Optimal Transport for Domain Adaptation

Domain adaptation from one data space (or domain) to another is one of the most challenging tasks of modern data analytics. If the adaptation is done correctly, models built on a specific data space become more robust when confronted to data depicting the same semantic concepts (the classes), but observed by another observation system with its own specificities. Among the many strategies proposed to adapt a domain to another, finding a common representation has shown excellent properties: by finding a common representation for both domains, a single classifier can be effective in both and use labelled samples from the source domain to predict the unlabelled samples of the target domain. In this paper, we propose a regularized unsupervised optimal transportation model to perform the alignment of the representations in the source and target domains. We learn a transportation plan matching both PDFs, which constrains labelled samples in the source domain to remain close during transport. This way, we exploit at the same time the few labeled information in the source and the unlabelled distributions observed in both domains. Experiments in toy and challenging real visual adaptation examples show the interest of the method, that consistently outperforms state of the art approaches

Optimal Transport for Domain Adaptation

International audienceDomain adaptation is one of the most chal- lenging tasks of modern data analytics. If the adapta- tion is done correctly, models built on a specific data representation become more robust when confronted to data depicting the same classes, but described by another observation system. Among the many strategies proposed, finding domain-invariant representations has shown excel- lent properties, in particular since it allows to train a unique classifier effective in all domains. In this paper, we propose a regularized unsupervised optimal transportation model to perform the alignment of the representations in the source and target domains. We learn a transportation plan matching both PDFs, which constrains labeled samples of the same class in the source domain to remain close during transport. This way, we exploit at the same time the labeled samples in the source and the distributions observed in both domains. Experiments on toy and challenging real visual adaptation examples show the interest of the method, that consistently outperforms state of the art approaches. In addition, numerical experiments show that our approach leads to better performances on domain invariant deep learning features and can be easily adapted to the semi- supervised case where few labeled samples are available in the target domain

Hierarchical Optimal Transport for Unsupervised Domain Adaptation

In this paper, we propose a novel approach for unsupervised domain adaptation, that relates notions of optimal transport, learning probability measures and unsupervised learning. The proposed approach, HOT-DA, is based on a hierarchical formulation of optimal transport, that leverages beyond the geometrical information captured by the ground metric, richer structural information in the source and target domains. The additional information in the labeled source domain is formed instinctively by grouping samples into structures according to their class labels. While exploring hidden structures in the unlabeled target domain is reduced to the problem of learning probability measures through Wasserstein barycenter, which we prove to be equivalent to spectral clustering. Experiments on a toy dataset with controllable complexity and two challenging visual adaptation datasets show the superiority of the proposed approach over the state-of-the-art

Unsupervised Domain Adaptation via Deep Hierarchical Optimal Transport

Unsupervised domain adaptation is a challenging task that aims to estimate a transferable model for unlabeled target domain by exploiting source labeled data. Optimal Transport (OT) based methods recently have been proven to be a promising direction for domain adaptation due to their competitive performance. However, most of these methods coarsely aligned source and target distributions, leading to the over-aligned problem where the category-discriminative information is mixed up although domain-invariant representations can be learned. In this paper, we propose a Deep Hierarchical Optimal Transport method (DeepHOT) for unsupervised domain adaptation. The main idea is to use hierarchical optimal transport to learn both domain-invariant and category-discriminative representations by mining the rich structural correlations among domain data. The DeepHOT framework consists of a domain-level OT and an image-level OT, where the latter is used as the ground distance metric for the former. The image-level OT captures structural associations of local image regions that are beneficial to image classification, while the domain-level OT learns domain-invariant representations by leveraging the underlying geometry of domains. However, due to the high computational complexity, the optimal transport based models are limited in some scenarios. To this end, we propose a robust and efficient implementation of the DeepHOT framework by approximating origin OT with sliced Wasserstein distance in image-level OT and using a mini-batch unbalanced optimal transport for domain-level OT. Extensive experiments show that DeepHOT surpasses the state-of-the-art methods in four benchmark datasets. Code will be released on GitHub.Comment: 9 pages, 3 figure

Semi-supervised Learning of Pushforwards For Domain Translation & Adaptation

Given two probability densities on related data spaces, we seek a map pushing one density to the other while satisfying application-dependent constraints. For maps to have utility in a broad application space (including domain translation, domain adaptation, and generative modeling), the map must be available to apply on out-of-sample data points and should correspond to a probabilistic model over the two spaces. Unfortunately, existing approaches, which are primarily based on optimal transport, do not address these needs. In this paper, we introduce a novel pushforward map learning algorithm that utilizes normalizing flows to parameterize the map. We first re-formulate the classical optimal transport problem to be map-focused and propose a learning algorithm to select from all possible maps under the constraint that the map minimizes a probability distance and application-specific regularizers; thus, our method can be seen as solving a modified optimal transport problem. Once the map is learned, it can be used to map samples from a source domain to a target domain. In addition, because the map is parameterized as a composition of normalizing flows, it models the empirical distributions over the two data spaces and allows both sampling and likelihood evaluation for both data sets. We compare our method (parOT) to related optimal transport approaches in the context of domain adaptation and domain translation on benchmark data sets. Finally, to illustrate the impact of our work on applied problems, we apply parOT to a real scientific application: spectral calibration for high-dimensional measurements from two vastly different environmentsComment: 19 pages, 7 figure

Joint Distribution Optimal Transportation for Domain Adaptation

This paper deals with the unsupervised domain adaptation problem, where one wants to estimate a prediction function $f$ in a given target domain without any labeled sample by exploiting the knowledge available from a source domain where labels are known. Our work makes the following assumption: there exists a non-linear transformation between the joint feature/label space distributions of the two domain $\mathcal{P}_s$ and $\mathcal{P}_t$. We propose a solution of this problem with optimal transport, that allows to recover an estimated target $\mathcal{P}^f_t=(X,f(X))$ by optimizing simultaneously the optimal coupling and $f$. We show that our method corresponds to the minimization of a bound on the target error, and provide an efficient algorithmic solution, for which convergence is proved. The versatility of our approach, both in terms of class of hypothesis or loss functions is demonstrated with real world classification and regression problems, for which we reach or surpass state-of-the-art results.Comment: Accepted for publication at NIPS 201

Functional optimal transport: map estimation and domain adaptation for functional data

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.Comment: 23 pages, 6 figures, 2 table