Journal ArticleDOI
Gromov–Wasserstein Distances and the Metric Approach to Object Matching
TLDR
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers.Abstract:
This paper discusses certain modifications of the ideas concerning the Gromov–Hausdorff distance which have the goal of modeling and tackling the practical problems of object matching and comparison. Objects are viewed as metric measure spaces, and based on ideas from mass transportation, a Gromov–Wasserstein type of distance between objects is defined. This reformulation yields a distance between objects which is more amenable to practical computations but retains all the desirable theoretical underpinnings. The theoretical properties of this new notion of distance are studied, and it is established that it provides a strict metric on the collection of isomorphism classes of metric measure spaces. Furthermore, the topology generated by this metric is studied, and sufficient conditions for the pre-compactness of families of metric measure spaces are identified. A second goal of this paper is to establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. This is done by proving explicit lower bounds for the proposed distance that involve many of the invariants previously reported by researchers. These lower bounds can be computed in polynomial time. The numerical implementations of the ideas are discussed and computational examples are presented.read more
Citations
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Book ChapterDOI
Metric structures on datasets: stability and classification of algorithms
TL;DR: It is described how using this formalism leads to an axiomatic description of many clustering algorithms, both flat and hierarchical, as well as several computational techniques that operate within the context of data/shape matching under invariances.
Proceedings ArticleDOI
Efficient and Robust Shape Correspondence via Sparsity-Enforced Quadratic Assignment
TL;DR: This work introduces a novel local pairwise descriptor and develops a simple, effective iterative method to solve the resulting quadratic assignment through sparsity control for shape correspondence between two approximate isometric surfaces.
Proceedings ArticleDOI
Geometry-aware domain adaptation for unsupervised alignment of word embeddings.
TL;DR: The authors proposed a manifold based geometric approach for learning unsupervised alignment of word embeddings between the source and the target languages, which formulates the alignment learning problem as a domain adaptation problem over the manifold of doubly stochastic matrices.
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Partial Gromov-Wasserstein with Applications on Positive-Unlabeled Learning.
TL;DR: This paper addresses the partial Gromov-Wasserstein problem and proposes an algorithm to solve it, and highlights that partial Wasserstein-based metrics prove effective in usual PU learning settings and demonstrates that partial Grosvenstein metrics is efficient in scenario where point clouds come from different domains or have different features.
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Gromov-Wasserstein Factorization Models for Graph Clustering
TL;DR: In this paper, a nonlinear factorization model based on Gromov-Wasserstein (GW) discrepancy is proposed, which estimates observed graphs as GW barycenters constructed by a set of atoms with different weights.
References
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