Balanced Coarsening for Multilevel Hypergraph Partitioning via Wasserstein Discrepancy

We propose a balanced coarsening scheme for multilevel hypergraph partitioning. In addition, an initial partitioning algorithm is designed to improve the quality of k-way hypergraph partitioning. By assigning vertex weights through the LPT algorithm, we generate a prior hypergraph under a relaxed balance constraint. With the prior hypergraph, we have defined the Wasserstein discrepancy to coordinate the optimal transport of coarsening process. And the optimal transport matrix is solved by Sinkhorn algorithm. Our coarsening scheme fully takes into account the minimization of connectivity metric (objective function). For the initial partitioning stage, we define a normalized cut function induced by Fiedler vector, which is theoretically proved to be a concave function. Thereby, a three-point algorithm is designed to find the best cut under the balance constraint

Comparing Morse Complexes Using Optimal Transport: An Experimental Study

Morse complexes and Morse-Smale complexes are topological descriptors popular in topology-based visualization. Comparing these complexes plays an important role in their applications in feature correspondences, feature tracking, symmetry detection, and uncertainty visualization. Leveraging recent advances in optimal transport, we apply a class of optimal transport distances to the comparative analysis of Morse complexes. Contrasting with existing comparative measures, such distances are easy and efficient to compute, and naturally provide structural matching between Morse complexes. We perform an experimental study involving scientific simulation datasets and discuss the effectiveness of these distances as comparative measures for Morse complexes. We also provide an initial guideline for choosing the optimal transport distances under various data assumptions.Comment: IEEE Visualization Conference (IEEE VIS) Short Paper, accepted, 2023; supplementary materials: http://www.sci.utah.edu/~beiwang/publications/GWMC_VIS_Short_BeiWang_2023_Supplement.pd

Image-to-Image Retrieval by Learning Similarity between Scene Graphs

As a scene graph compactly summarizes the high-level content of an image in a structured and symbolic manner, the similarity between scene graphs of two images reflects the relevance of their contents. Based on this idea, we propose a novel approach for image-to-image retrieval using scene graph similarity measured by graph neural networks. In our approach, graph neural networks are trained to predict the proxy image relevance measure, computed from human-annotated captions using a pre-trained sentence similarity model. We collect and publish the dataset for image relevance measured by human annotators to evaluate retrieval algorithms. The collected dataset shows that our method agrees well with the human perception of image similarity than other competitive baselines.Comment: Accepted to AAAI 202

Hybrid Gromov-Wasserstein Embedding for Capsule Learning

Capsule networks (CapsNets) aim to parse images into a hierarchy of objects, parts, and their relations using a two-step process involving part-whole transformation and hierarchical component routing. However, this hierarchical relationship modeling is computationally expensive, which has limited the wider use of CapsNet despite its potential advantages. The current state of CapsNet models primarily focuses on comparing their performance with capsule baselines, falling short of achieving the same level of proficiency as deep CNN variants in intricate tasks. To address this limitation, we present an efficient approach for learning capsules that surpasses canonical baseline models and even demonstrates superior performance compared to high-performing convolution models. Our contribution can be outlined in two aspects: firstly, we introduce a group of subcapsules onto which an input vector is projected. Subsequently, we present the Hybrid Gromov-Wasserstein framework, which initially quantifies the dissimilarity between the input and the components modeled by the subcapsules, followed by determining their alignment degree through optimal transport. This innovative mechanism capitalizes on new insights into defining alignment between the input and subcapsules, based on the similarity of their respective component distributions. This approach enhances CapsNets' capacity to learn from intricate, high-dimensional data while retaining their interpretability and hierarchical structure. Our proposed model offers two distinct advantages: (i) its lightweight nature facilitates the application of capsules to more intricate vision tasks, including object detection; (ii) it outperforms baseline approaches in these demanding tasks

Multi-Marginal Gromov-Wasserstein Transport and Barycenters

Gromov-Wasserstein (GW) distances are combinations of Gromov-Hausdorff and Wasserstein distances that allow the comparison of two different metric measure spaces (mm-spaces). Due to their invariance under measure- and distance-preserving transformations, they are well suited for many applications in graph and shape analysis. In this paper, we introduce the concept of multi-marginal GW transport between a set of mm-spaces as well as its regularized and unbalanced versions. As a special case, we discuss multi-marginal fused variants, which combine the structure information of an mm-space with label information from an additional label space. To tackle the new formulations numerically, we consider the bi-convex relaxation of the multi-marginal GW problem, which is tight in the balanced case if the cost function is conditionally negative definite. The relaxed model can be solved by an alternating minimization, where each step can be performed by a multi-marginal Sinkhorn scheme. We show relations of our multi-marginal GW problem to (unbalanced, fused) GW barycenters and present various numerical results, which indicate the potential of the concept

Sliced Multi-Marginal Optimal Transport

Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability in the number of measures, samples and dimensionality. In this work, we propose a multi-marginal optimal transport paradigm based on random one-dimensional projections, whose (generalized) distance we term the sliced multi-marginal Wasserstein distance. To construct this distance, we introduce a characterization of the one-dimensional multi-marginal Kantorovich problem and use it to highlight a number of properties of the sliced multi-marginal Wasserstein distance. In particular, we show that (i) the sliced multi-marginal Wasserstein distance is a (generalized) metric that induces the same topology as the standard Wasserstein distance, (ii) it admits a dimension-free sample complexity, (iii) it is tightly connected with the problem of barycentric averaging under the sliced-Wasserstein metric. We conclude by illustrating the sliced multi-marginal Wasserstein on multi-task density estimation and multi-dynamics reinforcement learning problems