On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers

A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if $h(x) = Af(x) +b$ where $A$ is a linear map and $f$ is the output of the penultimate layer of the network (after activation), then all data points $x_{i, 1}, \dots, x_{i, N_i}$ in a class $C_i$ are mapped to a single point $y_i$ by $f$ and the points $y_i$ are located at the vertices of a regular $k-1$-dimensional standard simplex in a high-dimensional Euclidean space. We explain this observation analytically in toy models for highly expressive deep neural networks. In complementary examples, we demonstrate rigorously that even the final output of the classifier $h$ is not uniform over data samples from a class $C_i$ if $h$ is a shallow network (or if the deeper layers do not bring the data samples into a convenient geometric configuration).