In this paper we approach the problem of unique and stable identifiability from a finite number of input-output samples of generic feedforward deep artificial neural networks of prescribed architecture with pyramidal shape up to the penultimate layer and smooth activation functions. More specifically we introduce the so-called entangled weights, which compose weights of successive layers intertwined with suitable diagonal and invertible matrices depending on the activation functions and their shifts. We prove that instances of entangled weights are completely and stably approximated by an efficient and robust algorithm as soon as $\mathcal{O}(D^2\times m)$ nonadaptive input-output samples of the network are collected, where D is the input dimension and m is the number of neurons of the network. Moreover, we empirically observe that the approach applies to networks with up to $\mathcal{O}(D\times m_L)$ neurons, where $m_L$ is the number of output neurons at layer L. Provided knowledge of layer assignments of entangled weights and of remaining scaling and shift parameters, which may be further heuristically obtained by least squares fitting, the entangled weights identify the network completely and uniquely. To highlight the relevance of the theoretical result of stable recovery of entangled weights, we present numerical experiments, which demonstrate that multilayered networks with generic weights can be robustly identified and therefore uniformly approximated by the presented algorithmic pipeline.

在本文中，我们从有限数量的通用前馈深度人工神经网络的输入输出样本中处理唯一且稳定的可识别性问题，该网络具有金字塔形状直至倒数第二层和平滑激活函数的规定架构。更具体地说，我们介绍了所谓的纠缠权重，它由连续层的权重组成，这些层的权重与合适的对角矩阵和可逆矩阵交织在一起，具体取决于激活函数及其偏移。我们证明了纠缠权重的实例可以通过有效且鲁棒的算法完全稳定地逼近$\mathcal{○}(D^2\times 米)$收集网络的非自适应输入输出样本，其中D是输入维度，m是网络的神经元数量。此外，我们凭经验观察到，该方法适用于高达$\mathcal{○}(D\times {米}_{\mathrm{大号}})$神经元，其中${米}_{\mathrm{大号}}$是L层输出神经元的数量。提供纠缠权重的层分配以及剩余缩放和移位参数的知识，可以通过最小二乘拟合进一步启发式地获得，纠缠权重可以完全且唯一地识别网络。为了强调纠缠权重稳定恢复的理论结果的相关性，我们提出了数值实验，这表明具有通用权重的多层网络可以被稳健地识别，因此可以通过所提出的算法管道统一近似。