Appendices

The appendix is organized as follows. We first introduce the basic definitions and inequalities used throughout the appendices. In Appendix A, we provide more details about the datasets, computational resources, and more experiment results on CIFAR10, CIFAR100 and miniImageNet datasets. In Appendix B, we prove that CE, FL and LS satisfy the contrastive property in Definition 1. In Appendix C, we provide a detailed proof for Theorem 1, showing that the Simplex ETFs are the only global minimizers, as long as the loss function satisfies the Definition 1. Finally, in Appendix D, we present the whole proof for Theorem 2 that the FL function is a locally strict saddle function with no spurious local minimizers existing locally and LS function is a globally strict saddle function with no spurious local minimizers existing globally. The following Lemma extends the standard variational form of the nuclear norm.