Material

In what follows, we give some details of content omitted in the paper due to space limit. The supplements are organized as follows. We give some proof of Lemma 1, 2, Proposition 1, Lemma 3, 4, and Theorem 2 in Section A.1 -A.6, respectively. We provide some training details in Section A.13 as well as experiment details and results in Section A.14. We compare polynomial CBFs with NCBF in A.15, compare NCBFs with different activation functions in A.16. A.1 Proof of Lemma 1 We prove by induction on L. If L =1, then x 2 X(S) if the pre-activation input to the (1,j) neuron is nonnegative for all j 2 S1 and nonpositive for all j/2 S1. We have that the pre-activation input is equal to WT1jx+r1j, establishing the result for L =1 .