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This paper advances the theory and practice of Domain Generalization (DG) in machine learning. We consider the typical DG setting where the hypothesis is composed of a representation mapping followed by a labeling function. Within this setting, the majority of popular DG methods aim to jointly learn the representation and the labeling functions by minimizing a well-known upper bound for the classification risk in the unseen domain. In practice, however, methods based on this theoretical upper bound ignore a term that cannot be directly optimized due to its dual dependence on both the representation mapping and the unknown optimal labeling function in the unseen domain. To bridge this gap between theory and practice, we introduce a new upper bound that is free of terms having such dual dependence, resulting in a fully optimizable risk upper bound for the unseen domain. Our derivation leverages classical and recent transport inequalities that link optimal transport metrics with information-theoretic measures. Compared to previous bounds, our bound introduces two new terms: (i) the Wasserstein-2 barycenter term that aligns distributions between domains, and (ii) the reconstruction loss term that assesses the quality of representation in reconstructing the original data. Based on this new upper bound, we propose a novel DG algorithm named Wasserstein Barycenter Auto-Encoder (WBAE) that simultaneously minimizes the classification loss, the barycenter loss, and the reconstruction loss. Numerical results demonstrate that the proposed method outperforms current state-of-the-art DG algorithms on several datasets.

Even though existence of non-convergent evolution of the states of populations in ecological and evolutionary contexts is an undeniable fact, insightful game-theoretic interpretations of such outcomes are scarce in the literature of evolutionary game theory. As a proof-of-concept, we tap into the information-theoretic concept of relative entropy in order to construct a game-theoretic interpretation for periodic orbits in a wide class of deterministic discrete-time evolutionary game dynamics, primarily investigating the two-player two-strategy case. Effectively, we present a consistent generalization of the evolutionarily stable strategy -- the cornerstone of the evolutionary game theory -- and aptly term the generalized concept: information stable orbit. The information stable orbit captures the essence of the evolutionarily stable strategy in that it compares the total payoff obtained against an evolving mutant with the total payoff that the mutant gets while playing against itself. Furthermore, we discuss the connection of the information stable orbit with the dynamical stability of the corresponding periodic orbit.

Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu and Shao (2004), using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.

Foundation models (e.g., CLIP or DINOv2) have shown their impressive learning and transferring capabilities on a wide range of visual tasks, by training on a large corpus of data and adapting to specific downstream tasks. It is, however, interesting that foundation models have not been fully explored for universal domain adaptation (UniDA), which is to learn models using labeled data in a source domain and unlabeled data in a target one, such that the learned models can successfully adapt to the target data. In this paper, we make comprehensive empirical studies of state-of-the-art UniDA methods using foundation models. We first demonstrate that, while foundation models greatly improve the performance of the baseline methods that train the models on the source data alone, existing UniDA methods generally fail to improve over the baseline. This suggests that new research efforts are very necessary for UniDA using foundation models. To this end, we propose a very simple method of target data distillation on the CLIP model, and achieves consistent improvement over the baseline across all the UniDA benchmarks. Our studies are under a newly proposed evaluation metric of universal classification rate (UCR), which is threshold- and ratio-free and addresses the threshold-sensitive issue encountered when using the existing H-score metric.

Domain gap between synthetic and real data in visual regression (\eg 6D pose estimation) is bridged in this paper via global feature alignment and local refinement on the coarse classification of discretized anchor classes in target space, which imposes a piece-wise target manifold regularization into domain-invariant representation learning. Specifically, our method incorporates an explicit self-supervised manifold regularization, revealing consistent cumulative target dependency across domains, to a self-training scheme (\eg the popular Self-Paced Self-Training) to encourage more discriminative transferable representations of regression tasks. Moreover, learning unified implicit neural functions to estimate relative direction and distance of targets to their nearest class bins aims to refine target classification predictions, which can gain robust performance against inconsistent feature scaling sensitive to UDA regressors. Experiment results on three public benchmarks of the challenging 6D pose estimation task can verify the effectiveness of our method, consistently achieving superior performance to the state-of-the-art for UDA on 6D pose estimation.