AI 中文总结
针对双曲空间注意力检索的有限精度确定性保证问题，提出HCC+理论框架，实现10%以内的注意力偏差、O(1/√n)的软注意力总变差衰减，并获6.1倍存储缩减，为非欧几何提供首个相关确定性检索证书。
AI 中文摘要
我们研究双曲空间中注意力检索的利普希茨稳定性。现有方法无法在有限精度表示下对注意力权重的保持性提供确定性保证。我们提出HCC+，一个利用庞加莱球三项性质的理论框架：指数体积增长支持与查询无关的边界截断；双曲1-中心的对数覆盖半径支持与维度无关的关键键识别；以及与嵌入维度无关常数的打包界。我们证明两项确定性保证：对于精确检索，每层注意力偏差被限制在其理想值的10%以内；对于软注意力，总变差距离以O(1/√n)速率衰减，这是有限样本方差的速率。作为该防护机制的结果，该框架相对于FP16实现了6.1倍的存储缩减因子。我们提供了非欧几里得几何中首个与查询无关的确定性检索证书。
英文摘要
We study the Lipschitz stability of attention retrieval in hyperbolic spaces. Existing methods lack deterministic guarantees on attention-weight preservation under finite-precision representations. We introduce HCC+, a theoretical framework exploiting three properties of the Poincaré ball: exponential volume growth enabling query-independent boundary truncation; logarithmic covering radius of hyperbolic 1-centers enabling dimension-independent critical-key identification; and a packing bound with constants independent of the embedding dimension. We prove two deterministic guarantees: for exact retrieval, the per-layer attention deviation is bounded by 10\% of its ideal value; for soft attention, the total variation distance decays as $O(1/\sqrt{n})$, the rate of finite-sample variance. As a consequence of the guarding mechanism, the framework achieves a storage reduction factor of $6.1\times$ relative to FP16. We provide the first deterministic, query-independent retrieval certificate in non-Euclidean geometry.
Comments9 pages, no figures, theoretical paper
