Duality induced by an embedding structure of determinantal point process

This paper investigates the information geometrical structure of a determinantal point process (DPP). It demonstrates that a DPP is embedded in the exponential family of log-linear models. The extent of deviation from an exponential family is analyzed using the $\mathrm{e}$-embedding curvature tensor, which identifies partially flat parameters of a DPP. On the basis of this embedding structure, the duality related to a marginal kernel and an $L$-ensemble kernel is discovered.

A Short Survey on Importance Weighting for Machine Learning

Importance weighting is a fundamental procedure in statistics and machine learning that weights the objective function or probability distribution based on the importance of the instance in some sense. The simplicity and usefulness of the idea has led to many applications of importance weighting. For example, it is known that supervised learning under an assumption about the difference between the training and test distributions, called distribution shift, can guarantee statistically desirable properties through importance weighting by their density ratio. This survey summarizes the broad applications of importance weighting in machine learning and related research.

Scalable Counterfactual Distribution Estimation in Multivariate Causal Models

We consider the problem of estimating the counterfactual joint distribution of multiple quantities of interests (e.g., outcomes) in a multivariate causal model extended from the classical difference-in-difference design. Existing methods for this task either ignore the correlation structures among dimensions of the multivariate outcome by considering univariate causal models on each dimension separately and hence produce incorrect counterfactual distributions, or poorly scale even for moderate-size datasets when directly dealing with such multivariate causal model. We propose a method that alleviates both issues simultaneously by leveraging a robust latent one-dimensional subspace of the original high-dimension space and exploiting the efficient estimation from the univariate causal model on such space. Since the construction of the one-dimensional subspace uses information from all the dimensions, our method can capture the correlation structures and produce good estimates of the counterfactual distribution. We demonstrate the advantages of our approach over existing methods on both synthetic and real-world data.

Information Geometrically Generalized Covariate Shift Adaptation

Many machine learning methods assume that the training and test data follow the same distribution. However, in the real world, this assumption is very often violated. In particular, the phenomenon that the marginal distribution of the data changes is called covariate shift, one of the most important research topics in machine learning. We show that the well-known family of covariate shift adaptation methods is unified in the framework of information geometry. Furthermore, we show that parameter search for geometrically generalized covariate shift adaptation method can be achieved efficiently. Numerical experiments show that our generalization can achieve better performance than the existing methods it encompasses.

Active Learning by Query by Committee with Robust Divergences

Active learning is a widely used methodology for various problems with high measurement costs. In active learning, the next object to be measured is selected by an acquisition function, and measurements are performed sequentially. The query by committee is a well-known acquisition function. In conventional methods, committee disagreement is quantified by the Kullback--Leibler divergence. In this paper, the measure of disagreement is defined by the Bregman divergence, which includes the Kullback--Leibler divergence as an instance, and the dual $\gamma$-power divergence. As a particular class of the Bregman divergence, the $\beta$-divergence is considered. By deriving the influence function, we show that the proposed method using $\beta$-divergence and dual $\gamma$-power divergence are more robust than the conventional method in which the measure of disagreement is defined by the Kullback--Leibler divergence. Experimental results show that the proposed method performs as well as or better than the conventional method.

Unsupervised Domain Adaptation for Extra Features in the Target Domain Using Optimal Transport

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 target domain; hence, the dimensionality of the target domain is higher than that of the source domain. To leverage the homogeneity of the common features, the adaptation between these source and target domains is formulated as an optimal transport (OT) problem. In addition, a learning bound in the target domain for the proposed OT-based method is derived. The proposed algorithm is validated using both simulated and real-world data.

Gaussian Process Koopman Mode Decomposition

In this paper, we propose a nonlinear probabilistic generative model of Koopman mode decomposition based on an unsupervised Gaussian process. Existing data-driven methods for Koopman mode decomposition have focused on estimating the quantities specified by Koopman mode decomposition, namely, eigenvalues, eigenfunctions, and modes. Our model enables the simultaneous estimation of these quantities and latent variables governed by an unknown dynamical system. Furthermore, we introduce an efficient strategy to estimate the parameters of our model by low-rank approximations of covariance matrices. Applying the proposed model to both synthetic data and a real-world epidemiological dataset, we show that various analyses are available using the estimated parameters.

Geometry of EM and related iterative algorithms

The Expectation--Maximization (EM) algorithm is a simple meta-algorithm that has been used for many years as a methodology for statistical inference when there are missing measurements in the observed data or when the data is composed of observables and unobservables. Its general properties are well studied, and also, there are countless ways to apply it to individual problems. In this paper, we introduce the $em$ algorithm, an information geometric formulation of the EM algorithm, and its extensions and applications to various problems. Specifically, we will see that it is possible to formulate an outlier-robust inference algorithm, an algorithm for calculating channel capacity, parameter estimation methods on probability simplex, particular multivariate analysis methods such as principal component analysis in a space of probability models and modal regression, matrix factorization, and learning generative models, which have recently attracted attention in deep learning, from the geometric perspective.

Gradual Domain Adaptation via Normalizing Flows

Conventional domain adaptation methods do not work well when a large gap exists between the source and the target domain. Gradual domain adaptation is one of the approaches to address the problem by leveraging the intermediate domain, which gradually shifts from the source to the target domain. The previous work assumed that the number of the intermediate domains is large and the distance of the adjacent domains is small; hence, the gradual domain adaptation algorithm by self-training with unlabeled datasets was applicable. In practice, however, gradual self-training will fail because the number of the intermediate domains is limited, and the distance of the adjacent domains is large. We propose using normalizing flows to mitigate this problem while maintaining the framework of unsupervised domain adaptation. We generate pseudo intermediate domains from normalizing flows and then use them for gradual domain adaptation. We evaluate our method by experiments with real-world datasets and confirm that our proposed method mitigates the above explained problem and improves the classification performance.

Information Geometry of Dropout Training

Dropout is one of the most popular regularization techniques in neural network training. Because of its power and simplicity of idea, dropout has been analyzed extensively and many variants have been proposed. In this paper, several properties of dropout are discussed in a unified manner from the viewpoint of information geometry. We showed that dropout flattens the model manifold and that their regularization performance depends on the amount of the curvature. Then, we showed that dropout essentially corresponds to a regularization that depends on the Fisher information, and support this result from numerical experiments. Such a theoretical analysis of the technique from a different perspective is expected to greatly assist in the understanding of neural networks, which are still in their infancy.