Fast Tree-Field Integrators: From Low Displacement Rank to Topological Transformers
This work addresses computational bottlenecks in graph and image processing tasks, offering incremental improvements in speed and accuracy for applications like graph classification and Transformer models.
The paper tackles the problem of efficiently integrating tensor fields on weighted trees by introducing fast tree-field integrators (FTFIs) based on low displacement rank theory, achieving 5.7-13x speedups for exact algorithms on graphs with thousands of nodes and 1.0-1.5%+ accuracy gains in Topological Transformers with minimal extra parameters.
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.