  While deep learning has enabled significant advances in many areas of
science, its black-box nature hinders architecture design for future artificial
intelligence applications and interpretation for high-stakes decision makings.
We addressed this issue by studying the fundamental question of how deep neural
networks process data in the intermediate layers. Our finding is a simple and
quantitative law that governs how deep neural networks separate data according
to class membership throughout all layers for classification. This law shows
that each layer improves data separation at a constant geometric rate, and its
emergence is observed in a collection of network architectures and datasets
during training. This law offers practical guidelines for designing
architectures, improving model robustness and out-of-sample performance, as
well as interpreting the predictions.
