Neural collapse under cross-entropy loss

We consider the variational problem of cross-entropy loss with n feature vectors on a unit hypersphere in R d . We prove that when d ≥ n − 1 , the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We also prove that, as n → ∞ with fixed d , the minimizing points will distribute uniformly on the hypersphere and show a connection with the frame potential of Benedetto & Fickus.