AI 中文总结
研究针对最大内积相似度单向量嵌入的维度下界问题。此前上下界在m指数上有差距，本文几乎弥合。证明结合方法得出，对固定δ及足够小ε、m≥(1/ε)^A_δ，单向量近似维度D≥m^(c_δ/ε^(2 - 2δ))，此结论对多种情况适用，指数接近上界。
AI 中文摘要
多向量嵌入通过点云表示项目，并使用倒角相似度比较查询和文档点云，而单向量嵌入使用普通内积。对于单例查询，倒角成为最大内积相似度（MAX-IP）。在我们的设置中，MUVERA给出维度\(m^{O(1/\varepsilon^2)}\)，而先前的下界\((\varepsilon^2m)^{\Omega(1/\varepsilon)}\)在\(m\)的指数中在\(1/\varepsilon\)和\(1/\varepsilon^2\)之间留下差距。我们几乎弥合了这个差距。对于每个固定的\(\delta\in(0,1)\)，存在常数\(A_\delta,c_\delta>0\)，使得对于所有足够小的\(\varepsilon>0\)和每个\(m\geq(1/\varepsilon)^{A_\delta}\)，存在单位查询向量和最多\(m\)个单位向量的文档点云，对于所有成对MAX-IP值到加性误差\(\varepsilon\)的每个单向量近似具有维度\[D\geq m^{c_\delta/\varepsilon^{2 - 2\delta}}\]。即使对于在看到数据集后选择的完全数据依赖表示也成立。它也适用于倒角，因为所有查询都是单例。由于\(\delta\)可以任意小，指数接近上界的\(O(1/\varepsilon^2)\)依赖性。证明将Sherstov的模式矩阵方法与计算近似度为\(\Omega(k^{1 - \delta})\)的函数的多项式大小、恒定宽度DNF公式相结合。均匀宽度填充和块编码创建一个\(\Omega(\varepsilon)\)差距。然后一个虚拟坐标均衡所有错误输入，产生一个单位球MAX-IP矩阵，它是DNF模式矩阵的精确二值仿射图像，差距至少为\(8\varepsilon\)。这允许应用近似秩界。
英文摘要
Multi-vector embeddings represent items by point clouds and compare query and document point clouds using Chamfer similarity, whereas single-vector embeddings use ordinary inner products. For singleton queries, Chamfer becomes maximum inner product similarity (MAX-IP). In our setting, MUVERA gives dimension $m^{O(1/ε^2)}$ [DHJ+24], whereas the previous lower bound $(ε^2m)^{Ω(1/ε)}$ [Jay26] left a gap between $1/ε$ and $1/ε^2$ in the exponent of $m$. We nearly close this gap. For every fixed $δ\in(0,1)$, there are constants $A_δ,c_δ>0$ such that, for all sufficiently small $ε>0$ and every $m\ge(1/ε)^{A_δ}$, there exist unit query vectors and document point clouds of at most $m$ unit vectors for which every single-vector approximation of all pairwise MAX-IP values to additive error $ε$ has dimension \[ D \ge m^{c_δ/ε^{2-2δ}}. \] This holds even for fully data-dependent representations chosen after seeing the dataset. It also applies to Chamfer because all queries are singletons. Since $δ$ can be arbitrarily small, the exponent approaches the $O(1/ε^2)$ dependence of the upper bound. The proof combines Sherstov's pattern matrix method with polynomial-size, constant-width DNF formulas computing functions of approximate degree $Ω(k^{1-δ})$. Uniform-width padding and a block encoding create an $Ω(ε)$ gap. A dummy coordinate then equalizes all false inputs, yielding a unit-sphere MAX-IP matrix that is an exact two-valued affine image of the DNF pattern matrix with gap at least $8ε$. This allows the approximate-rank bound to apply. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.
