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Grassmannian Geometry Meets Dynamic Mode Decomposition in DMD-GEN: A New Metric for Mode Collapse in Time Series Generative Models

Amime Mohamed Aboussalah, Yassine Abbahaddou

TL;DR

The paper tackles mode collapse in time-series generative models by defining a time-series specific collapse notion and proposing DMD-GEN, a training-free metric built on Dynamic Mode Decomposition, Grassmannian subspace distances, and Optimal Transport. It represents real and generated time-series dynamics via temporal modes derived from the top-DMD eigenvectors and compares their subspaces on Gr$(k,n)$ using principal angles, with an OT-based Wasserstein distance quantifying preserved dynamics. The approach yields an interpretable decomposition into dynamic modes and demonstrates consistency with established metrics across synthetic and real datasets, while offering computational efficiency. The work provides a foundation for evaluating and potentially guiding the training of time-series generative models to better capture diverse temporal dynamics.

Abstract

Generative models like Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) often fail to capture the full diversity of their training data, leading to mode collapse. While this issue is well-explored in image generation, it remains underinvestigated for time series data. We introduce a new definition of mode collapse specific to time series and propose a novel metric, DMD-GEN, to quantify its severity. Our metric utilizes Dynamic Mode Decomposition (DMD), a data-driven technique for identifying coherent spatiotemporal patterns, and employs Optimal Transport between DMD eigenvectors to assess discrepancies between the underlying dynamics of the original and generated data. This approach not only quantifies the preservation of essential dynamic characteristics but also provides interpretability by pinpointing which modes have collapsed. We validate DMD-GEN on both synthetic and real-world datasets using various generative models, including TimeGAN, TimeVAE, and DiffusionTS. The results demonstrate that DMD-GEN correlates well with traditional evaluation metrics for static data while offering the advantage of applicability to dynamic data. This work offers for the first time a definition of mode collapse for time series, improving understanding, and forming the basis of our tool for assessing and improving generative models in the time series domain.

Grassmannian Geometry Meets Dynamic Mode Decomposition in DMD-GEN: A New Metric for Mode Collapse in Time Series Generative Models

TL;DR

The paper tackles mode collapse in time-series generative models by defining a time-series specific collapse notion and proposing DMD-GEN, a training-free metric built on Dynamic Mode Decomposition, Grassmannian subspace distances, and Optimal Transport. It represents real and generated time-series dynamics via temporal modes derived from the top-DMD eigenvectors and compares their subspaces on Gr using principal angles, with an OT-based Wasserstein distance quantifying preserved dynamics. The approach yields an interpretable decomposition into dynamic modes and demonstrates consistency with established metrics across synthetic and real datasets, while offering computational efficiency. The work provides a foundation for evaluating and potentially guiding the training of time-series generative models to better capture diverse temporal dynamics.

Abstract

Generative models like Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) often fail to capture the full diversity of their training data, leading to mode collapse. While this issue is well-explored in image generation, it remains underinvestigated for time series data. We introduce a new definition of mode collapse specific to time series and propose a novel metric, DMD-GEN, to quantify its severity. Our metric utilizes Dynamic Mode Decomposition (DMD), a data-driven technique for identifying coherent spatiotemporal patterns, and employs Optimal Transport between DMD eigenvectors to assess discrepancies between the underlying dynamics of the original and generated data. This approach not only quantifies the preservation of essential dynamic characteristics but also provides interpretability by pinpointing which modes have collapsed. We validate DMD-GEN on both synthetic and real-world datasets using various generative models, including TimeGAN, TimeVAE, and DiffusionTS. The results demonstrate that DMD-GEN correlates well with traditional evaluation metrics for static data while offering the advantage of applicability to dynamic data. This work offers for the first time a definition of mode collapse for time series, improving understanding, and forming the basis of our tool for assessing and improving generative models in the time series domain.

Paper Structure

This paper contains 26 sections, 3 theorems, 48 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Mode Collapse for Time Series
  4. Dynamic Mode Decomposition
  5. DMD-GEN: Toward an Explainable Metric for Capturing Mode Collaspe in Time Series
  6. Notations
  7. Measuring the Similarity Between Time Series Using Their Respective DMD Modes
  8. Measuring Mode Collapse For Time Series
  9. Experiments
  10. Datasets
  11. Baseline Metrics
  12. Evaluation of Generative Models Using DMD Eigenvalues
  13. Consistency of DMD-GEN with Established Metrics
  14. Synthetically Analyzing Metrics Under Mode Collapse Settings
  15. Conclusion
  16. ...and 11 more sections

Key Result

Theorem 4

Let $\mathcal{M}_{k}(\mathbf{X}), \mathcal{M}_{k}(\widetilde{\mathbf{X}}) \in \mathbb{R}^{n \times k}$ be matrices whose columns form orthonormal bases of two $k$-dimensional subspaces of $\mathbb{R}^n$. Let $\Theta = \operatorname{diag}(\theta_1, \theta_2, \dots, \theta_k)$ be the diagonal matrix o Then, the geodesic linking $\mathcal{M}_{k}(\mathbf{X})$ and $\mathcal{M}_{k}(\widetilde{\mathbf{X}

Figures (7)

  • Figure 1: Comparison of DMD Eigenvalues between Original and Generated Time Series for DiffusionTS at Initial and Final Training Epochs on the dataset Sines.
  • Figure 2: Comparison of DMD Eigenvalues between Original and Generated Time Series for DiffusionTS at Initial and Final Training Epochs on the dataset Stock.
  • Figure 3: Examples of time series generated using the generators $\mathcal{G}_1$ and $\mathcal{G}_2$.
  • Figure 4: Comparison of DMD Eigenvalues between Original and Generated Time Series for DiffusionTS through Epochs on the dataset Energy.
  • Figure 5: Comparison of DMD Eigenvalues between Original and Generated Time Series for DiffusionTS through Epochs on the dataset ETTh.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Definition 1: Temporal Modes
  • Definition 2: Grassmannian manifold
  • Definition 3: Principal Angles Between Temporal Modes
  • Theorem 4: DMD Mode Geodesic
  • Theorem 5: DMD Mode Geodesic
  • proof
  • Theorem 6