Weighted Minwise Hashing Beats Linear Sketching for Inner Product Estimation

TL;DR

This paper introduces a weighted minwise hashing technique for inner product estimation that outperforms traditional linear sketching methods, especially for sparse vectors, with strong accuracy guarantees and empirical validation.

Contribution

The paper proposes a novel weighted MinHash-based sketching method that improves inner product approximation accuracy over existing linear sketching approaches, particularly for sparse data.

Findings

Outperforms CountSketch and Johnson-Lindenstrauss in accuracy for sparse vectors

Provides strong theoretical guarantees matching linear sketching for dense vectors

Empirically demonstrates superior performance on sparse data sets

Abstract

We present a new approach for computing compact sketches that can be used to approximate the inner product between pairs of high-dimensional vectors. Based on the Weighted MinHash algorithm, our approach admits strong accuracy guarantees that improve on the guarantees of popular linear sketching approaches for inner product estimation, such as CountSketch and Johnson-Lindenstrauss projection. Specifically, while our method admits guarantees that exactly match linear sketching for dense vectors, it yields significantly lower error for sparse vectors with limited overlap between non-zero entries. Such vectors arise in many applications involving sparse data. They are also important in increasingly popular dataset search applications, where inner product sketches are used to estimate data covariance, conditional means, and other quantities involving columns in unjoined tables. We complement our theoretical results by showing that our approach empirically outperforms existing linear sketches and unweighted hashing-based sketches for sparse vectors.