Matrix Product Sketching via Coordinated Sampling

We revisit the well-studied problem of approximating a matrix product, $\bv{A}^T\bv{B}$, based on small space sketches $\mathcal{S}(\bv{A})$ and $\mathcal{S}(\bv{B})$ of $\bv{A} \in \R^{n \times d}$ and $\bv{B}\in \R^{n \times m}$. We are interested in the setting where the sketches must be computed independently of each other, except for the use of a shared random seed. We prove that, when $\bv{A}$ and $\bv{B}$ are sparse, methods based on \emph{coordinated random sampling} can outperform classical linear sketching approaches, like Johnson-Lindenstrauss Projection or CountSketch. For example, to obtain Frobenius norm error $\epsilon\|\bv{A}\|_F\|\bv{B}\|_F$, coordinated sampling requires sketches of size $O(s/\epsilon^2)$ when $\bv{A}$ and $\bv{B}$ have at most $s \leq d,m$ non-zeros per row. In contrast, linear sketching leads to sketches of size $O(d/\epsilon^2)$ and $O(m/\epsilon^2)$ for $\bv{A}$ and $\bv{B}$. We empirically evaluate our approach on two applications: 1) distributed linear regression in databases, a problem motivated by tasks like dataset discovery and augmentation, and 2) approximating attention matrices in transformer-based language models. In both cases, our sampling algorithms yield an order of magnitude improvement over linear sketching.