Motonobu Kanagawa

Davit Gogolashvili, Matteo Zecchin, Motonobu Kanagawa et al.

This paper investigates when the importance weighting (IW) correction is needed to address covariate shift, a common situation in supervised learning where the input distributions of training and test data differ. Classic results show that the IW correction is needed when the model is parametric and misspecified. In contrast, recent results indicate that the IW correction may not be necessary when the model is nonparametric and well-specified. We examine the missing case in the literature where the model is nonparametric and misspecified, and show that the IW correction is needed for obtaining the best approximation of the true unknown function for the test distribution. We do this by analyzing IW-corrected kernel ridge regression, covering a variety of settings, including parametric and nonparametric models, well-specified and misspecified settings, and arbitrary weighting functions.

Kensuke Mitsuzawa, Motonobu Kanagawa, Stefano Bortoli et al.

We study two-sample variable selection: identifying variables that discriminate between the distributions of two sets of data vectors. Such variables help scientists understand the mechanisms behind dataset discrepancies. Although domain-specific methods exist (e.g., in medical imaging, genetics, and computational social science), a general framework remains underdeveloped. We make two separate contributions. (i) We introduce a mathematical notion of the discriminating set of variables: the largest subset containing no variables whose marginals are identical across the two distributions and independent of the remaining variables. We prove this set is uniquely defined and establish further properties, making it a suitable ground truth for theory and evaluation. (ii) We propose two methods for two-sample variable selection that assign weights to variables and optimise them to maximise the power of a kernel two-sample test while enforcing sparsity to downweight redundant variables. To select the regularisation parameter - unknown in practice, as it controls the number of selected variables - we develop two data-driven procedures to balance recall and precision. Synthetic experiments show improved performance over baselines, and we illustrate the approach on two applications using datasets from water-pipe and traffic networks.

Parastoo Pashmchi, Jérôme Benoit, Motonobu Kanagawa

Missing values are common in photovoltaic (PV) power data, yet the uncertainty they induce is not propagated into predictive distributions. We develop a framework that incorporates missing-data uncertainty into short-term PV forecasting by combining stochastic multiple imputation with Rubin's rule. The approach is model-agnostic and can be integrated with standard machine-learning predictors. Empirical results show that ignoring missing-data uncertainty leads to overly narrow prediction intervals. Accounting for this uncertainty improves interval calibration while maintaining comparable point prediction accuracy. These results demonstrate the importance of propagating imputation uncertainty in data-driven PV forecasting.

Parastoo Pashmchi, Jerome Benoit, Motonobu Kanagawa

We study a missing-value imputation method, termed kNNSampler, that imputes a given unit's missing response by randomly sampling from the observed responses of the $k$ most similar units to the given unit in terms of the observed covariates. This method can sample unknown missing values from their distributions, quantify the uncertainties of missing values, and be readily used for multiple imputation. Unlike popular kNNImputer, which estimates the conditional mean of a missing response given an observed covariate, kNNSampler is theoretically shown to estimate the conditional distribution of a missing response given an observed covariate. Experiments demonstrate its effectiveness in recovering the distribution of missing values. The code for kNNSampler is made publicly available (https://github.com/SAP/knn-sampler).

Motonobu Kanagawa, Philipp Hennig, Dino Sejdinovic et al.

This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. Connections and equivalences between them are reviewed for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as for sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.

Motonobu Kanagawa

We describe a fast computation method for leave-one-out cross-validation (LOOCV) for $k$-nearest neighbours ($k$-NN) regression. We show that, under a tie-breaking condition for nearest neighbours, the LOOCV estimate of the mean square error for $k$-NN regression is identical to the mean square error of $(k+1)$-NN regression evaluated on the training data, multiplied by the scaling factor $(k+1)^2/k^2$. Therefore, to compute the LOOCV score, one only needs to fit $(k+1)$-NN regression only once, and does not need to repeat training-validation of $k$-NN regression for the number of training data. Numerical experiments confirm the validity of the fast computation method.

Kensuke Mitsuzawa, Margherita Grossi, Stefano Bortoli et al.

Given a pair of multivariate time-series data of the same length and dimensions, an approach is proposed to select variables and time intervals where the two series are significantly different. In applications where one time series is an output from a computationally expensive simulator, the approach may be used for validating the simulator against real data, for comparing the outputs of two simulators, and for validating a machine learning-based emulator against the simulator. With the proposed approach, the entire time interval is split into multiple subintervals, and on each subinterval, the two sample sets are compared to select variables that distinguish their distributions and a two-sample test is performed. The validity and limitations of the proposed approach are investigated in synthetic data experiments. Its usefulness is demonstrated in an application with a particle-based fluid simulator, where a deep neural network model is compared against the simulator, and in an application with a microscopic traffic simulator, where the effects of changing the simulator's parameters on traffic flows are analysed.

Jonas Wacker, Motonobu Kanagawa, Maurizio Filippone

Dot product kernels, such as polynomial and exponential (softmax) kernels, are among the most widely used kernels in machine learning, as they enable modeling the interactions between input features, which is crucial in applications like computer vision, natural language processing, and recommender systems. We make several novel contributions for improving the efficiency of random feature approximations for dot product kernels, to make these kernels more useful in large scale learning. First, we present a generalization of existing random feature approximations for polynomial kernels, such as Rademacher and Gaussian sketches and TensorSRHT, using complex-valued random features. We show empirically that the use of complex features can significantly reduce the variances of these approximations. Second, we provide a theoretical analysis for understanding the factors affecting the efficiency of various random feature approximations, by deriving closed-form expressions for their variances. These variance formulas elucidate conditions under which certain approximations (e.g., TensorSRHT) achieve lower variances than others (e.g., Rademacher sketches), and conditions under which the use of complex features leads to lower variances than real features. Third, by using these variance formulas, which can be evaluated in practice, we develop a data-driven optimization approach to improve random feature approximations for general dot product kernels, which is also applicable to the Gaussian kernel. We describe the improvements brought by these contributions with extensive experiments on a variety of tasks and datasets.

Veit Wild, Motonobu Kanagawa, Dino Sejdinovic

We investigate the connections between sparse approximation methods for making kernel methods and Gaussian processes (GPs) scalable to large-scale data, focusing on the Nyström method and the Sparse Variational Gaussian Processes (SVGP). While sparse approximation methods for GPs and kernel methods share some algebraic similarities, the literature lacks a deep understanding of how and why they are related. This may pose an obstacle to the communications between the GP and kernel communities, making it difficult to transfer results from one side to the other. Our motivation is to remove this obstacle, by clarifying the connections between the sparse approximations for GPs and kernel methods. In this work, we study the two popular approaches, the Nyström and SVGP approximations, in the context of a regression problem, and establish various connections and equivalences between them. In particular, we provide an RKHS interpretation of the SVGP approximation, and show that the Evidence Lower Bound of the SVGP contains the objective function of the Nyström approximation, revealing the origin of the algebraic equivalence between the two approaches. We also study recently established convergence results for the SVGP and how they are related to the approximation quality of the Nyström method.

Motonobu Kanagawa, Philipp Hennig

Adaptive Bayesian quadrature (ABQ) is a powerful approach to numerical integration that empirically compares favorably with Monte Carlo integration on problems of medium dimensionality (where non-adaptive quadrature is not competitive). Its key ingredient is an acquisition function that changes as a function of previously collected values of the integrand. While this adaptivity appears to be empirically powerful, it complicates analysis. Consequently, there are no theoretical guarantees so far for this class of methods. In this work, for a broad class of adaptive Bayesian quadrature methods, we prove consistency, deriving non-tight but informative convergence rates. To do so we introduce a new concept we call weak adaptivity. Our results identify a large and flexible class of adaptive Bayesian quadrature rules as consistent, within which practitioners can develop empirically efficient methods.

Takafumi Kajihara, Motonobu Kanagawa, Yuuki Nakaguchi et al.

We propose a novel approach to model selection for simulator-based statistical models. The proposed approach defines a mixture of candidate models, and then iteratively updates the weight coefficients for those models as well as the parameters in each model simultaneously; this is done by recursively applying Bayes' rule, using the recently proposed kernel recursive ABC algorithm. The practical advantage of the method is that it can be used even when a modeler lacks appropriate prior knowledge about the parameters in each model. We demonstrate the effectiveness of the proposed approach with a number of experiments, including model selection for dynamical systems in ecology and epidemiology.

Keiichi Kisamori, Motonobu Kanagawa, Keisuke Yamazaki

We propose a novel calibration method for computer simulators, dealing with the problem of covariate shift. Covariate shift is the situation where input distributions for training and test are different, and ubiquitous in applications of simulations. Our approach is based on Bayesian inference with kernel mean embedding of distributions, and on the use of an importance-weighted reproducing kernel for covariate shift adaptation. We provide a theoretical analysis for the proposed method, including a novel theoretical result for conditional mean embedding, as well as empirical investigations suggesting its effectiveness in practice. The experiments include calibration of a widely used simulator for industrial manufacturing processes, where we also demonstrate how the proposed method may be useful for sensitivity analysis of model parameters.

Motonobu Kanagawa, Philipp Hennig, Dino Sejdinovic et al.

This paper is an attempt to bridge the conceptual gaps between researchers working on the two widely used approaches based on positive definite kernels: Bayesian learning or inference using Gaussian processes on the one side, and frequentist kernel methods based on reproducing kernel Hilbert spaces on the other. It is widely known in machine learning that these two formalisms are closely related; for instance, the estimator of kernel ridge regression is identical to the posterior mean of Gaussian process regression. However, they have been studied and developed almost independently by two essentially separate communities, and this makes it difficult to seamlessly transfer results between them. Our aim is to overcome this potential difficulty. To this end, we review several old and new results and concepts from either side, and juxtapose algorithmic quantities from each framework to highlight close similarities. We also provide discussions on subtle philosophical and theoretical differences between the two approaches.

Krikamol Muandet, Motonobu Kanagawa, Sorawit Saengkyongam et al.

Counterfactual inference has become a ubiquitous tool in online advertisement, recommendation systems, medical diagnosis, and econometrics. Accurate modeling of outcome distributions associated with different interventions -- known as counterfactual distributions -- is crucial for the success of these applications. In this work, we propose to model counterfactual distributions using a novel Hilbert space representation called counterfactual mean embedding (CME). The CME embeds the associated counterfactual distribution into a reproducing kernel Hilbert space (RKHS) endowed with a positive definite kernel, which allows us to perform causal inference over the entire landscape of the counterfactual distribution. Based on this representation, we propose a distributional treatment effect (DTE) that can quantify the causal effect over entire outcome distributions. Our approach is nonparametric as the CME can be estimated under the unconfoundedness assumption from observational data without requiring any parametric assumption about the underlying distributions. We also establish a rate of convergence of the proposed estimator which depends on the smoothness of the conditional mean and the Radon-Nikodym derivative of the underlying marginal distributions. Furthermore, our framework allows for more complex outcomes such as images, sequences, and graphs. Our experimental results on synthetic data and off-policy evaluation tasks demonstrate the advantages of the proposed estimator.

Takafumi Kajihara, Motonobu Kanagawa, Keisuke Yamazaki et al.

We propose a novel approach to parameter estimation for simulator-based statistical models with intractable likelihood. Our proposed method involves recursive application of kernel ABC and kernel herding to the same observed data. We provide a theoretical explanation regarding why the approach works, showing (for the population setting) that, under a certain assumption, point estimates obtained with this method converge to the true parameter, as recursion proceeds. We have conducted a variety of numerical experiments, including parameter estimation for a real-world pedestrian flow simulator, and show that in most cases our method outperforms existing approaches.

Motonobu Kanagawa, Bharath K. Sriperumbudur, Kenji Fukumizu

This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.

Damien Garreau, Wittawat Jitkrittum, Motonobu Kanagawa

In kernel methods, the median heuristic has been widely used as a way of setting the bandwidth of RBF kernels. While its empirical performances make it a safe choice under many circumstances, there is little theoretical understanding of why this is the case. Our aim in this paper is to advance our understanding of the median heuristic by focusing on the setting of kernel two-sample test. We collect new findings that may be of interest for both theoreticians and practitioners. In theory, we provide a convergence analysis that shows the asymptotic normality of the bandwidth chosen by the median heuristic in the setting of kernel two-sample test. Systematic empirical investigations are also conducted in simple settings, comparing the performances based on the bandwidths chosen by the median heuristic and those by the maximization of test power.

Motonobu Kanagawa, Bharath K. Sriperumbudur, Kenji Fukumizu

Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super-$\sqrt{n}$ convergence rates in numerical integration, and thus provide alternatives to Monte Carlo integration in challenging settings where integrands are expensive to evaluate or where integrands are high dimensional. These rules are based on the assumption that the integrand has a certain degree of smoothness, which is expressed as that the integrand belongs to a certain reproducing kernel Hilbert space (RKHS). However, this assumption can be violated in practice (e.g., when the integrand is a black box function), and no general theory has been established for the convergence of kernel quadratures in such misspecified settings. Our contribution is in proving that kernel quadratures can be consistent even when the integrand does not belong to the assumed RKHS, i.e., when the integrand is less smooth than assumed. Specifically, we derive convergence rates that depend on the (unknown) lesser smoothness of the integrand, where the degree of smoothness is expressed via powers of RKHSs or via Sobolev spaces.

Yu Nishiyama, Motonobu Kanagawa, Arthur Gretton et al.

Kernel Bayesian inference is a principled approach to nonparametric inference in probabilistic graphical models, where probabilistic relationships between variables are learned from data in a nonparametric manner. Various algorithms of kernel Bayesian inference have been developed by combining kernelized basic probabilistic operations such as the kernel sum rule and kernel Bayes' rule. However, the current framework is fully nonparametric, and it does not allow a user to flexibly combine nonparametric and model-based inferences. This is inefficient when there are good probabilistic models (or simulation models) available for some parts of a graphical model; this is in particular true in scientific fields where "models" are the central topic of study. Our contribution in this paper is to introduce a novel approach, termed the {\em model-based kernel sum rule} (Mb-KSR), to combine a probabilistic model and kernel Bayesian inference. By combining the Mb-KSR with the existing kernelized probabilistic rules, one can develop various algorithms for hybrid (i.e., nonparametric and model-based) inferences. As an illustrative example, we consider Bayesian filtering in a state space model, where typically there exists an accurate probabilistic model for the state transition process. We propose a novel filtering method that combines model-based inference for the state transition process and data-driven, nonparametric inference for the observation generating process. We empirically validate our approach with synthetic and real-data experiments, the latter being the problem of vision-based mobile robot localization in robotics, which illustrates the effectiveness of the proposed hybrid approach.

Motonobu Kanagawa, Yu Nishiyama, Arthur Gretton et al.

This paper addresses the problem of filtering with a state-space model. Standard approaches for filtering assume that a probabilistic model for observations (i.e. the observation model) is given explicitly or at least parametrically. We consider a setting where this assumption is not satisfied; we assume that the knowledge of the observation model is only provided by examples of state-observation pairs. This setting is important and appears when state variables are defined as quantities that are very different from the observations. We propose Kernel Monte Carlo Filter, a novel filtering method that is focused on this setting. Our approach is based on the framework of kernel mean embeddings, which enables nonparametric posterior inference using the state-observation examples. The proposed method represents state distributions as weighted samples, propagates these samples by sampling, estimates the state posteriors by Kernel Bayes' Rule, and resamples by Kernel Herding. In particular, the sampling and resampling procedures are novel in being expressed using kernel mean embeddings, so we theoretically analyze their behaviors. We reveal the following properties, which are similar to those of corresponding procedures in particle methods: (1) the performance of sampling can degrade if the effective sample size of a weighted sample is small; (2) resampling improves the sampling performance by increasing the effective sample size. We first demonstrate these theoretical findings by synthetic experiments. Then we show the effectiveness of the proposed filter by artificial and real data experiments, which include vision-based mobile robot localization.