Efficient algorithms speed up computations on tree structures with distance kernels
Structure-Adaptive Tree Field Integrators
Machine LearningComputer Vision and Pattern RecognitionData Structures and Algorithms
Summary
Calculating how elements affect each other in complex networks arranged like trees can be very slow, especially when distances matter. The authors introduce a new way to break down these tree structures into simpler parts and use fast math tricks to do these calculations much quicker. This approach works better than older methods and is shown to help in tasks like moving resources optimally on shapes and improving attention in vision AI. Their work includes both mathematical proofs and practical tests to show its usefulness.
What this means in practice
- •For graphics engineers: Accelerate computations related to optimal transport problems on 3D mesh structures, improving performance in mesh processing tasks.
- •For machine learning developers: Enhance efficiency of topological attention mechanisms in vision models through faster tree-based integrations.
Authors
Millend Roy, Soham Samal, Ivan Zelich, Krzysztof Marcin Choromanski
Abstract
We present a new class of near-linear algorithms for efficiently integrating general tensor fields defined on trees with distance dependent kernels, the Structure-Adaptive Tree Field Integrators (STAD-TFIs). STAD-TFIs exploit the tree's underlying structure through decompositions built around path backbones and single vertex separators, and use two-dimensional fast Fourier transforms to compute interactions jointly. By exploiting this structural information, STAD-TFIs achieve more computationally efficient integration than their regular efficient tree field integrators (TFI) counterparts. We provide a detailed theoretical analysis of our proposed approach and complement it with an exhaustive empirical evaluation, ranging from speed tests on synthetic trees, through accelerated Sinkhorn-based relaxations of the Optimal Transport algorithms on real meshes, to Topological Attention Transformers for vision tasks. To the best of our knowledge, we provide some of the first results showing that efficient to compute and accurate relaxations of the geodesic Sinkhorn-based solutions of the Optimal Transport problem can be derived by applying fast TFI methods.