Near-Optimal Dimension Lower Bounds for Single-Vector Embeddings of Maximum Inner Product Similarity

Abstract

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 $m^{O(1/\varepsilon^2)}$~\cite{dhulipala2024muvera}, whereas the previous lower bound $(\varepsilon^2m)^{Ω(1/\varepsilon)}$~\cite{jayaram2026expressive} left a gap between $1/\varepsilon$ and $1/\varepsilon^2$ 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 $\varepsilon>0$ and every $m\ge(1/\varepsilon)^{A_δ}$, 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 $\varepsilon$ has dimension \[ D \ge m^{c_δ/\varepsilon^{2-2δ}}. \] 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 $O(1/\varepsilon^2)$ 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 $Ω(\varepsilon)$ 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 $8\varepsilon$. This allows the approximate-rank bound to apply.