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Entropy based lower dimension bounds for finite-time prediction of Dynamic Mode Decomposition algorithms

Till Hauser, Julian Hölz

TL;DR

The paper investigates lower bounds on the dimension of finite-dimensional subspaces $F ⊆ L^2(X)$ necessary for accurate long-horizon prediction via Dynamic Mode Decomposition. It provides two complementary bounds: (i) a measure-theoretic-entropy bound for partition-based subspaces $F = L^2(X|α)$, linking required dimension to the partition entropy $h_μ(φ, α)$, and (ii) an approximation-entropy bound for general finite-dimensional $F$, linking dimension growth to the approximation entropy $h_{apr}(T_φ, F)$; this is complemented by a Hilbert-space treatment using delay observables. The work shows that the subspace size must grow at least exponentially with horizon when entropy is positive, and that for general $F$, the delay-extended subspaces $F_n$ obey a linear-in-$K$ lower bound involving $h_{apr}(T_φ, F)$. A spectral-theoretic perspective ties $h_{apr}$ to the absolutely continuous spectrum via $h_{apr}(T) = ∫ m \, d\mathfrak{m}$ on the relevant spectral support, and the paper discusses how positive entropy implies infinite approximation entropy through countable Lebesgue components, informing subspace design and the use of delay observables in DMD-type methods.

Abstract

Motivated by Dynamic Mode Decomposition algorithms, we provide lower bounds on the dimension of a finite-dimensional subspace $F \subseteq \mathrm{L}^2(\mathrm{X})$ required for predicting the behavior of dynamical systems over long time horizons. We distinguish between two cases: (i) If $F$ is determined by a finite partition of $X$ we derive a lower bound that depends on the dynamical measure-theoretic entropy of the partition. (ii) We consider general finite-dimensional subspaces $F$ and establish a lower bound for the dimension of $F$ that is contingent on the spectral structure of the Koopman operator of the system, via the approximation entropy of $F$ as studied by Voiculescu. Furthermore, we motivate the use of delay observables to improve the predictive qualities of Dynamic Mode Decomposition algorithms.

Entropy based lower dimension bounds for finite-time prediction of Dynamic Mode Decomposition algorithms

TL;DR

The paper investigates lower bounds on the dimension of finite-dimensional subspaces necessary for accurate long-horizon prediction via Dynamic Mode Decomposition. It provides two complementary bounds: (i) a measure-theoretic-entropy bound for partition-based subspaces , linking required dimension to the partition entropy , and (ii) an approximation-entropy bound for general finite-dimensional , linking dimension growth to the approximation entropy ; this is complemented by a Hilbert-space treatment using delay observables. The work shows that the subspace size must grow at least exponentially with horizon when entropy is positive, and that for general , the delay-extended subspaces obey a linear-in- lower bound involving . A spectral-theoretic perspective ties to the absolutely continuous spectrum via on the relevant spectral support, and the paper discusses how positive entropy implies infinite approximation entropy through countable Lebesgue components, informing subspace design and the use of delay observables in DMD-type methods.

Abstract

Motivated by Dynamic Mode Decomposition algorithms, we provide lower bounds on the dimension of a finite-dimensional subspace required for predicting the behavior of dynamical systems over long time horizons. We distinguish between two cases: (i) If is determined by a finite partition of we derive a lower bound that depends on the dynamical measure-theoretic entropy of the partition. (ii) We consider general finite-dimensional subspaces and establish a lower bound for the dimension of that is contingent on the spectral structure of the Koopman operator of the system, via the approximation entropy of as studied by Voiculescu. Furthermore, we motivate the use of delay observables to improve the predictive qualities of Dynamic Mode Decomposition algorithms.
Paper Structure (11 sections, 17 theorems, 73 equations)

This paper contains 11 sections, 17 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Lower bounds by measure-theoretic entropy
  3. Lower bounds by approximation entropy
  4. Structure of the article
  5. Lower bounds by measure-theoretic entropy
  6. Measure-theoretic entropy
  7. Projections and conditional expectations
  8. Proof of Theorem \ref{['thm:exponential_lower_bound_dimension']}
  9. Lower bounds by approximation entropy
  10. Interplay of the measure-theoretical and the approximation entropy
  11. A spectral theoretic point of view

Key Result

Theorem 1.1

Let $(\mathrm{X}, \varphi)$ be a measure-preserving system, $\varepsilon > 0$ and $\alpha$ be a finite $\Sigma$-measurable partition of $X$. Then there exists $K_0 \in \mathbb{N}$ and $\delta > 0$ that satisfy the following: Whenever for a linear operator $\hat{T} \colon \mathrm{L}^2(\mathrm{X} \mid holds for all $f \in F$ and $k \in \{0, \dots, K - 1\}$, we conclude that Here $\delta$ only depen

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2: Markov structure
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 26 more