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Considering the case where the response variable is a categorical variable and the predictor is a random function, two novel functional sufficient dimensional reduction (FSDR) methods are proposed based on mutual information and square loss mutual information. Compared to the classical FSDR methods, such as functional sliced inverse regression and functional sliced average variance estimation, the proposed methods are appealing because they are capable of estimating multiple effective dimension reduction directions in the case of a relatively small number of categories, especially for the binary response. Moreover, the proposed methods do not require the restrictive linear conditional mean assumption and the constant covariance assumption. They avoid the inverse problem of the covariance operator which is often encountered in the functional sufficient dimension reduction. The functional principal component analysis with truncation be used as a regularization mechanism. Under some mild conditions, the statistical consistency of the proposed methods is established. It is demonstrated that the two methods are competitive compared with some existing FSDR methods by simulations and real data analyses., Comment: There are some problems with the methodology or experimental design in the article that make it impossible to produce reliable conclusions, so we have decided to withdraw it

The most recent multi-source covariate shift algorithm is an efficient hyperparameter optimization algorithm for missing target output. In this paper, we extend this algorithm to the framework of federated learning. For data islands in federated learning and covariate shift adaptation, we propose the federated domain adaptation estimate of the target risk which is asymptotically unbiased with a desirable asymptotic variance property. We construct a weighted model for the target task and propose the federated covariate shift adaptation algorithm which works preferably in our setting. The efficacy of our method is justified both theoretically and empirically.

Online dimension reduction is a common method for high-dimensional streaming data processing. Online principal component analysis, online sliced inverse regression, online kernel principal component analysis and other methods have been studied in depth, but as far as we know, online supervised nonlinear dimension reduction methods have not been fully studied. In this article, an online kernel sliced inverse regression method is proposed. By introducing the approximate linear dependence condition and dictionary variable sets, we address the problem of increasing variable dimensions with the sample size in the online kernel sliced inverse regression method, and propose a reduced-order method for updating variables online. We then transform the problem into an online generalized eigen-decomposition problem, and use the stochastic optimization method to update the centered dimension reduction directions. Simulations and the real data analysis show that our method can achieve close performance to batch processing kernel sliced inverse regression.

Federated learning has become a popular tool in the big data era nowadays. It trains a centralized model based on data from different clients while keeping data decentralized. In this paper, we propose a federated sparse sliced inverse regression algorithm for the first time. Our method can simultaneously estimate the central dimension reduction subspace and perform variable selection in a federated setting. We transform this federated high-dimensional sparse sliced inverse regression problem into a convex optimization problem by constructing the covariance matrix safely and losslessly. We then use a linearized alternating direction method of multipliers algorithm to estimate the central subspace. We also give approaches of Bayesian information criterion and hold-out validation to ascertain the dimension of the central subspace and the hyper-parameter of the algorithm. We establish an upper bound of the statistical error rate of our estimator under the heterogeneous setting. We demonstrate the effectiveness of our method through simulations and real world applications.

Based on the theory of reproducing kernel Hilbert space (RKHS) and semiparametric method, we propose a new approach to nonlinear dimension reduction. The method extends the semiparametric method into a more generalized domain where both the interested parameters and nuisance parameters to be infinite dimensional. By casting the nonlinear dimensional reduction problem in a generalized semiparametric framework, we calculate the orthogonal complement space of generalized nuisance tangent space to derive the estimating equation. Solving the estimating equation by the theory of RKHS and regularization, we obtain the estimation of dimension reduction directions of the sufficient dimension reduction (SDR) subspace and also show the asymptotic property of estimator. Furthermore, the proposed method does not rely on the linearity condition and constant variance condition. Simulation and real data studies are conducted to demonstrate the finite sample performance of our method in comparison with several existing methods.

This article studies global testing of the slope function in functional linear regression model in the framework of reproducing kernel Hilbert space. We propose a new testing statistic based on smoothness regularization estimators. The asymptotic distribution of the testing statistic is established under null hypothesis. It is shown that the null asymptotic distribution is determined jointly by the reproducing kernel and the covariance function. Our theoretical analysis shows that the proposed testing is consistent over a class of smooth local alternatives. Despite the generality of the method of regularization, we show the procedure is easily implementable. Numerical examples are provided to demonstrate the empirical advantages over the competing methods., Comment: There are some mistakes in the paper

Due to the demand for tackling the problem of streaming data with high dimensional covariates, we propose an online sparse sliced inverse regression (OSSIR) method for online sufficient dimension reduction. The existing online sufficient dimension reduction methods focus on the case when the dimension $p$ is small. In this article, we show that our method can achieve better statistical accuracy and computation speed when the dimension $p$ is large. There are two important steps in our method, one is to extend the online principal component analysis to iteratively obtain the eigenvalues and eigenvectors of the kernel matrix, the other is to use the truncated gradient to achieve online $L_{1}$ regularization. We also analyze the convergence of the extended Candid covariance-free incremental PCA(CCIPCA) and our method. By comparing several existing methods in the simulations and real data applications, we demonstrate the effectiveness and efficiency of our method.

Based on the theories of sliced inverse regression (SIR) and reproducing kernel Hilbert space (RKHS), a new approach RDSIR (RKHS-based Double SIR) to nonlinear dimension reduction for survival data is proposed and discussed. An isometrically isomorphism is constructed based on RKHS property, then the nonlinear function in the RKHS can be represented by the inner product of two elements which reside in the isomorphic feature space. Due to the censorship of survival data, double slicing is used to estimate weight function or conditional survival function to adjust for the censoring bias. The sufficient dimension reduction (SDR) subspace is estimated by a generalized eigen-decomposition problem. Our method is computationally efficient with fast calculation speed and small computational burden. The asymptotic property and the convergence rate of the estimator are also discussed based on the perturbation theory. Finally, we illustrate the performance of RDSIR on simulated and real data to confirm that RDSIR is comparable with linear SDR method. The most important is that RDSIR can also extract nonlinearity in survival data effectively., Comment: 51 pages, 3 figures