# Metric entropy limits on recurrent neural network learning of linear dynamical systems

### Authors

Clemens Hutter, Recep Gül, and Helmut Bölcskei### Reference

*Applied and Computational Harmonic Analysis*, Vol. 59, pp. 198-223, July 2022, (

*invited paper*).

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### Abstract

One of the most influential results in neural network theory is the universal approximation theorem [1, 2, 3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems—including time-varying systems—by recurrent neural networks (RNNs). For the subclass of linear time-invariant (LTI) systems, we devise a quantitative version of this statement. Specifically, measuring the complexity of the considered class of LTI systems through metric entropy according to [4], we show that RNNs can optimally learn—or identify in system-theory parlance—stable LTI systems. For LTI systems whose input-output relation is characterized through a difference equation, this means that RNNs can learn the difference equation from input-output traces in a metric-entropy optimal manner.### Keywords

Recurrent neural networks, linear dynamical systems, metric entropy, Hardy spaces, universal approximation, system identification### Comments

In Definition 1.1, \(\mathcal{R}_{\Phi}: \ell_{\infty} \rightarrow \ell_{\infty}\)

can be replaced by the more general \(\mathcal{R}_{\Phi}: \ell_{\infty}(\mathbb{N}_{0}) \rightarrow \mathbb{R}^{\mathbb{N}_0}\)

.

The networks constructed in the proofs of Lemma 2.2 and Theorem 2.3 are applicable to input signals \(x\) with \(\lVert{x}\rVert_{\ell_\infty} \leq C\), where \(C \in\mathbb{R}^+\). Therefore, the sentence above equation (25) To this end, we first recall that RNNs according to Definition 1.1 accept input signals in \(\ell_\infty(\mathbb{N}_0)\) and set \(C={\lVert x \rVert}_{\ell_\infty}\).

should be replaced by To this end, we first recall that RNNs according to Definition 1.1 accept input signals in \(\ell_\infty\) and choose a \(C \in \mathbb{R}^+\) such that \({\lVert x \rVert}_{\ell_\infty} \leq C\).

Similarly, \(C = {\lVert x \rVert}_{\ell_\infty}\)

below equation (37) should be replaced by \(C \in \mathbb{R}^+\) is such that \({\lVert x \rVert}_{\ell_\infty} \leq C\)

.

The constant \(\frac{1}{b}\) in the scaling result \(\mathcal{E}(\epsilon; \mathcal{C}(a, b), \rho) \thicksim \frac{1}{b}\left(\log\left(\frac{a}{\epsilon}\right)\right)^2\) specified in Theorem 3.2 is incorrect and should be replaced by \(\frac{\gamma}{2b}\), i.e., \(\mathcal{E}(\epsilon; \mathcal{C}(a, b), \rho) \thicksim \frac{\gamma}{2b}\left(\log\left(\frac{a}{\epsilon}\right)\right)^2\), where \(\gamma := \log_2(e)\). To clarify matters, we provide a self-contained proof of this scaling result in arXiv:2211.15466. Similarly, the number of required bits specified as \(\frac{1}{b}\left(\log\left(\frac{a}{\epsilon}\right)\right)^2 + o\left(\left(\log\left(\frac{1}{\epsilon}\right)\right)^2\right)\) in Theorem 4.1 should be replaced by \(\frac{\gamma}{2b}\left(\log\left(\frac{a}{\epsilon}\right)\right)^2 + o\left(\left(\log\left(\frac{1}{\epsilon}\right)\right)^2\right)\).

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Copyright Notice: © 2022 C. Hutter, R. Gül, and H. Bölcskei.

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