\theta = 30^\circ$, input SNR $= 10$ dB, 500 Monte Carlo trials. All methods degrade monotonically; structured orderings outperform random by 7--8 dB. The dashed line marks $n = M = 10$.
  • Figure 2: Massive MIMO: AD vs. MMSE at SNR $= 15$ dB, $K = 4$ users. (a) Effective throughput vs. $M$ for three CDL channel models. Dashed: MMSE; solid: AD. (b) Percentage gain of AD over MMSE. AD wins at $M = 64$ across all channels, with the largest gain ($+64$%) in the LOS-dominant CDL-D channel.
  • Figure 3: Single-pulse chirp characterization via the chirp-adapted group ($M = 31$). (a) Spectral concentration vs. SNR: the adapted group (green) recovers $8.3\times$ higher concentration than the mismatched cyclic group (red) and exceeds the tone baseline (blue). (b) Blind chirp rate estimation via $\psi$ sweep: a sharp peak at the true rate $\mu = 0.5$ enables single-pulse parameter estimation.
  • Figure 4: SNR robustness of the chirp-adapted group ($M = 31$). (a) Concentration advantage ratio vs. SNR for three chirp rates; $\geq 2\times$ advantage maintained to $-2$ dB. (b) Blind chirp rate estimation RMSE vs. SNR; RMSE $< 0.05$ at SNR $\geq 2$ dB.
  • Figure 5: Four-class single-pulse waveform classification ($M = 31$). (a) Per-class and overall accuracy vs. SNR. Chirp identified at 2 dB; overall 90% accuracy at 14 dB. (b) Confusion matrix at the 90% threshold.
  • ...and 13 more figures

Theorems & Definitions (55)

  • Definition 1: Group Action on $\mathbb{C}^M$
  • Definition 2: Group-Averaged Estimator
  • Remark 1
  • Remark 2
  • Definition 3: Cayley Graph Autocorrelation Matrix
  • Remark 3: Consistency with Classical Spectral Analysis
  • Theorem 4: General Replacement Theorem
  • proof
  • Remark 4: Role of Group Size
  • Corollary 5: Sample Complexity of Group-Constrained Estimation
  • ...and 45 more