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

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.