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We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hilbert space of functions to another. For numerous machine learning tasks, data can be naturally viewed as samples drawn from spaces of functions, such as curves and surfaces, in high dimensions. Optimal transport for functional data analysis provides a useful framework of treatment for such domains. In this work, we develop an efficient algorithm for finding the stochastic transport map between functional domains and provide theoretical guarantees on the existence, uniqueness, and consistency of our estimate for the Hilbert-Schmidt operator. We validate our method on synthetic datasets and study the geometric properties of the transport map. Experiments on real-world datasets of robot arm trajectories further demonstrate the effectiveness of our method on applications in domain adaptation.
This article investigates the quality of the estimator of the linear Monge mapping between distributions. We provide the first concentration result on the linear mapping operator and prove a sample complexity of $n^{-1/2}$ when using empirical estima
Functional connectomes derived from functional magnetic resonance imaging have long been used to understand the functional organization of the brain. Nevertheless, a connectome is intrinsically linked to the atlas used to create it. In other words, a
Optimal transport is a machine learning problem with applications including distribution comparison, feature selection, and generative adversarial networks. In this paper, we propose feature-robust optimal transport (FROT) for high-dimensional data,
In recent years, manifold methods have moved into focus as tools for dimension reduction. Assuming that the high-dimensional data actually lie on or close to a low-dimensional nonlinear manifold, these methods have shown convincing results in several
Missing data is a crucial issue when applying machine learning algorithms to real-world datasets. Starting from the simple assumption that two batches extracted randomly from the same dataset should share the same distribution, we leverage optimal tr