Stable recovery of entangled weights: Towards robust identification of deep neural networks from minimal samples

Abstract

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 O(D2×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 O(D×mL) neurons, where mL 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.