Multi-Vector Embeddings are Provably More Expressive than Single Vector Embeddings

2026-06-22

AI Summary

This work investigates whether multi-vector embeddings are strictly more expressive than single-vector embeddings, particularly whether the former can be effectively approximated by the latter under identical representation budgets. By constructing hard instances based on Chamfer similarity—specifically, pattern matrices encoding NANDₖ Boolean functions—and leveraging the Pattern Matrix Method for complexity analysis, the paper establishes the first rigorous proof that no low-dimensional single-vector embedding can linearly approximate the similarity captured by multi-vector embeddings. Moreover, it demonstrates that any single-vector embedding approximating such a multi-vector representation requires dimensionality at least (ε²m)^Ω(1/ε), thereby revealing a fundamental gap in representational efficiency between the two paradigms.

Abstract

Multi-vector (MV) embeddings have become a powerful paradigm in neural information retrieval (IR), achieving high retrieval accuracy by representing data with multiple vectors and scoring them via the non-linear Chamfer similarity. Despite their widely perceived superiority over single-vector (SV) embeddings which use inner product similarity, to date there is no formal proof that SV similarities cannot approximate MV similarities with the same representation size. Specifically, we ask the following: for any bounded dataset size $n \leq 2^{poly(m)}$, what is the smallest dimension $D$ so that given any collection of MV embeddings $Q_1,\dots,Q_n,X_1,\dots,X_n \subset \mathbb{R}^d$ containing at most $m$ vectors each, there always exist $q_1,\dots,q_n$, $d_1,\dots,d_n \in \mathbb{R}^{D}$ satisfying $|\langle q_i, d_j \rangle - \texttt{Chamfer}(Q_i,X_j)| \leq ε$ for all $i,j$? Recently, the MUVERA algorithm demonstrated that $D = m^{O(1/ε^2)}$ is possible. If improved to $D = md$, this would imply that MV embeddings are no more expressive than SV embeddings. In this paper, we rule out this scenario. Specifically, we prove the existence of a collection of MV embeddings in $\mathbb{R}^d$, each containing at most $m$ vectors, which require single-vector dimension of $D =(ε^2 m)^{Ω(1/ε)}$ to approximate, establishing a strong separation in representation size between MV and SV embeddings. Our proof leverages the Pattern Matrix Method by constructing a hard instance whose Chamfer similarity matrix encodes the $NAND_k$ boolean function. Our results confirm a long-held belief in the IR community: at a fixed representation size, multi-vector embeddings can express similarities which cannot even be approximately represented by single vector embeddings.

Problem

• multi-vector embeddings
• single-vector embeddings
• Chamfer similarity
• representation expressiveness
• approximation hardness

Innovation

• multi-vector embeddings
• Chamfer similarity
• representation expressiveness
• lower bound
• neural information retrieval