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# How Can Deep Rectifier Networks Achieve Linear Separability and Preserve Distances?

*Proceedings of the 32nd International Conference on Machine Learning*, PMLR 37:514-523, 2015.

#### Abstract

This paper investigates how hidden layers of deep rectifier networks are capable of transforming two or more pattern sets to be linearly separable while preserving the distances with a guaranteed degree, and proves the universal classification power of such distance preserving rectifier networks. Through the nearly isometric nonlinear transformation in the hidden layers, the margin of the linear separating plane in the output layer and the margin of the nonlinear separating boundary in the original data space can be closely related so that the maximum margin classification in the input data space can be achieved approximately via the maximum margin linear classifiers in the output layer. The generalization performance of such distance preserving deep rectifier neural networks can be well justified by the distance-preserving properties of their hidden layers and the maximum margin property of the linear classifiers in the output layer.