提出超球空间注意力保真保障框架，实现可验证的高效检索。
HCC+: Hyperbolic Guarding for Certified Attention Retrieval
- 利用庞加莱球的几何特性构建确定性防护机制
- 层内注意力偏差<理想值10%，软注意力总变差以1/√n衰减
- 无需查询依赖，存储压缩6.1倍，适合高维检索场景
我们研究了超球空间中注意力检索的利普希茨稳定性。现有方法在有限精度表示下缺乏对注意力权重保持的确定性保证。本文提出HCC+，一个理论框架，利用庞加莱球的三个性质：指数体积增长支持查询无关的边界截断；对数覆盖半径使1-中心的临界键识别与维度无关；以及与嵌入维度无关的打包界常数。证明了两个确定性保证：对于精确检索，每层注意力偏差被限制在理想值的10%以内；对于软注意力，总变差距离以O(1/√n)速率衰减，达到有限样本方差率。由于防护机制，该框架相比FP16实现6.1倍的存储压缩。这是首个在非欧几里得几何中实现确定性、查询无关检索证书的工作。
原文摘要 · Abstract (English)
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.
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