Determining whether deep neural network (DNN) models can reliably recover
target functions at overparameterization is a critical yet complex issue in the
theory of deep learning. To advance understanding in this area, we introduce a
concept we term "local linear recovery" (LLR), a weaker form of target function
recovery that renders the problem more amenable to theoretical analysis. In the
sense of LLR, we prove that functions expressible by narrower DNNs are
guaranteed to be recoverable from fewer samples than model parameters.
Specifically, we establish upper limits on the optimistic sample sizes, defined
as the smallest sample size necessary to guarantee LLR, for functions in the
space of a given DNN. Furthermore, we prove that these upper bounds are
achieved in the case of two-layer tanh neural networks. Our research lays a
solid groundwork for future investigations into the recovery capabilities of
DNNs in overparameterized scenarios.

在深度学习理论中，我们引入了 “局部线性恢复”（LLR）的概念，证明了表达能力较弱的深度神经网络（DNN）可从更少数量的样本中恢复，而且对于两层 tanh 神经网络，我们还证明了这一结果的最优情况。我们的研究为进一步研究过参数化情况下 DNN 的恢复能力奠定了坚实的基础。