310 research outputs found
Using Gromov-Wasserstein distance to explore sets of networks
In many fields such as social sciences or biology, relations between data or variables are presented as networks. To compare these networks, a meaningful notion of distance between networks is highly desired. The aim of this Master thesis is to study, implement, and apply one Gromov-Wasserstein type of distance introduced by F.Memoli (2011) in his paper "Gromov-Wasserstein Distances and the Metric Approach to Object Matching" to study sets of complex networks. Taking into account theoretical underpinnings introduced in this paper we represent some real world networks as metric measure spaces and compare them on basis of Gromov-Wasserstein distance
Learning Generative Models across Incomparable Spaces
Generative Adversarial Networks have shown remarkable success in learning a
distribution that faithfully recovers a reference distribution in its entirety.
However, in some cases, we may want to only learn some aspects (e.g., cluster
or manifold structure), while modifying others (e.g., style, orientation or
dimension). In this work, we propose an approach to learn generative models
across such incomparable spaces, and demonstrate how to steer the learned
distribution towards target properties. A key component of our model is the
Gromov-Wasserstein distance, a notion of discrepancy that compares
distributions relationally rather than absolutely. While this framework
subsumes current generative models in identically reproducing distributions,
its inherent flexibility allows application to tasks in manifold learning,
relational learning and cross-domain learning.Comment: International Conference on Machine Learning (ICML
On Nonrigid Shape Similarity and Correspondence
An important operation in geometry processing is finding the correspondences
between pairs of shapes. The Gromov-Hausdorff distance, a measure of
dissimilarity between metric spaces, has been found to be highly useful for
nonrigid shape comparison. Here, we explore the applicability of related shape
similarity measures to the problem of shape correspondence, adopting spectral
type distances. We propose to evaluate the spectral kernel distance, the
spectral embedding distance and the novel spectral quasi-conformal distance,
comparing the manifolds from different viewpoints. By matching the shapes in
the spectral domain, important attributes of surface structure are being
aligned. For the purpose of testing our ideas, we introduce a fully automatic
framework for finding intrinsic correspondence between two shapes. The proposed
method achieves state-of-the-art results on the Princeton isometric shape
matching protocol applied, as usual, to the TOSCA and SCAPE benchmarks
Gromov-Monge quasi-metrics and distance distributions
Applications in data science, shape analysis and object classification
frequently require maps between metric spaces which preserve geometry as
faithfully as possible. In this paper, we combine the Monge formulation of
optimal transport with the Gromov-Hausdorff distance construction to define a
measure of the minimum amount of geometric distortion required to map one
metric measure space onto another. We show that the resulting quantity, called
Gromov-Monge distance, defines an extended quasi-metric on the space of
isomorphism classes of metric measure spaces and that it can be promoted to a
true metric on certain subclasses of mm-spaces. We also give precise
comparisons between Gromov-Monge distance and several other metrics which have
appeared previously, such as the Gromov-Wasserstein metric and the continuous
Procrustes metric of Lipman, Al-Aifari and Daubechies. Finally, we derive
polynomial-time computable lower bounds for Gromov-Monge distance. These lower
bounds are expressed in terms of distance distributions, which are classical
invariants of metric measure spaces summarizing the volume growth of metric
balls. In the second half of the paper, which may be of independent interest,
we study the discriminative power of these lower bounds for simple subclasses
of metric measure spaces. We first consider the case of planar curves, where we
give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver.
Our results on plane curves are then generalized to higher dimensional
manifolds, where we prove some sphere characterization theorems for the
distance distribution invariant. Finally, we consider several inverse problems
on recovering a metric graph from a collection of localized versions of
distance distributions. Results are derived by establishing connections with
concepts from the fields of computational geometry and topological data
analysis.Comment: Version 2: Added many new results and improved expositio
SHREC'16: partial matching of deformable shapes
Matching deformable 3D shapes under partiality transformations is a challenging problem that has received limited focus in the computer vision and graphics communities. With this benchmark, we explore and thoroughly investigate the robustness of existing matching methods in this challenging task. Participants are asked to provide a point-to-point correspondence (either sparse or dense) between deformable shapes undergoing different kinds of partiality transformations, resulting in a total of 400 matching problems to be solved for each method - making this benchmark the biggest and most challenging of its kind. Five matching algorithms were evaluated in the contest; this paper presents the details of the dataset, the adopted evaluation measures, and shows thorough comparisons among all competing methods
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
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