← 研踪 PaperTrailarXiv:2611.02237 · v1
Nonreciprocal Spherical Attention and Retrieval: Sharp Coarsening Clocks and Certified Score Correction
首次提交 2026-11-02 05:58
版本更新 2026-11-02 05:58
雷达观测 2026-11-04 23:42
摘要
We develop nonreciprocal spherical attention as a mathematical framework for coarsening and compressed nearest-neighbor retrieval. For two shrinking acute blocks whose centers have zero skew pairing and whose total mass ratio is subexponential in inverse temperature, we prove $\lim_{β\to\infty}β^{-1}\log T_β=1-\cosφ$, without restricting particle-count growth. An explicit three-body flow has two rigorously separated metastable exponents. For finite compression, mass centering yields a sharp $O(r^2)$ attention-output error bound at fixed temperature, with an explicit spherical-normalization correction. We formulate skew query-side score calibration as a strictly convex moment problem and derive a unique normal equation, residual-based optimization certificates, and bounded low-rank projections. A norm-aware margin objective calibrates native squared-distance rankings. Experiments use semantic retrieval data, one million visual descriptors with three codebook seeds, and a ten-million-vector FAISS IVF-PQ index with recomputed exact cosine ground truth. On SIFT1M, skew margin calibration raises native-L2 hit@10 from 0.8926 to 0.8948; symmetric calibration reaches 0.8996. The ten-million-vector index reaches 0.8827 hit@10 at 0.1794 ms per query in a four-thread batch benchmark, while validation selects no skew update. Exact streamed attention over ten million keys evaluates compressed memory on 1,024 held-out queries. Independent matrix solves, automatic differentiation, finite differences, interval arithmetic, full-flow tests, and nine Rocq-checked algebra lemmas support reproducibility. Directionality alone does not establish retrieval superiority.
作者备注：42 pages, 9 figures, 10 tables. Includes mathematical appendices and ancillary reproducibility code
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v1 · 2026-11-02 05:58 · 最新观测版本
Nonreciprocal Spherical Attention and Retrieval: Sharp Coarsening Clocks and Certified Score Correction
We develop nonreciprocal spherical attention as a mathematical framework for coarsening and compressed nearest-neighbor retrieval. For two shrinking acute blocks whose centers have zero skew pairing and whose total mass ratio is subexponential in inverse temperature, we prove $\lim_{β\to\infty}β^{-1}\log T_β=1-\cosφ$, without restricting particle-count growth. An explicit three-body flow has two rigorously separated metastable exponents. For finite compression, mass centering yields a sharp $O(r^2)$ attention-output error bound at fixed temperature, with an explicit spherical-normalization correction. We formulate skew query-side score calibration as a strictly convex moment problem and derive a unique normal equation, residual-based optimization certificates, and bounded low-rank projections. A norm-aware margin objective calibrates native squared-distance rankings. Experiments use semantic retrieval data, one million visual descriptors with three codebook seeds, and a ten-million-vector FAISS IVF-PQ index with recomputed exact cosine ground truth. On SIFT1M, skew margin calibration raises native-L2 hit@10 from 0.8926 to 0.8948; symmetric calibration reaches 0.8996. The ten-million-vector index reaches 0.8827 hit@10 at 0.1794 ms per query in a four-thread batch benchmark, while validation selects no skew update. Exact streamed attention over ten million keys evaluates compressed memory on 1,024 held-out queries. Independent matrix solves, automatic differentiation, finite differences, interval arithmetic, full-flow tests, and nine Rocq-checked algebra lemmas support reproducibility. Directionality alone does not establish retrieval superiority.
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