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Journal Articles Constructive Approximation Year : 2022

An Embedding of ReLU Networks and an Analysis of their Identifiability

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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Dates and versions

hal-03292203 , version 1 (20-07-2021)
hal-03292203 , version 2 (03-01-2022)
hal-03292203 , version 3 (14-01-2022)
hal-03292203 , version 4 (31-01-2022)

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Cite

Pierre Stock, Rémi Gribonval. An Embedding of ReLU Networks and an Analysis of their Identifiability. Constructive Approximation, In press, ⟨10.1007/s00365-022-09578-1⟩. ⟨hal-03292203v4⟩
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