Mehler’s formula, Branching process and Compositional Kernels of Deep Neural Networks


Hai Tran-Bach

University of Chicago
Fri, Sep 18 2020 - 10:00 am (GMT + 7)
About Speaker

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.

Related seminars

Trieu Trinh

Google Deepmind

AlphaGeometry: Solving IMO Geometry without Human Demonstrations
Fri, Jul 5 2024 - 10:00 am (GMT + 7)

Tat-Jun (TJ) Chin

Adelaide University

Quantum Computing in Computer Vision: A Case Study in Robust Geometric Optimisation
Fri, Jun 7 2024 - 11:00 am (GMT + 7)

Fernando De la Torre

Carnegie Mellon University

Human Sensing for AR/VR
Wed, Apr 24 2024 - 07:00 am (GMT + 7)

Anh Nguyen

Microsoft GenAI

The Revolution of Small Language Models
Fri, Mar 8 2024 - 02:30 pm (GMT + 7)