Diagnostic Certificates of Data Quality and Regression Identifiability for Koopman Identification

Technical Summary: Diagnostic Certificates of Data Quality and Regression Identifiability for Koopman Identification

Problem Statement
In the identification of nonlinear dynamical systems using the Koopman operator framework and Extended Dynamic Mode Decomposition with control (EDMDc), the quality of the identified model depends on a joint regression object comprising visited states, lifted dictionary features, and control inputs. Classical persistent excitation (PE) criteria, which assess input richness, are insufficient for this context. A signal may be input-rich while the state trajectory remains confined to a narrow region; state samples may appear well-spread while the chosen dictionary produces degenerate or highly correlated lifted features; and lifted features may be individually nondegenerate while the final concatenated regression matrix (lifted features + inputs) remains ill-conditioned due to redundancy between columns.

The paper addresses the lack of a structured framework to diagnose where data quality fails in Koopman identification. It posits that data quality is a multilayer problem involving state-space geometry, lifted-feature geometry, and regression-space geometry, which are related but not interchangeable.

Methodology
The authors propose a diagnostic certificate framework that separates data quality assessment into three distinct layers, moving from upstream sampling properties to the final estimation stability:

1. State-Space Certificates: These are indirect diagnostics assessing the geometry of the collected trajectory.

   • Directional Coverage ($C_{dir}$): Measures the dispersion of trajectory displacement directions.
   • Frostman-type Non-clustering ($C_{fr}$): Diagnoses local repeated sampling and multiscale clustering.
   • State Isotropy ($C_{state}$): Assesses the spectral condition of the active standardized state matrix.

2. Lifted-Feature Certificates: These assess the dictionary's behavior on the collected data.

   • Lifted-Feature Isotropy ($C_{lift}$): Measures the minimum eigenvalue of the standardized lifted-feature Gram matrix, detecting constant columns or high correlations induced by the specific state distribution.

3. Regression-Space Certificates: These assess the final EDMDc regression problem.

   • Regression Isotropy ($C_{reg}$): Defined as the minimum eigenvalue of the active standardized concatenated matrix (lifted features + inputs). This is the primary certificate with direct theoretical guarantees regarding estimation stability.
   • Composite Certificate ($C_{GPE}$): A bottleneck-style metric taking the minimum of the above layers (normalized by thresholds) to localize the primary data-quality failure.

Theoretical Framework
The paper establishes that $C_{reg}$ directly controls the smallest singular value of the active standardized design matrix ($\sigma_{min}(\bar{\Phi})$). Under a fixed design and conditional Gaussian noise, this certificate bounds the Fisher information and the expected error of the ridge/least-squares estimator. The authors prove a population-to-sample lower bound for the regression spectrum, showing that a positive spectral bound requires three conditions working in concert: state distribution coverage, dictionary nondegeneracy on that region, and conditional input excitation relative to lifted features.

Crucially, the paper provides a Schur-complement condition and counterexamples demonstrating that favorable state geometry does not imply favorable lifted or regression geometry (e.g., a state set where a specific polynomial dictionary term vanishes identically). Thus, upstream diagnostics cannot substitute for direct inspection of the final regression spectrum.

Experimental Validation
The framework is evaluated using the IGPE-DOPT algorithm, a certificate-driven sampling method that scores candidate trajectory segments based on a weighted mix of the proposed certificates. Experiments are conducted on Duffing, Van der Pol, and Lorenz systems, comparing IGPE-DOPT against:

• Input-oriented baselines (Adaptive PE, Optimal Input Design).
• State-oriented baselines (State-space k-center, GPE-STATE).
• Lifted-space baselines (Lifted D-optimal).
• Regression-space baselines (Regression D/E-optimal).

Key Results

1. Layer Separation: The experiments confirm that certificate layers are distinct. A method optimizing state coverage (e.g., STATE-KCENTER) does not necessarily optimize the regression spectrum, and vice versa.
2. Theoretical Correspondence: The regression certificate ($C_{reg}$) correlates with the minimum singular value and one-step estimation stability, validating the theoretical link.
3. Non-Monotonicity: Downstream performance (long-horizon prediction and closed-loop control) is not monotone with respect to any single certificate. A method with the highest regression certificate does not always yield the best control performance, as task metrics depend on dictionary bias, operating regions, and closed-loop controllability.
4. Sensitivity: Changing the weights in the IGPE-DOPT scoring function shifts the data-quality profile (e.g., prioritizing regression over state coverage), but no single weight configuration guarantees optimality across all downstream tasks.

Significance and Claims
The paper explicitly frames its contribution as a computable, interpretable, and ablatable diagnostic framework, not as a claim of universal optimality for data collection.

• Diagnostic Utility: The framework serves as a "triage workflow" for practitioners to identify whether a data-quality bottleneck lies in state coverage, dictionary degeneracy, or the final regression spectrum.
• Theoretical Boundaries: The authors emphasize that their theoretical guarantees apply strictly to one-step EDMDc regression stability in active standardized coordinates. They do not claim to control long-horizon rollout error, Koopman invariant-subspace recovery, or closed-loop control performance.
• Practical Guidance: The paper argues that relying solely on input PE or state coverage is insufficient. Practitioners must report active dimensions, active ranks, and spectral quantities for the concatenated regression matrix to ensure identifiability.

In summary, the work reframes Koopman data acquisition from a search for "rich inputs" to a structured diagnosis of multilayer geometric properties, providing tools to locate and repair specific failures in the data-to-model pipeline.