arXiv Analytics

Sign in

arXiv:2107.09370 [cs.LG]AbstractReferencesReviewsResources

An Embedding of ReLU Networks and an Analysis of their Identifiability

Pierre Stock, Rémi Gribonval

Published 2021-07-20Version 1

Neural networks with the Rectified Linear Unit (ReLU) nonlinearity are described by a vector of parameters $\theta$, and realized as a piecewise linear continuous function $R_{\theta}: x \in \mathbb R^{d} \mapsto R_{\theta}(x) \in \mathbb R^{k}$. Natural scalings and permutations operations on the parameters $\theta$ leave the realization unchanged, leading to equivalence classes of parameters that yield the same realization. These considerations in turn lead to the notion of identifiability -- the ability to recover (the equivalence class of) $\theta$ from the sole knowledge of its realization $R_{\theta}$. The overall objective of this paper is to introduce an embedding for ReLU neural networks of any depth, $\Phi(\theta)$, that is invariant to scalings and that provides a locally linear parameterization of the realization of the network. Leveraging these two key properties, we derive some conditions under which a deep ReLU network is indeed locally identifiable from the knowledge of the realization on a finite set of samples $x_{i} \in \mathbb R^{d}$. We study the shallow case in more depth, establishing necessary and sufficient conditions for the network to be identifiable from a bounded subset $\mathcal X \subseteq \mathbb R^{d}$.

Related articles: Most relevant | Search more
arXiv:2202.03841 [cs.LG] (Published 2022-02-08)
Width is Less Important than Depth in ReLU Neural Networks
arXiv:1809.07122 [cs.LG] (Published 2018-09-19)
Capacity Control of ReLU Neural Networks by Basis-path Norm
arXiv:2001.03040 [cs.LG] (Published 2020-01-09)
Deep Network Approximation for Smooth Functions