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High-dimensional, low sample-size (HDLSS) data problems have been a topic ofimmense importance for the last couple of decades. There is a vast literaturethat proposed a wide variety of approaches to deal with this situation, amongwhich variable selection was a compelling idea. On the other hand, a deepneural network has been used to model complicated relationships andinteractions among responses and features, which is hard to capture using alinear or an additive model. In this paper, we discuss the current status ofvariable selection techniques with the neural network models. We show that thestage-wise algorithm with neural network suffers from disadvantages such as thevariables entering into the model later may not be consistent. We then proposean ensemble method to achieve better variable selection and prove that it hasprobability tending to zero that a false variable is selected. Then, we discussadditional regularization to deal with over-fitting and make better regressionand classification. We study various statistical properties of our proposedmethod. Extensive simulations and real data examples are provided to supportthe theory and methodology.

ENNS: Variable Selection, Regression, Classification and Deep Neural Network for High-Dimensional Data