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

中文翻译：

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交叉熵损失下的神经崩溃

我们考虑在单位超球面上具有n 个特征向量的交叉熵损失的变分问题${\mathbb{电阻}}^d$. 我们证明当$d\ge n-1$，全局最小值由单纯形等角紧框架给出，这证明了神经崩溃行为的合理性。我们还证明，作为$n\to \infty$固定d 时，最小化点将均匀分布在超球面上，并显示出与 Benedetto & Fickus 的框架势的联系。