An Embedding of ReLU Networks and an Analysis of their Identifiability

• URL: http://arxiv.org/abs/2107.09370v1
• Date: Tue, 20 Jul 2021 09:43:31 GMT
• Title: An Embedding of ReLU Networks and an Analysis of their Identifiability
• Authors: Pierre Stock and R\'emi Gribonval
• Abstract summary: This paper introduces an embedding for ReLU neural networks of any depth, $Phi(theta)$, that is invariant to scalings. We derive some conditions under which a deep ReLU network is indeed locally identifiable from the knowledge of the realization.
• Score: 5.076419064097734
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\mathcal X \subseteq \mathbb R^{d}$.

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