arXiv subjects

Motonobu Kanagawa

Publications and source records attributed to Motonobu Kanagawa.

At least 19 recordsLinked to original sources

arXiv2026

Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences

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. We study connections and equivalences for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as 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.

stat.ML↗

arXiv2026

Machine-Learning Emulation of Satellite Greenhouse Gas Retrievals: Stability over Time

Retrieval algorithms are used to estimate atmospheric concentrations of greenhouse gases (GHGs), such as carbon dioxide (CO2) and methane (CH4), by solving inverse problems from high-spectral-resolution satellite radiance measurements. However, these algorithms are computationally expensive, which makes real-time estimation at scale difficult. Machine-learning models have therefore been proposed as fast emulators of retrieval algorithms. Most existing studies, however, evaluate them only on test data from the same period as the training data. We study the stability over time of such emulators using data from the Greenhouse Gases Observing SATellite (GOSAT). We show that prediction accuracy generally deteriorates when the test period moves away from the training period. We also show that including time as an input feature substantially improves XCH4 prediction for Lasso and neural-network models. Among the methods considered, a simple Lasso model performs as well as or better than more complex methods such as neural networks, and yields more stable predictions over time. We further validate the results using the Total Carbon Column Observing Network (TCCON), a ground-based observation network. On the TCCON-matched dataset, the time-augmented Lasso achieves errors against TCCON that are comparable to the disagreement between GOSAT and TCCON for both XCO2 and XCH4.

cs.LG↗

arXiv2026

Predictive Uncertainty in Short-Term PV Forecasting under Missing Data: A Multiple Imputation Approach

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.

cs.LG↗

arXiv2025

kNNSampler: Stochastic Imputations for Recovering Missing Value Distributions

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 illustrate the performance of kNNSampler. The code for kNNSampler is made publicly available (https://github.com/SAP/knn-sampler).

stat.ML↗

arXiv2025

Variable Selection in Maximum Mean Discrepancy for Interpretable Distribution Comparison

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.

stat.ML↗

arXiv2025

Comparing Scale Parameter Estimators for Gaussian Process Interpolation with the Brownian Motion Prior: Leave-One-Out Cross Validation and Maximum Likelihood

Gaussian process (GP) regression is a Bayesian nonparametric method for regression and interpolation, offering a principled way of quantifying the uncertainties of predicted function values. For the quantified uncertainties to be well-calibrated, however, the kernel of the GP prior has to be carefully selected. In this paper, we theoretically compare two methods for choosing the kernel in GP regression: cross-validation and maximum likelihood estimation. Focusing on the scale-parameter estimation of a Brownian motion kernel in the noiseless setting, we prove that cross-validation can yield asymptotically well-calibrated credible intervals for a broader class of ground-truth functions than maximum likelihood estimation, suggesting an advantage of the former over the latter. Finally, motivated by the findings, we propose interior cross validation, a procedure that adapts to an even broader class of ground-truth functions.

math.ST↗

arXiv2024

Variable Selection for Comparing High-dimensional Time-Series Data

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.

stat.ME↗

arXiv2024

Fast Computation of Leave-One-Out Cross-Validation for $k$-NN Regression

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.

stat.ML↗

arXiv2024

Improved Random Features for Dot Product Kernels

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.

stat.ML↗

arXiv2023

Intergenerational risk sharing in a Defined Contribution pension system: analysis with Bayesian optimization

We study a fully funded, collective defined-contribution (DC) pension system with multiple overlapping generations. We investigate whether the welfare of participants can be improved by intergenerational risk sharing (IRS) implemented with a realistic investment strategy (e.g., no borrowing) and without an outside entity (e.g., share holders) that helps finance the pension fund. To implement IRS, the pension system uses an automatic adjustment rule for the indexation of individual accounts, which adapts to the notional funding ratio of the pension system. The pension system has two parameters that determine the investment strategy and the strength of the adjustment rule, which are optimized by expected utility maximization using Bayesian optimization. The volatility of the retirement benefits and that of the funding ratio are analyzed, and it is shown that the trade-off between them can be controlled by the optimal adjustment parameter to attain IRS. Compared with the optimal individual DC benchmark using the life-cycle strategy, the studied pension system with IRS is shown to improve the welfare of risk-averse participants, when the financial market is volatile.

econ.GN↗

arXiv2023

When is Importance Weighting Correction Needed for Covariate Shift Adaptation?

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.

stat.ML↗

arXiv2023

Connections and Equivalences between the Nyström Method and Sparse Variational Gaussian Processes

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.

stat.ML↗

arXiv2021

Counterfactual Mean Embeddings

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.

stat.ML↗

arXiv2020

Simulator Calibration under Covariate Shift with Kernels

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.

stat.ML↗

arXiv2019

Convergence Guarantees for Adaptive Bayesian Quadrature Methods

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.

stat.ML↗

arXiv2019

On the positivity and magnitudes of Bayesian quadrature weights

This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g., Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing stable and robust Bayesian quadrature rules. Secondly, we show that magnitudes of the weights admit an upper bound in terms of the fill distance and separation radius if the RKHS of the kernel is a Sobolev space (e.g., Matérn kernels), suggesting that quasi-uniform points should be used. A number of numerical examples demonstrate that significant generalisations and improvements appear to be possible, manifesting the need for further research.

math.ST↗

arXiv2019

Model-based Kernel Sum Rule: Kernel Bayesian Inference with Probabilistic Models

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.

stat.ML↗

arXiv2019

Model Selection for Simulator-based Statistical Models: A Kernel Approach

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.

stat.ML↗