Mehler’s formula, Branching process and Compositional Kernels of Deep Neural Networks
Hai Tran-Bach
Hai Tran-Bach earned a Bachelor’s of Science in Mathematics and Data Science at the University of Michigan with high honors and high distinctions. Currently, he is a third year distinguished Neubauer Doctoral Fellow in the Department of Statistics at the University of Chicago. His research interest is to understand the empirical phenomena in modern machine learning and to use these insights to derive principled machine learning algorithms. He is currently working on identifying favorable properties of neural networks that lead to good generalization.
We utilize a connection between compositional kernels and branching processes via Mehler’s formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation functions in compositional neural networks. Concretely, we
study the unscaled and rescaled limits of the compositional kernels and explore the different phases of the limiting behavior, as the compositional depth increases.
investigate the memorization capacity of the compositional kernels and neural networks by characterizing the interplay among compositional depth, sample size, dimensionality, and non-linearity of the activation.
provide explicit formulas on the eigenvalues of the compositional kernel, which quantify the complexity of the corresponding RKHS.
propose a new random features algorithm, which compresses the compositional layers by devising a new activation function.
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