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Geometric Framework for Robust Order Detection in Delay-Coordinates Dynamic Mode Decomposition

Yoav Harris, Hadas Benisty, Ronen Talmon

Abstract

Delay-coordinates dynamic mode decomposition (DC-DMD) is widely used to extract coherent spatiotemporal modes from high-dimensional time series. A central challenge is distinguishing dynamically meaningful modes from spurious modes induced by noise and order overestimation. We show that model order detection and mode selection in DC-DMD are fundamentally problems of subspace geometry. Specifically, true modes are characterized by concentration within a low-dimensional signal subspace, whereas spurious modes necessarily retain non-negligible components outside any moderate overestimate of that subspace. This geometric distinction yields a perturbation-robust definition of true and spurious modes and yields fully data-driven selection criteria. This geometric framework leads to two complementary data-driven selection criteria. The first is derived directly from the geometric distinction and uses a data-driven proxy of the signal-subspace to compute a residual score. The second arises from a new operator-theoretic analysis of delay embedding. Using a block-companion formulation, we show that all modes exhibit a Kronecker-Vandermonde (KV) structure induced by the delay-coordinates, and true modes are distinguished by the degree to which they conform to it. Importantly, we also show that this deviation is governed precisely by the geometric residual. In addition, our analysis provides a principled explanation for the empirical behavior of magnitude- and norm-based heuristics, clarifying when and why they fail under delay-coordinates. Extensive numerical experiments confirm the theoretical predictions and demonstrate that the proposed geometric and structure-based methods achieve robust and accurate order detection and mode selection, consistently better than existing baselines across noise levels, spectral separations, damping regimes, and embedding lengths.

Geometric Framework for Robust Order Detection in Delay-Coordinates Dynamic Mode Decomposition

Abstract

Delay-coordinates dynamic mode decomposition (DC-DMD) is widely used to extract coherent spatiotemporal modes from high-dimensional time series. A central challenge is distinguishing dynamically meaningful modes from spurious modes induced by noise and order overestimation. We show that model order detection and mode selection in DC-DMD are fundamentally problems of subspace geometry. Specifically, true modes are characterized by concentration within a low-dimensional signal subspace, whereas spurious modes necessarily retain non-negligible components outside any moderate overestimate of that subspace. This geometric distinction yields a perturbation-robust definition of true and spurious modes and yields fully data-driven selection criteria. This geometric framework leads to two complementary data-driven selection criteria. The first is derived directly from the geometric distinction and uses a data-driven proxy of the signal-subspace to compute a residual score. The second arises from a new operator-theoretic analysis of delay embedding. Using a block-companion formulation, we show that all modes exhibit a Kronecker-Vandermonde (KV) structure induced by the delay-coordinates, and true modes are distinguished by the degree to which they conform to it. Importantly, we also show that this deviation is governed precisely by the geometric residual. In addition, our analysis provides a principled explanation for the empirical behavior of magnitude- and norm-based heuristics, clarifying when and why they fail under delay-coordinates. Extensive numerical experiments confirm the theoretical predictions and demonstrate that the proposed geometric and structure-based methods achieve robust and accurate order detection and mode selection, consistently better than existing baselines across noise levels, spectral separations, damping regimes, and embedding lengths.
Paper Structure (35 sections, 7 theorems, 130 equations, 9 figures, 2 tables, 4 algorithms)

This paper contains 35 sections, 7 theorems, 130 equations, 9 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. System and signal model
  4. Delay-coordinates DMD
  5. Signal space, order detection, and component selection
  6. Related work on order detection and mode selection
  7. Mode geometry, signal subspace residual, and true and spurious modes
  8. Kronecker--Vandermonde structure in modes
  9. Block-companion minimizer and compression
  10. Nested DMD
  11. Connection to existing order detection methods
  12. Spatiotemporal coupling ratio-based method
  13. Mode-norm
  14. Numerical Results
  15. Experimental setup
  16. ...and 20 more sections

Key Result

Lemma 5.1

The optimization problem eq:least_squares_objective admits the minimizer $\bm{C}_L$ with the following block-companion form: where $\bm B_\ell\in\mathbb C^{D\times D}$.

Figures (9)

  • Figure 1: Geometric illustration of signal-subspace residual (SSR) and estimated-subspace residual (ESR) in the case $m=1$ and $M=2$, where the true signal subspace $\mathcal{S}$ (green) lies inside the truncation subspace $\mathcal{U}_M = \operatorname{span}\{u_1,u_2\}$. True and spurious modes are drawn in blue and red, respectively, together with their projections onto $\mathcal{U}_M$ (dashed). (a) SSR corresponds to the in-plane deviation of a mode from $\mathcal{S}$, shown as violet arrows. (b) ESR corresponds to the out-of-plane component of a mode, shown as orange arrows. In this constructed example, the true mode has small SSR and ESR, whereas the spurious mode has larger residual in both senses.
  • Figure 2: Kronecker--Vandermonde structure of a DMD mode. A KV mode is formed by stacking $L$ lag segments of length $D$, where each segment is a scaled and rotated copy of a base spatial vector $\widehat{\bm{\phi}}^{(0)}$. The multipliers $\widehat{\lambda}^{\ell}$ form the Vandermonde vector $\bm v_L(\widehat{\lambda})=[1,\widehat{\lambda},\ldots,\widehat{\lambda}^{L-1}]^\top$, illustrated at the bottom. Reshaping the KV mode yields the rank-1 outer product $\widehat{\bm{\phi}}^{(0)}\,\bm v_L(\widehat{\lambda})^\top$, mirroring the rank-1 structure underlying order-1 DMD.
  • Figure 3: Spurious eigenvalue magnitude statistics versus embedding length $L$. For each $L$, we pool the magnitudes $|\widehat{\lambda}|$ of all spurious eigenvalues from all Monte--Carlo trials and plot the resulting empirical CDF (one curve per $L$). Overall, spurious eigenvalue magnitudes shift upward with increasing $L$, but the pooled distributions retain a non-negligible lower tail.
  • Figure 4: Order-hit probability vs. SNR under Gaussian noise. Panels show $m=2$ (left), $m=3$ (center), and $m=5$ (right). Working point: $N = 200$, $D = 45$, $L = 64$, $M = 15$; eigenvalues satisfy $\rho_j = 0.98$ with equally spaced phases and minimal separation $\Delta\theta = 0.01$; all amplitudes equal to one. Only the SNR varies.
  • Figure 5: Order-hit probability vs. phase separation $\Delta\theta$ (defined in \ref{['eq:delta_theta']}) under Gaussian noise. Panels show $m=2$ (left), $m=3$ (center), and $m=5$ (right). Working point matches Fig. \ref{['fig:snr_curves']}, with SNR fixed at $10\,\mathrm{dB}$; only $\Delta\theta$ varies.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Definition 4.1: Signal-subspace residual vector
  • Definition 4.2: True and spurious modes
  • Definition 4.3: Estimated-subspace residual vector
  • Lemma 5.1: Block-companion minimizer
  • Lemma 5.2: Eigenvectors of block-companion matrices (see, e.g., Ref. MackeyMackeyMehlMehrmann2006)
  • Proposition 5.3
  • Theorem 5.4
  • Lemma 3.1: Signal--subspace deviation
  • proof : Sketch
  • Lemma 5.1: Moore--Penrose projection identity
  • ...and 4 more