Derivations of Formulas

We have omitted a number of complicated formulas in the main text to provide clear intuition and concise proof sketch. We will list all mentioned formulas here for readers' reference. We consider the case where U = V = Aand Σ is symmetric and full-rank, and we use gradient flow. We can derive the dynamics of S = AA>as S:= (Σ S)S+ S(Σ S), which is a quadratic ordinary differential equation and it is hard to solve directly. For simplicity, define X:= X Σ 1. Then X = XΣ ΣX. (24) Solving this equation and we have And it is interesting to verify that S(t) + P(t) Σ by using the following lemma.