A Geometry-Aware Metric for Mode Collapse in Time Series Generative Models
Authors: Yassine ABBAHADDOU
Amine M. Aboussalah
Venue: NeurIPS 2025 poster ()
Keywords: Time series
Generative models
Mode Collapse
Primary area: deep_learning

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
Yassine Abbahaddou*
Amine Mohamed Aboussalah'
LIX. École Polytechnique
Department of Finance and Risk Engineering
Institut Polytechnique de Paris yassine. que edu
Tandon School of Engineering
New York University
ama10288@nyu.edt
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, 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 measure discrepancies in underlying dynamics. By representing the subspaces spanned by the DMD eigenvectors point structures Grassmann manifold, and comparing them via Wasserstein distances computed from principal angles, DMD-GEN enables principled geometric comparison between real and generated sequences. The metric is efficient, requiring no additional training, supports mini- batch evaluation, and 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 TimeGAN, VAE, and DiffusionTS show that DMD-GEB -EN aligns with existing metrics while providing first principled framework for detecting and interpreting mode collapse in time series. Our code is available at: here
datasets using
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 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 broad spectrum of variations [ 6 40). An illustration of this challenge can be seen in GANs, which often experience mode collapse, phenomenon where the generator focuses limited subset of the data distribution, leading the production of repetitive
meaningful latent representations of data [17 23
38
41
45
or similar samples rather than capturing the full diversity the training data B. 4. [5] 42). VAEs
also face phenomenon called posterior collapse, where the model tends to generate outputs that similar or indistinguishable for different inputs. This limitation reduces the model's ability to produce
"Equal contribution.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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diverse samples 18 [6].
Diffusion models, while generally robust against mode collapse compared 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 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 dieedeeendentt 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 a generative model preserves the diversity is 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 sime-siries-ppecific definition of a mode. Our approach based on Dynamic Mode Decomposition (DMD) [52] a spectral method that identifies dominant coherent structures in temporal dynamics. This allows to develop new metric that is interpretable, training-free -fre,, 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 new definition of mode collapse specifically for time series data, leveraging DMD to capture and analyze coherent dynamic patterns.
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.
Development of DMD-GEN Metric:
Enhanced Interpretability: The DMD-GEN metric provides increased interpretability by decomposing the underlying dynamics into distinct modes, allowing for a clearer under- standing 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 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 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 determine the beginning and end of a specific evolving pattern (mode). (ūi) 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 nigh-dimensional spaces, this can cause 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. 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) [77, address temporal dependencies by aligning sequences to minimize distance. However, if fails to capture the underlying modes or coherent dynamic patterns present the data. This inability to recognize and preserve the essential modes means DTW falls short in assessing mode collapse.
data rarely exhibit single clean pattern 226 30

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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 whole. However, the custom nottallomaiizaionn 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) [3], the model introduces an approach 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 nonlinear dynamical system directly from data, without requiring prior knowledge of the system. Given a dynamical system xx()) = fx((t), t: p)) where x(t) E R denotes the n-dimensional state vector, μ R" represents the system parameters, and f R" x R x R + R" defines the dynamics, DMD approximates the system locally by Ax, 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 '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 discrete-time approximation of the dynamical system. Given x : E IR + x(t) E R",
we collect m consecutive snapshots to construct two data matrices X, X' E Ruxm defined as
Xg
X1
X
X'
X1
X2
Xm
These snapshots are taken with a time-step At small enough to capture the highest frequencies in the system's 's dynamics, i.c., Vk E N, X₃ x(k:At). Assuming uniform sampling time, we approximate the dynamical system linearly as X & A*x,, where A E R=*"" is the best-fit operator, i.e., A+ arg minA ||X' - AX| 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( (AA At); 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 finiteddimensional approximation of the infiniteddimension. 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 lincarizing nonlinear dynamics, allowing DMD to capture coherent spatiotemporal structures and their evolution through the system's 's spectrum
X'Xi, where ||||| the Frobenius norm and X is
12 34 35| By analyzing the eigenvalues = diagg(A....... Ar) and the corresponding eigenvectors $
Φ ....... ................................ $,1 Ra of the DMD operator A where denotes the rank of X, can capture the dominant dynamic patterns that govern the system's '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 the magnitude of the eigenvalues, i.e., |λ]] 2... ....21. |-rl-
Throughout this paper, we distinguish between two related quantities: 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, A < denotes the number of dominant modes retained for reduced-ordes 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& (X) refer to the & leading DMD modes.
For a high-dimensional state vector x E TR. the matrix A comprises n² elements, making its representation and spectral decomposition computationally challenging. To address this, apply
3

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dimensionality reduction to efficiently compute the dominant eigenvalues and eigenvectors of A by constructing reduced-order approximation A Rexe. The DMD approximation at each time step can be expressed as follows,
11
Xx
03 AA
where Φ2 are the DMD modes (eigenvectors of the matrix A). X3 are the corresponding DMD eigenvalues, and by denotes the mode amplitude, given by b b T in matrix form. The detailed steps for this process are provided in the Appendix A.3 DMD modes can be interpreted basis vectors spanning a subspace that captures coherent spatiotemporal patterns among the components of x().. It decomposes a complex time series into collection of simpler, coherent modes, where each of them captures a specific aspect of the system's 's behavior, e.g., an oscillation, an exponential growth/decay, traveling wave [58). The DMD modes spatial fields that often reveal coherent structures within the flow. These structures are fully characterized by the DMD eigenvalues A and eigenvectors 4, which respectively encode the temporal frequencies and spatial patterns of the underlying dynamics. Specifically, the imaginary part of the cigenvalues A determines the oscillation frequency, while the real part indicates the rate of exponential growth or decay [TO] [54]. The corresponding DMD eigenvectors & capture the spatial coherence associated with each mode, providing an interpretable link between the system's 's temporal evolution and its spatial distribution.
3 Quantifying Mode Collapse in Time Series
Notations. We denote by G generative model specific to time series. This model is trained on dataset comprising N time series, each with fixed length, represented (X.) During the inference phase, we synthetically generate set of N time series, denoted as { using G. Both the original and generated time series are assumed have consistent length, denoted as and dimensionality, represented as n. Formally, for any pair of indices and j. the original and generated time series X; and X, are elements of the Euclidean space nx
3.1 Defining Temporal Modes
In this section, formalize the notion of temporal modes. Definition 3.1 captures the essence of mode in terms of the dominant cigenvalues and cigenvectors the DMD operator, highlighting the significant dynamic structures in the time series data.
Definition 3.1 (Temporal Modes). Given time series X [x1, - X E Rexn, define the set of temporal modes M& (X) as the & dominant eigenvectors {Øi,.... On of the associated DMD operator. These capture the primary dynamic patterns in the time series. We represent My(X)) as,
M&(X)
Φ1
Ø2
Ø&
R
Selecting number retained modes involves classical bias-variance trade-off: smaller yields robustness to noise but may underrepresent the full system dynamics, whereas larger & enables more accurate reconstruction at the risk of overfitting. To formalize this trade-off, let r = rank $ denote the rank of Φ, i.c., the concatenation of all DMD eigenmodes. The approximation error The exact expression for this error (X)ll
then be quantified using the Frobenius norm term is provided in Proposition 3.2
Proposition 3.2 CekkartYoung-Miinng
or be the sin- gular values of 4. Then the DMD eigenmodes M&(X) that uses the first k. modes satisfies,
theorem [14], Let σ1 2 02
$ MM +.. 2
1/2
common practical guideline is to select the smallest such that a prescribed proportion 7 E [0. o3. 0 In practice, typical choice is 0.95.
of the total energy is retained, i.c., Σ/= of E5=

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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; and Xj. 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 [ the dominant modes can be identified from the eigenvectors associated with the largest DMD eigenvalues. Consequently, we focus comparing the similarity between the corresponding temporal mode subspaces, M& (X) and Mr (X). However, directly measuring the distance between these two subspaces is mathematically challenging, as their bases are not
necessarily aligned. Although M and M& (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 n-dimensional vector space The Grassmann manifold Gr(k, n) is the set of all di-mmnnsionnal subspaces of V, where I k S n. Mathematically, it can be expressed as Gr(k, n) = (W f V dim(w) k).
The Riemannian distance between two subspaces 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.
Grassmann manifold is defined the length
Definition 3.4 (Principal Angles Between Temporal Modes). Let the columns of .M ( (X) and
Mr (X) represent two linear subspaces U and V, respectively. The principal angles 0 s 01 S S
0, S 7/2 between the two subspaces are defined recursively as follows:
{
uJTT --u=uu v-v=1 u u4 i 0,
cos O2 max wEU
max u VEV
8.l.
The work of [8] has shown that the principal angles can be efficiently computed via the Singular Value Decomposition (SVD) of Q
T Q, where QR and QR denote the QR factorizations of Mı and V M&(X (X), respectively. The SVD of Q Q can then be written as Q T Q U anng Eang V ang where Uang and V ang are orthogonal matrices containing the left and right singular vectors, respectively, and Lang is diagonal matrix whose entries correspond to the singular values associated with the principal angles between the two subspaces. If s denotes the rank of aang. then the principal angles correspond to the arccosine of the first singular values of Lang. i.c., + diag (cos 0i...., cos a.). Following (7). these principal angles can be used to define several Grassmannian metrics. One such metric the projection distance, defined as the Frobenius norm of the matrix sin O i.e.
1/2
dproj (Ma(X), = [[sin Σ sin? 2 Oz
The projection distance serves as similarity metric between the temporal mode subspaces (x) and (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, . n/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 33. 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 [xi....., xe] E RIxE and X
X XI..... x E Raxx be two sequences of state snapshots. Assume that both X and X admis
5

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DMD representation with the same initial condition Xa- Let Mx E Cuxk and (x(X)= E Caxk denote the respective DMD mode matrices, associated with diagonal eigenvalue matrices A, A E Ckxk, Then, for all time steps I, the reconstructed states satisfy
X = MM) A M&(X)*xo.
X; =M=Ma(X) (x(Xxxx.
Let E, denote the difference in dynamics between X and X, i.e., Vt, X; x E, Then, for all time steps t, the projection distance between the corresponding temporal mode subspaces satisfies dproj (Ma Mx, <
[. where ó, denotes the spectral gap of A and -| || represents
hs
the Frobenius norm. The proof of Theorem 3.5 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 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 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 given set of L sampled batches of real time series X' = {X,, X...... x
and generated time series = [X1,Xx X Xr). modes {M& (X:)) and {M& (X,)) which encapsulate the dominant temporal dynamics of each sequence. We then construct a cost matrix C, where each entry Cij quantifies the dissimilarity
we compute the corresponding sets DMD)
between the mode subspaces M (X;) and M&(X;) using a principal-angle-bae -ngle- metric. The OT problem is subsequently solved to obtain the transport plan Y that minimizes the total transportation cost, therefore identifying the optimal mapping between real and generated modes via the Wasserstein distance defined as follows:
(X, X) = Eis ( (Ma().) Mr (,(( = EE. min (Y, C%p YEN
(4)
where p denotes the order of the Wasserstein distance, II represents the set of all joint probability distributions, and )) denotes the p-order cost-weighted inner product used to compute the transport cost. The resulting distance provides robust measure of mode collapse: 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 7 in Equation 4is defined formally in Theorem
Theorem 3.6 (DMD Mode Geodesic). Let Mr M& E B k be matrices whose columns form orthonormal bases of two k-dimensional subspaces of IR".. Let = diag(0 O2..... 0g) denote the diagonal matrix of principal angles between the subspaces spanned by MM (X) and M& (X).
Further, let A E g be orthonormal matrix such that M& (X) MR(X] cos ( A (i). Then, the geodesic connecting M₁ (X) and M₁(28 on the Grassmann manifold Gr(k, n) is given by ((t)= = My(X cos(t@) + A sin(, for t E [0, 1], and the length of this geodesic corresponds exactly the projection distance:
1/2
ápros (M(I
0
(5)
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 dproo in Equation 5 is positively correlated with dproj in Equation 3 since all principal angles lie within the
interval [0, म 2/2.. This approach provides a quantitative and interpretable framework for evaluating the performance of generative models time series data. We approximate the metric in Equation

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Algorithm Computation of DMD-GEN Metric
Input: Number of batches B
Initialize: dDMD <- 0
foreach / 1...c.. 1..... B do
Sample batch of real time series generated time series X.
foreach X; X' do
foreach X, E , 2 do
Step Extract temporal modes MMx(X.) from X,,
Step 2. Extract temporal modes Ms (X,) from X,.
Step 3. Compute orthonormal bases QR factorization:
Mx(X,) Q.R,, Mx(X,)) Q R,
Step 4. Compute principal angles from the SVD:
Q Q U ang c lang-
where cos( = diag(cos OI...., cos 0,) and the principal angles are given by 0,- = cos (ar).
with {ox) being the singular values of Q Q
Step 5. Compute dissimilarity entry:
Ci = ( (M,,
end foreach
end foreach
Update metric estimate:
dDMD + dDMD minyeu'y, C)p-
end foreach
Output: dDMD
using the law large numbers, and the detailed computational procedure is outlined Algorithm T The values of the optimal transport matrix " arg min 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 [ Let B be the number of batches, S₁ be the batch size, be the data dimensionality, m be the time series length, & be the number of modes, and be the OT solver precision. We assume m 2 k. The total complexity is driven by the B outer loop iterations. In each iteration, we compute an S, x S, cost matrix and solve the Optimal Transport (OT) problem. Computing C requires S pairwise comparisons. The ) is dominated by the two DMD extractions (Steps 1-2). which, as detailed
cost for each pair (X,, X,, in Appendix A.3 (Algorithm 2). have complexity of DMD O(nmk + nuk2 mkk2 k-)). This cost is higher than the subsequent QR factorization (Step 3: O(nk²)) and geodesic computation (Step Okk2n k*)). After building the matrix, solving the OT problem with Sinkhorn's algorithm [44] costs (SS Therefore, the cost per batch is O TDMD + /c). The total complexity for B batches is: O(B S7. (nmk + nk? + mk? k3 + +/c))-
4 Experiments
We evaluate the diversity of generative models across one synthetic dataset and three real-world datasets. The statistics of each dataset well as the baselines metrics can be found Appendix D
Consistency of DMD-GEN with Established Metries. Table I demonstrates that DMD-GEN produces results consistent with established evaluation metrics such as the Predictive Score, Dis- criminative 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 their performance. A key advantage of DMD-GEN, however, lies in its efficiency: unlike other metrics, 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
7

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Table 1: Comparison of generative model performance across multiple time series datasets using four evaluation metrics. Highlighted values indicate the best performance for each dataset. All metries, including DMD-GEN, consistently identify the same best performing model, demonstrating strong agreement among evaluation methods. The symbol "_' denotes cases where computation failed or was not applicable.
Metric
Model
Sines
ETTh
Stock
Energy
TimeGAN
0.03
0.20=a.00
0.08
0.27= 04
Disc. Score
TimcVAE
0.33 oz
0.50+0.00
0.50=0.00
0.50=0 00
DitfusionTS
0.0210.01
0.5010.00
0.50.10.00
0.5011
TimeGAN
0.09 oo
12.39
6.40
24.01::0 00
Prod. Score
TAmcVAE
0.12 on
13.03+0.03
27.12=0.57
24.61= 06
DiffusionTS
0.09:0000
13. 18+0.00
17.78=0.08
24.49 07
TimeGAN
0.04= 01
0.40=
11. 11.92+1.05
Context-FID
VAE
5.0111.04
12. 12.22.1115
135 222.22222222
DitfusionT'
0.01
11.65:a0.76
127 (77.002413.00
TimeGAN
33.9111.75
20.9611.10
0.7310.1
44.57:17:4
DMD-GEN
TimeVAE
31.65=00551 51
98.91
4.02=0.08
164.48+0.44
DiffusionTS
29.6620.34
105.46+0.52
13.62:2.53
150.67:20.0 07.7
to time series requires flattening the sequences into independent samples, thereby discarding the essential temporal structure and feature dependencies. Nonetheless, for completeness, conducted experiments using Mio on the Energy, ETTh, and Sines datasets to compare the performance of Time 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 MivoDDv is not 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 eval- uation 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, 91 and G2, defined in Appendix E Time series produced by the same generator are considered belong to the same mode. The generator is chosen using Bernoulli random variable with parameter A [o, 1]: G1 is selected if A < X and G2 otherwise. The reference value Axef 0.5 corresponds a balanced mixture, where both modes are equally likely (co-exist) and no collapse occurs. We denote the resulting dataset by Dxx (A).
For each metric m, the non-collapse case is given by m(Dx (Awet), Dx (Ara)),, while the collapsed cases correspond to (D (Nret), D,v (A)) for À & Avet- Since the metrics have different numerical ranges, compare them using normalized performance measure:
m(D Nv (Aret), DN
(A) = mD (Xret). (>eee)
Table 2 reports Perf(A) for several metrics across different mode collapse levels. We observe that the benchmark metrics exhibit large fluctuations in both magnitude and sign as À varies, indicating sensitivity changes mode balance. In contrast, DMD-GEN remains stable and robust, effectively detecting even minor collapses. Both DMD-GEN and Context-FID increase rapidly as X deviates from Ames 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, T /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 iillustrates the evolution of each evaluation metric the mode collapse severity À 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 gradual mode imbalances. These results confirm that DMD-GEN offers both sensitivity and robustness in detecting varying degrees of mode collapse.
8

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Table 2: Relative performance of evaluation metrics in detecting increasing levels of synthetic mode collapse. DMD-GEN and Context- demonstrate strong sensitivity to collapse while maintaining stable behavior across severity levels.
Mode Collapse Severity (A)
Metric
10%
20%
30%
40%
60%
70%
Discr, Score
+586.79%
+4413.40%
+181.13%
-20.75%
-16.98%
+143.40%
Pred. Score
-0.54%
-0.71%
-0.83%
.0.40%
+0.35%
+0.54%
Context-FI FID
+3,,796 796.45%
+18,394.644%
+8,210.25%
+1,874.76%
+15855888%
+7.019.51%
DMD-GEN
+681.03%
+477.76%
+312.22%
+115.02%
+114.92%
1814.18%
Mode Dominates
Balanced (Both Modes Exist)
Mode 2 Dominates
Disc. Score
600
0.5
Mode
400
Mode
0.0
+
200
Mode
Mode
-0.5
Context-FF
DMD-GEN
30000
Metric
10000
200
0
10
20
30
40
50
60
70
10
20
30
40
50
60
70
Collapse Severity
Mode Collapse Severity
Comparison of metric sensitivity to varying mode collapse severity levels À in the synthetic dataset. Shaded regions indicate when one mode dominates (blue or red) versus when both modes coexist (gray). DMD-GEN and Context-FID show clear monotonic trends with X, effectively distinguishing increasing collapse severity, whereas Predictive and Discriminative Scores fluctuate, indicating lower robustness to changes in mode balance.
Figure 1:
Assessing DMD-GEN on Bootstrapped Time Series.
Stock Dataset
Previous experiments focused primarily deep learning- based generative models. To further validate the versatility of DMD-GEN, evaluate it on time series generated through the classical non parametric Moving Block Boot- strap (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 (introduc- ing 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 pat- terns are maintained. As shown in Figure 2 increasing the block size consistently reduces OMD-GEN distance and stabilizes its variability, indicating that larger blocks bet- ter preserve the underlying temporal dynamics, whereas smaller blocks tend to distort them through excessive re- sampling noise
0.18
Moan Std Dev
DMD-GEN
0.16
0.14
0.12
5.0
Block Size
Figure 2: Mean DMD-GEN distance as a function of bootstrap block size for the Moving Block Bootstrap (MBB) ex- periment. Error bars denote standard deviations across trials. As block size increases, the DMD-GEN distance de- creases and stabilizes, indicating im- proved preservation of temporal dynam- ics.

---

Original Data Sureples
Centroted
Dencoiia Strnned
Creniig Lianer
1.0
100
Original Samples
Groensted Tatee Sampless
Becogin Apxwee
Greniing Lawwer
Time Step
Tans Step
Geneented Data Samples
Docerios Charse
1.35
0.50
035
-225
-2.00
Stop
100
49
Stop
Figure 3: Visual interpretability of DMD-GEN across three representative cases: (top) Complete Mode Collapse, (middle) Dynamic Frequency Mismatch, and (bottom) Spurious Mode Injection. Each row illustrates how DMD- GEN responds to progressively more complex distortions in temporal structure, assigning higher distances to collapsed or over-generatee modes and moderate, interpretable values frequency mismatches. These examples demonstrate DMD-GEN's ability to quantify both the degree and the nature of dynamical discrepancies, with the following measured scores: Complete Mode Collapse 0.7708, Dynamic Frequency Mismatch 0.6833, and Spurious Mode Injection 0.9114.
Interpreting Mode Behavior Through the DMD Spectrum. Figure illustrates how the DMD spectrum reveals interpretable changes in system dynamics. In the top row, mode collapse concentrates dynamical activity into 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 transparent view of how temporal structures deform across different dynamic perturbations.
Conclusions and Limitations
We introduced DMD-GEN, 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-GBR requires no additional training, making it computationally efficient and easy to apply. Its mode-basee formulation also enhances interpretability by showing how key dynamical patterns are preserved or distorted in generated sequences. A current limitation is that DMD provides only linear approximation nonlinear dynamics: although Koopman theory allows such systems to be represented by infiite-dimeesiioall linear operator, practical approximations rely on finite number of modes. Capturing stronger nonlinearities would therefore require expanding this set, increasing computational cost and reducing efficiency.
metric for evaluating generative models of time series and detecting
10

---

Acknowledgments
The authors thank Kritaporn (Lune) Nitjaphanich for insightful discussions that helped shape this work.
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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 (t) ((x()),t;1)) we linearly approximate the dynamics using DMD using the operator A E mx i.e.
Vt. (t) Ax.
Discretizing time into intervals of At and capturing snapshots accordingly, we establish the relation- ship between consecutive time steps in the following equation:
Vk, Xk+1 = X + Ax = (I + AtA) X
(6)
For time-step At that is sufficiently small, we can employ the first-order Taylor expansion of the matrix exp( At)), expressed as:
erp(At. R= + AtA
Therefore, from Equations 6 and conclude that:
Vk, X eap( (LAA)x-
Thus,
A ~ etp(AtA).
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 DMD expansion
We prove the closed formula Ak,
Xk = - oiAlbb + Ab, using recursion.
For = 0, we have,
Xo Ixo
$
4b
Let's now consider the equation hold for k = O,... m. we have,
Xk+1 A 'x
A
PAAkb
-AK-Ib.
Therefore, the equality holds for all & N.
15

---

Algorithm 2 Dynamic Mode Decomposition
From collected snapshots the system, build pair of data matrices (X, X').
X
X
X2
Xm
The closed formula of optimal DMD operator is
X'x''
Compute the compact singular value dccomposition (SVD) of X:
X UEV
where U E COXT, ME CTXT, V Cm and I min(m, n) is the rank of X. Therefore,
A = u'v
Define a matrix
= =
since U is a unitary matrix.
À TR" defines low-dimensional linear model of the dynamical system on proper orthogonal decomposition (POD) coordinates.
Compute the eigen-decomposition of A:
ÁW WA,
where columns of W x eigenvectors and A diag(Ar...... E R is diagonal matrix containing the corresponding eigenvalues.
5.
Return DMD modes +:
$ X'VE "W. Each column of + is eigenvector of A meaning DMD mode corresponding eigenvalue A
B Proof of Theorem 3.6 DMD Mode Geodesic
Theorem 3.6 [MD Mode Geodesic]
Let Mx pa be matrices whose columns form orthonormal bases of k- dimensional subspaces of R Let @ (ii, .z...., 0s) be the diagonal matrix of principal angles between the subspaces spanned by (X) and (XX).. Let A R be an orthonormal matrix such that
My X) = MM XX) cos(O) + A sin(O).
Then, geodesic linking M&(X) and M& (X on the Grassmann manifold Gr(k, n)) given by
(t) Mr (X) (19 + A sin(10), for [0, 1},
and the length of this geodesic corresponds exactly to the projection distance defined by
1/2
Tpooj (Mx())
Σ
03
Proof. Preliminaries and Definitions
1. Grassmann Manifold Gr(k, m):: The set of all k-dimensional linear subspaces of R".
2. Orthonormal Bases: For k-dimensional subspace S C RRa orthonormal basis is
represented by an n & matrix Q with columns satisfying Q Q = Tk, where Te is the x identity matrix.
16

---

3. Principal Angles and Vectors: Given two subspaces S₁ and S2 with orthonormal bases Qi and Q2. the principal angles recursively by
0, S 02 I 0g. S between them are defined
cOs(O,) max max u v'v, subject to u uu, 0, v v, = =0 j = ,.. 1 (8) WESt vESa |
4. Projection Distance: The projection distance between S1 and S2 is defined as
1/2
S2)
Σ
Computation of the Principal Angles
Let Q1 (X) and Q2 (X). Both Qı and Q2 are n x k matrices with orthonormal columns.
We construct the matrix C follows:
C Q Q2 E R
(10)
Since Q I and Q Ig. C captures the pairwise inner products between the basis vectors of Q1 and Q2.
We then perform the Singular Value Decomposition (SVD) of C:
C= -VVV
(11)
where
U.V V E Rkxk are orthogonal matrices, i.c., U = V =
diag(o) 01, o2...., σK) with a; 2 0.
The singular values o, of C' are the cosines of the principal angles between the subspaces:
o, cos((,)) (,), 0g [0, 1/2, i -,....,k k.
(12)
This result stems from the fact that the SVD aligns the basis vectors U and V to maximize the projections in the directions of the principal angles, which correspond to the largest cosines.
Since principal angles 0₂ are defined in the range [0, π 4/2], 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 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:
A QUQ, B= Q2V.
Verification of Orthonormality:
A A = -Q = UUU U UU= I
BT B (022))(Q22) = = Q2 =V VVV= =
We then compute A B:
A A
J
-U = 11E4=

---

Thus, A Σ = (iag)cos(O1)...... cos(
3. Decomposition of B in Terms of A and A
We aim to express B as a linear combination of A and another orthonormal matrix that is orthogonal to A.
Let us define A:
A = (B- - A (Os( sin(
(13)
where cos(@)= = Σ and ()) = diag(sin(0))) ................................ sin( and sin(O)-1 denotes the diagonal matrix with entries sin(0.)
Verification that A is Orthogonal to A:
AT A
A (B cos( sin(
(A B - A A cos( sin(
(E 1 cos( (i)
(cos cos( sin( 0.
Verification that is Orthonormal:
First, we compute A ATA: A:
T A = ((B - (O)) (in ((B - A cos( sin(9)
-in(O)- - A cos ()) (B - A cos(O)) sin
We compute the inner term:
(B - co)) (B-Aco(())) =BB cos [ [B
A cos(())
= =B B - B A Aco((e) - cos (os())A) BB
+ cos( cos(e).
Since A Ix, BT B Ix. and AT A E cos(O) TO) () BT
(B- ==occo(O)) (B- A coss()))
1 cos(O) cos(t) cos(O) cos(O)
+ cosl ()) cos(@)
= 1 - -cos² (0) cos2 (() cos (
= T cos2 cOs)(
(e).
Since: (i) + cos2(() Ig.
Thus,
A sin( sin)(() sin( (Ix).(I&)) It.
Therefore, A is orthonormal.
Expressing B in Terms of and A:
Using Equation 13 we have:
B = Aco((O) AO).
(14)
4. Define the Geodesic Path
On the Grassmann manifold, the geodesic 7() from A to is given by: 7(1) A cos(10) + Aii((1O), E 10.
(15)
18

---

Verification of Endpoints:
At / 0:
7(0) A cos(0 6) + A sin(0 6) = AI + A 0 = A.
At 1:
7(1) A cos(O) + A sin(@) B.
(16)
Thus, 7t) is continuous path on Gr(k, n) connecting A and B. Relate Back to Original Bases:
Recall that A = QUU MM and B Q2V - MM()VV
Since and are orthogonal matrices, the subspaces spanned by Q1 and A. and by Q2 and B, are identical. Therefore, we can express the geodesic in terms of M2 (X) and A.
Expressing the Geodesic in Original Terms:
Let us redefine A accordingly to absorb U and V. so that we can write:
(() Ma (X) (IS + A (in(.
5. Compute the Length the Geodesic
The length L of the geodesic y(t) is given by:
L = S. [|](1)] dt,
(17)
where I denotes the Frobenius norm.
Compute the Derivative +(): Since (t) MM (X) cos (1) + A sin() we have:
+() -M (in())(()))))))))))))))))))))))) + AO cos(1e),
where we used the fact that the derivative of cos(t@) with respect to f - sin(tO), and similarly for sin(
Compute the Squared Norm 1+(0)
IK(1))1 Tr (6(()T<(t))
Tr (TMM((((X)))i))(() + cOco (-Ma(X)(ssn((9)) A0 cos(t))))
=Tr (Or (1o) Mx + cos ( A sin(10) (os( ATM)
Since (x = Th. A In. and M& [X] T 0, the cross terms vanish, and we
have:
1H011 = Tr (0² (sin + cos (10)(x))
= =Tr It21,)
Σ R.
1=1
19

---

Compute the Length L:
Since [|+(()]l is constant with respect I, we have:
l+(t)ll L.' dt
[+(t)) F dt
0
1/2
0
1
i=1
1/2
0
6. Length Equals the Projection Distance
Comparing the computed length L with the projection distance defined in Equation 9 we find:
h
1/2
L (M =
0,
(18)
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 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 s norm is constant, the total length of the geodesic over the interval E [o, 1] is precisely the projection distance.
Therefore, the length the geodesic (t) connecting Ma (x and Mx 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
[xx,...,xe] E Rx and X [X,,.... Xy] E Rexn be two sequences of state snapshots. Suppose that both X and X with M& (X) Bank and Ma(X) E B as the respective DMD eigenvectors, and À as the respective DMD eigenvectors, and admit DMD form with the same initial condition Xo Xo. i.c.
Vt, x (&)
Xo: X, Mc Mxx
Let E, be the difference in dynamics between X and X, i.c. Vt, X, X, = Eg We have, || where & is the spectral gap of A and ||-ll p the Frobenius be
Vt, (((()) < Norm.
Proof. Let X = [x],.... rtl.
X [e,..., ūz] E R and denote their k-dominant DMD Q = Me(X) E Bank We define the time-/ linear propagators
bases by Q = (X) E Bank that generate the snapshots
4; QN'! Āー so that It = A,No, 3, = A,zo-
Because is arbitrary, identify the perturbation matrix A, A. E..
20

---

Wedin's 's theorem for diagonalizable matrices states that if E perturbs a matrix A whose spectrum splits into two clusters separated by gap 6, then || sin Ø | where θ collects the principal angles between the invariant subspaces associated with the chosen spectral clusters.
Applying Wedin with A = A,, E Es,, and the dominant invariant subspace span(Q), the gap is 5 5, The left-hand side is exactly the Grassmann projection distance dproj Q) = || sin θ F-
Hence dproj (Q,
Frlle which the desired bound.
D Datasets and Implementation Details
D.I Basic Statisties on the Datasets
Sine waves. We generated synthetic dataset consisting of two sets of sine waves to represent bimodal distributed data. The data were generated using the following formula:
yt) A sin + 6),
(19)
where A is the amplitude. of is the frequency, t is the time variable and e is the phase angle of the sine wave. Each mode consists of 2000 samples with phases being randomly chosen between 0 and 2T. For all the samples, the duration is 2 seconds and the sample rate is 12, making the length of each sequence be 24. = 0.5 and f = 1 Hz. for the first mode, and = 5 and f = 0.5 Hz for the second mode.
Stock price. To test our framework on complex multimodal dataset, we used Google stocks data from 2004 to 2019, which was used in [60]. The data consists of 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].
consisting 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 February 2005 and consists of 28 features. Unlike the previous datasets, this one has an unimodal distribution. The data into several sequences of length 7.
Energy. We conducted experiments on UCT's air quality dataset (55
Electricity Transformer Temperature and humidity (ETTh). The ETTh dataset focuses on temperature and humidity data from electricity transformers 622 includes 2 years of data an hourly granularity, providing detailed temporal information about transformer conditions.
Table provides 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.
Table 3: Statistics of the four datasets used in our experiments.
Dataset
Sine
Stock
Energy
ETTh
#Samples
10,000
3,773
19,711
17,420
Dimension
28
8
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 model the generated data and assessing its performance on a real dataset. A lower predictive score indicates that the generated data contains patterns that more representative of the temporal patterns found in the original data.
Discriminative Score. 660 The discriminative score measures the similarity between real and generated time series data by training binary classifier to distinguish between them.
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Contextual Frechet Inception Distance (context-FID). [22 Context-FID is an adaptation of the Frechet Inception Distance (FID), metric used to assess the quality of images created by a generative model [19] For time series, context-FIID 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 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 TimeGAN and VAF
E Synthetic Generators
To evaluate the ability of DMD-GER to detect Mode Collapse, we generate synthetic time series using two parametric functions, denoted 91 and G2. 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,
x cos ((c + 2.3) t) | x e [-5, 5], t E [0. (a, b, c) ~ u
9 (t,z) + cosh(æ + b 3)
+ a cosh(x)
G2- = (t,z) H
x tanh(x) x sin ((2.8 + b) f) æ E [-5,5], 5], & E [0, 4T], (a, ~ -
where U4 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 illlustraees examples of time series generated using 91 and G2. Generator 91 produces smooth, localized wave patterns with oscillations that gradually decay in space, resulting in broader and less frequent peaks over time. In contrast, G2 generates sharper, more structured wave patterns with higher frequency oscillations
X1 - 91
X2 ~ 92
12
035
12
033
05
005
Time
0.85
-2.
-0.10
Space
Space
Figure 4: Examples of time series generated using the generators G1 and G2.
F Comparaison of Generative Models Using MTopDiv Metric
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
http::/myydmd..tttbbbii///DDD
http://ghthhiccom////duussssssst TS
hutp://gthbbboommzzawwwww/n/n/nt
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deviations, limiting meaningful comparisons and suggesting that MTToDivv is not well-suited for evaluating time series generative models.
Dataset
Time VAE
TimeGAN
DiffusionTS
Energy
424.34 +19.75
467.35 +-49.10
402.42 +
ETTh
116.23 +44.6
130.85 +88.0
116.51 +:5.52
Sines
7.72 +0018
4.92++0.22
4.53 +0.13
Table 4: Comparison of different models on various datasets.
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.
Eael K06
Figure 5: Comparison DMD Eigenvalues between Original and Generated Time Series for DiffusionTS through Epochs on the dataset ETTh.
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Tou 1956
Figure 6: Comparison of DMD Eigenvalues between Original and Generated Time Series for DiffusionTS through Epochs on the dataset Sines.
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 improve the reliability and interpretability of such models, promoting the development of more transparent and trustworthy generative systems. We encourage responsible of these methods and careful consideration of ethical implications in applied domains.
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Reviews:
[{"review_id": "rAuxYHsMhh", "reviewer": "Reviewer_CvCr", "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.", "questions": "- How many samples are required for DMD-GEN to converge reliably?\n- 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", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The use of eigendecomposition provides interpretability, which can help uncover patterns in time series data.\n- The paper addresses an important gap between image generation metrics and time series evaluation, contributing to an underexplored area in the literature.\n\nWeaknesses:\n- 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.\n- The paper lacks illustrative toy examples, which could clarify the method and demonstrate alignment with human intuition.\n\n[1] Jalali et al. \" An Information-Theoretic Evaluation of Generative Models in Learning Multi-modal Distributions\", in NeurIPS 2023\n\n[2] Ospanov et al. \"Towards a Scalable Reference-Free Evaluation of Generative Models\", in NeurIPS 2024", "was_revised": true, "final_justification": "The authors have addressed my concerns regarding the literature on image generation evaluation metrics, sample convergence, and the interpretability of their method. I have no further concerns and maintain my positive score.", "rebuttal": "We sincerely thank Reviewer CvCr for their positive assessment and insightful feedback. We are encouraged that the reviewer found our work to be a technically solid contribution that addresses an important and underexplored area. We address the reviewer's valuable suggestions below.\n\n**[Weakness 1] Relation to Image Generation Evaluation Metrics.**\nWe thank the reviewer for recommending these references. While our main focus is on evaluating temporal generative models, we agree that incorporating a discussion of image-based diversity metrics such as RKE and FKEA-Vendi would enhance our work and helps situate our contribution within the broader context of diversity evaluation. In the revised manuscript, we have added a discussion of these methods in our related work section (Section 2).\n\n**Conceptual Distinction (FKEA-Vendi):** A key distinction is that FKEA-Vendi is a reference-free metric, designed to measure the intrinsic diversity of a generated set. While valuable, our work specifically targets mode collapse, a phenomenon that, by definition, requires a comparison between the generated samples and the ground-truth data distribution. Therefore, a reference-based metric like DMD-GEN is not only appropriate but essential for detecting and identifying mode collapse, as it directly quantifies how well the diversity of the original data is preserved.\n\n**Experimental Comparison (RRKE):** To directly address the reviewer’s request, we conducted new experiments comparing DMD-GEN with RRKE. To apply RRKE to time series, we flattened each multivariate sample into a single vector. While this process unfortunately discards the temporal structure that DMD-GEN is designed to analyze, it provides a valuable point of comparison. As shown in the table below, the model rankings produced by RRKE are consistent with DMD-GEN on the ETTh and Sines datasets. This consistency with an established eigendecomposition-based metric further validates our approach. We will include the complete RRKE results for all datasets in the final manuscript.\n\n\n| **Model** | **RRKE (ETTh Dataset)** | **RRKE (Sines Dataset)** |\n| :--- | :---: | :---: |\n| TimeGAN | $308.84 \\pm 15.97$ | $0.84 \\pm 0.12$ |\n| TimeVAE | $378.99 \\pm 0.00$ | $4.15 \\pm 0.00$ |\n| Diffusion-TS | $392.35 \\pm 22.20$ | $0.46 \\pm 0.01$ |\n\n**[Weakness 2] Illustrative Toy Example.**\nThank you for the valuable suggestion. In our revised version, we added a new figure featuring an intuitive toy example. This example shows a ground-truth dataset with two distinct, easily visible dynamic modes (e.g., a decaying sinusoid and a growing linear trend). We then compare this to a ‘‘mode-collapsed’’ generated set that only captures one of the modes. This figure will visually demonstrate how the DMD eigenvalues of the generated set fail to match the full spectrum of the original, and how DMD-GEN assigns a high (poor) score, directly aligning with visual intuition. This new visual example will be a simple introduction to the more detailed tests in Section 4 (Table 2 and Figure 2), where we rigorously test DMD-GEN’s sensitivity across a spectrum of synthetically-controlled mode collapse severities.\n\n**[Question 1] Required Number of Samples for Convergence.**\nThis is an excellent question regarding the practical application and efficiency of our metric. To demonstrate the sample efficiency of DMD-GEN, we performed an ablation study on the Stock dataset using TimeGAN, varying the number of samples per batch.\n\n| **Model \\ Num Samples** | **128** | **64** | **32** | **16** |\n| :--- | :---: | :---: | :---: | :---: |\n| TimeGAN | $0.73 \\pm 0.19$ | $0.71 \\pm 0.20$ | $0.72 \\pm 0.22$ | $0.69 \\pm 0.23$ |\n\nThe results demonstrate the stability of DMD-GEN. The mean value remains highly consistent even when the batch size is reduced to just 16 samples, with only a minor, acceptable increase in variance. This highlights that our metric is efficient and can provide reliable estimates without requiring a large number of evaluation samples. We added this analysis to the appendix to provide clear guidance for future users.\n\n**[Question 2] Visualization of Samples for Interpretability.**\nThank you for this constructive suggestion, as it directly ties into a core contribution of our work: the interpretability of DMD-GEN. Visualizing the modes is a perfect way to make this concrete. In the revised manuscript, we will add a new figure to illustrate this. Specifically, our analysis will: 1) Compute the DMD modes for a batch of original time series. 2) Use the optimal transport plan $\\\\gamma^\\\\star$ (from the DMD-GEN calculation) to cluster these original time series based on the similarity of their dynamic mappings to the generated data. 3) For each resulting cluster, we will visualize a representative time series sample. 4) Alongside the sample, we will plot its dominant underlying DMD modes (those with the largest eigenvalues). This visualization will provide a clear, interpretable link between the raw time series and the specific dynamic patterns that are (or are not) being successfully captured by the generative model, showcasing the diagnostic power of our metric.\n\nWe thank the reviewer once again for their time and valuable feedback, which has helped us to significantly improve the clarity and completeness of our paper."}, {"review_id": "rtdHEO8FTg", "reviewer": "Reviewer_jdT8", "summary": "This 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.", "questions": "1. 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?", "limitations": "While 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.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "Many thanks for the response. My main concerns on the experimental evaluations have been addressed, especially on the sensitivity part. I believe this metric could evaluate the mode collapse in regular and extreme settings. So I would like to keep my score.", "rebuttal": "We sincerely thank Reviewer jdT8 for their detailed and highly constructive review. The feedback has helped us identify a few areas where our experimental validation and explanations can be significantly strengthened. We have run new experiments to directly address your concerns, and we are confident these additions improve the paper.\n\n**1. On Hyperparameter Choices and Sensitivity**\n\n**• [Question 1] Choice of τ=0.95:** We chose τ=0.95 as it is a standard and well-established practice in spectral analysis literature for capturing a system’s dominant energy/variance without overfitting to noise. Our experiments (e.g., Table 1) confirm that this value provides stable and meaningful results across diverse datasets. A dynamic, data-dependent selection of τ is non-trivial, as it depends on the singular value spectrum from SVD, a non-differentiable operation that cannot be easily optimized. We will clarify this rationale in Section 3.1.\n\n**• [Limitations] Sensitivity to Sequence Length:** We agree that the metric’s sensitivity to sequence length is an important limitation to discuss. Shorter sequences provide fewer snapshots for DMD to reliably estimate modes. To quantify this, we ran a new experiment on the Stock dataset (with DiffusionTS) varying sequence length. The results confirm that as sequence length increases, the DMD-GEN score becomes more stable (lower variance), reflecting the metric’s improved ability to capture richer dynamics from more structured data. We will add this analysis to the limitations section.\n\n| **Sequence Length** | **DMD-GEN Score** |\n|:--------------------|:-----------------:|\n| 5                   | $10.46 \\pm 5.22$  |\n| 10                  | $12.59 \\pm 2.43$  |\n| 24                  | $13.62 \\pm 2.53$  |\n\n\n**2. Strengthening Experimental Validation and Presentation**\n\n**• [Question 2] Extreme Mode Collapse Scenarios:** This is an excellent suggestion. We have run the synthetic experiment for more extreme mode collapse cases (λ=80% and 90%) and will add them to Table 2. The results further validate our claims: DMD-GEN and Context-FID remain highly sensitive, while the Predictive score is far less responsive. (Note: These settings are symmetric to the 20% and 10% cases already in the paper).\n\n| **Metric** | **80%** | **90%** |\n|-------------|--------------|----------------|\n| Discr. Score| +413.20 %     | +566.03   %     |\n| Pred. Score | +2.17%       | +22.80%        |\n| Context-FID | +7205.79%    | +31424.17%     |\n| DMD-GEN     | +489.74%     | +696.34%       |\n\n**• [Question 4 & Concern 1] Clarifying Baselines and Failures:**\n\n**o [Question 4] MTopDiv:** We agree that the rationale for not including MTopDiv in the main comparison should be clearer. MTopDiv is not designed for sequential data; it flattens time series, ignoring temporal dependencies. Our experiments in the supplementary material show it produces scores with excessively high variance, making it unreliable for this task. We will clarify this in the main text.\n\n**o [Concern 1] Missing Values in Table 1:** The dashes (\"—\") denote computational failures. For certain datasets such as Stock Dataset, training the auxiliary models required by metrics like Context-FID was numerically unstable. We will add a note to Table 1 explaining this. This highlights a key advantage of DMD-GEN: its training-free nature makes it more robust and broadly applicable.\n\n**3. On the Interpretation of Visual Results (Figures 1 & 2)**\n\n**• [Question 3] Eigenvalue Overlap in Figure 1:** The reviewer correctly observes the significant overlap post-training. This is not a sign of insensitivity, but rather a visual confirmation of **successful learning**. At Epoch 0, the eigenvalues are distinct because the model’s dynamics are random. The post-training overlap signifies that the generator has learned to reproduce the characteristic frequencies and growth/decay rates of the real data. We will revise the text to make it clear that this overlap is the desired outcome.\n\n**• [Concern 2] High Variance in Figure 2:** The reviewer correctly notes the high variance for small block sizes. This is not a flaw in our metric but an expected outcome that **DMD-GEN correctly captures**. The high variance stems directly from the Moving Block Bootstrap (MBB) process itself, not our metric: when the block size is small, each bootstrapped time series becomes a highly randomized shuffle of the original patterns. Because this resampling introduces significant variability, the resulting samples have very different dynamic structures from one another. Our metric’s high variance in this regime accurately reflects this induced randomness. Conversely, once the block size is large enough to preserve the underlying temporal dynamics, the DMD-GEN estimates stabilize. This confirms that the metric is reliable and robust, provided the resampling window is appropriate for the data’s structure. We will ensure this full explanation is included in our discussion of Figure 2.\n\nOnce again, we thank the reviewer for their valuable insights, which we believe have helped us improve the quality of our work."}, {"review_id": "fd04ylM4qx", "reviewer": "Reviewer_zvbY", "summary": "The 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.", "questions": "See above", "limitations": "See above", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "Based on my original review and the authors' rebuttal, I would like to support the acceptance of this paper.", "rebuttal": "We are very grateful to Reviewer zvbY for their thorough review and positive assessment. We are especially encouraged that they recognized the novelty of our approach and the importance of the problem. We appreciate the insightful weaknesses raised, which will help us improve the paper significantly.\n\n**[Weakness 1] Discrepancy in Benchmark Results (Table 1)**\n\nThank you for this observation. The reviewer is correct that the absolute scores for the baseline metrics (Predictive/Discriminative Score) on TimeGAN may differ from those in other papers. This is not an inconsistency in our metric but a result of ensuring a **fair and controlled comparison within our study**.\n\nTo eliminate confounding variables, we implemented and trained all baseline models (TimeGAN, TimeVAE, DiffusionTS) and all evaluation metrics under a single, unified experimental setting (i.e., identical data splits, preprocessing, and training hyperparameters). While this ensures our internal comparisons are rigorous and fair, it can lead to different absolute scores compared to papers that may use slightly different experimental setups.\n\nWe will add a footnote to Table 1 to clarify this point, stating that all models and metrics were evaluated in our controlled environment to ensure a fair head-to-head comparison.\n\n**[Weakness 2] Ambiguity and Apparent Contradiction in Figure 1** \n\nThe reviewer makes an excellent point about the potential confusion caused by Figure 1. Our core argument is that a robust quantitative comparison requires using the DMD subspaces (eigenvectors), as raw eigenvalues are not invariant. We recognize that Figure 1 was unclear and will revise it.\n\nThe only purpose of Figure 1 is to provide a **high-level**, **qualitative intuition** for how a model’s dynamics evolve during training. The visual clustering of the generated eigenvalues (blue dots) around the original ones (red dots) suggests that the model is learning the correct range of frequencies and decay rates. It is an illustration, not a part of our proposed metric.\n\nTo resolve this ambiguity, we will revise the caption of Figure 1 to explicitly state: ‘‘This figure provides a qualitative illustration of spectral properties. It is intended for intuition only. Our proposed metric, DMD-GEN, is based on the principled and robust comparison of DMD subspaces, not raw eigenvalues.’’\n\n**[Weakness 3] Demonstrating Novel Insights Provided by DMD-GEN**\n\nThis is a crucial point. A new metric should ideally offer insights that existing ones do not. DMD-GEN provides unique advantages in both **sensitivity** and **interpretability**, which we will emphasize more clearly in the revised manuscript.\n\n**1. Superior Sensitivity to Emerging Mode Collapse:** DMD-GEN can detect subtle mode collapse that other metrics miss. This is demonstrated in our synthetic experiment in **Table 2**, which shows the metric’s high sensitivity to mode collapse, and is further supported by our bootstrapping experiment in **Figure 2**, which shows its responsiveness to the preservation of temporal structure. When we introduce a minor 10% mode imbalance:\n\n- **DMD-GEN**'s score changes dramatically (**+681%**), clearly flagging the problem.\n\n- **Predictive Score** barely moves (**-0.54%**), completely failing to detect the issue. This shows that DMD-GEN provides a much clearer and more sensitive signal, which is a significant practical advantage over existing aggregate metrics.\n\n**2. Diagnostic Interpretability (Pinpointing a Failure Mode):** Beyond a single score, DMD-GEN can help diagnose what kind of dynamics are being lost. As we will add in a new appendix figure (in response to reviewer feedback), the optimal transport plan from our metric can identify specific original (real) samples that are poorly matched by the generator. By visualizing the DMD modes of these ''unmatched'' samples, a user can see exactly which dynamic patterns (e.g., a high-frequency oscillation, a slow decay) were missed. This diagnostic capability is a novel insight that aggregate scores from metrics like Predictive Score or Context-FID cannot provide.\n\nWe will add a paragraph and an illustrative figure to our experiment section to explicitly highlight these two unique insights demonstrated by our results.\n\nWe thank the reviewer again for their supportive and constructive feedback. We are confident that these clarifications and additions will strengthen our paper and more clearly articulate the novel contributions of DMD-GEN."}, {"review_id": "taAM2jVMdF", "reviewer": "Reviewer_VZKP", "summary": "This 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.", "questions": "Please see weaknesses above to address my concerns about writing, formatting, and experiments.", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "since most of my concerns have been addressed, I raise my score to 4.", "rebuttal": "We sincerely thank Reviewer VZKP for their detailed and constructive review. Your feedback has highlighted a few key areas where we can significantly improve the paper’s clarity, and we appreciate the opportunity to clarify them. We have taken the feedback seriously and believe the proposed revisions will substantially strengthen the paper.\n\n**[Question 1] Clarifying the Theoretical Motivation and Formulation of DMD-GEN.** We have revised Section 3.4 to better explain the metric’s design. Our approach addresses two fundamental questions: first, how to compare the dynamics of any two individual time series, and second, how to scale this comparison to entire sets (batches) of time series to detect mode collapse.\n\n**A. The Core Component: A Geometric Distance Between Two Time Series**\n\nThe first challenge is to define a meaningful distance between the dynamics of a single original time series ($X_i$) and a single generated one ($X̃_j$).\n\n• **The Problem:** The key dynamics of each series are captured by its DMD modes, which form a mathematical subspace. Because these subspaces can be oriented differently, they do not share a common basis, making a simple Euclidean distance invalid.\n\n• **Our Solution ($d_{\\text{proj}}$):** We use the principled, geometry-aware **geodesic distance on the Grassmann manifold** (defined in Equation 3 and Theorem 3.6). This measures the shortest path between the two dynamic subspaces, providing a mathematically sound way to compare their core dynamics.\n\n**B. The Main Goal: Comparing Distributions to Detect Mode Collapse**\n\nMode collapse is a distributional problem: a model fails if the distribution of its generated dynamics does not match the distribution of the original data's dynamics.\n\n• **The Problem:** How do we compare the entire set of original dynamics to the entire set of generated dynamics?\n\n• **Why Simple Averaging is Insufficient:** Just averaging the pairwise $d_{\\text{proj}}$ distances can be misleading. A generator might create a single ''average'' sample that is reasonably close to all original samples but fails to reproduce any of their specific, distinct dynamic modes.\n\n• **Our Solution (Optimal Transport):** Optimal Transport (OT) is the established mathematical framework for comparing two distributions. It finds the \"minimal effort\" required to transform one distribution into the other by finding the best possible matching between their respective samples.\n\n**Putting It All Together: The DMD-GEN Metric ($d_{\\text{DMD}}$)**\n\nOur final metric integrates these two solutions into a single, robust framework:\n\n1.\tWe use our geometric distance, $d_{\\text{proj}}$, as the **cost** to match any two time series.\n\n2.\tWe build a cost matrix $C$, where each entry $C_{ij}$ is the pairwise geodesic distance $d_{\\text{proj}}$($X_i$,$X̃_j$).\n\n3.\tOptimal Transport uses this cost matrix to find the minimum total effort to align the generated distribution with the original one. This minimum effort is our final metric, $d_{\\text{DMD}}$.\n\nA low $d_{\\text{DMD}}$ signifies that the distributions of dynamics are similar (low mode collapse), while a high cost indicates a poor match (severe mode collapse). This principled, two-level approach ensures that DMD-GEN is both geometrically sound at the sample level and distributionally aware at the set level. We will incorporate this clearer explanation into the revised manuscript.\n\n**[Question 2] Experimental Evidence for DMD-GEN’s Advantages and Efficiency.** We acknowledge that Table 1 only shows consistency and does not, by itself, demonstrate the primary advantages of DMD-GEN. The key advantages are **sensitivity**, **efficiency**, and **interpretability**, which we will highlight more clearly with the following new additions to the paper.\n\n• **Superior Sensitivity to Mode Collapse:** Our primary claim is that DMD-GEN is more sensitive to mode collapse than existing metrics. This is experimentally proven in **Table 2**. In our controlled synthetic experiment, a minor 10% mode imbalance causes DMD-GEN to increase by +681%, signaling a strong detection. In contrast, the Predictive Score barely changes (<1%) in absolute value, and the Discriminative Score gives a noisy signal that is not stable across severities (flips sign). We will explicitly state this in the main text as a key takeaway.\n\n• **Quantitative Comparison of Computational Complexity:** DMD-GEN’s ‘‘training-free’’ nature provides a significant practical advantage. Additionally, we report below the average time needed to compute DMD-GEN w.r.t different number of samples.\n\n**Computational time (in seconds) on the stock dataset.**\n| **Model \\ Num Samples** | **128** | **64** | **32** | **16** |\n| :--- | :---: | :---: | :---: | :---: |\n| TimeGAN | $168.36 \\pm 0.35$ | $41.27 \\pm 0.08$ | $10.36 \\pm 0.02$ | $2.65 \\pm 0.01$ |\n\n**[Question 3] Proving the Interpretability of DMD-GEN.** We agree that the claim of interpretability requires direct experimental proof. We thank the reviewer for raising this point. To validate this claim, we added a new section to the appendix with visualizations that demonstrate how DMD-GEN identifies which modes are poorly captured.\n\n• **Our Visualization Strategy:** As you suggested, we will visualize the captured patterns. Our process is as follows:\n\n1. For a batch of original and generated data, we compute the DMD-GEN score and the optimal transport plan $\\gamma^\\star$.\n\n2. The transport plan $\\gamma^\\star$ tells us exactly how the original modes are ‘‘mapped’’ to the generated modes. We can identify a cluster of original time series that are poorly matched (i.e., have a high transport cost to all generated samples). These are the ‘‘poorly captured modes.’’\n\n3. We will then create a plot showing a representative time series from this ‘‘poorly-captured’’ cluster alongside a visualization of its dominant DMD mode (e.g., a specific high-frequency oscillation). This provides direct, visual proof that DMD-GEN has identified a specific dynamic pattern in the original data that the generative model failed to reproduce.\n\nThis new analysis will provide the experimental evidence for our interpretability claim.\n\n**[Question 4] Typos.**  Thank you for your careful reading and for catching these errors. You are correct about both typos. We have corrected the reference in line 248 to point to the correct equation, and we have numbered the previously unnumbered equation and updated the reference in line 253. We appreciate your attention to detail.\n\nWe are confident that these clarifications and new experimental results directly address your concerns and have significantly improved the quality and clarity of our paper. We hope these changes will convince the reviewer of our work’s merit. We thank you again for your valuable guidance."}]

Author rebuttal:

Meta-review: 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.

Strengths: Addressing the problem of mode collapse in time-series is novel. The theoretical framework for mode collapse is solid. The metric employs OT over DMD subspace representations, enabling a flexible, structure-preserving comparison between time series. 

Weaknesses: The experiments lack depths and diversity, and the numerical results are inconsistent with existing results. 

The reviewers' concerns are all addressed in the rebuttal, and this is the reason for my decision together with the fact that this seems to be the first paper addressing mode collapse in time-series.
Decision: Accept (poster)