{"paper":{"title":"Algebraic Diversity: Group-Theoretic Spectral Estimation from Single Observations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Group averaging applied to a single observation can match the subspace estimation of traditional multi-observation covariance methods.","cross_cats":["cs.IT","eess.SP","math.IT"],"primary_cat":"cs.LG","authors_text":"Mitchell A. Thornton","submitted_at":"2026-04-04T08:08:34Z","abstract_excerpt":"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 "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A General Replacement Theorem proves that a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the data object possesses representation-theoretic symmetry under a chosen group G such that the group action preserves the relevant subspace structure without requiring prior knowledge of that symmetry.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Group-averaged estimators from one observation achieve equivalent subspace decomposition to multi-observation covariance methods, with variance scaling by group order and unifying DFT, DCT, and KLT as special cases.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Group averaging applied to a single observation can match the subspace estimation of traditional multi-observation covariance methods.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5434fbe3e9805a2caa55b5c7749aaa1df68b4eec3af28a60a6b5088884472c4c"},"source":{"id":"2604.03634","kind":"arxiv","version":5},"verdict":{"id":"18625041-7901-45a2-9e86-a9f8aabd239b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T18:49:28.900340Z","strongest_claim":"A General Replacement Theorem proves that a group-averaged estimator from one snapshot achieves equivalent subspace decomposition to multi-snapshot covariance estimation.","one_line_summary":"Group-averaged estimators from one observation achieve equivalent subspace decomposition to multi-observation covariance methods, with variance scaling by group order and unifying DFT, DCT, and KLT as special cases.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the data object possesses representation-theoretic symmetry under a chosen group G such that the group action preserves the relevant subspace structure without requiring prior knowledge of that symmetry.","pith_extraction_headline":"Group averaging applied to a single observation can match the subspace estimation of traditional multi-observation covariance methods."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.03634/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d4f75e331eb1c3358e16ac3c7518203c7da1dbc60b4cd3802e92e24d7235279d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}