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

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 $m^{O(1/\epsilon^2)}$ [DHJ+24], whereas the previous lower bound $(\epsilon^2m)^{\Omega(1/\epsilon)}$ [Jay26] left a gap between $1/\epsilon$ and $1/\epsilon^2$ in the exponent of $m$. We nearly close this gap. For every fixed $\delta\in(0,1)$, there are constants $A_\delta,c_\delta>0$ such that, for all sufficiently small $\epsilon>0$ and every $m\ge(1/\epsilon)^{A_\delta}$, 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 $\epsilon$ has dimension \[ D \ge m^{c_\delta/\epsilon^{2-2\delta}}. \] This holds even for fully data-dependent representations chosen after seeing the dataset. It also applies to Chamfer because all queries are singletons. Since $\delta$ can be arbitrarily small, the exponent approaches the $O(1/\epsilon^2)$ 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 $\Omega(k^{1-\delta})$. Uniform-width padding and a block encoding create an $\Omega(\epsilon)$ 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 $8\epsilon$. 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.