67 research outputs found
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
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
Learning Graphons via Structured Gromov-Wasserstein Barycenters
We propose a novel and principled method to learn a nonparametric graph model
called graphon, which is defined in an infinite-dimensional space and
represents arbitrary-size graphs. Based on the weak regularity lemma from the
theory of graphons, we leverage a step function to approximate a graphon. We
show that the cut distance of graphons can be relaxed to the Gromov-Wasserstein
distance of their step functions. Accordingly, given a set of graphs generated
by an underlying graphon, we learn the corresponding step function as the
Gromov-Wasserstein barycenter of the given graphs. Furthermore, we develop
several enhancements and extensions of the basic algorithm, , the
smoothed Gromov-Wasserstein barycenter for guaranteeing the continuity of the
learned graphons and the mixed Gromov-Wasserstein barycenters for learning
multiple structured graphons. The proposed approach overcomes drawbacks of
prior state-of-the-art methods, and outperforms them on both synthetic and
real-world data. The code is available at
https://github.com/HongtengXu/SGWB-Graphon
Distances and Isomorphism between Networks and the Stability of Network Invariants
We develop the theoretical foundations of a network distance that has
recently been applied to various subfields of topological data analysis, namely
persistent homology and hierarchical clustering. While this network distance
has previously appeared in the context of finite networks, we extend the
setting to that of compact networks. The main challenge in this new setting is
the lack of an easy notion of sampling from compact networks; we solve this
problem in the process of obtaining our results. The generality of our setting
means that we automatically establish results for exotic objects such as
directed metric spaces and Finsler manifolds. We identify readily computable
network invariants and establish their quantitative stability under this
network distance. We also discuss the computational complexity involved in
precisely computing this distance, and develop easily-computable lower bounds
by using the identified invariants. By constructing a wide range of explicit
examples, we show that these lower bounds are effective in distinguishing
between networks. Finally, we provide a simple algorithm that computes a lower
bound on the distance between two networks in polynomial time and illustrate
our metric and invariant constructions on a database of random networks and a
database of simulated hippocampal networks
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