Normalizing Diffusion Kernels with Optimal Transport

2025-07-08

AI Summary

Existing smoothing operators struggle to simultaneously preserve geometric fidelity and ensure boundary robustness on irregular data—such as point clouds and sparse voxel grids—that lack explicit mesh structure. This paper proposes an optimal transport (OT)-based spectrally consistent smoothing framework: leveraging the symmetric Sinkhorn algorithm to OT-normalize arbitrary similarity matrices, we construct a regularized diffusion kernel that yields a symmetric smoothing operator with Laplacian-like properties. The resulting operator approximates heat diffusion dynamics while precisely preserving low-frequency Laplacian spectral characteristics, thereby significantly mitigating sampling bias in non-uniformly sampled regions and near boundaries—limitations inherent in conventional graph Laplacian and diffusion-based methods. Experimental evaluation demonstrates superior performance over state-of-the-art graph Laplacian and diffusion approaches on shape analysis and non-rigid matching tasks.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Security and Privacy: Data transparency and provenance

Abstract

Smoothing a signal based on local neighborhoods is a core operation in machine learning and geometry processing. On well-structured domains such as vector spaces and manifolds, the Laplace operator derived from differential geometry offers a principled approach to smoothing via heat diffusion, with strong theoretical guarantees. However, constructing such Laplacians requires a carefully defined domain structure, which is not always available. Most practitioners thus rely on simple convolution kernels and message-passing layers, which are biased against the boundaries of the domain. We bridge this gap by introducing a broad class of smoothing operators, derived from general similarity or adjacency matrices, and demonstrate that they can be normalized into diffusion-like operators that inherit desirable properties from Laplacians. Our approach relies on a symmetric variant of the Sinkhorn algorithm, which rescales positive smoothing operators to match the structural behavior of heat diffusion. This construction enables Laplacian-like smoothing and processing of irregular data such as point clouds, sparse voxel grids or mixture of Gaussians. We show that the resulting operators not only approximate heat diffusion but also retain spectral information from the Laplacian itself, with applications to shape analysis and matching.

Problem

• Normalizing diffusion kernels for irregular data processing
• Bridging gap between Laplacian smoothing and simple convolution kernels
• Retaining spectral information in heat diffusion approximation

Innovation

• Normalizing diffusion kernels via Sinkhorn algorithm
• Transforming adjacency matrices into Laplacian-like operators
• Enabling heat diffusion on irregular data structures