A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems
Authors: Thibaut Germain
Rémi Flamary
Vladimir R Kostic
Karim Lounici
Venue: ICLR 2026 Poster (2026)
Keywords: dynamical system
optimal transport
transfer operator
koopman operator
Primary area: learning on time series and dynamical systems
TL;DR: A state-of-the-art operator-based metric for dynamical systems, robust to sampling, opening up to machine learning applications, and meaningful interpolation.

Abstract: The geometry of dynamical systems estimated from trajectory data is a major challenge for machine learning applications. Koopman and transfer operators provide a linear representation of nonlinear dynamics through their spectral decomposition, offering a natural framework for comparison. We propose a novel approach that represents each system as a distribution over its joint operator eigenvalues and spectral projectors and defines a metric between systems leveraging optimal transport. The proposed metric is invariant to the sampling frequency of trajectories. It is also computationally efficient, supported by finite-sample convergence guarantees, and enables the computation of Fréchet means, providing interpolation between dynamical systems. Experiments on simulated and real-world datasets show that our approach consistently outperforms standard operator-based distances in machine learning applications, including dimensionality reduction and classification, and provides meaningful interpolation between dynamical systems.

Published as a conference paper ICLR 2026
A SPECTRAL-GRASSMANN WASSERSTEIN METRIC FOR OPERATOR REPRESENTATIONS OF DYNAMICAL SYSTEMS
Thibaut Germain
Rémi Flamary CMAP, Ecole Polytechnique
CMAP, Ecole Polytechnique
t www.gem.i.nnnoooleeeiiiiueeeeeee
remi.
Vladimir R. Kostic
Karim Lounici
Istituto Italiano di Tecnologia & University of Novi Sad vladimir www.iiiiiii..........
CMAP, Ecole Polytechnique
kar airim.ma
ABSTRACT
The geometry of dynamical systems estimated from trajectory data is major challenge for machine learning applications. Koopman and transfer operators provide a linear rep- resentation of nonlinear dynamics through their spectral decomposition, offering natural framework for comparison. We propose novel approach that represents each system as distribution over its joint operator eigenvalues and spectral projectors and defines metric between systems leveraging optimal transport. The proposed metric is invariant also computationally efficient, supported
to the sampling frequency of trajectories. by finite-sample convergence guarantees, and enables the computation of Fréchet means, providing interpolation between dynamical systems. Experiments on simulated and real- world datasets show that our approach consistently outperforms standard operator- distances in machine learning applications, including dimensionality reduction and classi- fication, and provides meaningful interpolation between dynamical systems.
1 INTRODUCTION
Dynamical systems widely used across scientific and engineering disciplines to model state variables' evolution over time (Lasota & Mackey (013). Nonlinear ordinary or partial differential equations typically govern these systems and may incorporate stochastic components (Meyn & Tweedie 2012). However, in many practical situations, analytical models are unavailable or intractable, motivating the use of data- driven approaches infer the underlying dynamics from sampled trajectories. In this context, Koopman and transfer operator regressions have emerged as a powerful framework for learning and interpreting dynamical systems from data (Brunton al. 2022). Rather than directly modeling the evolution of state variables, these operators advance observables (scalar functions defined on the state space) by mapping each to its expected future value conditioned on the current state. Crucially, these operators are linear even when the underlying systems are not linear. Under suitable conditions, they admit spectral decomposition that provides insight into the system's 's long-temn term behavior, stability, and modal structure (Mauroy et al. 2020). These properties have made the aperator-centric framework particularly appealing for both theoretical analysis and practical applications across various domains, including chemistry for molecular kinetics explainability (Wu et al. 2017), robotics for control (Bruder et al., 2020), and fluid dynamics for prediction (Lange al. 2021).
Koopman and transfer operators for dynamical systems. From a learning standpoint, Koopman and transfer operators provide compact and structured representation of dynamical systems, making them

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well-suited for machine learning applications requiring system comparison, such as time series classifica- tion (Surana 2020) and dynamical graph clustering (Klus & Djurdjevac Conrad 2023). However, in order leverage these representations in standard statistical and machine learning pipelines, one must first define a meaningful metric between them. Unfortunately, despite recent advances in operator estimation (Colbrook)
ee al. 2023; Kostic et al., 2023; 2024a, [evannaa et al., 2023). the development of similarity measures be- tween operator representations of dynamical systems remains relatively underexplored despite the growing need for interpretable metrics on dynamical systems in machine learning applications (Ishikawa et al. 2018))
Comparing dynamical systems. We succinctly review existing similarity measures on dynamical systems; a detailed account is given in Appendix [ The case of (stochastic) linear dynamical systems (LDSs) and linear state-space models was first addressed in the literature (Afsari & Vidal 2014). While early metries are theoretically sound and leverage the manifold structure of LDS spaces, they suffer from high compu- tational cost, making them impractical in most machine learning settings (Hanzon & Marcus [982] Gray] 2009). Originally designed for ARMA models, the Martin pseudo-metric (Martin) (002) offers a practical alternative and has later been extended to general LDS spaces and inspired kernel- variants (Chaudhry & Vidal 2013). These measures have been generalized to nonlinear systems through the Koopman/transfer operator framework (Fujii et al. 2017 Ishikawa et al. 2018). More recent work considers topological conju- gacy, where similarities can be defined Via alignment methods (Ostrow et al. 2023 Glaz 2025 or Optimal Transport (OT) between operator spectra (Redman al., 2024 ZZhag et al. (025). related line of research studies Wasserstein-type metrics functional spaces such as Antonini & Cavalletti (2021). introducing OT between measures derived from the eigenvalues of normal operators.
The above approaches face key limitations. Norm-based measures and the Martin pseudo pseudo-metric are noise- sensitive and lack interpretability.. OT-based similarities improve interpretability by comparing spectral ge- ometry, but they restricted to self-adjoint operators and define pscudo-metrice rather than metries. As a result, no existing method combines theoretical soundness, robustness, and computational efficiency, and defining principled metric for dynamical systems remains an open challenge.
Contributions. In Section 3 we introduce a novel representation of transfer operators as joint distribu- tions over eigenvalues and eigenspaces. Building on tools from optimal transport and Grassmann geometry, we propose a new Wasserstein metric, named Spectral-Grassmann Optimal Transport (SGOT), that com- pares transfer operators through their joint spectral distribution. We show that SGOT is theoretically well- founded, computationally efficient, and broadly applicable, as it is compatible with any operator estimation method. Assuming operator estimation via reduced-rank regression, we further strengthen the theory of non-parametric spectral learning by establishing learning bounds for Koopman eigenvalues and eigenfunc- tions under weaker regularity assumptions, which in turn yield finite-sample convergence guarantees for SGOT. We then exploit SGOT to design scalable algorithm for computing Fréchet barycenters of dynami- cal systems, enabling new forms of system averaging and interpolation. Finally, in Section 4 we empirically validate the advantages of our approach over existing metrics and demonstrate its utility in system interpola- tion and machine learning tasks, using operator estimators derived from kernel methods and deep learning.
2 BACKGROUND
Linear evolution operators. Let (Xt)teT be the flow in some state space X' whose governing laws are temporally invariant, where the time index can be either discrete (T = No) or continuous (T = [0, +o)).. While the flows of many important dynamical systems are nonlinear and possibly stochastic, under quite general assumptions they admit linear operator representations on a suitably chosen space of realevalued functions F C R henceforth referred to as observable space. Namely, letting t T, the transfer operator, also known as Koopman operator for deterministic systems, Ar: F + F evolves an observable f X' + IR for time l via conditional expectation
[A,())(()) Elf(x,) 2). TEA X.
(1)
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Clearly, since A,A, Ar+s, in the discrete-time setting the process can be studied only through the trans- fer operator A := A1 of one unit of time, typically a second. On the other hand, when time is con- tinuous, the process is characterized by the infinitesimal generator of the semigroup (At)120 defined as L := lim (A, - Id)/t / that is a differential operator with domain in F. that encodes the equations of motion and generate dynamics as A, = ett, see Lasota & Mackey (1994); Ross (1995).
Spectral decomposition. The utility of transfer operator representations stems from its linearity on suit- ably chosen F that is invariant space under the action of A,, that [, S F for all t (a property that we tacitly assumed above), and rich enough to represent the flow of the process, ie.., it contains observables from which we can reconstruct all the relevant information of the state (e.g. in the case of a stochastic sys- tem distribution (4 any time t ). Namely, using the spectral theory of linear operators (Kato, 2013), under suitable assumptions, one can spectrally decompose generator L= [)E P3 + N2)+ into distinct complex scalars E called cigenvalues, forming point-spectrum and mutually commuting Riesz spec- tral projectors P3 that satisfy satisfy equations T.Pz=AşP, Pj and P Σ, Py. P, being of finite rank my (geometric multiplicity), N being nilpotent, and jEJ being countably many. Assuming for simplicity that F is a separable Hilbert space and L is a non-defective operator with purely discrete spectrum, e.g. stable diffusion processes, see Ross 1995). we have that I== 9j 8 Jj. with - L gj gド=スン and (fj- 95) where (Az, Jj. 95) jeN are eigen-triplets consisting of an eigenvalue, left and right eigenfunc- tion, respectively. This, in turn, allows one to decouple the evolution of an arbitrary observable fEF
Blf(X.) Xo mo] = [A.f(())- "(EnE>"(()1)))== = EjeNe t euuws m (2r),
(2)
into modes m³ 9f X R that evolve as scalar oscillators at timescales given by reciprocals of r, R(2,) and frequencies wa 3(2)) / 2m in Hz (assuming time in seconds).
Learning transfer operators. In machine learning applications, dynamical systems are only observed, and neither A nor its domain, such the space of square integrable functions w.r.t. the equilibrium mea- sure, is known, providing a key challenge to learn them from data. The most popular algorithms (Brunton let al., (002)) aim learn the action of A: F F F on predefined, possibly infinite dimensional, Repro- ducing Kernel Hilbert Space (RKHS), resulting in estimating the restriction of A on H. C J. by projection, that is PH H + H. typically via empirical risk minimization (Kawahara 2016
Kostic et al. 2022
meaning it IS dense in F. such techniques have strong spectral estimation guarantees (Kostic et al.] 2023), can forecast well the states (Bevanda et al.) [2023] Alexander & Giannakis (020)), and evolve distributions of stochastic processes via kernel mean embeddings (Kostic et al. 2024c). As an alternative, finite-dimensional H spaces can be used with these methods (Kutz et al. (016) or be learned from data in the form of rich neural represen- tations (Liu et al. 2024). that can be also trained to minimize the projection error 4| |F==F (Kostic ctt al., (0244)). In these settings, major limitation of the existing statistical learning guarantees is assum- ing well-specifiedness, i.e., the existence of an exact RKHS representation of A. A more realistic learning scenario requires that only the most relevant spectral part of A lives in a suitable universal RKHS space.
When H. is given by a universal reproducing kernel (Steinwart & Christmann 2008).
Discrete optimal transport. Optimal Transport (OT) is a well-defined framework to compare probability distributions, with many applications in machine learning (Peyré al., 2019). In discrete OT, one seeks a transport plan mapping samples from a source distribution to those of a target distribution while minimizing a transportation cost. Formally, consider 2g {z, E 2 | i [ks]] and 2T {z/ Z | E [Kr]] the sets of source and target samples in space Z. We associate to these sets the probability distributions μs = Eielke a,d, and μr- = Eie(kr) byō_ with (a. b) A ks ArT and A = {p Ry Σiels) P: = 1}
the n-simplex. Let C TR X be the cost matrix with Cjj = c(2,, =/}) being the transport cost between 2; and 2), given by the cost function c. The Monge-Kantorove problem aims at identifying a coupling matrix,
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also denoted OT plan P E RRsxxk, that is solution of the constrained linear problem:
min (C.P), s.t [[ss pT) = (P E Rkakkr P1 = a, P1 = b) PeN(as.pr))
(3)
where II(a, b) is the set of joint-distributions over 25 x 2r with marginals a and In what follows, we denote (us, JUT) the application returning the optimal value of problem (3) where e indicates the cost function. A fundamental property of OT is that, under suitable conditions on the cost function, the Wasserstein distance defined as W,,,,,) 2 (Lar (H, v)) is a metric on the space of probability measures,
see Villani et 20088 Theorem 68).
3
OPTIMAL TRANSPORT (SGOT)
Problem setting and assumptions. Machine learning tasks on (stochastic) dynamical systems, such comparing trajectories, identifying regimes, or clustering dynamics, require a diseriminative and compu- lationally efficient notion of distance between observed processes. We address this by representing each system through its associated Koopman/transfer operator and then introducing the SGOT metric to compare them. To that end, us formalize the problem setting (AT) and make the main learnability assumptions (A2) (A3) necessary to obtain strong statistical learning guarantees.
(A1, Dynamical system sampling) Consider E N' time homogeneous, Markovian dynamical systems defined on a common state space X' and characterized by their generators Le: dom(La) C FR-FR defined on the respective spaces F& of observables X X-R ke[N]. For every k=[N], J, let (x y/ ( hejma)) be a dataset of observations of the of the k-th system, consisting of consecutive states separated by time-lag Ale- Notably, in the case of a single trajectory y = =
Since data-driven methods distinguish between systems only up to the temporal resolution at which the observations are made (Zayed 2018), recalling equation 2 the systems differing in spectral components be- yond the observable range of timescales 1/T, and frequencies w, are undistinguishable from measurements. Therefore, we focus below on spectral projections of dynamics that can be learned from finite data.
(A2, Low rank) For every [=[N] there exists N such that rı eigenvalues of La closest to the origin are separated from the rest of the spectrum, and let Psr, : FF denote the corresponding spectral projector.
Recalling the case of dynamical systems sampled equilibrium, i.e. F& C2 (X) with The being the invariant measure of the k-th system, a central conceptual difficulty in introducing distance between systems is that transfer operators for different systems naturally act on different spaces, and therefore cannot be compared directly. To resolve this, we restrict each operator to a common reproducing kernel Hilbert space H. that included the domain of all the transfer operators.
(A3, Common functional space) Let H. be a separable RKHS associated with kernel x, such that for all kE[N] it holds Im(Psrg Lx) C H C FR. Hence, there exists representation Ti e Lu)x H-H with spectral decomposition Tx= Σ jjjee ei Qf where lg is the number of distinct eigenvalues.
First, we remark that this assumptions significantly relaxed compare to the most related prior work (Kostic et al. 2023). In fact, for typical Langevin dynamics, under suitable assumptions on the potential, universal Gaussian RBF RKHS with properly chosen landscale parameter contains a finite number of leading eigen- functions of generators La defined C2, (X) spaces weighted by Boltzmann distributions πą. Thus, (A3) holds true, while aforementioned is violated. Morcover, one can formally build a finiteddimensional space H by choosing exactly the basis of such generator's eigenfunctions; the complexity of the problem is then transferred to learning H. Beyond this case, one can similarly work in other domains, see e.g. Colbrook et al. (2025): Alexander & Giannakis (2000: Bevanda al. (2023).
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Finally, if Assumptions (A2) and (A3) hold, we obtain an exact spectral representation in H. which, we will show, enables unbiased operator comparison. When these assumptions are violated, the comparison incurs a bias that can be assessed through the metric distortion between the chosen common subspace and the true operator domain. This phenomenon is analyzed in (Kostic et al. 2023), which also provides an efficient empirical estimator of the distortion.
Spectral Grassmanian Wasserstein metric. Since the spectral decompasition of a non-defective operator Th into its eigenvalues and spectral projectors uniquely defined up to permutation, any meaningful com- parison approach based on operators' spectral decomposition must be invariant to permutations and change of basis in which spectral projectors are expressed. While discrete optimal transport naturally provides in- variance to permutations through the minimizing coupling matrix eq. (3), need to design a ground metric that takes into account both spectral and subspace aspects to obtain a true OT metric. This is done below, where we define a Wassertein metric on the set of non-defective operators (complete proof in Appendix (C.
Theorem 1. Let H be a separable C-Hillerrt space and S, (H) the set of non-defective operators with rank most D. Let (G, dg) be the Grassmanian manifold of the space of Hilbert-Schmidl operators on H. Given PEN and (E(0,1). 1). let p.: (.(P))+Ppl x 9) and do; (C x g) -R+ be given by
((T) Ljj mist
mi ((A), and
V") X1+ dg
(4)
with| |. applied on polar coordinates X, X, Mhat = -Evetoj m, Vj the my-dimensional vector space in (H. H) spanned by the rank one operators of the right/left eigenfinctions associated with the eigenvalue e>s Tf (same notation for )".. Then, (H), ds) is metric space, where ds: (.) given by ds(T,T') = Wa (p(T),
(5)
First, recalling equation 2 and (A1) note that while typically in data-driven methods datasets are sampled at some frequency w
Atg to estimate eigenvalues eAt of transfer operators A we build a metric using the difference in the generator cigenvalues. This is compare Koopman modes' eigenvalues as physical quantities, since for the k-th system the observed time-scales are T //[ and the oscillating
frequencies w/ /wxe/. So, by re-normalizing eigenvalues, time-scales in the universal time units. Further, we remark that assuming non-defective operators is not a major bottleneck, since Theorem cean be extended to the space of general linear operators with rank most r by leveraging the Dunford-Jordae decomposition (Dunford & Schwartz) 1988). In this case, the cost metric in d, compares the spectrum and subspaces of Jordan blocks. As well, depending on the prior geometry one wishes to emphasize on G, one can consider other metrics as depicted in Appendix C.3
compare systems observed different
Metric computation. In order to evaluate the SGOT metric. one needs to compute the cost matrix (see section 2). i.e., d, for each pair of spectrals. Following (A1) (A3) T be an operator estimated from samples {( M1) Heeni with a kernel based method. Suppose that T admits eigenvalues, each with multi- plicity my- Let Br, ar (Cax))² be the control parameters of the left/right eigenfunctions related to the th eigenvalue and preprocessed to form an orthonormal basis. Let 7" be another estimated operator, and M. A (k(e,, (()e,ee)))lseeee) with yx. be the cross-kernel matrices. For p the cost matrix C E CERl"" is given by:
Ct - m/>: (1- - 211((B1,,
(6)
With the rank 2 ma) the time complexity of ds is in 0 + log()) respectively due to the cost matrix computation and the OT solver (that is negligible for small r). Consequently, ds and the kernel metrie computation are asymptotically equivalent, overcoming the usual computational drawbacks of OT-based methods relative to kernel ones. If needed, both metrics can further benefit from standard kernel scaling techniques (Meanti et al. 2023)
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Statistical guarantees. In the following, we show how using RRR estimators yields unbiased estimation of the SGOT. To that end, consider T. (0% [ +[(0++)))==* where C Eie(ms) Miz Kizz
C% zy Te Eie(ma] Mag @ Kigg 7 > 0 and [[I denoting best rank-r approximation. As discussed above, one can efficiently compute ds( T2) so that the following holds.
Theorem 2. Let (AI) (A3) hold with k [2], Fx-La and x(x,)) < 00 a.s. for x ~ The Let B|C*] C and assume that for some aE 2] and BE(0, 1) it holds that [I[((*)) || Tall--- and A,(C.) S 1/9 for E. Given 8 E (0,1), if is large enough and A 2(a+P))
alog then w.p.a.l 1-6
in the i..dd draw of samples D1 and D2 it holds (T, T2) ds(Ti, (1,,77 T2)1 S n a(++)) In(25"1).
e-1
Sketch of Proof. To obtain this result, we needed to overeome the overly strong assumption of well- specifiedness of H made in Kostic et al. (2023), which significantly reduces the applicability of those bounds to estimate the distance between true generators La with high probability. By carefully treating the approximation errors originating from rank reductions, Tj being population version of Tx. we obtain IT& 7all S 7 +ea*k under realistic assumption (A3) Furthermore, we derive an upper bound on the operator norm [Tk Tellu- S Vy-8-1n-1 log(
w.p.a.l. 5. Balancing the two terms gives the bound on [ - Txll- Next, we apply standard polar analysis and Kahin perturbation analysis to derive the bound on (Tx, Te). Finally, the stability property of the Wasserstein distance gives the final bound. Full proof is available in appendix
Spectral Grassmann OT barycenter, parametric model and optimization. Computation of barycenters is fundamental for many unsupervised methods; it is known as the Fréchet mean problem metric spaces. consists in identifying element that minimizes weighted sum of distances to the observations. Formally, given the importance weights YEA assuming (A1) (A3) and p=2 in Theorem we aim to solve:
arg min (T,
TES,(H) KE(N]
By construction of ds. problem 7 corresponds to free-support Wassertein barycenter estimation problem which aims at optimizing the support of the atoms parametrizing the barycenter, our case, its spectral de- composition. State-of-thh-aad art algorithms typically rely on coordinate descent scheme, alternating between transport plan computation and measure optimization (Cuturi & Doucet 2014; Claici et al.) 20188)
Whenever the RKHS H is infinite dimensional, the Fréchet mean problem (eq. 7 is intractable. So we restrain the optimization over set of parametrized operators defined such that for any 0 - (,, x):
To : E H + As (1x,,, E H
(8)
where X x X representer functions Nxx = {x(., }))]]][ with к the kernel of H, i.c. kg Σje(nl Kx, ajt- While these operators compact with rank at most r, further constraints the control points and parameters are required to ensure a spectral decomposition (see eq. (2). Together with the definition of discrete optimal transport (see Section 2). it leads the constrained optimization problem:
E
are state space control points, and ac, 3 E C"xr control parameters acting on the
(O), s.t. { SE[N]
a²K3 = K,K3, = E
K = [K(2,, 2 Hagiea
arg min O.P
(9)
P. (I ((.)). We
where P (P.lve[N) T (Ti such that (C,(0), P.) are the cost and transport matrices associated the Wasserstein metric ds defined in Theorem between the parametric operator Tg and T,
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Hilbert-Schmidt
Operator
Martin
SOT
GOT
SGOT (n= 0.5)
WmM
10
Metric value
0.8-
0.-
0.-
0.
1.0
1.5 2.0 2.5
80
400 80
100
100
160
200
250
300
Fraquency (Hz)
Decay rate
(c) Operator rank
(d) Sampling frequency (Hz)
Figure I: Similarity measures' behaviors under four scenarios of shifts of a linear oscillatory system: (a)
frequency shift, (b) decay rate shift, (c) operator rank/subspace shift, (d) sampling frequency variation. In scenarios (a,b,c), metric values are normalized by their maximum.
Following Cuturi & Doucel (2014) and considering a differentiable kernel w.r. the control points, we pro- pose an inexact coordinate descent scheme with a cyclic update rule for optimizing problem | Each cycle begins with the computation of the optimal transport plans, then the subsequent coordinate updates are per- formed with few gradient descent steps and closed-form projection scheme to enforce the constraints. In Appendix D we provide more detail about the computational and theoretical aspects of the barycenters.
4 NUMERICAL EXPERIMENTS
We now illustrate the benefits of our metric and barycenter through numerical experiments dynamical systems. We first study the behavior of different similarity measures under various shifts and compare them on unsupervised and supervised machine learning tasks. Finally, we demonstrate the properties of operator barycenters using our proposed algorithm on two simulated examples.
Compared similarity measures. In addition to our proposed metric SGOT, we compare other OT-based similarities that focus solely on the eigenvalues (SOT) (Redman et al.) (022)) or solely on the eigenspaces using a Grassmannian metric (GOT) (Antonini Cavalletti] 2021). We also include metrics induced by the Hilbert-Schmidt and operator norms, well as the Martin similarity (Martin 2002).
4.1 COMPARISON WITH OTHER SIMILARITY MEASURES
Simulated system and shifts. First, we illustrate the behavior of different similarity measures between dynamical systems with regard variations of the spectral decompositions of their Koopman operators. We consider a referent linear oscillatory system that is the sum of two simple harmonic oscillators with frequencies 0.5Hz and 1.0Hz, respectively, with trajectory sampled at 200Hz. Considering the linear kernel, we compare the Koopman operator of the referent system with those of shifted systems according to four scenarios: (a) Frequency shift, changes the IHz harmonic frequency. (b) Decay rate shift, changes the IHz harmonic decay rate. (c) Subspace shift (rank) gradually transforms the IHz sine wave into a 1Hz square wave signal using a Fourier decomposition
square wave signal with increasing order. (d) Sampling frequency shift where the system is sampled at different sampling frequencies instead of the reference 200Hz. In each scenario, Koopman operators are estimated from sampled trajectories with the RRR method (Kostic et al. 2022) with rank fixed to twice the number of harmonic oscillators.
Results & interpretation. Values of the different metrics as function the shifts are shown Figure [ In scenarios (a,b,c), our metric SGOT increases continuously (near linearly) with the shifts almost everywhere. contrast, other similarities tend to saturate quickly, and some even oscillate as shifts increase. In partic- ular, OT-based competitors exhibit extreme behaviors: the pseudo-metris metric SOT oscillates in the frequency scenario, while GOT saturates fastest overall. Likewise, the Hilbert-Schmice and operator metrics present both saturating and an oscillating behavior, introducing many local minima. When changing the sampling
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Table 1:
Classification rank per kernel type. Deep: kernel based on learned deep features. Best and
second best performers are highlighted (lower is better). Ranks are denoted: (mean) + (std).
Hilbert
Hilbert-Schmidt Schmidt
Operator
Martin
SOT
GOT
SGOT
Linear
3.29 + 1.02
3.92 1.1
5.30 1.31
4.49
1.15
2,66
1.18
1.34 + 0.79
RBF
3.74 + 1.27
NA
4.02 + 0.98
3.28
+ 1.15
2.48 + 1.19
1.48 + 0.70
Deep
3.33 + 1.56
4.14 + 1.27
5.06 + 1.48
3.84 + 1.34
2.94 4 1.33
1.71 + 0.77
Table 2: Classification accuracy for operators estimated with RBF kernels. Datasets on rows and similarities on columns. Best and second best performers are highlighted. Accuracy scores are denoted: (mean) + (std)-
Hilbert-Schan Schmidt
Martin
SOT
GOT
SGOT
BusicMotions
0.26 0.17
0.77 0.06
0.87 + 0.05
0.69
0.14
0.95 + 0.02
ERing
0.74 + 0.07
0.22 + 0.05
0.38 + 0.05
0.96 - 0.01
0.98 + 0.02
Epilepsy
0.31 0.02
0.80 0.01
0.77 0.02
0.93 0.02
0.95 = 0.02
FingerMovements
0.53 + 0.06
0.50 0.03
0.53 0.05
0.50
0.06
0.53 + 0.01
NATOPS
0.59 = 0.06
0.25 + 0.02
0.35 = 0.02
0.78 = 0.03
0.80 = 0.05
frequency in scenario (d), only GOT and our metric SGOT are robust and remain low and almost constant. Appendix and G.5 provides details and sensitivity analysis the r) parameter in SGOT.
4.2 MACHINE LEARNING OF DYNAMICAL SYSTEMS
Experimental setup. We now illustrate and study the usability of our metric SGOT in machine learning applications, both unsupervised (dimensionality reduction) and supervised (classification), when sequential data are embedded by estimated operators governing their dynamics. In both experiments, we considered 14 multivariate time series datasets from the UEA database (Ruiz et al.] (02T), and time series are represented with Koopman operators estimated with the RRR method (Kostic al. 2022).
Dimensionality reduction. We first explore the dimensional- ity reduction capabilities of the diffeerntt similarity measures. Considering linear kernel, 5 datasets and all similarities, the estimated operators embedded as 2D vector with the T-
Hilbert-Schmidt
GOT
SGOT
ElgenWorms
distributed Stochastic Neighbor Embedding (T-SNE) (Maaten
& Hinton 2008) method fitted on the cross-distance matrix estimated with the similarity. Figure 2 illustrates the em- beddings for the most discriminative similarities on datasets Eigen Worms (motion) and Epilepsy (biomedical). TSNE em- bedding for all datasets and metrics are available in Ap-
Epilepsy
pendix G.2 Figure The Hilbert-Schmidt distance is too con- servative, and no clusters or classes can identified. For OT- based metrics, GOT better identifies classes; however, they do not form distinct clusters as is obtained with our metric SGOT.
Figure 2: T-SNE embeddings. Datasets on rows, metrics on columns, classes in colors.
Classification setup. We now quantify similarities' performances three types of kernels for operator estimation via RRR Kostic et al. (2022): lincar, RBF, and kernels based Monte-Carlo nested cross-validation
classification task. We consider
learned deep features. For each kernel type and dataset, procedure with a (0.7, 0.3) train/test split and no data preprocessing. We perform 10 iterations for the linear case and 5 iterations for the RBF and deep-feature cases. For every similarity measure, train
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1,0
5GOT better
SGOT betterr
SGOT betthe
SGOT
better
0.6
0.6
0.4
0.2
040Vaaae 3000
0.
aVo per dalnee
0.0
0.
0.4
0.6
0.6
1.0
1.0
0.0
0.2
0.4
0.6
0.
100
Hilbert-S Hilbert-chrmict Schmiot
Operator
SOT
GOT
Figure 3: Classification performance (accuracy) comparison between SGOT and competitive metrics. Each
point represents a dataset accuracy, with SGOT the y-axis and the competing metrics on the x-axis.
(8) Hibert-Schmidt
(b) Constrained Hilbert-Schmict
(e) SGOT
10
Mwy
Ratlo
Time
Time
Time (9)
T (y= 0.0) y= 0.2 0.4
y= 0.6
T (y= 1.0)
Figure 4: Predictions of interpolated systems between two linear oscillating systems from the same ini- tialization. Interpolated systems correspond to weighted Fréchet barycenter for three different metrics: (a) SchberttSShhiit, (b) Hilbert-Schmidt with spectral decomposition constraints, and (c) metric SGOT. The interpolation controlled by a ratio parameter E [0, 1] which sets operators' weights.
k-NN classifier and tune the hyperparameters- k (and 7) for SGOT) with a 5-fold inner cross- cross-validene validation. The RBF experiments restricted to the five smallest datasets, and the operator metric is excluded due computational constraints. For the deep-feature experiments, an additional preprocessing step is required: features learned from the training data using Multi-Layer Perceptron (MLP), following the strategy Kostic et al. (2024b) for learning invariant representations of hom-hhomogeeeuss stochastic dynamical systems. Further details the experimental protocol provided in Appendix G
Classification results. Table I presents the average rank per metric/kemnel combination, while Table provides the accuracy scores in the RBF setting and Figure compares the accuracy between metric SGOT and other metrics in the linear case. Overall, SGOT consistently outperforms other metrics across most datasets and for any kernel type. These results indicate that SGOT is robust and well-behaved metric for operator comparison, independent of the underlying estimation method. In particular, SGOT surpasses both SOT and GOT jointly leveraging information from eigenvalues and eigensubspaces. Detailed results for each kernel type, including comparison plots, full performance tables, critical-differee difference diagrams, and execution times, are provided in Appendix G.2 G.3 and G.4
4.3 BARYCENTERS AND INTERPOLATION OF DYNAMICAL SYSTEMS
Interpolation between 1D DS. In this experiment, we compare the interpolation between dynamical sys- tems through the weighted Fréchet barycenters of their Koopman operators, estimated with a linear kernel, for different metrics. The two systems are linear oscillatory systems, each being the sum of two simple harmonic oscillators with different frequencies and decay rates, and additive Gaussian noise. The interpo- lation is controlled by a ratio parameter E [o. 1] with weights (1 - 7, 7) in the Fréchet mean problem equation 7 We compare (a) the Cilbert--Schide metric without spectral decomposition constraints given

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T( (y= 0.0)
SGOT barycenter (y= 0.5)
T (y= 1.0)
Eig fot
Eig fot
Elg. fot
Figure 5: SGOT barycenter of Koopman operators of flows past static objects: a cylinder ( and a triangle T(1, Each operator's 's three leading right eigenfunctions are displayed and can be associated with the vortex- shedding phenomenon of the fluids flowing from left to right.
by Toor = (1- + +T(() (b) the Hilhert-Schmidt metric with spectral decomposition constraints, and (c) our proposed metric SGOT, For (b) and (c), barycentric operators are estimated with the proposed optimization scheme, and experimental settings are detailed in Appendix H
The interpolated predictions, starting from an identical initialization signal (in gray) containing all four fre- quencies, are illustrated in Figure 4 for all three metrics. In the Hilbert-Schnin case (fig. 4a) leads to over- damped systems V' E (0, 1). Adding spectral decomposition constraints on the
barycenter (fig. 4 (b) mitigates the damping effects; however, the oscillatory frequencies and decay rate converge to local minimum close to initialization, as expected by the saturating behavior the Hilhert-Schmidt metric (see fig. [T. D.. Only SGOT barycenters naturally interpolate between the systems, notably by retrieving the frequencies and the decay rates.
Interpolating fluid dynamics. We aim to compute the barycenter of fluid dynamics systems. To that end, we consider the Flow past bluff object dataset (Tali et al. 2025), which gathers trajectories of time- varying 2D velocity and pressure fields of incompressible Navier- Stokes fluids flowing around static objects. We select two trajectories, one with a cylinder object and the other with a triangular object. We only kept the velocity field along the flowing direction for each trajectory, leading to trajectories containing 242 samples of 1024x256 grids, which we down-sampled to grids with 256x64 resolution. We estimate Koopman operator with linear kernel using the RRR method from each trajectory: ( for the cylinder and (1) for the triangle. The operators are restricted the fourth leading eigenvalues and eigenfunctions. We compute the SGOT barycenter with the optimization scheme described in Appendix (ratxxxDDwiii with an initialization being the average of eigenvalues and eigenfunctions. Figure 5 illustrates the non-conjugated right eigenfunctions of all three Koopman operators (cylinder, barycenter, triangle). By symmetry of boundary conditions and the cylinder, the eigenfunctions in the cylinder case have an axial symmetry that is lost with the triangle. SGOT by interpolating between both introduces the asymmetry the eigenfunctions the barycenter.
Appendix we detail the experimental settings.
5 CONCLUSION
this paper, we proposed SGOT, a novel optimal transport metric between distributional representations of transfer operators in the joint spectral Grantmmnnrasaaann space. The metric has strong theoretical properties, in- duces a meaningful geometry for barycenters and interpolation, and can be computed efficiently. Numerical experiments demonstrate the superiority of the proposed metric for machine learning tasks and system in- terpolation. Our method opens the door to machine learning applications on dynamical systems, with future work including dictionary learning and conditional prediction accelerate numerical simulations.
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6 ACKNOWLEDGMENTS
This project received funding from the European Union's Horizon Europe research and innovation program under grant agreement 101 101120237 (ELIAS), Fondation de I'Rcole Polytechnique, Hi! PARIS, the French National Research Ageney (ANR) through France 2030 program (ANR-23-IACI-0005 and ANR-25-PEIA- 0005), NextGeenrrationEU and MUR PNRR project PE0000013 CUP 153022003300006 "Future Artificial Intelligence Research (FAIR)"
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A RELATED WORK
Metric for linear dynamical systems. A substantial body of research addresses the comparison of (stochastic) linear dynamical systems (LDSSs) and linear state-space models (Afsari & Vidal 2014). Early methods exploit the Riemannian manifold structure of LDS spaces to define meaningful metries (Hanzon & Marcus| 1982), with related developments in power spectral density spaces (Georgiou) (207)) including approaches based Wasserstein metrics (Gray 2009). However, these methods suffer from high compu- tational cost. The Martin distance (Martin 2002) offers a practical alternative, comparing ARMA models via their cepstrum. It has been generalized state-space models and shown equivalent metrics based
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on angles between observability subspaces (De Cock & De Moor 2002 Sinha et al. 2024). Other ap-
proaches include kernel-basee metrics derived from the Binet-Cauchy theorem (Vishwanathan et al. 2007
Kullback-Leibler divergence (Chan & Vasconcelos, 2005), and moment matching (Bissacco et al. 2007 However, compared to the Martin distance, these metrics are sensitive to trajectory initial conditions, so extensions have been proposed to address this issue (Chaudhry & Vidal, 2013).
Extension to nonlinear dynamical systems. For nonlinear dynamical systems, most work leverages the Koopman framework to linearize dynamics. The Binet-Cauchh kernel has been extended to nonlinear sys- tems (Fujii et al. 2017) within this context. Another kernel leverages Koopman representation to compare observability subspaces (Ishikawa et al. 2018). The latest has been used alongside a deep learning method for estimating Koopman operators (ResDMD Colbrook et al. (2023) in the case of continuous spectrum (Sakata & Kawahara 2024). However, both kernels are sensitive to trajectory initial conditions like the lin- car case. In Mezić & Banaszuk (2004), the authors propose metrics to compare the asymptotic dynamics of measure- pressevvipe systems via Koopman representations, later extended to dissipative systems over finite time MMezic 2016))
Metric for topologically conjugated dynamical systems. Recently, interest has grown comparing neu- ral network dynamics in neuroscience and deep learning (Klabunde al. 2025). Such comparisons often consider topological conjugacy, leading to metrics on quotient spaces. [Redman et al. (2022 2024) show that topologically conjugate systems share identical Koopman spectra and propose a pseudo-metrey based on optimal transport. [Ostrow et al.] (2023) extends Procrustes analysis to compare Koopman representations up orthogonal transformations, extending earlier work the LDS setting (Afsari & Vidal 2013). Glaz
(2025) further generalizes these metrics to accommodate broader transformation classes.
Optimal transport on functional spaces. A related direction studies measures on functional spaces. Some works have studied measures on Gaussian processes (Masarotto et al. 2019 Mallasto & Feragen] 2017], for which there exists closed-form of the metric. In Antonini & Cavalletti (2021), the authors propose a theoretical Wasserstein metric between measures derived from the spectral decomposition of nor- mal operators. More recently. Zhu et al. (2024) introduced a computable approximation of the Wasserstein metric between measures on infniteddimensional Hilbert spaces, obtained by restriction to linear mappings.
B LEARNING KOOPMAN TRANSFER OPERATORS WITH KERNEL METHODS
many practical scenarios, A, is unknown, but data from system trajectories are available. For such cases, Koopman operator regression in reproducing kernel Hilbert spaces (RKHS) provides a learning framework estimate A, on 2 (X)) (Kostic et al. 2022). Let be a RKHS with a bounded kernel i and feature
map é such that k(v.v) v) (Ø(r), ( J. We recall that the injection operator S H + L (2') is
Hilbert-Schmidt (Caponnetto & De Vito 2007 Steinwart & Christmann 2008), and thus so is the restricted Koopman operator Z. := A, Sx H
The goal is to approximate Z.- S₂ by minimizing the risk R(G) (Gh:) Xx, 2r| over Hilbert-Schmidt operators G' E HS (H), where (hi)iEN is an orthonormal basis of H. This risk admits decomposition R(G') = R, + {Gs( where
= -ENN Σien (X((())+11)
R₂ = S.S llass 2r fas 2 0 and EGs(G) = || - G = A S Giissss..cc( (10) are the irreducible risk and the excess risk, respectively. Using universal kernels, the excess risk can be made arbitrarily small: inf Ens(G) = 0.
common approach is to solve the Tikhonov-regullrizeed problem min Rd (G))-)((() + + GEHS(H)
(11)
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with 7 > Defining the covariance operator Og := S Ez~rd(z) <(x) and the cross-covariance
operator Czy == Z E -(z,v)~p(( (x) @ Φ(y) (where p is the joint measure of consecutive states), the
unique solution to equation 11 is the Kernel Ridge Regression (KRR) estimator G := Cny. where
C == C2 + -Id..
To approximate the leading eigenvalues of Ar, low-rank estimators are used. The Reduced Rank Regression (RRR) estimator (Kostic et al., 2022) is the solution to equation I1 under a rank-r constraint:
C 12[C, 1/2 Cayl
arg min RG),), GEB,(H)
(12)
where B, (H) denotes the set of rank-r HS operators and [.] is the r-truncated SVD.
Given data D { vi) hejm) empirical estimators derived by minimizing the regularized empirical
risk R2 (G):= Eieim| ((v))) G (<x;))1 + >lGl Introducing the sampling operators for data D and
RKHS 7. by
3: H R"
s.l. f (1AA)lle and
s.t. f J. (1WO)ll)))) Z: H + R"
and their adjoints by
S R" H
s.t. w vn
wrD(x;)
and
2": R² - H
s.t. W v
Σ
wab(v:);
VE(N]
(E(M))
we obtain (C)= +
The empirical covariance and cross-covariance operators are:
: -= 2. :zy:= : 5-
(13)
The corresponding regularized empirical covariance is C2 == Õ₂ + 7 Idy. The kernel Gram matrices are:
K K:= := "2"
(14)
The empirical RRR estimator then 1/70 These empirical estimators can be expressed in the form G V, 2 for matrices U,, V. E Raxr (Kostic et al. 2022), enabling the computation of spectral decompositions in infnnitedimmnsional RKHS.
Theorem 1 (Kostic et al. 2. ). Let 1 S r S and G SU.V. Z, where U,, Vr E |Raxr If V. MU, E "R""" for M n "Mye, s,)s.e is full rank and non-defective, the spectral decomposition (A,, E, Vs))e()) of can be expressed in terms of the spectral decomposition (Re, U4 Ds)ie(r) of V, MU, &
X,2 V //Ag and 3. = U,U for all i E [r].
RKHS embedddings into (x). We recall some facts on the injection operator S.. Note first that S₂ E HS (H, (x))). Then according to the spectral theorem for positive self-adjoint operators, S, has an SVD, i.e. there exists at most countable positive sequence (ø, ))y,,, where J == f1.2..... N, and ortho- normal systems (G))Ee and (hz)jeJ of cl(Im(S,)) and Ker(S,)* rrespectively such that Snh ojl, and S = oyh3, j E
Now, given a 2 0, let us define sealed injection operator S.: H + C2 as
S. = Σ
Jg 2 hi-
(15)
JEJ
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Clearly, we have that S St, while Im So cl(Im(S))). Next, we equip Im(Sa) with norm || lla build an interpolation space:
[H], { / Im( M1 Eo 00 )
JEJ
C SPECTRAL-GRASSMANN WASSERSTEIN METRIC (SGOT) PROOF
C.I MAIN PROOF
In this section prove that (H) can be endowed with Wasserstein metric based operator spectral decomposition as summarized by the following theorem:
Theorem 3. Let H be separable C-Hilbert space and (H) the set of non-defective operators with rank at most r E 'D. Let (G, dg) be Grassmanian manifold of the space of fiilbert-Schmite operators on H. Given PEN and m/ (0, 1), let p.:. (,H
9) and d,; (C x G)2-RR be given by
( JE( meot my (1y)))) and d, (X, / (1+(1-)) - do(,
(16)
with |. applied on polar coordinates N, Mrot = Eiely my, Vj the my-dimensionall vector space in HS(H, H) spanned by the rank one operators of the right/left eigenfunctions associated with the eigenvalue e> of (same notation for T'). Then. (H,( is metric space, where ds: (,) is given by
ds(T,T") v (p((T)
(17)
Discrete Optimal transport. For conciseness, we first recall discrete OT where one seeks a transport plan mapping samples from source distribution to those of target distribution while minimizing transporta-
tion cost. Formally, consider 2ş = {& E Z | i E [ks]] and 27- = {2} E Z i E [kr]] as the sets of source and target samples in space Z. We associate with these sets the probability distributions ps
Eie(ks) (2552 and pr = Eie(k,) by0gg with (a, b) E A ks x AKr and A = (p E Ry Σie(n) Pi = 1}
the n-simplex. Let E R x be the cost matrix with Ctj- = c(z,, 2/)) being the transport cost between 2; and zj given by the cost function c. The Monge- Kantorovich problem aims at identifying a coupling matrix, also denoted OT plan P* E TRksx that is solution of the constrained linear problem:
min (C,P),
ll PT) (P Rksxkr P1 a, P = bb.
(18)
Pell(as,P)
where [T(a, b) is the set of joint-distributions over 2s X 27 with marginals a and b. In what follows, we denote (Jus,, HT) the application returning the optimal value of problem (18) where indicates the cost function. A fundamental property of OT is that, under suitable conditions on the cost function, the Wasserstein distance is a metric on the space of probability measures:
Theorem 4 (Theorem 6.18 in Villani et al. 2008)). Let (2,d) d) be separable complete metric space dowed with its Borel set. Let p E N and Pp (2) the set of probability distributions on Z admitting moments of order p. Consider the application:
Wp: (H,V) E P(Z) P(Z) (La (H, E RR,.
(19)
Then, (P,(Z), W2) defines a separable complete metric space, known as a Wasserstein space.
Main proof. For proof correctness, restrict the Grassmann manifold on Hilbert-Schmidt operators & to the set of operators with rank at most r, denoted by G.- This restricted space endowed with the Hilbert- Schmidt norm is complete metric space detailed in Appendix C.2 The next two propositions detail
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the essential building blocks to derive a Wasserstein metric on S,HH). Proposition Seneatiiionsspeeeeeeeee map from S,(H) to space of probability measures while proposition 2 defines a metric on the measures' support space with sufficient topological properties to derive Wassertein metric on S.(H).
specifies an inclusion
Proposition 1. Consider p E N and the embedding map:
p: E (7)) + Σ 4
(A. E P
(20)
Lot iEM1
where Vj the my-dimensional vector space in HS(H, H) spanned by the rank one operators of the right/left eigenfunctions associated with the eigenvalue e² of T. and Miot = -Eey Then, is a one-to-one inclusion map.
Proof. Let T + E S,H)), both operators differ by at least one pair (A,; V.) E Gr. by symmetry the pair is associated to T. Since (C, and (g., dg) metric spaces, the singleton {(),, V:)} belongs the Borel set. Therefore, pr((A,, Vi)) m,/ Meot while pe ((),, Vi) = O. ie pr * pr.
Proposition 2. Consider E (0, 1), wres E R. and the application:
d, V), ()) x (C) π|> XI (1 m)dg(V V') E.
(21)
Then, (C x Gr, dn)) is separable complete metric space.
Proof. By proposition 4. (g.,dg) separable complete metric space. Hence, for any n (0,1), (C Gr, dn) is a separable complete metric space as (C, deat) is homeomorphic to (-)).)).
I
Note that we introduce metric, deat, that compares Koopman modes' eigenvalues from physics-infonen quantities, namely the time-scales p and the ocsillating frequencies w. The previous two propositions lead to our main contribution, a Wasserstein metric on the space of non hon-defective finite rank operators S, S,(H): Proposition 3. Consider E 10, E N and the application:
ds (T,T") E (,H) S,HH) Wa,, (u(T), R+
(22)
Then, (H), ds) is metric space.
Proof. Application of theorem with Propositions D and
C.2 GRASSMAN METRIC
Grassmann manifold is a collection of vector subspaces of given vector space. Such manifolds ap- pear in a handful of applications whenever subspaces must be compared. The particular case of Grassman manifolds gathering all equidimensional subspaces of unite-dimensional real vector space has been ex- tensively studied, see [Bendokat et al. (2024) for thorough review. In our context, this particular setting is limiting as we must consider a manifold including subspaces of various dimensions over possibly infinite- dimensional complex vector space. On such manifolds, classical metric compares subspaces through the associated orthogonal projectors with the operator norm (Andruchow) 2014). Unfortunately, this metric computationally expensive, and the topology it induces does not provide the necessary conditions derive Wasserstein metrics, namely, the separability. the following proposition, we define Grassmann manifold with the necessary conditions to derive Wasserstein metries.
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Proposition 4. Let N be fixed, and (H) denote the set of all closed vector subspaces of a (possibly separable Hilbert space H having dimension at most r. Endow G-(H) with the well-
defined metric:
dg (., E (H) x G.(HL) - || - Pyllus E RR+
(23)
where Pu is the orthogonal projector onto U, and ||-||xs the Cilbeet-Schhidi norm. Then (G, dg) a separable complete metric space.
Proof. Before the main proof, we investigate the properties of an inclusion map, which is useful for deter-
mining the metric and completeness properties.
The map i : V E (H) + Pv HS(H),, which associates to any subspace the orthogonal projector onto itself, is well defined and a one-to-one inclusion.
Lemma 1.
Proof. Since any V E G.(H) is finite dimensional, it is closed, and the orthogonal projector Py is a well- defined bounded linear operator by consequence of the Hilbert projection theorem. Furthermore, since H is separable. it admits an orthogonal basis, respecting the orthogonal decomposition H. = V D V Since dim(V) S and by invariance of the Hilbert-Schmidt norm to change of basis, || |Py ||x is finite, more precisely: | = dim(V) < r. Furthermore, for any V + V E (H),, Pv # Per due to the orthogonal decomposition H = vn v @ VV/VVV) @ //VVVV) @ (VUV)..
Lemma 2. P, (Pv V e (+))) is closed subspace of HSS(H) for the topology induced by the Hilbert-Schmid norm.
Proof. First notice that, P, c HS(H) and HS(() is Hilbert space, thus complete. Consider a sequence (Pa)nEN P, converging an element P HS(H) (i.e. ||Pn pllns 0), let's prove that P P,..
Since P E HSHH), follows that the adjoint operator P E HS(H) exists and since ||P P Ils || Pallss + O, the operator P is self-adjoint / = Pa Furthermore by composition p E HSS(H), and:
|| - plax < p2 2 HS + || Palls + +I - Pl|æs
(24)
< p2 P l + ||,, - Pllus
(25)
S P, Pllss(1 + || + ||Pr||H))
(26)
Since 1|,,, - pllxs + 0, it follows that p2 P. meaning that is an orthogonal projector. Let V denote the closed vector subspace associated to P. Since P is an orthogonal projector with a finite Hilbert-Schmidt norm, V is finite dimentional, and dim(V) |||| S r, as I I for any n N. Thus P E Pr. indicating that P,, is closed subset of H.S(H).
Main proof. Since the map i, defined in Lemma 1 is a one-to-one inclusion into the space HSHH), the metric derived from the Hilbert-Schmidt norm (Il-I induces metric onto the space G-H) Furthermore, since P, = (Py VE E G, (?t) is a closed subset of a complete space by Lemma 2 it is complete. Hence, the metric space (H), dg) is complete. Lastly, the space H(S(H) is separable it is homeomorphic H @ H. which is separable space as the tensor product of the separable space H. Hence P, C HS) is also separable by inclusion. Finally, the metrie space (G, dg) is separable and complete, which concludes the proof.
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Table 3: Metrics on Grassmann manifold nk) n): angle-based and matrix-based formulations.
Here
M U =U.,=UU", Q= VV and $ = VM where U, V are orthonormal bases.
Metric
Angle formulation
Matrix formulation
1/2
Geodesic (canonical)
dgeo
0
dgeo arccos (S)
1/2
Chordal
dehord
Σ
sin 0:
dehord v2 [P - -Q11r
1/2
Procrustes
d dproc
2k 2
Σ
cos 0₃
dproe /2k: -rt(()
Binct-Cauchy
dBC
II
cos 0g
dBC VI 2
Martin
dartii
log(cos 0:)
d
log del
Fubini-Study
dys = arccos
COs 0
des = arceos( (eerco))(dee(M)))
1=1
Spectral (max)
dmax
max 02
dmna arccos((S)
k
Nuclear (sum)
dnuc
0₂
dnuc tr(arccos(S))
C.3 ALTERNATIVE METRICS ON GRASSMANM MANIFOLD
Principal angle definition. Considering the real vector space (R",(.,)), let G(k, n) denotes the Grass- mann manifold of all vector subspace of dimension k. Many metrics the Grassmann manifold are based on the principal angles between subspaces. Formally, let U1 V C Rº be two k-dimensional subspaces, the principal angles, 0 0, I 02 I ... 0z 214 between 2/ and V are defined recursively by
u LT, v U.
cos 0g
u v, s.t aell VEV |||||= |v|==1
Ei
where uy and by are the previously chosen principal vectors.
Computation. Let U, V E Bank be orthonormal basis matrices of U1 V. Let M UTV and W EZ be the singular decomposition of M, the the principal angles are given by
cos 0; (M) E.: Vi E [K].
and the corresponding principal vectors are
u
Uw, V. Vz,,
where w, and 2; the i-th left and right singular vectors of M.
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Metric based on principal angles. Table 3 provides provides a non-exhaustive list of metrics to compare subspaces of identical dimensions from principal angles. For each metric, angle-based and matrix- based formulations are provided. These metrics can be extended compare subspaces of different dimensions following the methods proposed in Ye & Lim (2066).
D SPECTRAL GRASSMAN BARYCENTER
D.1 PROBLEM FORMULATION
Computing barycenters is fundamental problem for many unsupervised methods. When data lie in a metric space, it is known as the Fréchet mean problem. It involves identifying an element that minimizes weighted sum of distances to the observations. Formally, given the importance weights Y N assuming (A1) (A3)
for p=2 in Theorem 1 we aim to solve:
arg min
Σ 7ads(T,Ty) ?
(27)
(E).(H) RE(N]
By construction of ds. problem corresponds to the estimation Wasserstein barycenter over a set of finite measures with support a manifold embedded in a (possibly infnite-dimensional) Hilbert space. From a theoretical standpoint, the existence (and uniqueness) of Wasserstein barycenters has been established in several settings, including continuous measures (Aguch & Carlier 2011), and discrete measures finite- dimensional Euclidean spaces (Anderes et al. 2016). and measures on geodesic spaces (Le Gouic & Loubes, 2017). In Han et (2024), the authors address the case of continuous measures ric spaces. In our settings, we assume the existence of a barycenter in the closure of S,),) see discussion on the extension general operators in section 3 A formal proof would require extending previous works to finite measures on manifolds in infiniteddimensionaa Hilbert spaces.
infnite-dimmenionnll met-
From computational standpoint, problem 27 is closely related the free-support Wassertein barycen- ter estimation, which aims at optimizing the support and, optionally, the mass of the atoms parametrizing the barycenter. State-of-the-art algorithms typically rely on coordinate descent scheme (Wright 2015))
alternating between transport plan computation and measure optimization with strategies including gradi-
ent descent (Cuturi & Doucet 2014), fixed point iteration Selvarez-Estebaa et al. 2016 Lindheim 2023).
stochastic optimization (Claici et al. 2018 Li al., 2020), or proximal operators (Qian Pan [2021). In
our context, on the measure's 's support, i.e., the barycenter's 's spectral decomposition, must be optimized as the eigensubspaces' dimensions condition the masses according the embedding map in eq. [T6).
A parametric problem formulation. Whenever the RKHS H. is infinite dimensional (the finite case is discussed in appendix [D3),, the Fréchet mean problem (eq. (27)) is intractable in its original form. We restrained the optimization over a set of parametrized operators defined such that for any 0 2 aA, x):
To h E + (Ha,
(28)
where A E C", x E X" are state space control points, and a. B E Cuxr control parameters acting on the representer functions these operators are compact with rank at most r, further constraints on the control points and parameters are required to ensure a spectral decomposition (see Equation (2). Together with the definition of discrete optimal transport (see Section 2). it leads to the constrained optimization problem:
= {x(,, [j)]j( with x the kernel of H, i.c. Kga; ΣjEini Kr, ajt- While
a K3 I {
K = (K(2,, 1i) Maneen
arg min Σ ((C. s.t. O.P
(29)
3,K3, = 1, Vje [r]
P. E Il(T (T.), Vi É [N]
N
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where P (Pi)ee T T= (TN such that (C)( are the cost and transport matrices associated to the Wasserstein metric, ds defined in proposition between the parametric operator To and T,
D.2 BARYCENTER ESTIMATION METHOD
An inexact coordinate descent scheme. In what follows, let X' be bounded open set of Id with d N' and k : X' x X' + be a differentiable kernel. Following Cuturi & Doucet (2014), we propose an inexact coordinate descent scheme with cyclic update rule designed converge lem 29 Each cycle begins with the computation of the exact optimal transport plans P to enforce sparsity. This step is carried with the algorithm of Bonneel al. (201T), whose complexity depends on the num- ber of eigenvalues, typically small in practice (Brunton et al. (022). The subsequent coordinate updates are performed using few gradient descent steps with first-order optimizer such as ADAM (Kingma & Ba 2014). It starts with the cigenvalues X, optionally followed by the state spaces control points x, for which no optimization constraints exist. Next, the right eigenfunctions, 3, are updated only considering the normalization constraints: K3 = J E [r]. Finally, the left eigenfunctions,
stationary point of prob-
are updated considering the affine constraints: a = I, leading to an iterated closed- form projection scheme detailed in Equa- tion (33). Algorithm [ summarizes the full procedure, and further implementation details are provided in the next paragraphs. We usually repeat 10 gradient descent steps in experiments when updating X, ax, G and
x at each cycle.
Algorithm 1 Spectral Barycenter
Require: T (f.heIN & &,HH)*
1: (A, a. x) - Initialization
Operator parameters, see eq.
2: while not converged do
3: P ComputeT romputee[rampprrtt Plans (Te.
See Theorem and Section
4:
+ Update Eigent VagenValues((( P. T)
5:
if optimize control points then
6:
+- L PonloteeCootrooooiinsst P.
7:
Update EigenF unctions Fnntton 0 P, T)
Detailed in Algorithmm
8:
unctions a f Update Le f unctions 0, P,T))
D Detailed in Algorithm
9:
P - ComputeT TocaateTTranssaee vansport
return O.P
Update right eigenfunctions. We detail the Upalateiighliigenfumiiiii step of Algorithm Let X, x, and be fixed; we aim to perform minimization steps of problem 29 with regard to B3 the parameters controlling the right eigenfunctions. Each optimization step consists of first-order gradient descent step followed by a projection of each eigenfunction on the RKHS unit sphere as described in Algorithm 2
Update left eigenfunctions. We detail the UpdaateLffiiigeenunttttn step of Algorithm [ Let X, x, G and P be fixed; aim to perform minimization steps of problem 29 with regard to the parameters controlling the left eigenfunctions. Each optimization step consists of a first-order gradient descent step followed by projection onto the manifold induced by the spectral decomposition constraint: a K'K = I. Algorithm describes the optimization procedure, and the next paragraph discusses the projection.
Projection step. Let & E Caxr be the estimated parameters controlling the left eigenfunctions after a gra- dient descent step without constraints. Hence, these parameters might not verify the spectral decomposition
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Algorithm 2 UlddackighhEigge//nctcooo
Require: 0 0= (A, x, a, ,), P,T
1: while stopping criteria not met do
2: 3 Gradient descent step of J(0, P. w.r.t
3:
for i E [r] do
b. being the number of eigenfunctions Projection on the RKHS unit sphere
- 3./v 3.3
return B
Algorithm 3 UpdaceLeilEiggeaFuuciioo
Require: 0 = (A, x, a, P, T 1: while stopping criteria not met do
2: Gradient descent step of J(0,P; w.r.Q.
3:
ā (ā (ā* I) (B*K3)
> Manifold Projection, see below.
return a
constraint. We aim to identify the closest parameters, proj(ā:) E Caxr, for the RKHS metric and lying on the manifold induced by the spectral decomposition constraint. problem:
leads constrained optimization
(roj(ā)
arg min Tr ((a - a) â))
(30)
s.t. a K
Considering the real representation of the problem, it becomes convex problem, and since r & n, it is strictly feasible. Strong duality holds by Slater's constraint qualification. As the optimization function J and constraints are differentiable with respect to the KKT conditions are necessary and sufficient conditions to characterize the optimum. The Lagrangian can be expressed as:
LL, Tr ((a á) K(( @)) + n² (Re(a"K,3) K33) - 1) + (Imm("(()))
(31)
Taking Wirtinger derivative notation, the optimal primal-dual variables verify:
VaL(a, v.v) K(a â + ((p--(((T)) V v, = Re(a K33) I= V.laa, Im(a K,3)
(32)
Regarless of the rank of K, a â + B(u -iv)T 0 always verifies the first optimality equation. leads the projector:
(roj(@) = ā 3 ( -I(K3))))
(33)
D.3 CASE OF UNITE-DIMENSIONAL RKHS
Consider H. be finite d-dimensional RKHS with the orthonormal basis = (efleej---- For instance, H.
based on functional dictionary as used in extended DMD (Kutz et al. 2016). Let (H) be the set of non-defective compact operators acting H with rank most 3 d. We aim to solve:
arg min (E2, [E[M]
(Ts (
(34)
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with 7 A the importance weights, {ī, E Z, e [[]] estimated operators, and ds defined in Theorem given (0, 1) and p 2. For any compact operator acting on H with rank at most r, there exists coeffcients À E C', and control parameters of the functional basis a, B e Cdxr such that:
To h E
Σ xE|ri
Xe(fa,, NYHJa, E H,
(35)
where Sa. EEjd Ajifj Ej Pzlly. and (A, az 3) To ensure non-defectiveness of Ty further constraints are imposed on the control parameters, which leads to the constrained optimization problem:
arg min 2, s.t. a =I 3733 1, e
ΣiENN Mds(To,
(36)
Note that this optimization problem is related to the infinite-dimensionall problem defined eq. y by assuming the control points to be fixed such that the kernel matrix is the identity matrix, i.e. K I. It follows that the optimization procedure described in the case of infnite-diimensioall RKHS in Appendix D.2 also handles the finite-dimensional case.
E PROOFS OF STATISTIC RESULTS
We now prove the main statistical results in this section.
Theorem 2. Let (AI) (A3) hold with k E [2], Fx=La (X) and k(x,z) < 00 a.s. for ~ The Let
B[CH] = C. and assume that for some aE(1, 2] and (E(0, 1] it holds that [|[((C THll--HHOO and Ar(Ca) S 1/3 for E E. Given E (0, 1), if n is large enough and Are 2(++8))
algg then w.p.a.l 1-8
in the ii.dd draw of samples Dj and Dz it holds (ds(Ti, T2) ds(Ti, T2)) S 2(+ (26
Proof of Theorem 2 Without loss of generality, we can assume that the operators eigenvalues are of multi- plicity Then the discrete distribution representation of the operator Th provided equation becomes
((TT) E, j(r ((2)())))30))))
(37)
Similarly
jj - (2,
(( - Tk
(38)
Stability of the ds metric. to the Wasserstein metric:
Next by definition of ds in equation and the triangular inequality applied
|ds(fi, F2) dT( T2)l s (T(T))) + (u(T))) ((
(39)
We recall that W, ()(T))) (f+))) is defined as:
1/8
W, ((((T) =
min Pellanifem (rk)
2 Po.
1=1
where the cost matrix C = (ci,s) defined as
Gi = (() V((s),, (,,1)) 2 0,
Vi,j E [rd].
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and the set of uniform transport plans is:
uniiform (Fx) = { (PERTAY
P1 -1,
P
1 }
Tk
Tk
Then note that the transport plan Ti,i 1/rl for any i [r*] and vijj set IT uniifrrm (r Consequently
0 for any i + j belongs the
Fk 7k
1
2
TH.
(40)
TE
==1
view of equation 49 below, we prove w.p.a.l. that
max (cr,) S (x(
2
In 12(8))))
bk E [2].
iE(ra]
Then we get with the same probability
Wp(n(TT), S zāao. In(28-1)
Wk E [2].
and consequently obtain the final bound [ds T2) (Ti T2) in view of equation 39
Bounding the learning error I|T - For brevity we set || || for the operator norm ||H- and -| || for the Hilbert- Schhmrt-Schhidt norm || [|HS(H)))) For any k; we introduce the population RRR and Ridge
operators as
Tk = (C [(C + Cylle
Th (C* -
Then we have the following Bias- Variance decomposition in operator norm:
1 Tall S Th Tkil
Th - Txll
(41)
"Bias"
"Variance"
Bias term d1.
We have
01 S [T TEy + 1T T.
(42)
Next since
A&S. ST TRi get
St P Ar C+ Th.
and
T. = (C -11++~1100000 C + (c+ + + St Anz Sax
= TTh (C + (ct)
1)/2 ((t) + (C* - P.L., Ar, SmL
Therefore,
[T - Th S 7ll(C= + 711 (Ch) ||[[Ct]]]] Tilll +
p Src Are Vl
< {=
+ x | Azall
(43)
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We tackle now the second term in the right-han side of equation 42
We note first that
T Tk,y (C + 7f) C 1.
Hence we get
1 V7
VC V7
IT Tkyll S rratil(C* +>1)
|| P Azall-
(44)
Combining equation 42 equation and equation 44
with the assumption
||I[(C*) T
Tellx-- <00, we
obtain the following control on the bias:
Tk.i s 11 11/2 1 2VCH
Q1 [TT
|| Pr Azell
V7
S ///0-1)/2+ 2 2HH P Ama
(45)
Variance term 02.
We note first
Tk,y Tx (c -1)-1122 (1)112(TTe)) Txx)
Taking the operator norm, get
02 |Tk,, - Tull C + + 17) TT)1s J I| +/1)1/2(TTx, Tx)
Define B₁ (C+ + + An analysis of the variance of the RRR estimation (see Sections D.3.4. and D.4 and more specifically Lemma and the proof of Proposition 18 Kostic et al. (2023)) gives for any 5 E (0, 1) w.p.a.l. 1
a2 S - I| + (TT) - -Thill
n 1/22(1+11)22 In 1n
Vyn (1+
(Bk) 0z, (B²) o² (B&)
In8-1.
(46)
Combining the previous display with equation 45
we get w.p.a.l. 1 1
Th Fell S 7 (0-1)/2 1)/2
n 1122700+11 1)/2 1)/2 In5-1
2 VCH P Aull + Van
(B() 1+ o2 (Bx) oz (Bk)
Ing-1
(47)
Since we assumed that the spectrum of A₁ decreases exponentially fast to 0, that is Ar. log
4a+ and
assuming in addition that the gap gapr. is bounded away from O. Then, for E (0, 1) small, the dominating terms the above display are the first two terms and we propose to balance y using only those two. Hence
we get for y x n aाश w.p.a.l 1 1-8 5
[Te Tx S 20+35 1
(48)
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Perturbation bounds. For simplicity we assume here that the all the eigenvalues admit multiplicity By standard Davis Kahhs perturbation argument, we get
104 D s ||
7
71
17 T
[|&: - Edll - ||a
gap;
17 71
11 - T S ||
gap;
Final Bound on the metric. An union combining equation 48 for any l E [N]. we get w.p.a.l. 8 that the condition in equation 50 in Lemma 5 is satisfied with
Eo E1 = Ta In(N8")=:: =: En).
Proposition guaranttess w.p.a.l. 1- 8 that for any E [N], the operators Te, T. with corresponding spectral (((k)
decomposition (1! (,(k) and (D! p!()): Vi E [ra].
11 (k)
[d (1v) p(()),, (D. p()))) < 2
(o),
Vi [rx]. Ak E [N].
(49)
gap,
(R)
|
E.I AUXILIARY RESULTS
We propose a control on the metric (.,... For simplicity, we assume that all the eigenvalues of the Koop- man transfer operators are of multiplicity 1.
Proposition 5. Lef So, 81 E (0, be absolute constant such that, for any E [r].
11: 0,
|s Eo and [P. P:ll S lus
||
E1-
(50)
|
gap;
Then have for any + j [r]
[d,(1;, Pt), (vg, 12)) - d, (Dz, P.), (2,, P3D1
| walll
Ile, ||ll |z
||ا
$ 222
Eo +
E1
gap;
gap,
Similarly for any E [r]
|| [ii
| gap,
dn,(,, P.), (D,, s 2v2
Eo
+
E1
Proof of Proposition The metric d, a convex combination of two parts.
We focus on the distance between generator cigenvalues, which is the same as the polar distance doot be- tween transfer operator eigenvalues. Similarly as above, we have by the triangular inequality
[dea drat(D, < [I(T,w) (f,0)lla II(T',w w/) (f',"') (2))))l2
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Using Lemma we get that
1w-1 6/2
[I(T,w) - (f, 112 S I0 812 + aresin
(51)
4/2/10/
Assume the relative eigenvalue error is small:
|// E
1 2
(52)
|
Under equation 52 we have |0] 2 (1 - E)|pl and therefore
D12
E2
u =
4|/1|/1
4(1 -
1,
so the argument of arcsin(-) lies in (0, 1) as required. Morcover, since arcsin(z) Lipschitz near 0 and arcsin(x)) S (r/2)w for all x E u), we get
|1 15-0je 6/3
π 10 D12 S
aresin
4|//188
2 4|21|01
Hence
[I(T,w) - 1), 10 8)p(1 + 8|0|188 1u 2ip(1
8(1 E)|u1
and therefore
[I(T,w) (f, (1, S | -- O 1 8(1 5)1-2 S v2 v D1 since |0| < for all transfer operator eigenvalues.
(53)
|
Apply the same bound to (v', 0') and combine with the first display obtain, for each matched pair of eigenvalues,
[dn - dona (o, 0,0') V 01
[v]
|
Now apply this inequality to every eigenvalue pairs + j E [r]. In view of equation 50 we get
(usat vÿ) - doal (o,, 23)1 S
en.
Vi E [r].
(54)
For the Grassmanian part, we have for any i + j [r]
pP: Py11 - PI: P31| II,, P. - (P, P3)l| 2
| ||
|ks ||ااا| gap,
E1.
gap,
Combining the last two displays gives the first result. The second result follows from a similar and actually simpler argument.
Lemma 1. Let 2, ric and 22 rgei02 be complex numbers in polar form with r1,72 2 0 and
0,, O2 E [0, 2Tr. Then
2 - 221 = (r1 - r22) 2r1r2 (1- - cos(O: - 02)) = (r1 - r2)² + 4r ai (0) 02
2
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Proof. Write the difference and compute its squared modulus:
21 22 2 lr r2e (rie²) r22 (ri rge -12))
Expanding yields
121 11-20 z2|² = - r172 (e (0(00-0))) te- -((11)0))))
Using edd = @ with 81 f2 gives
2
221 22 2 = rî + +r2 2r cos - -22).
Rearrange the first two terms as perfect square plus a correction:
rf r 2r1r2 cos o (rî 3 2rir2) + (1 cos ()) (ri r22)² + 2rir2(1 cos $).
Finally, apply the trigonometric identity - cos r = 2 sin 22) to obtain
2r1r2(1 - cos Φ) 4r sin 1
2
which yields the claimed expression.
F COMPARISON WITH OTHER SIMILARITY MEASURES
F.1 EXPERIMENT PROTOCOL
Simulated system and shifts. We consider a referent linear oscillatory system that is the sum of two simple harmonic oscillators with frequencies 0.5Hz and 1.0Hz, respectively, with noisy trajectory of length 4001 samples sampled at 200Hz, which is an additive Gaussian noise with standard deviation of le 2. We compare the Koopman operator of the referent system with those of shifted systems according to four scenarios:
(a) Frequency shift, changes the IHz harmonic frequency from 0.6Hz to 2.5Hz in 39 evenly spaced frequencies.
(b) Decay rate shift, changes the 1Hz harmonic decay rate from -0.3 (diverging) to 3.0 (converging) in 67 evenly spaced rates.
(c) Subspace shift (rank) gradually transforms the HHz sine wave into HHz square wave signal using Fourier Decomposition of square wave signal with increasing order up to 50. Series formulation of a square wave signal: xt) = Ex 2n+I sin ((2n + 1)t).
(d) Sampling frequency shift where the system is sampled different sampling frequencies ranging from 100Hz to 300Hz instead of the reference 200Hz. Performed in 19 evenly spaced sampling frequencies.
Koopman operators are estimated from sampled trajectories in each scenario with the RRR method (Kostic et al. 2022). We consider the linear kernel, the context (sliding window) set to one second, the operators' rank is always fixed to twice the number of harmonic oscillators, and the Tikhonov regularization is set to le
Compared similarity measures. We consider our proposed metric SGOT set with = 0.5. SOT, an OT-based similarity comparing eigenvalues (Redman et al. 2024). GOT, an OT-based similarity compar- ing eigensubspaces with Grassmannian metric and weighted by the normalized eigenvalues (Antonini & Cavalletti 2021). Note that compared to its theoretical definition, we extend the similarity to non-norma operators by taking the absolute value of eigenvalues. We also included the metrics induced by the Hilbert- Schmidt and Operator norms, and the Martin similarity (Martin 2002), which compares poles LDS trans- fer functions.
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0.0.1
0.2
7= 0.3
0.4
7=0.5
7=0.6
7= 0.7
0.8.8
0.9
Mebrc value
0.8
0.8-
0.4
0.4
02-
0.0
0.0
1.0
1.5 2.0
25
20
40
80
80
100
100
150
200
250
300
(a) Frequency (Hz)
(b) Decay rate
Operator rank
(d) Sampling frequency (Hz)
Figure 6: Influence of the y parameter in SGOT under four scenarios of shifts of linear oscillatory system: (a) frequency shift, (b) decay rate shift, (c) operator rank/subspace shift, (d) sampling frequency variation. scenarios (a,b,)), metric values are normalized by their maximum.
F.2 ABLATION STUDY FOR PARAMETER y OF SGOT
Following the same protocol presented in the previous paragraph, we compare our proposed metric SGOT with the parameter controlling the balance between eigenvalues and eigensubspaces 1) ranging in [0.1, 0.2, ...,.......... wise linearly, where the ascent gets steeper for scenario (a.b) as decreases (eigensubspaces have more weights the cost function). For scenario (c), SGOT behaves similarly for all 7,. Finally, SGOT becomes slightly more sensitive to the sampling frequency 17 decreases. In (d), the metric scale is not normalized, and the metric values remain relatively small.
Results are presented in Figure 6 In scenarios (a,b,c) for any 7, SGOT behaves piece-
G MACHINE LEARNING ON DYNAMICAL SYSTEMS SYSTEMM
G.1 EXPERIMENTAL PROTOCOL
We evaluate similarity performances on a time series classification task. We selected 14 multivariate datasets from the UEA database (Ruiz al. 2021 whose main characteristics are described in Table 4 For each dataset, we estimate operators for individual time series of samples with the RRR method (oos- [ie al. 2022), with the linear kernel, Tikhonov regularization 1e an arbitrary sampling frequency Ssamp min(100, (n/2) 0.2) and a context window Wten & min(50, n/2).. Once all operators are es- timated, we perform 10-iteration Monte-Carlo cross-validatio with 0.7/0.3 train/test split without any preprocessing step. perform classification, we consider K- K-Nearest Neighbors (K-NN) estimators defined with similarities: Hilbert-Schmidt, Operator, Martin (Martin 2002), SOT (Redman et al. 2024), GOT (An- tonini & Cavalletti 202T), and our metric SGOT. Note that the initialization-invan larity has been excluded from this experiment as it relates to the Martin distance. At each cross- coos-valiiation iteration, the number of neighbors (K) and the parameter for SGOT metric are set by grid search with a 5-fold noss-validation
Binet-Cauchy simi-
the train K peaked between 1 and 10 and n (0.001, 0.01, 0.1, 0.5, 0.9, 0.99). Scores are evaluated in terms of accuracy, and training time limit has been set to 5 hours per dataset/metric pair. The experiment has been seeded for reproducibility.
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Table Datasets main characteristics: Size: number of time series, Channels: number dimensions per time series, Length: time series length, Classes: number of classes.
#Size
#Channels
Length
#Classes
ArrilFiibrillation
30
2
640
BasicMotions
80
6
100
Cricket
180
6
1197
12
EigenWorme Worms
259
6
17984
Epilepsy
275
3
206
HRing
300
4
65
FingerMovements
28
HandMovemenntirection
10
400
Handwriting
1000
3
152
26
Heartbeat
409
61
405
NATOPS
360
24
51
SellRegulationSCPI
561
6
896
StandWalkJJum
27
4
2500
WWWaeGGesturel.ibraa
440
3
315
G.2 LINEAR KERNEL: ADDITIONAL RESULTS
Classification accuracy table. addition to scores comparison plots between our metric SGOT and com- petitive similarities in the main body (see Figure 3). Table provides mean and standard deviation of accu- racy scores per dataset and metric computed over the iterations. Our metric SGOT is the best performer all datasets, followed by GOT, another OT-based metric that only refers to eigensubspaces in its cost func- tion. Also, SOT, a third OT-based similarity comparing operator, only from eigenvalues, performs poorly. Incorporating eigenvalues and eigensubspaces within the cost function improves performance numerous datasets. By being more conservative (see Figure D.. Hilbert-Schhidt and Operator underperform compared times Heartbeat. Lastly, Martin distance performs poorly and
SGOT. Note that the Operator fails on some datasets due to its ill-definedness in some settings.
Table Classification accuracy scores. Transfer operators are estimated with the finite dimensional linear kernel. Datasets on rows and similarities on columns. Best and second best performers are highlighted. Accuracy scores are denoted: < mean + std >.
Hilbert-Schmid
Operator
Martin
SOT
GOT
SGOT
AtrialFibrillation
0.31 + 0.07
0.32+
0.27 + 0.09
0.24 + 0.14
0.4 + 0.12
0,44 + 0.13
BasicMotioos
0.48 + 0.15
0.51
0.3 0.06
0.35 0.1
0,8 0.07
0,93 0.05
Cricket
0.33 0.05
0
0.07
0.03
0.11 = 0.04
0.63 = 0.04
0.85
0.05
ERing
0.79 -
0.04
0.15 0.04
0.39 - 0.04
- 0.01
0.87 0.03
WigenWorms
0,6 0.04
0.04
0.57 0.06
0.04
0.03
Epilepsy
0.46
0.02
0.34 0.04
0.78 + 0.04
0.93 0.03
FingerMovements
0.51 0.05
0.04
0.51 + 0.05
0.51 + 0.05
0.57 0.03
HandMovemen Direcciion
0.23 0.04
0.00
0.27
0.04
0.21 = 0.05
0,24 0.05
0.29 0.03
Handwriting
0.12
0.12
0.05
0.01
0.05 - 0.01
0,21 0.02
0.42 0.02
Heartbeat
0.7
6
0.71
0.04
0.69 = 0.02
0,7
0.73 + 0.04
NATOPS
0.76
0.73
0.25 1 0.04
0.41 0.04
0.74
0.77 + 0.04
SelfRegulationSCPP
0.57 + 0.02
0.56 +
0
0.57 + 0.05
0.0++003
0.61 0.02
JuandWalkJump
0.5 + 0.15
0.41 +013
0.39 + 0.13
0.3
0.69 0.16
U UWaveGestureLibrary
0.24 0.04
0.21 0.05
0.13 - 0.02
0.47 0.03
0.64 0.04
avg. rank (lower is better)
3.29 1.02
3.92 1.1
5.3 1.31
4.49 - 1.15
2.66 + 1.18
1.34 0.79
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Computation times. During the classification experiment, kept track all metric computation time, which we average per metric in Table Operator norm is the least efficient metric, followed by the Hilbert- Schmidt. The most efficient similarities are Martin and SOT; however, they performed poorly. SGOT and GOT are slightly less effective than Martin and SOT but much more efficient than Hilbert-Schmide and Operator.
Table 6: Average computation time per similarity all validation folds. Hilbert-Schmidt
Operator
Martin
SOT
GOT
SGOT
4.96ms
13.04ms
0.02ms
0.03ms
0. 04ms
0. 1.ms
Critical diagram difference. Considering results from all 10 iterations of the Monte Carlo cross- validation, we compute the critical diagram difference to statistically compare all metric performances based rank. The diagram depicted in Figure The test significance level is set to 0.05. We use Friedman's test to reject the null hypothesis (All metrics' performances are similar) and compute the critical differences using the Nemenyi post-hoe Results show that SGOT is the best performer and statistically different from the second performer (SGOT).
3
2
Martin SOT
2.64
1.3393
SGOT
4.4829
2.6571
GOT Hilbert-Schmidt
Operator
3.9214
3.2929
Figure 7: Critical diagram difference for comparing metrics' performances on a classification task. The classifiers are K-NN defined with the metrics: Hilbert-Schmidu, Operator, Martin, SOT, GOT, and SGOT (ours). Computed from the performance of all 10 iterations of the Monte Carlo cross-validation. The test significance level is set to 0.05. the null hypothesis is rejected with Friedman's 's test, and the critical differences are computed using Nemenyi post-hoc test.
2D T-SNE embeddings. We illustrate the dimensionality reductions capabilities of the different similarity measures. We selected 5 datasets from fields including human activity recognition, motion recognition, and biomedical applications. For the 5 selected datasets and all similarities, dataset samples are embedded as 2D vector with the T-distributed Stochastic Neighbor Embedding (T-SNE) Maaten & Hinton (2008) method fitted the cross-distance matrix estimated with the similarities: Hilbert-Schmicc Operator, Martin, SOT, GOT, and SGOT. Figure displays the embeddings for all datasets and metrics. On the Eigenworms and Epilepsy datasets, Martin ill-defined, and the similarity values cannot be computed. No clusters classes can be identified for Hilbert-Schmidt, Operator, Martin, and SOT. Regarding other OT- metrics, GOT better identifies classes; however, they do not form distinct clusters obtained with metric SGOT.
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Hilbert-Schmidt
Operator
Martin
SOT
GOT
SGOT
BasicMotions
Cricket
ERRMA
EigenWorms
Epilepsy
Figure T-SNE 2D-embeddings the classification datasets: BasicMotions, Cricket, Ering, Worms, Epilepsy based on similarities: Hilbert-Schmidl, Operator, Martin, SOT, GOT and SGOT (ours). Each point represents a dataset sample (a time series) whose color corresponds to its class. The Martin similarity ill-defined on Eigen Worms and Epilepsy datasets; thus, the corresponding T-SNEs are missing.
G.3 GAUSSIAN KERNEL: RESULTS
The experimental protocol follows the same procedure as for the linear kernel de- scribed in appendix G.1 with the following modifications:
Experimental setup.
1. Estimating transfer operators with Gaussian kernels: The linear kernel is replaced by the Gaus- sian kernel, x(x, y) e exp([|> - /o²) such that for each datasets the kernel's scale parameter is set according to the heuristic:
/ (number of dimension) (context window length)
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1.0
SGOT
SGOT better
better
better
better
SGOT
0.6
0.2
fo aoore
0.2
Hilbert-Schmiat
Martin
SOT
GOT
Figure 9: Classification performance (accuracy) comparison between SGOT and competitive metrics for transfer operators estimated with Gaussian kernels. Each point represents dataset accuracy, with SGOT on the y-axis and the competing metrics on the x-axis.
2. Experiment scalability. Due to the computational cost of estimating transfer operators in infinite- dimensional kernel spaces, experiments are restricted the five smallest datasets: BASICMO- TIONS, ERING, EPILEPSY, FINGER Carlo cross- validatai is limited to 5 iterations, and the Operator metric is omitted.
and NATOPS. Additionally, the nested Monte-
Table Classification accuracy scores. The transfer operators are estimated with the infinite dimensional Gaussian kernel. Datasets on rows and similarities on columns. Best and second best performers are high- lighted. Accuracy scores are denoted: < mean > < std >.
Hilbert-Schmidt
Martin
SOT
GOT
SGOT
BasicMotions
0.26 + 0.17
0.77 0.06
0.87 + 0.05
0.69 = 0.14
0.95 + 0.02
ERing
0.74 + 0.07
0.22 0.05
0.38 0.05
0.96 0.01
0.98 0.02
Epilepsy
0.31 + 0.02
0.01
0.02
=
0.95 = 0.02
FingerMovements
0.53 0.06
0.03
0.53
0.06
0.53 = 0.01
NATOPS
0.59 + 0.06
0.25 0.02
0.35 - 0.02
0.78
0.03
0.8 0.05
avg. rank (lower is better)
3.74 + 1.27
4.02 + 0.98
3.28 + 1.15 2.48 1.19
1.48 + 0.7
Results. Table 7 reports the average accuracy over the 5-iterations of Monte-Carie cross-validation for each dataset/metric pair. Figure summarizes the comparative performance of our metric, SGOT, against several competitive alternatives: Hillert-Schhide Martin, SOT, and GOT. Finally, Figure T0] presents the critical difference diagram obtained using a significance level of 0.05; the null hypothesis was rejected using Friedman's 's test, and pairwise comparisons were performed with the Nemenyi post-hoe test.
As in the linear-kernel setting, SGOT achieves the best performance across all five datasets, followed by GOT, which compares eigen-subspaces critical difference diagram, which shows that SGOT statistically outperforms the other metrics. Moreover, in comparison to the linear kernel case, when transfer operators are estimated in an infmite-dimensional kernel space, classification accuracy increases with SGOT but decreases with the HfllerttSchhiie metric. This highlights the limitations the Hilbert-Schhide metric for high-dimensional representations, in contrast to the greater robustness exhibited by SGOT.
optimal transport. This ranking is further supported by the
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Martin 4.0200
1,4800
SGOT GOT
Hilbert-Schmic
3.7400
2,4800
3,2800
SOT
Figure 10:
Critical diagram difference between metrics when transfer operators are estimated with the infinite dimensional Gaussian kernel. The classifiers are K-NN defined with the metries: Hiilerrt-Schmid, Martin, SOT, GOT, and SGOT (ours). Computed from the performance of all 5 iterations of the Monte Carlo cross-valie cross-valiattio. The test significance level is set to 0.05, the null hypothesis is rejected with Friedman's test, and the critical differences are computed using Nemenyi post-hoc test.
G.4 DEEP-LEARNING EMBEDDINGS: RESULTS.
Experimental setup.
The experimental protocol follows the same procedure as for the linear kernel de-
scribed appendix G.T with the following modification:
Transfer operator estimation using kernels defined by learned deep features: The linear kernel is replaced by kernel of the form k(z, y) = (0o(x), (3)) where Øg X' + Rd is an embedding map parameterized by a neural network. The network architecture is a Multi-Layer Perceptron (MLP) with two hidden layers of dimension 128, 32-dimensional output layer, and LeakyReLU activations with negative slope 0.01. For each dataset, after data augmentation using the context window, embedding map is trained on the training set by following the strategy of (Kostic ct al. (024b) which designed to learn invariant representations of mmm-hooogeneous stochastic dynamical systems. Gradient descent is for 4000 iterations; each iteration, window of length (200, n.samples/2) is randomly extracted from a time series sampled uniformly from the training dataset.
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1.0
SGOT better
SGOT better
SGOT better
90
better
SGOT
0.6
0.2
fo aoore
0.2
1.0
0.2
Hilbert-Schmiat
Operator
SOT
GOT
Classification performance (accuracy) comparison between SGOT and competitive metrics for transfer operators estimated with kernels defined by deep-functions. Each point represents dataset accuracy, with SGOT on the y-axis and the competing metrics on the x-axis.
Figure 11:
Table 8:
Classification accuracy scores. The transfer operators estimated with finite kernel defined with deep-functions. Datasets on rows and similarities on columns. Best and second best performers are highlighted. Accuracy scores denoted: mean + < std >.
Hillert-SShhiic
Operator
Martin
SOT
GOT
SGOT
AtrialFibrillation
0.22 + 0.21
0.18 0.1
0.24 t 0.14
0.24 : 0.09
0.2 0.21
0.21
0.36 =: 0.14
BusicMotions
0.28 + 0.05
0.34 0.07
0.81 # 0.06
0.68 = 0.11
0.92 0.06
0.9 # 0.08
Cricket
0.82 + 0.05
0.73 + 0.04
0
0.4 0.1
0.72 0.07
0.72 = 0.07
ERing
0.31 + 0.07
0.31
0.39 0.04
0.55 0.06
0.57 0.04
EigenWonee Worms
0.78 +: 0.03
0.74
0.72 = 0.06
0.8 - 0.06
0.82 0.05
Epilepsy
0.7 0.06
0.05
0.91 0.06
0.88 0.04
0.92 0.05
FingerMovements
0.52 0.04
0.05
0.48
0.51 0.04
0.51 0.03
0.56 0.04
HandMovement
0.25
0.05
0.23 0.05
0.28 + 0.03
0.26 0.04
0.04
Handwriting
0.07 0.02
0.00
0.07 = 0.01
+ 0.02
+ 0.01
Heartbeat
0.7 0.02
0.09
0.72 0.04
0 0.04
0.71 = 0.03
0.73 0.04
NATOPS
0.41 1 0.08
0.00
0.3 133000 0.06
0.33 - 0.07
0.45 - 0.05
0.47 0.03
SeIfRegulationSCPI
0.77 + 0.04
0.07
0.51 + 0.03
0,52 + 0.04 0.04
0.77 + 0.03
0.78 0.03
Suam
0.16
0.17
0.27 0.1
0.22 - 0.08
0.36 0.09
UWaveGestureLibrryy
0.74 0.05
0.59 + 0.02
0.29 = 0.04
0.66 = 0.05
0.66 0.05
avg. rank (lower is better)
3.33 1.56
1.27
5.06 1.48
3.84 +
1.33
1.71 0.77
Table reports the average accuracy over the 5-iterations of Monte-Carie cross-validation for each dataset/metric pair. Figure [T summarizes the comparative performance of our metric, SGOT, against several competitive alternatives: Hilberr-Schmine Operator, SOT, and GOT. Finally, Figure 12 presents the critical difference diagram obtained using significance level of 0.05; the null hypothesis was rejected using Friedman's 's test, and pairwise comparisons were performed with the Nemenyi post-hoe test.
Results.
When transfer operators are estimated using kernels built from learned deep features, the performance of all metries decreases compared to linear-kernel setting. This drop attributable the limited size of the datasets, which constrains the training of sufficiently representative deep features, well-documented issue in operator embedding learning Lusch et al. (2018). Nevertheless, SGOT still achieves the highest performance 11 out of the 14 datasets and ranks second or third on the remaining ones. This advantage is further confirmed by the -iffecal-iiferrecce diagram, which indicates that SGOT statistically outperforms the competing metrics.
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3
2
Martin
5.0571
1.7071
SGOT
Operator SOT 3,8357
4.1357
2,9357
GTT
Hilbert-Schmidtt
Figure 12: Critical diagram difference between metrics when transfer operators are estimated with finite dimensional kernel defined with deep-functions. The classifiers are K-NN defined with the metrics: Hilbert- Schmidt, Martin, SOT, GOT, and SGOT (ours). Computed from the performance of all 5 iterations of the Monte Carlo cross-validation. The test significance level is set 0.05, the null hypothesis is rejected with Friedman's 8 test, and the critical differences are computed using Nemenyi post-hoe test.
G.5
SENSITIVITY ANALYSIS TO THE COST WEIGHTING PARAMETER
Goal. We evaluate how the weighting parameter (0, affects the classification performances. This parameter controls the trade-off between the eigenvalue term and the eigensubspace term in the cost the Wasserstein SGOT metric defined in Theorem
Experimental setup. The experimental protocol follows the procedure described in appendix G.1 Trans- fer operators estimated using linear kernels, and classification scores computed for SGOT metrics with 7) E {0.001, 0.01, 001,0.1... 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99). All results are obtained using a 5- iteration Monte-Carlo cross-validation.
reports the mean classification accuracy for each value of The vertical dashed line marks the heuristic value îj = (1+ = (1+ (222))) where fo = fsamp/ 2 is the Nyquist frequency. This value corresponds to equal weighting of the eigenvalue and eigensubspace terms the cost of the Wasserstein SGOT metric (see Theorem I1.
Results. Figure 13
Overall, Figure 13 shows that SGOT's performance varies smoothly with y), with optimal values typically favoring stronger emphasis on the eigensubspace cost. This trend suggests that the search range for can be substantially reduced in practice. In particular, the heuristic v consistently lies near regions of high accuracy, and optimal values generally fall within (0, 7) across datasets. Consequently, provides both practical initial choice and a principled bound for restricting the grid-search budget.
H BARYCENTER OF DYNAMICAL SYSTEMS
H.1 INTERPOLATION BETWEEN 1D DYNAMICAL SYSTEMS
Experimental settings. In this experiment, we compare the interpolation between dynamical systems through weighted Fréchet barycenters of their Koopman operators, estimated with linear kernel, for dif- ferent metrics. The two systems are linear oscillatory systems, each being the sum of two simple harmonic oscillators with different frequencies and decay rates, and additive Gaussian noise. The first system (O) combines a convergent low frequency oscillator (w = 1.7Hz, = -0.2, amplitude=1.0) with a divergent high frequency oscillator (w 4.7Hz, p 0.2, amplitude=0.2). The second system T()) is reversed; it combines a divergent low frequency oscillator (w 0. 0.7Hz, p 0.2, amplitude=1) with convergent high
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AtrialFibrillation
BasicMations
Cricket
EigenWorms
1.0
accuracy
0.0
0.5
0.5
1.0
Epilepsy
ERing
FingerMovements
HandMMovementiirection
1.0
1.0
1.0
0.8
0.8
0.8
0.8
accuracy
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.0
0.0
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Handwriting
Heartbeat
NATOPS
SelfRegulationSCP1
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
accuracy
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0.0
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
StandWalkJump
UWaveGestureLibrary
accuracy
avg. accuracy std. accuracy heuristic
0.4
0.0
0.5
1.0
n
Figure 13: Sensitivity of SGOT's classification accuracy to the parameter 1 across all datasets. The dark blue curve shows the mean accuracy over the 5-iteration Monte-Carle crossvvaidattonn and the light blue band indicates one standard deviation. The vertical dotted red lines mark the heuristic values n = (1 + fsamp/ (2v 2) which give equal weight to the eigenvalue and eigensubspace cost terms in the SGOT definition (see Theorem T].
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frequency oscillator (w 11. p = 0.2,, amplitude=1). Both systems are noisy with a Gaussian noise with variance o² = le - 4. The systems Koopman operators are estimated with the RRR methods (Kostic jet all (022) from trajectories of length 5000 samples at 800Hz. RRR estimator is set to estimate a rank 4 operator with context window of 400 samples, a linear kernl,, and Tikhonov regularization of le 8. The interpolation controlled by a ratio parameter "/ going from 0 to 1 in 0.1 steps. At each interpolation step, the weights in the Fréchet mean (see equation 7 are (1 7, 7). We compare (a) the Hilbert-Seee -chhett metric without spectral decompasition constraints given by Thr = (1 - -)) + -T(() (b) the Hilbert-Schmid metric with spectral decomposition constraints, and (c) our proposed metric SGOT. For (b) and (c), the barycentric operators are estimated with the proposed optimization scheme described in appendix In both cases, the initialization of the barycenter corresponds to the average of eigenvalues and eigenfunctions. For the Hilbert-Schhide (b), the barycenter optimizer is set with a 3e 5 learning rate, a maximal number of it- erations of 2000, with gradient descent per coordinate at each iteration, the stopping criteria corresponds to a consecutive metric error lower than 1e- - 6. For the SGOT (c), 17 0.9, and the barycenter optimizer is set with a l 1 2 learning rate, a maximal number of iterations of 200, with 10 gradient descent per coordinate at each iteration, the stopping criteria corresponds
consecutive metric error lower than le 6. Finally, for displaying the predicted signals from the interpolated barycenter in Figure 4 all predictions started with the same initialization set, being the first 400 samples of linear system that is the sum of 4 harmonic oscillators of the systems to interpolate.
Additional results. For all constrained Seihbert-Shhmitt (b) and SGOT (c) interpolated barycenters, we kept track of the decay rate and frequency of the two associated harmonic oscillators, the loss values, and the computation time. Figure 14 displays the normalized losses decrease per gradient descent step for cach metric and interpolation step. The representation is in gradient descent step as the number of iterations and gradient descent step per cycle differ from metric to metric. In Figure [3] we display the decays and frequen- cies of the interpolated barycenters. In particular, Figure 14 shows that the barycenter algorithm has con- verged for any metric and interpolation step. However, Figure 13 shows that constrained Hilbert-barycenter (b) remains stuck in a local minima close to the initialization. In contrast, the SGOT barycenter perfectly (linearly) interpolates the decay and frequency between the source and target systems. Furthermore, the average computation time per gradient descent step is 13. for the constrained Hillert-Schhiid while being 2.29ms for our metric SGOT, meaning that the barycenter algorithm is approximately 6x faster with the SGOT metric compared to the Hilbbet-SSchidt.
H.2 FLUID DYNAMIC INTERPOLATION
Experimental settings. We aim to compute the barycenter of two fluid dynamics systems. To that end, we consider the Flow past a bluff object dataset (Tali et al. 2025), which gathers trajectories of time-varying 2D velocity and pressure fields of incompressible Navier-Stokes fluids flowing around static objects. We select two trajectories, one with a cylinder object (Huggingface dataset file: harmonic/93) and the other with a triangular object (Huggingface dataset file: skeleton/48). For each trajectory, we only kept the velocity field along the flow direction, leading to trajectories containing 242 samples of 1024x256 grids, which we down-sampled grids with a 256x64 resolution. We estimate a Koopman operator with a linear kernel using the RRR method from each trajectory sampled at 100Hz with a context window of 1, and a Tikhonov regularization of 1. The operators are restricted to the fourth leading eigenvalues and eigenfunctions. We compute the SGOT barycenter with the optimization scheme described in Appendix D with an initialization being the average of cigenvalues and eigenfunctions. For the SGOT (c), 7 = 0.0I, and the barycenter optimizer is set with a le - 4 learning rate, a maximal number of iterations of 100, with 10 gradient descent per coordinate at each iteration, the stopping criteria corresponds to a consecutive metric error lower than le- 6.
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Published as conference paper ICLR 2026
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Figure 15: Decay rates and frequencies of the two harmonic oscillators associated with the interpolated barycenters for the constrained Hilbert-Schmiit (b) and the SGOT (c) metries. The source system harmonic oscillator is in red, and the target system harmonic oscillator is in green.
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Reviews:
[{"review_id": "l5Ni3W1Pft", "reviewer": "Reviewer_Tstr", "summary": "This paper introduces a novel metric, named the Spectral-Grassmann Optimal Transport (SGOT) metric, for comparing dynamical systems represented by Koopman or transfer operators. The core idea is to represent each dynamical system as a probability distribution over its joint spectral data, specifically its eigenvalues and corresponding spectral projectors (viewed as points on a Grassmann manifold). The distance between two systems is then defined as the Wasserstein distance between these two distributions. The ground metric for the optimal transport problem is a weighted combination of the distance between eigenvalues and a Grassmannian metric between the projector subspaces. The authors provide theoretical guarantees for the metric, including finite-sample convergence rates for its estimation. They also propose an algorithm to compute the Fréchet mean (barycenter) of a set of dynamical systems under this metric, enabling interpolation. The method's effectiveness is demonstrated through experiments on simulated and real-world datasets, where it is shown to outperform other operator-based metrics in tasks like dimensionality reduction and classification, and provides more meaningful interpolations.", "questions": "1.  How can a practitioner verify or enforce the \"Common functional space\" assumption (A3) when given a collection of time-series datasets from potentially very different underlying systems? What are the consequences for the metric if this assumption is violated (i.e., the relevant eigenspaces do not lie in the chosen RKHS)?\n2.  Your statistical guarantees are derived for the RRR estimator. Could you comment on the challenges of extending these guarantees to Koopman operator estimators based on deep neural networks, which are widely used for complex systems? Does the metric `ds` remain well-behaved if the estimated operators `T_k` come from different model classes?\n3.  Regarding scalability: What is the practical limit on the number of samples (`n`) and the rank (`r`) for which the SGOT metric and its barycenter can be computed in a reasonable amount of time? How does this compare to the scale of problems typically found in fields like fluid dynamics or climate science?\n4.  In Figure 1, the SGOT metric shows a desirable linear-like response. However, this is dependent on the hyperparameter `η`. The paper mentions a sensitivity analysis in the appendix but could you elaborate on how `η` should be chosen in practice? Does it require extensive cross-validation, and does the optimal `η` vary significantly across different tasks and datasets?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.  **Principled Metric Definition:** The SGOT metric is well-founded in theory, combining optimal transport with the geometry of Grassmann manifolds. It offers a holistic comparison of operators by considering both their spectral values (eigenvalues) and their geometric structure (eigenspaces).\n2.  **Theoretical Guarantees:** The paper provides finite-sample convergence guarantees (Theorem 2), which is a significant theoretical result that adds rigor to the proposed data-driven metric.\n3.  **Key Invariances:** The metric is designed to be invariant to the sampling frequency of trajectories and permutations of the spectral decomposition, which are desirable properties for a robust similarity measure.\n4.  **Enables Geometric Operations:** The framework naturally allows for the computation of barycenters, providing a principled way to average and interpolate between dynamical systems. The experimental results show this leads to more meaningful interpolations than simpler methods.", "weaknesses": "1.  **Strong and Potentially Impractical Assumptions:** The requirement of a \"Common functional space\" (A3) is a major limitation. It is not clear how one would verify this assumption in practice or how to choose a suitable RKHS when comparing a diverse set of dynamical systems. The paper does not address the sensitivity of the method to violations of this assumption.\n2.  **Limited Generality of Guarantees:** The statistical guarantees are tied to a specific operator estimation method (RRR) and kernel type. It is unclear if these guarantees would extend to other popular and powerful estimation methods, such as those based on neural networks.\n3.  **Scalability Concerns:** The metric computation involves calculating a cost matrix and solving an OT problem. While the paper claims it is computationally efficient for small ranks (`r`), the complexity `O(n^2 * r^2)` can be prohibitive for high-resolution data (large `n`) or when a large number of modes are needed to accurately represent the system (large `r`). The barycenter computation adds another layer of iterative optimization.\n4.  **Limited Experimental Scope for Barycenters:** The interpolation experiments are primarily demonstrated on simple 1D linear systems. The single fluid dynamics example is more compelling, but a more extensive evaluation on a wider range of complex, nonlinear systems is needed to fully validate the claimed superiority of SGOT for interpolation and averaging tasks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pCDeXbmGw8", "reviewer": "Reviewer_WDpQ", "summary": "The paper introduces Spectral-Grassmann Optimal Transport (SGOT), a metric that represents each linear operator by its eigenvalue subspace atoms and compares operators by a Wasserstein distance. The authors show how to compute this metric via kernel representations, provide statistical guarantees for finite-sample estimation, and propose a parametric formulation to compute barycenters in practice. Empirical results on synthetic systems and fluid flow data illustrate that SGOT responds to changes in frequency, decay and subspace structure compared with the baselines, better performances in ML tasks, and can produce visually meaningful interpolated modes.", "questions": "1. I agree that summing an eigenvalue distance and a subspace distance in Eq. (4) is reasonable, but it’s just one possible choice. For example, distances based on Koopman trace/determinant kernels (Fujii et al., 2017) could also be defined. Please explain why you chose the current form and mention other viable alternatives.\n\n2. Similarly, the second term in Eq. (6) is only one concrete realization of a subspace distance. Eq. (6) evaluates subspace differences as a quadratic form with kernel matrices and effectively uses a trace kernel-like representation. Why did you select this particular form? Please discuss alternatives (e.g. principal angles) and comment on related prior work (e.g. Kawahara et al., 2016 [a]), which is not cited but is relevant to distances in Koopman spectral decomposition.\n\n3. The results in Fig. 1 can be interpreted, but the current explanation lacks enough information for a reader to judge whether SGOT is truly better. In particular, please justify the claim that “grows linearly with the shifts almost everywhere” is a desirable property and explain why this behavior indicates superiority.\n\n4. Figure 5 may visualize the intermediate modes computed by SGOT, but a barycenter does not necessarily correspond to a physically realizable intermediate system. To make Fig. 5 more convincing, please evaluate the barycenter against concrete metrics that relate to physical realism. For example, mode frequencies, decay rates, and reconstruction/forecast RMSE.\n\n5. The experiments report that Koopman operators are estimated with a linear kernel (including Appendix H). Please state why you chose a linear kernel (interpretability, computational reasons, or data suitability) and discuss the relationship with non-linear kernels such as RBF.\n\nMinor points\n6. From Section 2 onward you use \\cite where \\citep would be appropriate.\n\n7. Line 308 contains two periods in a row.\n\n8. Figure 2 appears after Fig. 5 and a label seems to be “ilbert”.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "The paper proposes a coherent, high-level framework that combines spectral information (eigenvalues) with subspace geometry (Grassmann) under an optimal-transport view, which is an original and conceptually clean way to compare linear dynamical operators.\nMethodologically, the authors introduce a concrete, computable metric (SGOT), give a parametric barycenter formulation and an optimization scheme that bridges the abstract definition and practical computation.\nExperiments demonstrate the metric’s sensitivity to controlled changes, its usefulness for ML tasks (classification/embedding), and the interpolation behavior on physical flow data.", "weaknesses": "The paper’s choice of distances is only one possible design: Eqs. (4)  and (6) are reasonable, but the authors do not justify them or compare to clear alternatives. The empirical claims may need clear explanations. Finally, there are some poor presentation issues. Please see the questions below in detail.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6vBt8H9MLP", "reviewer": "Reviewer_vMWN", "summary": "This paper addresses the problem of quantifying and comparing the geometry of dynamical systems from trajectory data, a central challenge in machine learning for dynamical systems. The authors build on the Koopman and transfer operator framework, which provides a linear representation of nonlinear dynamics via spectral decomposition.\nThey propose a new operator-based metric that represents each system as a joint distribution over operator eigenvalues and spectral projectors, and defines distances between systems using optimal transport theory. The resulting metric is invariant to the sampling frequency of trajectories, computationally efficient, and theoretically supported by finite-sample convergence guarantees. Furthermore, it allows the computation of Fréchet means, enabling smooth interpolation between dynamical systems.\nComprehensive experiments on both synthetic and real-world datasets demonstrate that the proposed method consistently outperforms standard operator-based distances in machine learning tasks such as dimensionality reduction and classification, while also yielding meaningful interpolations between dynamical systems.", "questions": "1.\tTo what extent do the theoretical properties of the Koopman operator, such as boundedness or continuity, play a critical role in the proposed SGOT framework?\n2.\tRegarding the domain and codomain of the Koopman operator, what kind of function spaces are suitable for ensuring the validity of the spectral–Grassmann representation? For example, does the analysis require specific regularity, integrability, or compactness assumptions on observables?\n3.\tCan the proposed approach be extended to highly complex dynamical systems, such as stochastic nonlinear partial differential equations (SPDEs), where both randomness and infinite-dimensional state spaces come into play? If so, what are the main theoretical or computational challenges expected in such settings?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper tackles a fundamental and timely problem in machine learning for dynamical systems — defining a theoretically sound, interpretable, and computationally tractable metric between data-driven operator representations of nonlinear and stochastic dynamics. The proposed Spectral-Grassmann Optimal Transport (SGOT) framework elegantly combines spectral theory, Grassmannian geometry, and optimal transport, yielding a true metric that is invariant to sampling frequency and robust to operator estimation errors. The method is mathematically principled, supported by finite-sample convergence guarantees, and remains computationally efficient, overcoming the typical trade-off between theoretical rigor and practicality. Extensive experiments demonstrate that SGOT outperforms existing operator-based similarities in both unsupervised and supervised learning tasks, and its ability to compute Fréchet means enables meaningful interpolation between dynamical systems — a capability with broad implications for model comparison, system identification, and generative modeling of dynamics. Overall, the paper makes a significant and original contribution to bridging operator-theoretic dynamical systems and modern machine learning.", "weaknesses": "While the proposed SGOT framework is theoretically elegant and empirically strong, the paper could be further strengthened by a clearer characterization of its practical limits. In particular, it remains unclear how strongly nonlinear systems the approach can accurately handle, and how robust the method is under varying levels of stochastic noise in trajectory data. Although the finite-sample convergence guarantees are valuable, the mathematical conditions quantifying robustness to nonlinearity and noise are not explicitly derived. Providing such analyses — for example, bounds on operator estimation errors or perturbation stability with respect to noise amplitude — would help clarify the method’s applicability range and further enhance its theoretical completeness.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bG4DVfqAzD", "reviewer": "Reviewer_AGY8", "summary": "The authors proposed a representation of each dynamical system as a probability distribution over its joint operator eigenvalues and spectral projectors on the Grassmann manifold. They defined a metric between systems using optimal transport with a cost function that balances spectral information and geometric information, i.e., the eigenspace distances on the Grassmann manifold.", "questions": "(1) In Theorem 2, how would you verify the conditions $[(C_x^k)^{\\dagger}]^{\\frac{\\alpha - 1}{2}}T_k$ and $\\lambda_i(C_x^k)^{\\dagger}$ by data?\n(2) Theo theory replies on assumptions A1-A3, which is convenient to compare between the operator from different systems, but are they assured in practical cases?\n(3) When $\\lambda_j$ has multiplicity $m_j>1$, how is the space $V_j$ uniquely defined?\n(4) Do we also need to show that $V_j$ is independent of basis choice?\n(5) Is $d_{\\mathcal{S}}(T, T')$ independent of the choice of bi-orthogonal basis used to represent spectral decompositions? It is unclear whether $d_{\\mathcal S}(T, T')$ is intrinsic, i.e., independent of the choice of bi-orthogonal basis used to represent the spectral decomposition when eigenvalues have multiplicity > 1.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper introduces a new metric called Spectral-Grassmann Optimal Transport (SGOT) distance for comparing dynamical systems through their Koopman or transfer operator representations. Its main strength lies in integrating spectral and subspace information within an optimal transport framework, which is an original formulation that helps address several limitations of prior operator-based metrics. The theoretical analysis includes finite-sample guarantees, and the presentation is generally clear. The experiments provide evidence supporting the method’s potential usefulness in practical applications.", "weaknesses": "(1) All experiments use linear kernels. There is no exploration of RBF, polynomial, or learned kernels;\n(2) Assumption (A3) requires a shared RKHS $\\mathcal{H}$ that simultaneously contains the low-rank images of all Koopman operators. It is unclear how one could construct or validate such a space in real applications. Different dynamical systems may have very different scales and smoothness properties;", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)