Title: A Geometry-Aware Metric for Mode Collapse in Time Series Generative Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Time series

Generative models

Mode Collapse
Abstract: Generative models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and diffusion models often suffer from mode collapse, failing to reproduce the full diversity of their training data. While this problem has been extensively studied in image generation, it remains largely unaddressed for time series. We introduce a formal definition of mode collapse for time series and propose DMD-GEN, a geometry-aware metric that quantifies its severity. DMD-GEN leverages Dynamic Mode Decomposition (DMD) to extract coherent temporal structures and uses Optimal Transport between DMD eigenvectors to measure discrepancies in underlying dynamics. By representing the subspaces spanned by the DMD eigenvectors as point structures on a Grassmann manifold, and comparing them via Wasserstein distances computed from principal angles, DMD-GEN enables a principled geometric comparison between real and generated sequences. The metric is efficient, requiring no additional training, supports mini-batch evaluation, and is easily parallelizable. Beyond quantification, DMD-GEN offers interpretability by revealing which dynamical modes are distorted or missing in the generated data. Experiments on synthetic and real-world datasets using TimeGAN, TimeVAE, and DiffusionTS show that DMD-GEN aligns with existing metrics while providing the first principled framework for detecting and interpreting mode collapse in time series.

A Geometry-Aware Metric for Mode Collapse in Time Series Generative Models

Abstract

Generative models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and diffusion models often suffer from mode collapse , failing to reproduce the full diversity of their training data. While this problem has been extensively studied in image generation, it remains largely unaddressed for time series. We introduce a formal definition of mode collapse for time series and propose DMD-GEN, a geometry-aware metric that quantifies its severity. DMD-GEN leverages Dynamic Mode Decomposition (DMD) to extract coherent temporal structures and uses Optimal Transport between DMD eigenvectors to measure discrepancies in underlying dynamics. By representing the subspaces spanned by the DMD eigenvectors as point structures on a Grassmann manifold, and comparing them via Wasserstein distances computed from principal angles, DMD-GEN enables a principled geometric comparison between real and generated sequences. The metric is efficient, requiring no additional training, supports minibatch evaluation, and is easily parallelizable. Beyond quantification, DMD-GEN offers interpretability by revealing which dynamical modes are distorted or missing in the generated data. Experiments on synthetic and real-world datasets using TimeGAN, TimeVAE, and DiffusionTS show that DMD-GEN aligns with existing metrics while providing the first principled framework for detecting and interpreting mode collapse in time series. Our code is available at: here.

1 Introduction

Generative models have gained significant attention in recent years, driven by recent advancements in computational power, the availability of extensive datasets, and breakthrough developments in machine learning algorithms. Notably, models like GANs and VAEs excel at capturing rich and meaningful latent representations of data [17, 23, 38, 41, 45]. These models are applied in various ways, such as generating realistic samples that mimic real-world data distributions [51], modeling complex probability distributions through density estimation [59], and augmenting datasets with synthetic data to improve model generalization [25, 61], among others. However, recent studies have revealed that generative models sometimes fail to produce diverse samples, leading to reduced effectiveness in applications that require a broad spectrum of variations [1, 6, 40]. An illustration of this challenge can be seen in GANs, which often experience mode collapse, a phenomenon where the generator focuses on a limited subset of the data distribution, leading to the production of repetitive or similar samples rather than capturing the full diversity of the training data [3, 4, 15, 42]. V AEs also face a phenomenon called posterior collapse, where the model tends to generate outputs that are similar or indistinguishable for different inputs. This limitation reduces the model's ability to produce

diverse samples [18, 56]. Diffusion models, while generally robust against mode collapse compared to GANs and VAEs, are not entirely immune to difficulties in covering the full data distribution. These challenges often emerge under strong classifier-free guidance or limited data regimes [20, 46, 50].

The issue of diversity in generative models has received significant attention in fields such as computer vision and natural language processing [11, 21, 29, 32, 49], however, it remains relatively underexplored for time series data. The inherently time-dependent and dynamic nature of time series makes traditional definitions of mode collapse insufficient, highlighting the need for a framework tailored to time-dependent data. Defining modes in time series is particularly challenging, as it requires capturing evolving temporal patterns rather than simply avoiding repetitive or static sequences. A natural way to determine if a generative model preserves the diversity is to evaluate the similarity between original and generated time series. However, widely used existing evaluation metrics, such as, Predictive and Discriminative Scores [60], and Contextual FID [22], suffer from key limitations. They are often computationally expensive, since they rely on training auxiliary models that capture temporal dependencies in the data. More importantly, these metrics provide only aggregate performance indicators and fail to reveal which dynamic modes have been preserved or lost in the generated sequences.

To address these challenges, we introduce a time-series-specific definition of a mode. Our approach is based on Dynamic Mode Decomposition (DMD) [52], a spectral method that identifies dominant coherent structures in temporal dynamics. This allows to develop a new metric that is interpretable, training-free, and explicitly quantifies the preservation of temporal modes in generative models. The main contributions of this work are as follows,

New Definition of Mode Collapse for Time Series: We introduce a new definition of mode collapse specifically for time series data, leveraging DMD to capture and analyze coherent dynamic patterns.

Development of DMD-GEN Metric: We propose DMD-GEN, a new metric to detect mode collapse, which consistently aligns with traditional generative model evaluation metrics while offering unique insights into time series dynamics.

Enhanced Interpretability: The DMD-GEN metric provides increased interpretability by decomposing the underlying dynamics into distinct modes, allowing for a clearer understanding of the preservation of essential time series characteristics.

Efficiency in Time Complexity: Our approach offers significant computational efficiency as it requires no additional training, making it highly scalable for real-time applications.

2 Background and Related Work

Mode Collapse for Time Series. Before delving into the details of our new evaluation metric that incorporates the concepts mentioned above, it is worth highlighting the challenges to be addressed in order to measure mode collapse when dealing with time series. (i) Capturing Modes: For time series, we need to consider modes that represent different evolving patterns over time. Real-world time series data rarely exhibit a single clean pattern [26, 30]. Instead, time series data often exhibit multiple patterns simultaneously, e.g. layered over long-term trends and shorter-term fluctuations representing evolving modes. This makes it difficult to isolate and identify the specific mode of interest. Moreover, in contrast to data modalities with well-defined discrete structures, such as images or text, time series data exhibit inherent temporal continuity. This makes it challenging to determine the beginning and end of a specific evolving pattern (mode). (ii) Similarity Measurement: Time series data often have different characteristics that make standard distance metrics such as Euclidean distance less effective as a similarity measure. For instance, Euclidean distance is sensitive to variations in feature scales across dimensions. In high-dimensional spaces, this can cause a few dimensions to dominate, leading to distorted similarity measurements. In addition, Euclidean distance does not take into account the temporal dependencies present in the time series data. It assumes that each timestamp is independent, which is inappropriate for time series data where the ordering and temporal dependencies between observations are essential. More advanced similarity measures, such as Dynamic Time Warping (DTW) [37], address temporal dependencies by aligning sequences to minimize distance. However, if fails to capture the underlying modes or coherent dynamic patterns present in the data. This inability to recognize and preserve the essential modes means DTW falls short in assessing mode collapse.

Unlike in images, the mode collapse issue in time series cannot be easily distinguished with human eyes. Therefore, this area remains relatively under-explored. Few studies have formally addressed this problem and proposed solutions within the time series domain. [31] introduced an auto-normalization heuristic that normalizes each time series separately rather than the dataset as a whole. However, the custom auto-normalization heuristic addresses only mode collapse resulting from offset differences between time series, without considering whether the model captures the global trends and seasonality of the dataset. Additionally, DC-GAN [36] is the first time series GAN capable of generating all temporal features within a multimodal distributed time series. Based on directed chain stochastic differential equations (DC-SDEs) [13], the model introduces an approach to temporal generation. Although the authors did not explicitly discuss mode collapse, it would be valuable to investigate whether DC-GAN successfully captures the full range of trends and seasonal patterns present in the dataset.

Dynamic Mode Decomposition (DMD). DMD [43, 52] is a data-driven and model-free method used for analyzing the underlying dynamics of complex systems such as fluid dynamics. It is used to extract modal representations of a nonlinear dynamical system directly from data, without requiring prior knowledge of the system. Given a dynamical system ˙ x ( t ) = f ( x ( t ) , t ; µ ) , where x ( t ) ∈ R n denotes the n -dimensional state vector, µ ∈ R p represents the system parameters, and f : R n × R × R p → R n defines the dynamics, DMD approximates the system locally by ˙ x ≈ A x , where A is the best-fit linear operator obtained via regression to approximate f . This linear approximation allows for the representation of the system's behavior in a simplified framework and helps construct reduced-order models that capture the essential dynamics of the systems. This is particularly useful for systems with large state spaces, such as fluid dynamics [24, 28]. Analogously, we consider a discrete-time approximation of the dynamical system. Given x : t ∈ R ↦→ x ( t ) ∈ R n , we collect m consecutive snapshots to construct two data matrices X , X ′ ∈ R n × m defined as

These snapshots are taken with a time-step ∆ t small enough to capture the highest frequencies in the system's dynamics, i.e., ∀ k ∈ N , x k = x ( k ∆ t ) . Assuming uniform sampling in time, we approximate the dynamical system linearly as x k +1 ≈ A ⋆ x k , where A ⋆ ∈ R n × n is the best-fit operator, i.e., A ⋆ = arg min A ∥ X ′ -AX ∥ F = X ′ X † , where ∥ . ∥ F is the Frobenius norm and X † is the Moore-Penrose generalized inverse of X . The optimal discrete-time operator A ⋆ is related to the continuous-time operator A , defined earlier, through the exponential mapping A ⋆ = exp( A ∆ t ) ; see Appendix A. The DMD operator A ⋆ is deeply rooted in Koopman operator theory [9, 33, 39, 47], which provides a linear perspective on nonlinear dynamical systems. Specifically, DMD can be viewed as a finite-dimensional approximation of the infinite-dimensional Koopman operator that advances observables of the system forward in time. This connection, originally established by [48], enables spectral analysis of nonlinear dynamics through linear algebraic techniques. In essence, Koopman theory offers a principled framework for globally linearizing nonlinear dynamics, allowing DMDto capture coherent spatiotemporal structures and their evolution through the system's spectrum [12, 34, 35]. By analyzing the eigenvalues Λ = diag( λ 1 , . . . , λ r ) and the corresponding eigenvectors Φ = [ ϕ 1 , . . . , ϕ r ] ∈ R n × r of the DMD operator A ⋆ , where r denotes the rank of X , we can capture the dominant dynamic patterns that govern the system's evolution. These spectral components, the eigenvalues encoding temporal behavior and the eigenvectors capturing spatial coherence, together provide a compact yet expressive representation of the underlying dynamics. Conventionally, the eigenvectors and their corresponding eigenvalues are arranged in descending order based on the magnitude of the eigenvalues, i.e., | λ 1 | ≥ . . . ≥ | λ r | .

Throughout this paper, we distinguish between two related quantities: r denotes the full numerical rank of the snapshot matrix X , corresponding to the total number of available DMD modes obtained from the spectral decomposition of A ⋆ . In contrast, k ≤ r denotes the number of dominant modes retained for reduced-order approximation or comparison. Hence, r characterizes the complete spectral dimension of the system, while k represents the truncated subspace capturing the most significant dynamics. This distinction will be maintained consistently in subsequent sections, where all definitions and metrics involving M k ( X ) refer to the k leading DMD modes.

For a high-dimensional state vector x ∈ R n , the matrix A ⋆ comprises n 2 elements, making its representation and spectral decomposition computationally challenging. To address this, we apply

dimensionality reduction to efficiently compute the dominant eigenvalues and eigenvectors of A ⋆ by constructing a reduced-order approximation ˜ A ∈ R r × r . The DMD approximation at each time step can be expressed as follows,

where ϕ j are the DMD modes (eigenvectors of the matrix A ), λ j are the corresponding DMD eigenvalues, and b j denotes the mode amplitude, given by b = Φ † x 0 in matrix form. The detailed steps for this process are provided in the Appendix A.3. DMD modes can be interpreted as basis vectors spanning a subspace that captures coherent spatiotemporal patterns among the components of x ( t ) . It decomposes a complex time series into a collection of simpler, coherent modes, where each of them captures a specific aspect of the system's behavior, e.g., an oscillation, an exponential growth/decay, or a traveling wave [58]. The DMD modes are spatial fields that often reveal coherent structures within the flow. These structures are fully characterized by the DMD eigenvalues Λ and eigenvectors Φ , which respectively encode the temporal frequencies and spatial patterns of the underlying dynamics. Specifically, the imaginary part of the eigenvalues Λ determines the oscillation frequency, while the real part indicates the rate of exponential growth or decay [10, 54]. The corresponding DMD eigenvectors Φ capture the spatial coherence associated with each mode, providing an interpretable link between the system's temporal evolution and its spatial distribution.

3 Quantifying Mode Collapse in Time Series

Notations. We denote by G a generative model specific to time series. This model is trained on a dataset comprising N time series, each with a fixed length, represented as { X i } N i =1 . During the inference phase, we synthetically generate a set of ˜ N time series, denoted as { ˜ X j } ˜ N j =1 using G . Both the original and generated time series are assumed to have a consistent length, denoted as ℓ , and dimensionality, represented as n . Formally, for any pair of indices i and j , the original and generated time series X i and ˜ X j are elements of the Euclidean space R n × ℓ .

3.1 Defining Temporal Modes

In this section, we formalize the notion of temporal modes. Definition 3.1 captures the essence of a mode in terms of the dominant eigenvalues and eigenvectors of the DMD operator, highlighting the significant dynamic structures in the time series data.

Definition 3.1 ( Temporal Modes ) . Given a time series X = [ x 1 , . . . , x ℓ ] ∈ R ℓ × n , we define the set of temporal modes M k ( X ) as the k dominant eigenvectors { ϕ 1 , . . . , ϕ k } of the associated DMD operator. These capture the primary dynamic patterns in the time series. We represent M k ( X ) as,

Selecting the number of retained modes k involves a classical bias-variance trade-off: a smaller k yields robustness to noise but may underrepresent the full system dynamics, whereas a larger k enables more accurate reconstruction at the risk of overfitting. To formalize this trade-off, let r = rank( Φ ) denote the rank of Φ , i.e., the concatenation of all DMD eigenmodes. The approximation error can then be quantified using the Frobenius norm ∥ Φ -M k ( X ) ∥ F . The exact expression for this error term is provided in Proposition 3.2.

Proposition 3.2 ( Eckart-Young-Mirsky theorem [14] ) . Let σ 1 ≥ σ 2 ≥ · · · ≥ σ r be the singular values of Φ . Then the DMD eigenmodes M k ( X ) that uses the first k modes satisfies, ∥ Φ -M k ( X ) ∥ = ( σ 2 + . . . + σ 2 ) 1 / 2 .

A common practical guideline is to select the smallest k such that a prescribed proportion τ ∈ [0 , 1] of the total energy is retained, i.e., ∑ k j =1 σ 2 j ≥ τ ∑ r j =1 σ 2 j . In practice, a typical choice is τ = 0 . 95 .

3.2 Measuring the Similarity Between Time Series Using Their Respective DMD Modes

We are interested in quantifying the similarity of the underlying dynamics between X i and ˜ X j . Comparing their respective DMD eigenvectors provides a principled way to assess this similarity over time, as these eigenvectors capture and reveal the dominant dynamic patterns of the system. According to Equation 1, the dominant modes can be identified from the eigenvectors associated with the largest DMD eigenvalues. Consequently, we focus on comparing the similarity between the corresponding temporal mode subspaces, M k ( X ) and M k ( ˜ X ) . However, directly measuring the distance between these two subspaces is mathematically challenging, as their bases are not necessarily aligned. Although M k ( X ) and M k ( ˜ X ) have the same dimensionality, their respective eigenvector subspaces are not necessarily aligned and may be expressed in different bases, making direct comparison nontrivial. To address this, we draw upon the concept of Grassmann manifolds from Information Geometry [2], which provides a natural and principled framework for comparing subspaces [27].

Definition 3.3 ( Grassmann manifold ) . Let V be an n -dimensional vector space The Grassmann manifold Gr ( k, n ) is the set of all k -dimensional subspaces of V , where 1 ≤ k ≤ n . Mathematically, it can be expressed as Gr ( k, n ) = { W ⊆ V : dim( W ) = k } .

The Riemannian distance between two subspaces on a Grassmann manifold is defined as the length of the shortest geodesic connecting them. This distance can be computed using the principal angles between the subspaces, which in our case correspond to the temporal modes.

Definition 3.4 ( Principal Angles Between Temporal Modes ) . Let the columns of M k ( X ) and M k ( ˜ X ) represent two linear subspaces U and V , respectively. The principal angles 0 ≤ θ 1 ≤ · · · ≤ θ r ≤ π/ 2 between the two subspaces are defined recursively as follows:

The work of [8] has shown that the principal angles can be efficiently computed via the Singular Value Decomposition (SVD) of Q ⊤ ˜ Q , where QR and ˜ Q ˜ R denote the QR factorizations of M k ( X ) and M k ( ˜ X ) , respectively. The SVD of Q ⊤ ˜ Q can then be written as Q ⊤ ˜ Q = U ang Σ ang V ⊤ ang , where U ang and V ang are orthogonal matrices containing the left and right singular vectors, respectively, and Σ ang is a diagonal matrix whose entries correspond to the singular values associated with the principal angles between the two subspaces. If s denotes the rank of Σ ang , then the principal angles correspond to the arccosine of the first s singular values of Σ ang , i.e., Θ = diag ( cos -1 σ 1 , . . . , cos -1 σ s ) . Following [7], these principal angles can be used to define several Grassmannian metrics. One such metric is the projection distance , defined as the Frobenius norm of the matrix sin Θ , i.e.,

The projection distance serves as a similarity metric between the temporal mode subspaces M k ( X ) and M k ( ˜ X ) . Smaller principal angles indicate that the subspaces are closer to each other, reflecting a higher degree of dynamical similarity. It is known that multiple geodesics can connect two points on the Grassmann manifold Gr( k, n ) . However, when all principal angles lie within the interval [0 , π/ 2] , the corresponding geodesic is unique [53, 57].

3.3 Robustness of the DMD Mode Geodesic Distance

To verify that the proposed geodesic distance effectively captures mode collapse, it is essential to assess its stability under small perturbations in the system dynamics. Theorem 3.5 establishes an upper bound on this distance, showing that minor perturbations in the time series result in only small deviations in the subspace of DMD eigenvectors, thereby reinforcing the stability and reliability of the proposed metric.

Theorem 3.5 (DMD reconstruction consistency) . Let X = [ x 1 , . . . , x ℓ ] ∈ R n × ℓ and ˜ X = [ ˜ x 1 , . . . , ˜ x ℓ ] ∈ R n × ℓ be two sequences of state snapshots. Assume that both X and ˜ X admit

a DMD representation with the same initial condition x 0 = ˜ x 0 . Let M k ( X ) ∈ C n × k and M k ( ˜ X ) ∈ C n × k denote the respective DMD mode matrices, associated with diagonal eigenvalue matrices Λ , ˜ Λ ∈ C k × k . Then, for all time steps t , the reconstructed states satisfy

Let E t denote the difference in dynamics between X and ˜ X , i.e., ∀ t, x t -˜ x t = E t x 0 . Then, for all time steps t , the projection distance between the corresponding temporal mode subspaces satisfies d proj ( M k ( X ) , M k ( ˜ X ) ) ≤ ∥ E t ∥ F δ t , where δ t denotes the spectral gap of Λ t , and ∥ · ∥ F represents the Frobenius norm. The proof of Theorem 3.5 is provided in Appendix C. This result demonstrates that small perturbations in the system dynamics lead to only minor variations in the subspace of DMD eigenvectors, ensuring that the geodesic distance remains a stable and reliable measure of dynamical similarity. In simpler terms, small differences in the time series translate into proportionally small differences in their DMD-based representations, making the proposed metric particularly well-suited for evaluating time-series generative models.

3.4 Measuring Mode Collapse For Time Series

To quantify mode collapse in generative models for time-series data, we propose a new approach based on Optimal Transport (OT) to assess the similarity and preservation of modes between real and generated samples. In this framework, DMD is first applied to both real and generated time series to extract their dominant modes, therefore capturing the key dynamical patterns underlying each dataset. For a given set of L sampled batches of real time series X = { X 1 , X 2 , . . . , X L } and generated time series ˜ X = { ˜ X 1 , ˜ X 2 , . . . , ˜ X L } , we compute the corresponding sets of DMD modes {M k ( X i ) } L i =1 and {M k ( ˜ X j ) } L j =1 , which encapsulate the dominant temporal dynamics of each sequence. We then construct a cost matrix C , where each entry C ij quantifies the dissimilarity between the mode subspaces M k ( X i ) and M k ( ˜ X j ) using a principal-angle-based metric. The OT problem is subsequently solved to obtain the transport plan γ ⋆ that minimizes the total transportation cost, therefore identifying the optimal mapping between real and generated modes via the Wasserstein distance defined as follows:

where p denotes the order of the Wasserstein distance, Π represents the set of all joint probability distributions, and ⟨· , ·⟩ p denotes the p -order cost-weighted inner product used to compute the transport cost. The resulting distance provides a robust measure of mode collapse: a smaller Wasserstein distance indicates better preservation of the original modes in the generated data, hence reflecting the effectiveness of the generative model in maintaining the intrinsic dynamical patterns of the time series. The geodesic γ in Equation 4 is defined formally in Theorem 3.6.

Theorem 3.6 ( DMDMode Geodesic ) . Let M k ( X ) , M k ( ˜ X ) ∈ R n × k be matrices whose columns form orthonormal bases of two k -dimensional subspaces of R n . Let Θ = diag( θ 1 , θ 2 , . . . , θ k ) denote the diagonal matrix of principal angles between the subspaces spanned by M k ( X ) and M k ( ˜ X ) . Further, let ∆ ∈ R n × k be an orthonormal matrix such that M k ( ˜ X ) = M k ( X ) cos( Θ )+ ∆ sin( Θ ) . Then, the geodesic connecting M k ( X ) and M k ( ˜ X ) on the Grassmann manifold Gr( k, n ) is given by γ ( t ) = M k ( X ) cos( t Θ ) + ∆ sin( t Θ ) , for t ∈ [0 , 1] , and the length of this geodesic corresponds exactly to the projection distance :

Theorem 3.6 characterizes the geodesic path between two sets of temporal modes in time series data, showing that the transformation between these modes can be expressed precisely through trigonometric combinations of the principal angles (see proof in Appendix B). The distance ˜ d proj in Equation 5 is positively correlated with d proj in Equation 3, since all principal angles lie within the interval [0 , π/ 2] . This approach provides a quantitative and interpretable framework for evaluating the performance of generative models on time series data. We approximate the metric in Equation 4

Algorithm 1 Computation of DMD-GEN Metric

using the law of large numbers, and the detailed computational procedure is outlined in Algorithm 1. The values of the optimal transport matrix γ ⋆ = arg min γ ∈ Π ⟨ γ , C ⟩ p in Equation 4 quantify the extent to which the modes of each training time series are preserved in the generated samples.

Time Complexity. We analyze the computational complexity of Algorithm 1. Let B be the number of batches, S b be the batch size, n be the data dimensionality, m be the time series length, k be the number of modes, and ϵ be the OT solver precision. We assume n, m ≥ k . The total complexity is driven by the B outer loop iterations. In each iteration, we compute an S b × S b cost matrix C and solve the Optimal Transport (OT) problem. Computing C requires S 2 b pairwise comparisons. The cost for each pair ( X i , ˜ X j ) is dominated by the two DMD extractions (Steps 1-2), which, as detailed in Appendix A.3 (Algorithm 2), have a complexity of T DMD = O ( nmk + nk 2 + mk 2 + k 3 ) . This cost is higher than the subsequent QR factorization (Step 3: O ( nk 2 ) ) and geodesic computation (Step 4: O ( k 2 n + k 3 ) ). After building the matrix, solving the OT problem with Sinkhorn's algorithm [44] costs O ( S 2 b /ϵ ) . Therefore, the cost per batch is O ( S 2 b · T DMD + S 2 b /ϵ ) . The total complexity for B batches is: O ( B · S 2 b · ( nmk + nk 2 + mk 2 + k 3 +1 /ϵ )) .

4 Experiments

We evaluate the diversity of generative models across one synthetic dataset and three real-world datasets. The statistics of each dataset as well as the baselines metrics can be found in Appendix D.

Consistency of DMD-GEN with Established Metrics. Table 1 demonstrates that DMD-GEN produces results consistent with established evaluation metrics such as the Predictive Score, Discriminative Score, and Context-FID across all datasets. In each case, the rankings induced by DMD-GEN align closely with those from other metrics, effectively distinguishing between generative models based on their performance. A key advantage of DMD-GEN, however, lies in its efficiency: unlike other metrics, it requires no additional training to evaluate the generated time series. This makes DMD-GEN computationally efficient while maintaining consistent and reliable assessments of generative model quality.

As part of our ablation study, we also evaluated generic metrics not originally designed for time-series data but potentially applicable, such as MTopDiv [5], which measures divergence between original and generated samples by comparing data manifolds approximated as point clouds. Applying MTopDiv

to time series requires flattening the sequences into independent samples, thereby discarding the essential temporal structure and feature dependencies. Nonetheless, for completeness, we conducted experiments using MTopDiv on the Energy , ETTh , and Sines datasets to compare the performance of TimeVAE, TimeGAN, and DiffusionTS (see Appendix F). The results revealed substantial variability in standard deviations across models and datasets, making meaningful comparison difficult. This instability suggests that MTopDiv is not a reliable metric for evaluating multivariate time-series generative models, as it fails to capture the intrinsic temporal dependencies of the data.

Evaluating Metric Robustness Under Controlled Mode Collapse. To study how different evaluation metrics behave under varying degrees of mode collapse, we created a synthetic dataset that allows direct control over the collapse severity. Specifically, we generated N = 1000 time series, each drawn from one of two distinct generators, G 1 and G 2 , defined in Appendix E. Time series produced by the same generator are considered to belong to the same mode . The generator is chosen using a Bernoulli random variable with parameter λ ∈ [0 , 1] : G 1 is selected if λ < λ ref, and G 2 otherwise. The reference value λ ref = 0 . 5 corresponds to a balanced mixture, where both modes are equally likely (co-exist) and no collapse occurs. We denote the resulting dataset by D N ( λ ) .

̸

For each metric m , the non-collapse case is given by m ( D N ( λ ref ) , D N ( λ ref )) , while the collapsed cases correspond to m ( D N ( λ ref ) , D N ( λ )) for λ = λ ref. Since the metrics have different numerical ranges, we compare them using a normalized performance measure:

Table 2 reports Perf ( λ ) for several metrics across different mode collapse levels. We observe that the benchmark metrics exhibit large fluctuations in both magnitude and sign as λ varies, indicating sensitivity to changes in mode balance. In contrast, DMD-GEN remains stable and robust, effectively detecting even minor collapses. Both DMD-GEN and Context-FID increase rapidly as λ deviates from λ ref, signaling their sensitivity to emerging imbalances. However, unlike metrics such as Context-FID that can produce arbitrarily large values as discrepancies grow, DMD-GEN is naturally bounded by the geometry of the Grassmann manifold: the principal angles that define its distance lie within [0 , π/ 2] . This bounded structure prevents extreme variations, ensuring numerical stability and making DMD-GEN a reliable and computationally efficient metric for detecting mode collapse in practice. Figure 1 illustrates the evolution of each evaluation metric as the mode collapse severity λ increases. While Context-FID and DMD-GEN both exhibit a clear, monotonic increase reflecting stronger detection of collapse, DMD-GEN remains numerically stable across the entire range due to its bounded geometric formulation. In contrast, the Discriminative and Predictive Scores fluctuate considerably and fail to provide consistent trends, highlighting their limited sensitivity to gradual mode imbalances. These results confirm that DMD-GEN offers both sensitivity and robustness in detecting varying degrees of mode collapse.

Metric

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-0.71%

-0.83%

-0.40%

+0.35%

+0.54%

Context-FID

+36,796.45%

+18,394.64%

+8,210.25%

+1,874.76%

+1,855.58%

+7,019.51%

DMD-GEN

+681.03%

+477.76%

+312.22%

+115.02%

+114.92%

+314.18%

Mode 1 Dominates

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Assessing DMD-GEN on Bootstrapped Time Series.

Previous experiments focused primarily on deep learningbased generative models. To further validate the versatility of DMD-GEN, we evaluate it on time series generated through the classical non parametric Moving Block Bootstrap (MBB) method [16]. MBB preserves short-term temporal dependencies by resampling consecutive data blocks rather than individual points. To study how the choice of block size affects dynamic consistency, we apply MBB with three configurations: small blocks (introducing high randomness and weaker temporal coherence), medium blocks (partially retaining structure), and large blocks (preserving most temporal dependencies). We then compute the DMD-GEN distance between the original and bootstrapped time series to quantify how well dynamic patterns are maintained. As shown in Figure 2, increasing the block size consistently reduces DMD-GEN distance and stabilizes its variability, indicating that larger blocks better preserve the underlying temporal dynamics, whereas smaller blocks tend to distort them through excessive resampling noise.

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Interpreting Mode Behavior Through the DMD Spectrum. Figure 3 illustrates how the DMD spectrum reveals interpretable changes in system dynamics. In the top row, mode collapse concentrates dynamical activity into a few dominant components; the middle row shows frequency shifts indicating mild temporal distortion; and the bottom row displays additional spurious modes caused by injected noise. These spectral patterns provide a transparent view of how temporal structures deform across different dynamic perturbations.

5 Conclusions and Limitations

We introduced DMD-GEN, a new metric for evaluating generative models of time series and detecting mode collapse. By combining Dynamic Mode Decomposition with Optimal Transport, DMD-GEN provides a principled way to measure the similarity of temporal dynamics between real and generated data. Experiments show that DMD-GEN is more sensitive to mode collapse than existing metrics such as the Discriminative and Predictive Scores, while remaining consistent with their rankings. Unlike most existing metrics, DMD-GEN requires no additional training, making it computationally efficient and easy to apply. Its mode-based formulation also enhances interpretability by showing how key dynamical patterns are preserved or distorted in generated sequences. A current limitation is that DMDprovides only a linear approximation of nonlinear dynamics: although Koopman theory allows such systems to be represented by an infinite-dimensional linear operator, practical approximations rely on a finite number of modes. Capturing stronger nonlinearities would therefore require expanding this set, increasing computational cost and reducing efficiency.

Acknowledgments

The authors thank Kritaporn (Lune) Nitjaphanich for insightful discussions that helped shape this work.

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Thus,

Supplementary Material: A Geometry-Aware Metric for Mode Collapse in Multivariate Time Series Generative Models

A Dynamical Mode Decomposition: Details and Proofs

A.1 The Link Between the DMD operators in Continuous and Discrete Cases

Given a dynamical system ˙ x ( t ) = f ( x ( t ) , t ; µ ) , we linearly approximate the dynamics using DMD using the operator A ∈ R n × n , i.e.

Discretizing time into intervals of ∆ t and capturing snapshots accordingly, we establish the relationship between consecutive time steps in the following equation:

For a time-step ∆ t that is sufficiently small, we can employ the first-order Taylor expansion of the matrix exp (∆ t A ) , expressed as:

Therefore, from Equations 6 and 7, we conclude that:

A.2 Feasible Spectral Decomposition of the DMD Operator using Dimensionality Reduction

Algorithm 2 presents the steps to compute the eigenvectors and eigenvalues of the DMD operator A ⋆ using Singular Value Decomposition (SVD) for dimensionality reduction.

A.3 DMDexpansion

We prove the closed formula ∀ k, x k = ∑ j =1 ϕ j λ k j b j = ΦΛ k b , using recursion. For k = 0 , we have,

r

Let's now consider the equation hold for k = 0 , . . . , m , we have,

Therefore, the equality holds for all k ∈ N .

Algorithm 2 Dynamic Mode Decomposition

From collected snapshots of the system, build a pair of data matrices ( X , X ′ ) .

The closed formula of optimal DMD operator is

Compute the compact singular value decomposition (SVD) of X :

where U ∈ C n × r , Σ ∈ C r × r , V ∈ C m × r and r ≤ min ( m,n ) is the rank of X . Therefore,

Define a matrix

since U is a unitary matrix.

˜ A ∈ R r × r defines a low-dimensional linear model of the dynamical system on proper orthogonal decomposition (POD) coordinates.

Compute the eigen-decomposition of ˜ A :

where columns of W ∈ R r × r are eigenvectors and Λ = diag ( λ 1 , . . . , λ r ) ∈ R r × r is a diagonal matrix containing the corresponding eigenvalues.

Return DMD modes Φ :

Each column of Φ is an eigenvector of A meaning a DMD mode ϕ k corresponding to eigenvalue λ k

B Proof of Theorem 3.6 - DMD Mode Geodesic

Theorem 3.6 [DMD Mode Geodesic]

Let M k ( X ) , M k ( ˜ X ) ∈ R n × k be matrices whose columns form orthonormal bases of two k -dimensional subspaces of R n . Let Θ = diag( θ 1 , θ 2 , . . . , θ k ) be the diagonal matrix of principal angles between the subspaces spanned by M k ( X ) and M k ( ˜ X ) . Let ∆ ∈ R n × k be an orthonormal matrix such that

Then, the geodesic linking M k ( X ) and M k ( ˜ X ) on the Grassmann manifold Gr( k, n ) is given by

and the length of this geodesic corresponds exactly to the projection distance defined by

Proof. Preliminaries and Definitions

Grassmann Manifold Gr( k, n ) : The set of all k -dimensional linear subspaces of R n .

Orthonormal Bases : For a k -dimensional subspace S ⊂ R n , an orthonormal basis is represented by an n × k matrix Q with columns satisfying Q ⊤ Q = I k , where I k is the k × k identity matrix.

Principal Angles and Vectors : Given two subspaces S 1 and S 2 with orthonormal bases Q 1 and Q 2 , the principal angles 0 ≤ θ 1 ≤ θ 2 ≤ · · · ≤ θ k ≤ π 2 between them are defined recursively by

Projection Distance : The projection distance between S 1 and S 2 is defined as

1. Computation of the Principal Angles

Let Q 1 = M k ( X ) and Q 2 = M k ( ˜ X ) . Both Q 1 and Q 2 are n × k matrices with orthonormal columns.

We construct the matrix C as follows:

where

Since Q ⊤ 1 Q 1 = I k and Q ⊤ 2 Q 2 = I k , C captures the pairwise inner products between the basis vectors of Q 1 and Q 2 .

We then perform the Singular Value Decomposition (SVD) of C :

U, V ∈ R k × k are orthogonal matrices, i.e., U ⊤ U = V ⊤ V = I k .

Σ = diag( σ 1 , σ 2 , . . . , σ k ) with σ i ≥ 0 .

The singular values σ i of C are the cosines of the principal angles between the subspaces:

This result stems from the fact that the SVD aligns the basis vectors of U and V to maximize the projections in the directions of the principal angles, which correspond to the largest cosines.

Since principal angles θ i are defined in the range [0 , π/ 2] , their cosines naturally lie in [0 , 1] , matching the range of the singular values of C . Thus, the singular values encode the geometric relationship between the subspaces U and V in terms of the principal angles. This connection is fundamental to Grassmannian geometry, as it allows the distances and alignments between subspaces to be analyzed using the principal angles and their cosines.

2. Construction of Orthonormal Bases Aligned with Principal Directions

Define new orthonormal bases:

Verification of Orthonormality:

We then compute A ⊤ B :

3. Decomposition of B in Terms of A and

∆

Weaim to express B as a linear combination of A and another orthonormal matrix ∆ that is orthogonal to A .

Let us define ∆ :

where cos(Θ) = Σ and sin(Θ) = diag(sin( θ 1 ) , . . . , sin( θ k )) , and sin(Θ) -1 denotes the diagonal matrix with entries sin( θ i ) -1 .

Verification that ∆ is Orthogonal to A :

Verification that ∆ is Orthonormal :

First, we compute ∆ ⊤ ∆ :

We compute the inner term:

Since A ⊤ A = I k , B ⊤ B = I k , and A ⊤ B = Σ = cos(Θ) = cos(Θ) ⊤ = ( A ⊤ B ) ⊤ = B ⊤ A :

Since: sin 2 (Θ) + cos 2 (Θ) = I k

Thus,

Therefore, ∆ is orthonormal.

Expressing B in Terms of A and ∆ :

Using Equation (13), we have:

.

4. Define the Geodesic Path

On the Grassmann manifold, the geodesic γ ( t ) from A to B is given by:

Verification of Endpoints :

At t = 0 :

At t = 1 :

Thus, γ ( t ) is a continuous path on Gr( k, n ) connecting A and B .

Relate Back to Original Bases :

Since U and V are orthogonal matrices, the subspaces spanned by Q 1 and A , and by Q 2 and B , are identical. Therefore, we can express the geodesic in terms of M k ( X ) and ∆ .

Expressing the Geodesic in Original Terms :

Let us redefine ∆ accordingly to absorb U and V , so that we can write:

5. Compute the Length of the Geodesic

The length L of the geodesic γ ( t ) is given by:

where ∥·∥ F denotes the Frobenius norm.

Compute the Derivative ˙ γ ( t ) :

Since γ ( t ) = M k ( X ) cos( t Θ) + ∆sin( t Θ) , we have:

where we used the fact that the derivative of cos( t Θ) with respect to t is -Θsin( t Θ) , and similarly for sin( t Θ) .

Compute the Squared Norm ∥ ˙ γ ( t ) ∥ 2 F :

Since M k ( X ) ⊤ M k ( X ) = I k , ∆ ⊤ ∆ = I k , and M k ( X ) ⊤ ∆ = 0 , the cross terms vanish, and we have:

Compute the Length L :

Since ∥ ˙ γ ( t ) ∥ F is constant with respect to t , we have:

6. Length Equals the Projection Distance

Comparing the computed length L with the projection distance defined in Equation (9), we find:

On the Grassmann manifold, the geodesic distance between two subspaces is given by the length of the shortest path connecting them. This distance is intrinsically linked to the principal angles between the subspaces. The projection distance quantifies the separation between subspaces in terms of these principal angles.

By computing the squared norm of the derivative of the geodesic, we find that it equals the sum of the squares of the principal angles, which is the squared projection distance. Since the derivative's norm is constant, the total length of the geodesic over the interval t ∈ [0 , 1] is precisely the projection distance.

Therefore, the length of the geodesic γ ( t ) connecting M k ( X ) and M k ( ˜ X ) on the Grassmann manifold equals the projection distance between these two subspaces.

This completes the proof of Theorem 3.6.

C Proof of Theorem 3.5 - Metric Robustness

Theorem 3.5 Let X = [ x 1 , . . . , x ℓ ] ∈ R ℓ × n and ˜ X = [ ˜ x 1 , . . . , ˜ x ℓ ] ∈ R ℓ × n be two sequences of state snapshots. Suppose that both X and ˜ X with M k ( X ) ∈ R n × k and M k ( ˜ X ) ∈ R n × k as the respective DMD eigenvectors, Λ and ˜ Λ as the respective DMD eigenvectors , and admit a DMD form with the same initial condition x 0 = ˜ x 0 , i.e.,

Let E t be the difference in dynamics between X and ˜ X , i.e., ∀ t, x t -˜ x t = E t x 0 . We have, ∀ t, d proj ( M k ( X ) , M k ( ˜ X )) ≤ ∥ E t ∥ F δ t where δ t is the spectral gap of Λ t , and ∥·∥ F is the Frobenius Norm.

Proof. Let X = [ x 1 , . . . , x ℓ ] , ˜ X = [˜ x 1 , . . . , ˜ x ℓ ] ∈ R n × ℓ , and denote their k -dominant DMD bases by Q := M k ( X ) ∈ R n × k , ˜ Q := M k ( ˜ X ) ∈ R n × k . We define the timet linear propagators that generate the snapshots

Because x 0 is arbitrary, we identify the perturbation matrix A t -˜ A t = E t .

Wedin's theorem for diagonalizable matrices states that if E perturbs a matrix A whose spectrum splits into two clusters separated by a gap δ , then ∥ sin Θ ∥ F ≤ ∥ E ∥ F δ , where Θ collects the principal angles between the invariant subspaces associated with the chosen spectral clusters.

Applying Wedin with A = A t , E = E t , and the dominant invariant subspace span( Q ) , the gap is δ = δ t . The left-hand side is exactly the Grassmann projection distance d proj ( Q, ˜ Q ) = ∥ sin Θ ∥ F . Hence d proj ( Q, ˜ Q ) ≤ ∥ E t ∥ F δ t which is the desired bound.

D Datasets and Implementation Details

D.1 Basic Statistics on the Datasets

Sine waves. We generated a synthetic dataset consisting of two sets of sine waves to represent a bimodal distributed data. The data were generated using the following formula:

where A is the amplitude, f is the frequency, t is the time variable and ϕ is the phase angle of the sine wave. Each mode consists of 2000 samples with phases being randomly chosen between 0 and 2 π . For all the samples, the duration is 2 seconds and the sample rate is 12, making the length of each sequence be 24. A = 0 . 5 and f = 1 Hz for the first mode, and A = 5 and f = 0 . 5 Hz for the second mode.

Stock price. To test our framework on a complex multimodal dataset, we used Google stocks data from 2004 to 2019, which was used in [60]. The data consists of 6 features which are daily open, high, low, close, adjusted close, and volume. The time series were then cut into sequences with length 24, following the setup in the work done by [60].

Energy. We conducted experiments on UCI's air quality dataset [55] consisting of hourly averaged responses from an array of 5 metal oxide chemical sensors embedded in an Air Quality Chemical Multisensor Device in an Italian city. Data was recorded from March 2004 to February 2005 and consists of 28 features. Unlike the previous datasets, this one has an unimodal distribution. The data is cut into several sequences of length 7.

Electricity Transformer Temperature and humidity (ETTh). The ETTh dataset focuses on temperature and humidity data from electricity transformers [62]. It includes 2 years of data at an hourly granularity, providing detailed temporal information about transformer conditions.

Table 3 provides an overview of the datasets used in our experiments, including Sine, Stock, Energy, and ETTh. These datasets vary in both the number of samples and feature dimensions, offering a diverse evaluation setting for generative models.

D.2 Baseline Metrics

We compared our proposed metric DMD-GEN with well-established time series evaluation metrics. Specifically, this comparison includes three key metrics:

Predictive Score. [60] The predictive score evaluates how well a generative model captures the temporal dynamics of the original data. It involves training a model on the generated data and assessing its performance on a real dataset. A lower predictive score indicates that the generated data contains patterns that are more representative of the temporal patterns found in the original data.

Discriminative Score. [60] The discriminative score measures the similarity between real and generated time series data by training a binary classifier to distinguish between them.

Contextual Frechet Inception Distance (context-FID). [22] Context-FID is an adaptation of the Frechet Inception Distance (FID), a metric used to assess the quality of images created by a generative model [19]. For time series, context-FID measures the similarity between the real and generated data distributions by computing the Frechet distance between feature representations extracted from a time series feature encoder.

D.3 Implementation Details

The experiments were conducted on an NVIDIA A100 GPU. We utilized the pyDMD package 2 in Python to compute the DMD eigenvalues and eigenvectors. For generating synthetic time series, we used the original settings and the official implementation of DiffusionTS 3 , TimeGAN 4 and TimeVAE 5 .

E Synthetic Generators

To evaluate the ability of DMD-GEN to detect Mode Collapse, we generate synthetic time series using two parametric functions, denoted G 1 and G 2 . These generators produce diverse temporal patterns by incorporating nonlinear transformations and oscillatory components. Each function is parameterized by randomly sampled variables from a uniform distribution, ensuring variability across generated samples. Below, we give the expressions of these generators,

where U denotes the uniform distribution over [0 , 1] . Each time series is discretized to a length of T = 129 and a dimensionality of d = 65 . Figure 4 illustrates examples of time series generated using G 1 and G 2 . Generator G 1 produces smooth, localized wave patterns with oscillations that gradually decay in space, resulting in broader and less frequent peaks over time. In contrast, G 2 generates sharper, more structured wave patterns with higher frequency oscillations

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As part of our ablation study, we evaluated the MTopDiv metric, originally designed for general generative models, on time series data (Table 4). The results show high variability in standard

deviations, limiting meaningful comparisons and suggesting that MTopDiv is not well-suited for evaluating time series generative models.

G Evolution of the DMD eigenvalues During Training

In Figures 5, and 6, we plot the imaginary and real parts of the DMD eigenvalues of a 500 sample original and generated time series for datasets ETTh and Sines.

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H Broader Impact

This work aims to advance research in machine learning, particularly in the evaluation of generative models for time series data. Our goal is to improve the reliability and interpretability of such models, promoting the development of more transparent and trustworthy generative systems. We encourage responsible use of these methods and careful consideration of ethical implications in applied domains.

NeurIPS Paper Checklist

The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.

Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:

You should answer [Yes] , [No] , or [NA] .

[NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.

Please provide a short (1-2 sentence) justification right after your answer (even for NA).

The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.

The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While '[Yes] ' is generally preferable to '[No] ', it is perfectly acceptable to answer '[No] ' provided a proper justification is given (e.g., 'error bars are not reported because it would be too computationally expensive' or 'we were unable to find the license for the dataset we used'). In general, answering '[No] ' or '[NA] ' is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.

IMPORTANT, please:

Delete this instruction block, but keep the section heading 'NeurIPS Paper Checklist' ,

Keep the checklist subsection headings, questions/answers and guidelines below.

Do not modify the questions and only use the provided macros for your answers .

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: See the abstract and the introduction, c.f., Section 1.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We discuss the main limitations of our method in Section 5.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate 'Limitations' section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: The proofs were provided in the Appendices.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We released the code.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: We released the code in the supplementary material, we also used public datasets.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We provided s details of the experimental setup, including details about 889 the train/val/test used folds and the values of all hyperparameters.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We run each experiment 10 times, and we released the mean and standard deviation of each value

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer 'Yes' if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: See the experimental setup section and the appendices.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: Our work adheres to the NeurIPS Code of Ethics. We use public datasets and publicly available models and report on the limitations of our work.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We have included a Broader impacts section in the Appendix.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: Our paper does not release any data or models that have a high risk for misuse.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: All datasets, models and repositories were cited appropriately

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: Our paper does not release new assets, only rely on public datasets and models.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: Our paper does not involve crowdsourcing nor research with human subjects, all experiments are performed on publicly available datasets

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: There is no potential risks incurred by study participants.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: No LLM was used in this work for the core methods.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new definition of mode collapse specific to time series and proposes a new geometry-aware metric, DMD-GEN, to detect mode collapse. This metric consistently aligns with traditional generative model evaluation metrics, but requires no additional training, making it more computationally efficient. DMD-GEN allows for a clearer understanding of the preservation of essential time series characteristics by highlighting which modes are poorly captured in the generated data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1.  This paper introduces a new definition of mode collapse for time series data from a view of the underlying dynamic patterns of complex systems, which is novel and interesting.\n2. The proposed method uses optimal transport to measure mode collapse. For the cost matrix, the geodesic distance between subspaces in a Grassmann manifold is calculated based on principal angles, which is theoretically reasonable and interesting. Experiments validate that this design works well. \n\n\nWeaknesses:\n1. The relationship between $d_{DMD}$ and $d_{proj}$ is unclear, in page 6, it doesn't ellaborate the relation between the cost $C$ and $d_{proj}$, but in Algorithm 1, it seems $C =d_{proj}$. Why this geodesic can be seen as a cost in optimal transport and why we can use optimal transport to measure the mode collapse should be clarified.\n2. Experiments in Table 1 only show that DMD-GEN is consistent with other metrics, but no experiment directly shows the advantage of DMD-GEN over others. Moreover, a quantified comparison of computational complexity should be added, although DMD-GEN doesn't require additional training, it needs QR decomposition, SVD operation, and OT computation, all of which are costly when the dataset's dimension is large.\n3. The authors highlight that DMD-GEN allows for a clearer understanding of the preservation of essential time series characteristics and tell which modes are poorly captured. But no experiments prove this; it's better to visualize how DMD-GEN captures these underlying patterns and what patterns or characteristics are captured. Moreover, experimental validation should be supplemented to demonstrate which modes can be identified by the proposed method.\n4. typo: in line 248, the distance in Equation 8, should it be Equation 3? In line 253, it shouldn't be equation 3, but the equation for $\\gamma*$ is unnumbered.\n\nQUESTIONS:\nPlease see weaknesses above to address my concerns about writing, formatting, and experiments.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe manuscript introduces DMD-GEN, a novel, geometry-aware evaluation metric designed to detect and quantify mode collapse in generative models for multivariate time series. Leveraging Dynamic Mode Decomposition (DMD) and Optimal Transport (OT), the authors propose a principled framework that represents time series as subspaces on a Grassmann manifold, enabling interpretable and training-free comparisons between original and generated data. The authors further define a domain-specific notion of mode collapse and demonstrate the metric’s effectiveness across both synthetic and real-world datasets, benchmarking it against TimeGAN, TimeVAE, and DiffusionTS. DMD-GEN shows strong alignment with traditional evaluation metrics while providing unique insights into temporal dynamics.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\t1.\tImportant Problem Focus:\nThe paper targets an underexplored yet important challenge: evaluating generative time series models with respect to mode collapse. Most prior work focuses on image or text modalities, making this contribution highly relevant for time series domains.\n\t2.\tNovel Use of DMD Representations:\nThe authors innovatively leverage DMD to represent dominant dynamic patterns in time series, which are then embedded into a Grassmann manifold. This provides a unique geometric perspective for evaluating similarity across sequences.\n\t3.\tGeometry-Aware Optimal Transport Framework:\nThe metric employs OT over DMD subspace representations, enabling a flexible, structure-preserving comparison between time series. This geometry-aware design is well-grounded in mathematical theory and offers both interpretability and robustness.\n\t4.\tNo Auxiliary Training Required:\nUnlike common metrics that rely on discriminators or predictive models, DMD-GEN does not require training auxiliary networks. This simplifies the evaluation pipeline and avoids introducing confounding factors or additional computational costs.\n\t5.\tComputational Efficiency Governed by SVD:\nThe main computational overhead is from SVD and OT solvers, both of which are well-understood and efficiently implementable. The analysis in Section 3.4 clearly outlines the cost breakdown, showing practical feasibility for real-world use.\n\n⸻\n\nWeaknesses\n\t1.\tInconsistencies in Benchmark Results (Table 1):\nThe DMD-GEN scores reported for TimeGAN differ notably from values previously published using standard datasets and metrics (e.g., Predictive and Discriminative Scores). The manuscript does not explain why these values deviate from existing benchmarks, raising questions about experiment reproducibility or methodological differences.\n\t2.\tAmbiguity in Fig. 1 Comparison:\nFig. 1 presents direct comparisons between DMD eigenvalues of the real and generated datasets. However, such eigenvalues are not necessarily aligned or directly comparable due to differences in mode bases. This appears contradictory to the authors’ argument for comparing subspaces using Grassmannian metrics (Eq. 2), making the choice of this illustration somewhat unclear.\n\t3.\tLimited Novel Insight Derived from Metric:\nWhile DMD-GEN tracks closely with established metrics, the manuscript does not provide examples where DMD-GEN reveals insights that other metrics miss. This diminishes the novelty of the proposed approach. A stronger case would be made by highlighting scenarios where DMD-GEN uniquely detects or interprets mode collapse effects.\n\nQUESTIONS:\nSee above\n\nLIMITATIONS:\nSee above", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the issue of mode collapse in multivariate time series generative models, an area not widely explored before. It introduces a novel method, DMD-GEN, which uses Dynamic Mode Decomposition (DMD) to define temporal modes and quantifies their preservation using optimal transport over Grassmann manifolds. The method is training-free, computationally efficient, and offers interpretability by identifying dynamic patterns that are poorly captured. Experimental results on synthetic and real-world datasets show the proposed method aligns well with traditional evaluation metrics.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\tThe paper introduces the concept of mode collapse in time series, which is novel. The theoretical framework is well-constructed, leveraging DMD modes, Grassmann manifolds, and Wasserstein distance to measure dynamic mode preservation, providing stability guarantees. The method is efficient, training-free, and interpretable, offering valuable insights into which dynamic patterns are lost in generative models.\n\nWeaknesses:\n\tThe main weakness of the paper lies in the experimental section, which lacks depth and diversity. The synthetic data validation is relatively simple, and real-world experiments, such as in Figure 1, show minimal changes in DMD eigenvalues, limiting the metric's sensitivity. Additionally, some design choices, such as the fixed $\\tau =0.95$ and the omission of $\\lambda$ in Table 2, are not well justified. Furthermore, the method's applicability is limited to multivariate time series, excluding important univariate cases.\n\nQUESTIONS:\n1. Parameter Sensitivity: Why is $\\tau =0.95$ used consistently in Section 3.1? Was sensitivity tested for other values (e.g., 0.9, 0.99)? Could a dynamic selection of $\\tau$ improve robustness?\n2. Result Presentation: In Table 1, it will be more clear and readable to clarify whether a high value for each metric is better or worse. In Table 2, given the baseline $\\lambda =50%$, could the results for $\\lambda = 80%$ and $\\lambda = 90%$ be added to validate the performance of the method in extreme mode collapse scenarios?\n3. Feature Discriminability: In Figure 1 (Stock dataset), eigenvalues overlap significantly post-training. Does this suggest that DMD-GEN is insensitive to subtle dynamics, or is this typical for this dataset?\n4. Baseline Comparison: Why were topology-based metrics (e.g., MTopDiv) not compared beyond the supplementary material? Could such metrics provide additional insights into the evaluation of mode collapse?\n\nLIMITATIONS:\nWhile the authors have addressed the limitations regarding univariate time series, further exploration on sensitivity to hyperparameters and real-world scalability would strengthen the paper’s impact.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper bridges a gap between image generation evaluation literature and time series diversity assessment methods. It first introduces a new definition of mode collapse and then proposes a metric called DMD-GEN to detect collapsed modes. The method is subsequently evaluated across multiple time-series datasets.

STRENGTHS AND WEAKNESSES:
Strengths:
- The use of eigendecomposition provides interpretability, which can help uncover patterns in time series data.
- The paper addresses an important gap between image generation metrics and time series evaluation, contributing to an underexplored area in the literature.

Weaknesses:
- Although the paper references image generation evaluation, it does not clearly relate its contributions to that community. For example, recent works such as RKE [1] and FKEA-Vendi [2] also use eigendecomposition for diversity evaluation. Including a comparison, particularly with metrics like RRKE and Projection Distance, would strengthen the paper.
- The paper lacks illustrative toy examples, which could clarify the method and demonstrate alignment with human intuition.

[1] Jalali et al. " An Information-Theoretic Evaluation of Generative Models in Learning Multi-modal Distributions", in NeurIPS 2023

[2] Ospanov et al. "Towards a Scalable Reference-Free Evaluation of Generative Models", in NeurIPS 2024

QUESTIONS:
- How many samples are required for DMD-GEN to converge reliably?
- Can the authors visualize the samples corresponding to the top eigenvalues and highlight the resulting clusters to better demonstrate the interpretability of the method?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03