We study the high-dimensional asymptotics of empirical risk minimization (ERM) in over-parametrized two-layer neural networks with quadratic activations trained on synthetic data. We derive sharp asymptotics for both training and test errors by mapping the $\ell_2$-regularized learning problem to a convex matrix sensing task with nuclear norm penalization. This reveals that capacity control in such networks emerges from a low-rank structure in the learned feature maps. Our results characterize the global minima of the loss and yield precise generalization thresholds, showing how the width of the target function governs learnability. This analysis bridges and extends ideas from spin-glass methods, matrix factorization, and convex optimization and emphasizes the deep link between low-rank matrix sensing and learning in quadratic neural networks.

翻译：我们研究了在合成数据上训练的、具有二次激活函数的过参数化两层神经网络中经验风险最小化（ERM）的高维渐近性。通过将$\ell_2$正则化学习问题映射到带有核范数惩罚的凸矩阵感知任务，我们推导了训练误差和测试误差的尖锐渐近表达式。这表明此类网络中的容量控制源于学习特征映射中的低秩结构。我们的结果刻画了损失的全局最小值，并给出了精确的泛化阈值，揭示了目标函数的宽度如何支配可学习性。该分析融合并拓展了自旋玻璃方法、矩阵分解和凸优化的思想，强调了低秩矩阵感知与二次神经网络学习之间的深刻联系。