A Law of Data Separation in Deep Learning

Multilayer neural networks have achieved superhuman performance in many artificial intelligence applications. However, their black-box nature obscures the underlying mechanism for transforming input data into labels throughout all layers, thus hindering architecture design for new tasks and interpretation for high-stakes decision makings. We addressed this problem by introducing a precise law that governs how real-world deep neural networks separate data according to their class membership from the bottom layers to the top layers in classification problems. This law shows that each layer roughly improves a certain measure of data separation by an \textit{equal} multiplicative factor. This law manifests in modern architectures such as AlexNet, VGGNet, and ResNet in the late phase of training. This law together with the perspective of data separation offers practical guidelines for designing network architectures, improving model robustness and out-of-sample performance during training, as well as interpreting deep learning predictions.