发表机构
Indiana University（印第安纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对向量流定义余弦密度矩，将其推广至余弦相似度场景，给出单遍流算法及紧下界，并提出等和矩隔离技术证明非负情形下界。
AI 中文摘要
对于非零向量 $x_1,\ldots,x_n\in\mathbb{R}^d$ 的流，令 $u_i=x_i/\\|x_i\\|_2$。我们定义第 $i$ 个流元素的余弦密度为 $D_i:=\sum_{j\in[n]}\langle u_i,u_j\rangle$，并在有符号余弦和非负余弦两种情形下研究密度矩 $M_p:=\sum_{i\in[n]}D_i^p$。这些量是经典频率矩的相似性感知类比：将余弦相似度替换为等值（即 $D_i = \sum_{j\in[n]} \mathbf{1}\{u_j = u_i\}$）则得到 $M_p=F_{p+1}$，特别地，$M_{-1}=F_0$，即不同元素的数量。我们给出单遍流算法以及在对维度 $d$ 的依赖上紧或近乎紧的下界。我们的结果因此将经典数据流文献中的若干基本统计量推广到余弦相似度——一种在现代AI系统中广泛用于比较向量嵌入的度量。证明非负余弦空间下界的主要挑战在于消除不想要的贡献，而不依赖于相反向量对。我们通过一种称为“等和矩隔离”的构造来解决这一问题：两个仅插入的前缀具有相同的基数和向量和，有限差分比较抵消了它们的共同基线，同时隔离出所需的高阶信号。这一证明框架可能对其他无法直接抵消的仅插入流下界有用。
英文摘要
For a stream of nonzero vectors $x_1,\ldots,x_n\in\mathbb{R}^d$, let $u_i=x_i/\|x_i\|_2$. We define the cosine density of the $i$-th stream element by $D_i:=\sum_{j\in[n]}\langle u_i,u_j\rangle$ and study the density moments $M_p:=\sum_{i\in[n]}D_i^p$ in both the signed- and nonnegative-cosine regimes. These quantities are similarity-aware analogues of classical frequency moments: replacing cosine similarity by equality (that is, $D_i = \sum_{j\in[n]} \mathbf{1}\{u_j = u_i\}$) gives $M_p=F_{p+1}$ and, in particular, $M_{-1}=F_0$, the number of distinct elements. We give one-pass streaming algorithms and lower bounds that are tight or nearly tight in their dependence on the dimension $d$. Our results thus extend several fundamental statistics from the classical data stream literature to cosine similarity, a widely used measure for comparing vector embeddings in modern AI systems. The main challenge in proving a space lower bound for nonnegative cosine is to eliminate unwanted contributions without relying on pairs of opposite vectors. We address this through a construction that we call \emph{equal-sum moment isolation}: two insertion-only prefixes have the same cardinality and vector sum, and a finite-difference comparison cancels their common baseline while isolating the desired higher-order signal. This proof framework may be useful for other insertion-only streaming lower bounds, where direct cancellation is not possible.
Comments41 pages
