Sublinear Time Approximation of Text Similarity Matrices

TL;DR

This paper introduces a sublinear time algorithm for approximating pairwise similarity matrices in NLP, including indefinite matrices, enabling efficient computation for large datasets with high accuracy in downstream tasks.

Contribution

It generalizes the Nyström method to indefinite similarity matrices and achieves sublinear time complexity, improving efficiency in NLP similarity approximations.

Findings

High accuracy in document classification tasks

Effective approximation of indefinite similarity matrices

Sublinear time complexity achieved

Abstract

We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a similarity matrix for $n$ data points requires $\Omega(n^2)$ similarity computations. This quadratic scaling is a significant bottleneck, especially when similarities are computed via expensive functions, e.g., via transformer models. Approximation methods reduce this quadratic complexity, often by using a small subset of exactly computed similarities to approximate the remainder of the complete pairwise similarity matrix. Significant work focuses on the efficient approximation of positive semidefinite (PSD) similarity matrices, which arise e.g., in kernel methods. However, much less is understood about indefinite (non-PSD) similarity matrices, which often arise in NLP. Motivated by the observation that many of these matrices are still somewhat close to PSD, we introduce a generalization of the popular Nystr\"{o}m method to the indefinite setting. Our algorithm can be applied to any similarity matrix and runs in sublinear time in the size of the matrix, producing a rank-$s$ approximation with just $O(ns)$ similarity computations. We show that our method, along with a simple variant of CUR decomposition, performs very well in approximating a variety of similarity matrices arising in NLP tasks. We demonstrate high accuracy of the approximated similarity matrices in the downstream tasks of document classification, sentence similarity, and cross-document coreference.