Deep Optimal Transport for Domain Adaptation on SPD Manifolds
College of Computing and Data Science, Nanyang Technological University（计算与数据科学学院，南洋理工大学）；Université Paris-Saclay, Inria, CEA, Palaiseau, France（巴黎-萨克雷大学，法国国家信息与自动化技术研究所，法国原子能委员会，Palaiseau）
Ce Ju, Cuntai Guan
备注：
18 pages, 4 figures, 3 tables. This work has been accepted by the Artificial Intelligence. Copyright will be transferred without notice, after which this version may no longer be accessible
Recent progress in geometric deep learning has drawn increasing attention from the machine learning community toward domain adaptation on symmetric positive definite (SPD) manifolds, especially for neuroimaging data that often suffer from distribution shifts across sessions. These data, typically represented as covariance matrices of brain signals, inherently lie on SPD manifolds due to their symmetry and positive definiteness. However, conventional domain adaptation methods often overlook this geometric structure when applied directly to covariance matrices, which can result in suboptimal performance. To address this issue, we introduce a new geometric deep learning framework that combines optimal transport theory with the geometry of SPD manifolds. Our approach aligns data distributions while respecting the manifold structure, effectively reducing both marginal and conditional discrepancies. We validate our method on three cross-session brain computer interface datasets, KU, BNCI2014001, and BNCI2015001, where it consistently outperforms baseline approaches while maintaining the intrinsic geometry of the data. We also provide quantitative results and visualizations to better illustrate the behavior of the learned embeddings.
Technion, Israel Institute of Technology（以色列理工学院）；Technical University of Munich（慕尼黑工业大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
In this paper, we address the problem of Domain Adaptation (DA) using Optimal Transport (OT) on Riemannian manifolds. We model the difference between two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for DA in a well-defined sense, up to a volume preserving map. We then focus on the manifold of Symmetric and Positive-Definite (SPD) matrices, whose structure provided a useful context in recent applications. We demonstrate the polar factorization theorem on this manifold. Due to the uniqueness of the weighted Riemannian mean, and by exploi…
SPDIM: Source-Free Unsupervised Conditional and Label Shift Adaptation in EEG
Nara Institute of Science and Technology (NAIST)（奈良科学技术大学）；Department of Dynamic Brain Imaging, ATR（动态脑成像部门，ATR）；Center for Advanced Intelligence Project, RIKEN（先进智能项目中心，RIKEN）
RIKEN Center for Advanced Intelligence Project (RIKEN AIP)（理化学研究所先进智能项目中心）；Advanced Telecommunications Research Institute International (ATR)（国际电气通信基础技术研究所）
Electroencephalography (EEG) provides access to neuronal dynamics non-invasively with millisecond resolution, rendering it a viable method in neuroscience and healthcare. However, its utility is limited as current EEG technology does not generalize well across domains (i.e., sessions and subjects) without expensive supervised re-calibration. Contemporary methods cast this transfer learning (TL) problem as a multi-source/-target unsupervised domain adaptation (UDA) problem and address it with deep learning or shallow, Riemannian geometry aware alignment methods. Both directions have, so far, fa…
Thales Air Systems（泰雷兹空中系统公司）；Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）
Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the Riemannian manifold of symmetric positive definite (SPD) matrices they belong to. In the context of deep networks, several architectures for these matrices have recently been proposed. In our article, we introduce a Riemannian batch normalization (batchnorm) algorithm, which generalizes the one used in Euclidean nets. This novel layer makes use of geometric oper…
SPDAlign: Interpretable Riemannian Alignment for EEG Forward Modeling Shifts
BIFOLD（柏林智能数据与机器学习研究所）；Technical University of Berlin（柏林工业大学）；Advanced Telecommunications Research Institute International（国际电气通信基础技术研究所）；RIKEN AIP（理化学研究所先进智能研究中心）；Nanyang Technological University（南洋理工大学）
ITMO University（ITMO大学）；Artificial Intelligence Research Institute（人工智能研究院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；Eberhard Karls Universität Tübingen（蒂宾根大学）
We present a novel algorithm for domain adaptation using optimal transport. In domain adaptation, the goal is to adapt a classifier trained on the source domain samples to the target domain. In our method, we use optimal transport to map target samples to the domain named source fiction. This domain differs from the source but is accurately classified by the source domain classifier. Our main idea is to generate a source fiction by c-cyclically monotone transformation over the target domain. If samples with the same labels in two domains are c-cyclically monotone, the optimal transport map bet…
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 …
The Institute of Statistical Mathematics（统计数学研究所）；RIKEN AIP（理化学研究所先进智能项目中心）
Domain adaptation aims to transfer knowledge of labeled instances obtained from a source domain to a target domain to fill the gap between the domains. Most domain adaptation methods assume that the source and target domains have the same dimensionality. Methods that are applicable when the number of features is different in each domain have rarely been studied, especially when no label information is given for the test data obtained from the target domain. In this paper, it is assumed that common features exist in both domains and that extra (new additional) features are observed in the targe…
Univ Lyon（里昂大学）；INSA-Lyon（里昂国立应用科学学院）；Université Claude Bernard Lyon 1（里昂第一大学克洛德·贝尔纳大学）；UJM-Saint Etienne（圣艾蒂安大学（让·莫奈大学））；CNRS（法国国家科学研究中心）；Inserm（法国国家健康与医学研究院）
In this paper, we propose a new feature selection method for unsupervised domain adaptation based on the emerging optimal transportation theory. We build upon a recent theoretical analysis of optimal transport in domain adaptation and show that it can directly suggest a feature selection procedure leveraging the shift between the domains. Based on this, we propose a novel algorithm that aims to sort features by their similarity across the source and target domains, where the order is obtained by analyzing the coupling matrix representing the solution of the proposed optimal transportation prob…
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 i…
Domain adaptation arises as an important problem in statistical learning theory when the data-generating processes differ between training and test samples, respectively called source and target domains. Recent theoretical advances show that the success of domain adaptation algorithms heavily relies on their ability to minimize the divergence between the probability distributions of the source and target domains. However, minimizing this divergence cannot be done independently of the minimization of other key ingredients such as the source risk or the combined error of the ideal joint hypothes…
University of Maryland（马里兰大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Optimal Transport (OT) distances such as Wasserstein have been used in several areas such as GANs and domain adaptation. OT, however, is very sensitive to outliers (samples with large noise) in the data since in its objective function, every sample, including outliers, is weighed similarly due to the marginal constraints. To remedy this issue, robust formulations of OT with unbalanced marginal constraints have previously been proposed. However, employing these methods in deep learning problems such as GANs and domain adaptation is challenging due to the instability of their dual optimization s…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Microsoft Research Asia（微软亚洲研究院）
It is expensive and time-consuming to collect sufficient labeled data for human activity recognition (HAR). Domain adaptation is a promising approach for cross-domain activity recognition. Existing methods mainly focus on adapting cross-domain representations via domain-level, class-level, or sample-level distribution matching. However, they might fail to capture the fine-grained locality information in activity data. The domain- and class-level matching are too coarse that may result in under-adaptation, while sample-level matching may be affected by the noise seriously and eventually cause o…
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 estimates of first and second order moments. This result is then used to derive a generalization bound for domain adaptation with optimal transport. As a consequence, this method approaches the performance of theoretical Bayes predictor under mild conditions on the covariance structure of the problem. We also discuss the computational complexity of the l…
We present a 2-step optimal transport approach that performs a mapping from a source distribution to a target distribution. Here, the target has the particularity to present new classes not present in the source domain. The first step of the approach aims at rejecting the samples issued from these new classes using an optimal transport plan. The second step solves the target (class ratio) shift still as an optimal transport problem. We develop a dual approach to solve the optimization problem involved at each step and we prove that our results outperform recent state-of-the-art performances. W…
National Institute of Information and Communications Technology（日本国立信息与通信技术研究所）；Research Center for Information Technology Innovation, Academic Sinica（中央研究院资讯科技创新研究中心）
Due to the mismatch of statistical distributions of acoustic speech between training and testing sets, the performance of spoken language identification (SLID) could be drastically degraded. In this paper, we propose an unsupervised neural adaptation model to deal with the distribution mismatch problem for SLID. In our model, we explicitly formulate the adaptation as to reduce the distribution discrepancy on both feature and classifier for training and testing data sets. Moreover, inspired by the strong power of the optimal transport (OT) to measure distribution discrepancy, a Wasserstein dist…
Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same min…
In this paper, we propose a new strategy for learning inertial robotic navigation models. The proposed strategy enhances the generalisability of end-to-end inertial modelling, and is aimed at wheeled robotic deployments. Concretely, the paper describes the following. (1) Using precision robotics, we empirically characterise the effect of changing the sensor position during navigation on the distribution of raw inertial signals, as well as the corresponding impact on learnt latent spaces. (2) We propose neural architectures and algorithms to assimilate knowledge from an indexed set of sensor po…
Winterlight Labs（温特莱特实验室）；Massachusetts Institute of Technology（麻省理工学院）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Microsoft Research（微软研究院）
Multi-language speech datasets are scarce and often have small sample sizes in the medical domain. Robust transfer of linguistic features across languages could improve rates of early diagnosis and therapy for speakers of low-resource languages when detecting health conditions from speech. We utilize out-of-domain, unpaired, single-speaker, healthy speech data for training multiple Optimal Transport (OT) domain adaptation systems. We learn mappings from other languages to English and detect aphasia from linguistic characteristics of speech, and show that OT domain adaptation improves aphasia d…
University of Genova（热那亚大学）；Istituto Italiano di Tecnologia（意大利理工学院）；École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；Criteo AI Lab（科韬人工智能实验室）；University College London（伦敦大学学院）
The problem of domain adaptation on an unlabeled target dataset using knowledge from multiple labelled source datasets is becoming increasingly important. A key challenge is to design an approach that overcomes the covariate and target shift both among the sources, and between the source and target domains. In this paper, we address this problem from a new perspective: instead of looking for a latent representation invariant between source and target domains, we exploit the diversity of source distributions by tuning their weights to the target task at hand. Our method, named Weighted Joint Di…
Univ.Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne CNRS, Inserm, CREATIS UMR 5220, U1206（里昂大学，里昂国立应用科学学院，里昂第一大学，圣埃蒂安让·莫内大学，法国国家科学研究中心，法国国家健康与医学研究院，CREATIS实验室）；Univ. Lyon, UJM-Saint-Etienne CNRS, Lab. Hubert Curien UMR 5516（里昂大学，圣埃蒂安让·莫内大学，法国国家科学研究中心，于贝尔·居里安实验室）
Domain adaptation (DA) is an important and emerging field of machine learning that tackles the problem occurring when the distributions of training (source domain) and test (target domain) data are similar but different. Current theoretical results show that the efficiency of DA algorithms depends on their capacity of minimizing the divergence between source and target probability distributions. In this paper, we provide a theoretical study on the advantages that concepts borrowed from optimal transportation theory can bring to DA. In particular, we show that the Wasserstein metric can be used…
Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；McGill University（麦吉尔大学）；Université de Montréal（蒙特利尔大学）
It is common in computer vision to be confronted with domain shift: images which have the same class but different acquisition conditions. In domain adaptation (DA), one wants to classify unlabeled target images using source labeled images. Unfortunately, deep neural networks trained on a source training set perform poorly on target images which do not belong to the training domain. One strategy to improve these performances is to align the source and target image distributions in an embedded space using optimal transport (OT). However OT can cause negative transfer, i.e. aligning samples with…
Optimal Transport has recently gained interest in machine learning for applications ranging from domain adaptation, sentence similarities to deep learning. Yet, its ability to capture frequently occurring structure beyond the "ground metric" is limited. In this work, we develop a nonlinear generalization of (discrete) optimal transport that is able to reflect much additional structure. We demonstrate how to leverage the geometry of this new model for fast algorithms, and explore connections and properties. Illustrative experiments highlight the benefit of the induced structured couplings for t…
Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校贝克曼高级科学技术研究所）；Microsoft Research（微软研究院）
Histopathology is critical for the diagnosis of many diseases, including cancer. These protocols typically require pathologists to manually evaluate slides under a microscope, which is time-consuming and subjective, leading to interest in machine learning to automate analysis. However, computational techniques are limited by batch effects, where technical factors like differences in preparation protocol or scanners can alter the appearance of slides, causing models trained on one institution to fail when generalizing to others. Here, we propose a domain adaptation method that improves the gene…
Criteo AI Lab（Criteo人工智能实验室）；Ecole Polytechnique（巴黎综合理工学院）；INSA de Rouen Normandie（鲁昂国立应用科学学院）；Université Technologique de Compiègne（贡比涅技术大学）；Université de Rouen Normandie（鲁昂诺曼底大学）；Université de Bretagne Sud（南布列塔尼大学）
We address the problem of unsupervised domain adaptation under the setting of generalized target shift (joint class-conditional and label shifts). For this framework, we theoretically show that, for good generalization, it is necessary to learn a latent representation in which both marginals and class-conditional distributions are aligned across domains. For this sake, we propose a learning problem that minimizes importance weighted loss in the source domain and a Wasserstein distance between weighted marginals. For a proper weighting, we provide an estimator of target label proportion by blen…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；VinAI Research（VinAI研究院）
Mini-batch optimal transport (m-OT) has been widely used recently to deal with the memory issue of OT in large-scale applications. Despite their practicality, m-OT suffers from misspecified mappings, namely, mappings that are optimal on the mini-batch level but are partially wrong in the comparison with the optimal transportation plan between the original measures. Motivated by the misspecified mappings issue, we propose a novel mini-batch method by using partial optimal transport (POT) between mini-batch empirical measures, which we refer to as mini-batch partial optimal transport (m-POT). Le…
The Cooperative Institute for Research in Environmental Sciences (CIRES)（环境科学合作研究所）；University of Colorado（科罗拉多大学）；NOAA Physical Sciences Laboratory（美国国家海洋和大气管理局物理科学实验室）；Research Center for Information Technology Innovation（资讯科技创新研究中心）；Academia Sinica（中央研究院）；National Institute of Information and Communications Technology (NICT)（日本信息通信研究机构）
This paper presents a novel discriminator-constrained optimal transport network (DOTN) that performs unsupervised domain adaptation for speech enhancement (SE), which is an essential regression task in speech processing. The DOTN aims to estimate clean references of noisy speech in a target domain, by exploiting the knowledge available from the source domain. The domain shift between training and testing data has been reported to be an obstacle to learning problems in diverse fields. Although rich literature exists on unsupervised domain adaptation for classification, the methods proposed, esp…
Kyoto University（京都大学）；The University of Electro-Communications（电气通信大学）；Microsoft（微软）
In recent years, stochastic variance reduction algorithms have attracted considerable attention for minimizing the average of a large but finite number of loss functions. This paper proposes a novel Riemannian extension of the Euclidean stochastic variance reduced gradient (R-SVRG) algorithm to a manifold search space. The key challenges of averaging, adding, and subtracting multiple gradients are addressed with retraction and vector transport. For the proposed algorithm, we present a global convergence analysis with a decaying step size as well as a local convergence rate analysis with a fixe…
Ferdowsi University of Mashhad（马什哈德菲尔多西大学）；University of Waterloo（滑铁卢大学）；University of Tehran（德黑兰大学）
This work concentrates on optimization on Riemannian manifolds. The Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm is a commonly used quasi-Newton method for numerical optimization in Euclidean spaces. Riemannian LBFGS (RLBFGS) is an extension of this method to Riemannian manifolds. RLBFGS involves computationally expensive vector transports as well as unfolding recursions using adjoint vector transports. In this article, we propose two mappings in the tangent space using the inverse second root and Cholesky decomposition. These mappings make both vector transport and adjoin…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Optimal transport aims to estimate a transportation plan that minimizes a displacement cost. This is realized by optimizing the scalar product between the sought plan and the given cost, over the space of doubly stochastic matrices. When the entropy regularization is added to the problem, the transportation plan can be efficiently computed with the Sinkhorn algorithm. Thanks to this breakthrough, optimal transport has been progressively extended to machine learning and statistical inference by introducing additional application-specific terms in the problem formulation. It is however challengi…
University of Massachusetts（马萨诸塞大学）；Stanford University（斯坦福大学）；Microsoft（微软）；SRI International（斯坦福国际研究院）
We present a novel framework for domain adaptation, whereby both geometric and statistical differences between a labeled source domain and unlabeled target domain can be integrated by exploiting the curved Riemannian geometry of statistical manifolds. Our approach is based on formulating transfer from source to target as a problem of geometric mean metric learning on manifolds. Specifically, we exploit the curved Riemannian manifold geometry of symmetric positive definite (SPD) covariance matrices. We exploit a simple but important observation that as the space of covariance matrices is both a…
Geodesic Optimization for Predictive Shift Adaptation on EEG data
Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Université Paris-Saclay（巴黎萨克雷大学）；TAU（特拉维夫大学）；LISN-CNRS（法国国家科学研究中心南巴黎大学实验室）；Roche Pharma Research and Early Development（罗氏制药研发与早期研发部）；Neuroscience and Rare Diseases（神经科学与罕见病部）；Roche Innovation Center Basel（罗氏巴塞尔创新中心）；F. Hoffmann–La Roche Ltd.（霍夫曼-拉罗什有限公司）
Aalto University（阿尔托大学）；University of Manchester（曼彻斯特大学）
Sample-efficient domain adaptation is an open problem in robotics. In this paper, we present affine transport -- a variant of optimal transport, which models the mapping between state transition distributions between the source and target domains with an affine transformation. First, we derive the affine transport framework; then, we extend the basic framework with Procrustes alignment to model arbitrary affine transformations. We evaluate the method in a number of OpenAI Gym sim-to-sim experiments with simulation environments, as well as on a sim-to-real domain adaptation task of a robot hitt…
Learning long term climate-resilient transport adaptation pathways under direct and indirect flood impacts using reinforcement learning
Department of Technology, Management and Economics, Technical University of Denmark（技术、管理与经济系，丹麦技术大学）；School of Management, Technical University of Munich（管理学院，慕尼黑技术大学）；J.E. Cairnes School of Business & Economics, University of Galway（J.E. Cairnes商学院与经济学院，Galway大学）
Curriculum Reinforcement Learning (CRL) aims to create a sequence of tasks, starting from easy ones and gradually learning towards difficult tasks. In this work, we focus on the idea of framing CRL as interpolations between a source (auxiliary) and a target task distribution. Although existing studies have shown the great potential of this idea, it remains unclear how to formally quantify and generate the movement between task distributions. Inspired by the insights from gradual domain adaptation in semi-supervised learning, we create a natural curriculum by breaking down the potentially large…
