Fast Tree-Field Integrators: From Low Displacement Rank to Topological Transformers

TL;DR

This paper introduces fast algorithms based on structured matrices for tensor field integration on trees, with applications in graph metrics, classification, mesh modeling, and topological transformers, achieving significant speedups and accuracy improvements.

Contribution

It develops a new class of polylog-linear algorithms using low displacement rank matrices for efficient tensor field integration on trees, including novel RPE mechanisms for transformers.

Findings

Achieves 5.7-13x speedups on large graphs.

Provides 1.0-1.5% accuracy improvements with new RPE masking.

Most methods are exact, ensuring numerical equivalence to brute-force approaches.

Abstract

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular low displacement rank) for integrating tensor fields defined on weighted trees. Several applications of the resulting fast tree-field integrators (FTFIs) are presented, including (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) Topological Transformers (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as three extra learnable parameters per Transformer layer, leading to 1.0-1.5%+ accuracy gains. Importantly, most of FTFIs are exact methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide 5.7-13x speedups. We also provide an extensive theoretical analysis of our methods.