A contribution to Optimal Transport on incomparable spaces
arXiv:2011.04447
Abstract
Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from this theory, at the frontier between mathematics and optimization. This thesis proposes to study the complex scenario in which the different data belong to incomparable spaces. In particular we address the following questions: how to define and apply Optimal Transport between graphs, between structured data? How can it be adapted when the data are varied and not embedded in the same metric space? This thesis proposes a set of Optimal Transport tools for these different cases. An important part is notably devoted to the study of the Gromov-Wasserstein distance whose properties allow to define interesting transport problems on incomparable spaces. More broadly, we analyze the mathematical properties of the various proposed tools, we establish algorithmic solutions to compute them and we study their applicability in numerous machine learning scenarii which cover, in particular, classification, simplification, partitioning of structured data, as well as heterogeneous domain adaptation.
References in corpus (31)
- Distributed Representations of Words and Phrases and their Compositionality
- Sequence to Sequence Learning with Neural Networks
- Semi-Supervised Classification with Graph Convolutional Networks
- Geometric deep learning: going beyond Euclidean data
- DeCAF: A Deep Convolutional Activation Feature for Generic Visual Recognition
- Group Equivariant Convolutional Networks
- A Survey on Metric Learning for Feature Vectors and Structured Data
- Fast Computation of Wasserstein Barycenters
- A Survey on Graph Kernels
- Learning with a Wasserstein Loss
- On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
- Graph Matching Networks for Learning the Similarity of Graph Structured Objects
- Optimal Transport for structured data with application on graphs
- Sample Complexity of Sinkhorn divergences
- Interpolating between Optimal Transport and MMD using Sinkhorn Divergences
- Convergence Rate of Frank-Wolfe for Non-Convex Objectives
- Sliced Wasserstein Kernel for Persistence Diagrams
- A Deep Generative Model for Graph Layout
- Deep Graph Matching Consensus
- Subspace Robust Wasserstein Distances
- Geoopt: Riemannian Optimization in PyTorch
- Semi-dual Regularized Optimal Transport
- Policy Optimization as Wasserstein Gradient Flows
- McTorch, a manifold optimization library for deep learning
- Learning Autoencoders with Relational Regularization
- Recognizing Cuneiform Signs Using Graph Based Methods
- A Structured Prediction Approach for Label Ranking
- Fast and Robust Comparison of Probability Measures in Heterogeneous Spaces
- (Probably) Concave Graph Matching
- Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation
- Cournot-Nash equilibrium and optimal transport in a dynamic setting