Problem
Research questions and friction points this paper is trying to address.
Tree Field Integration
Tensor Fields
Optimal Transport
Computational Efficiency
Distance Dependent Kernels
Structure-Adaptive Tree Field Integrators
🤖 AI Summary
This study addresses the high computational complexity of tensor field integration (TFI) on trees, which limits its applicability in problems such as optimal transport. To overcome this, we propose the STAD-TFI algorithm, which achieves near-linear integration complexity by decomposing tree structures via path-trunk decomposition and single-vertex separators, combined with two-dimensional fast Fourier transforms (FFT). Furthermore, it integrates the Sinkhorn algorithm to compute exact relaxed solutions for optimal transport. This work significantly outperforms conventional TFI methods and, for the first time, derives an exact relaxation formulation of optimal transport based on fast TFI. Experiments demonstrate that the proposed approach exhibits superior performance across synthetic tree acceleration, grid-based optimal transport, and visual Topological Attention Transformers.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Structure-Adaptive Tree Field Integrators
Near-linear algorithms
Fast Fourier transforms
Optimal Transport
Topological Attention Transformers
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