A Simple Approach to Online Sparse Sliced Inverse Regression
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Sliced inverse regression is an efficient approach to estimate the central subspace for sufficient dimension reduction. Due to the demand for tackling the problem of sparse high dimensional data, several methods of online sufficient dimension reduction has been proposed. However, as far as we know, all of these methods are not well suitable for high dimensional and sparse data. Hence, the purpose of this paper is to propose a simple and efficient approach to online sparse sliced inverse regression (OSSIR). Motivated by Lasso-SIR and online SIR, we implement the Lasso-SIR in an online fashion. There are two important steps in our method, one is to iteratively obtain the eigenvalues and eigenvectors of matrix cov(E(x|Y)), the other is the online L_1 regularization. For the former problem, we expand the online principal component analysis and summarize four different ways. While in the online fashion, truncated gradient has been shown to be an online counterpart of L_1 regularization in the batch setting, so we apply the truncated gradient in the online sliced inverse regression for the latter problem. The theoretical properties of this online learner are established. By comparing with several existing methods in the simulations and real data applications, we demonstrate the effectiveness and efficiency of our algorithm.
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