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Register a new userENNS: Variable Selection, Regression, Classification and Deep Neural Network for High-Dimensional Data
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Kaixu Yang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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High-dimensional, low sample-size (HDLSS) data problems have been a topic of immense importance for the last couple of decades. There is a vast literature that proposed a wide variety of approaches to deal with this situation, among which variable selection was a compelling idea. On the other hand, a deep neural network has been used to model complicated relationships and interactions among responses and features, which is hard to capture using a linear or an additive model. In this paper, we discuss the current status of variable selection techniques with the neural network models. We show that the stage-wise algorithm with neural network suffers from disadvantages such as the variables entering into the model later may not be consistent. We then propose an ensemble method to achieve better variable selection and prove that it has probability tending to zero that a false variable is selected. Then, we discuss additional regularization to deal with over-fitting and make better regression and classification. We study various statistical properties of our proposed method. Extensive simulations and real data examples are provided to support the theory and methodology.
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Neural networks are usually not the tool of choice for nonparametric high-dimensional problems where the number of input features is much larger than the number of observations. Though neural networks can approximate complex multivariate functions, they generally require a large number of training observations to obtain reasonable fits, unless one can learn the appropriate network structure. In this manuscript, we show that neural networks can be applied successfully to high-dimensional settings if the true function falls in a low dimensional subspace, and proper regularization is used. We propose fitting a neural network with a sparse group lasso penalty on the first-layer input weights. This results in a neural net that only uses a small subset of the original features. In addition, we characterize the statistical convergence of the penalized empirical risk minimizer to the optimal neural network: we show that the excess risk of this penalized estimator only grows with the logarithm of the number of input features; and we show that the weights of irrelevant features converge to zero. Via simulation studies and data analyses, we show that these sparse-input neural networks outperform existing nonparametric high-dimensional estimation methods when the data has complex higher-order interactions.
This paper proposes a new robust smooth-threshold estimating equation to select important variables and automatically estimate parameters for high dimensional longitudinal data. A novel working correlation matrix is proposed to capture correlations within the same subject. The proposed procedure works well when the number of covariates p increases as the number of subjects n increases. The proposed estimates are competitive with the estimates obtained with the true correlation structure, especially when the data are contaminated. Moreover, the proposed method is robust against outliers in the response variables and/or covariates. Furthermore, the oracle properties for robust smooth-threshold estimating equations under large n, diverging p are established under some regularity conditions. Extensive simulation studies and a yeast cell cycle data are used to evaluate the performance of the proposed method, and results show that our proposed method is competitive with existing robust variable selection procedures.
We consider regression in which one predicts a response $Y$ with a set of predictors $X$ across different experiments or environments. This is a common setup in many data-driven scientific fields and we argue that statistical inference can benefit from an analysis that takes into account the distributional changes across environments. In particular, it is useful to distinguish between stable and unstable predictors, i.e., predictors which have a fixed or a changing functional dependence on the response, respectively. We introduce stabilized regression which explicitly enforces stability and thus improves generalization performance to previously unseen environments. Our work is motivated by an application in systems biology. Using multiomic data, we demonstrate how hypothesis generation about gene function can benefit from stabilized regression. We believe that a similar line of arguments for exploiting heterogeneity in data can be powerful for many other applications as well. We draw a theoretical connection between multi-environment regression and causal models, which allows to graphically characterize stable versus unstable functional dependence on the response. Formally, we introduce the notion of a stable blanket which is a subset of the predictors that lies between the direct causal predictors and the Markov blanket. We prove that this set is optimal in the sense that a regression based on these predictors minimizes the mean squared prediction error given that the resulting regression generalizes to unseen new environments.
Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high-dimensional assumptions. We propose a novel MCMC sampler using an informed proposal scheme, which we prove achieves a much faster mixing time that is independent of the number of covariates, under the same assumptions. To the best of our knowledge, this is the first high-dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called two-stage drift condition to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high-dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis.
Penalized (or regularized) regression, as represented by Lasso and its variants, has become a standard technique for analyzing high-dimensional data when the number of variables substantially exceeds the sample size. The performance of penalized regression relies crucially on the choice of the tuning parameter, which determines the amount of regularization and hence the sparsity level of the fitted model. The optimal choice of tuning parameter depends on both the structure of the design matrix and the unknown random error distribution (variance, tail behavior, etc). This article reviews the current literature of tuning parameter selection for high-dimensional regression from both theoretical and practical perspectives. We discuss various strategies that choose the tuning parameter to achieve prediction accuracy or support recovery. We also review several recently proposed methods for tuning-free high-dimensional regression.
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