327af0f71f7acdfd882774225f04775f-Supplemental.pdf

We will now derive continuous dynamics (2) in the main paper. Let 1m = 1 if class 1 is selected at iteration mand 1m = 0 otherwise. Likewise, we can obtain the dynamics of X2j similarly. We will next prove the separation theorem in binary classification, Theorem 2.1. Given the feature vectors X1i(t), X2j(t) for i,j [n], as t and large n, 1. if α > β, they are asymptotically separable with probability tending to one, 2. if α β, they are asymptotically separable with probability tending to zero. This also aligns with our intuition that the intra-class effect should be stronger than its inter-class counterpart. On the other hand, when α>β, ignoring a null set we may assume c1 >c2 without loss of generality.