AI summaryⓘ
The authors study how well complex point cloud embeddings can be approximated by simpler single-vector embeddings when comparing data using a measure called maximum inner product similarity (MAX-IP). They improve previous math results by nearly closing a gap in the required dimensions needed for good approximation, showing that the dimension must grow roughly like m to the power of a constant over epsilon squared (where epsilon measures approximation error). Their proof uses advanced techniques from complexity theory and logic to build hard examples that resist low-dimensional approximations, even when the approximation depends on the data. This result also applies to Chamfer similarity because queries are single points.
Multi-vector embeddingsSingle-vector embeddingsChamfer similarityMaximum inner product similarity (MAX-IP)Approximation dimensionPattern matrix methodDNF formulasApproximate degreeData-dependent representation
Authors
Rajesh Jayaram, Honghao Lin, Vahab Mirrokni, David P. Woodruff
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.