Dimension reduction for path signatures

Christian Bayer; Martin Redmann

REVIEW 2 major objections 5 minor 5 references

Dimension reduction for path signatures

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Model order reduction can compress high-dimensional truncated signature models to a handful of states while preserving implied-volatility accuracy.

desk verdict A clean, honest application of balanced truncation to signature SDEs, with a nice nilpotency trick and strong numerics; the reduction criterion is a heuristic, not a theorem, and the paper should say so. read the letter →

arxiv 2412.14723 v3 pith:G7TETBTM submitted 2024-12-19 math.PR cs.NAmath.NA

Christian Bayer , Martin Redmann This is my paper · ORCID

classification math.PRcs.NAmath.NA MSC 60H1060L1060L9065C3093A15

keywords pathsignaturessignaturemodelsroughpathsmodelorderreductionbalancedtruncationstochasticdifferentialequationsBergomifinancial

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the main practical obstacle to signature-based models — the explosive growth of the signature's dimension with the path dimension and truncation level — can be removed by model order reduction. The setting is a signature model of a financial market, where an asset price is a linear functional of a truncated signature and the truncated signature solves a high-dimensional linear SDE. The paper constructs reduced linear SDEs of much lower dimension whose outputs approximate the original output, and demonstrates on Bergomi and rough Bergomi examples that reduced models of dimension 5 to 55 reproduce option-implied volatilities with relative errors from $10^{-2}$ down to machine precision, starting from state spaces of dimension 1365 and 3280. This would make signature models computationally tractable for simulation, calibration, and path-dependent pricing.

What carries the argument

The machinery is balanced truncation applied to the linear SDE for the truncated signature. The truncated signature $X^{\le m}_{0,t}$ of a $d$-dimensional time-extended Brownian motion solves $dX_t = A X_t \, dt + \sum_{i=2}^d N_i X_t \, dB^i_t$, where the nilpotent matrices $N_i$ encode left multiplication in the truncated tensor algebra. The paper defines the time-averaged Gramians $P = \int_0^T \mathbb{E}[X_u X_u^\top]\, du$ and $Q = \int_0^T \mathbb{E}[\Phi(u,0)^\top L^\top L \Phi(u,0)]\, du$, and constructs a balancing transformation $T = \Sigma^{1/2} U^\top L_P^{-1}$ from the factorization $P = L_P L_P^\top$ and spectral decomposition $L_P^\top Q L_P = U \Sigma^2 U^\top$, which makes the transformed Gramians both equal to $\Sigma = \mathrm{diag}(\sigma_1,\dots,\sigma_n)$. The numbers $\sigma_k = \sqrt{\mathrm{eig}_k(PQ)}$ are the Hankel singular values that rank the importance of balanced state components; dropping those below a threshold yields the reduced system (3.16). $P$ and $Q$ are computable in closed form as finite sums over powers of the Lyapunov operator because the $N_i$ are nilpotent.

What would settle it

Take a signature model with the same setup as Section 4 but with a slowly decaying spectrum — for instance, increase the dimension $d$ of the driving Brownian motion or raise the truncation level $m$ until the $\sigma_k$ values no longer drop below $10^{-8}$ by $k = 27$ — and check whether the reduced system selected at the $\sigma$-threshold still reproduces the full model's implied volatilities; if the error at the suggested reduced dimension is orders of magnitude above the paper's reported $10^{-2}$ to machine-precision range, the $\sigma$-based truncation criterion is not reliable.

Watch

Extended reading notes

Core claim

The central claim is that a truncated signature model of dimension $n$ can be replaced, with essentially no loss of accuracy, by a balanced-truncation reduced model of dimension $\tilde{n} \ll n$. The paper shows that the truncated signature of a time-extended Brownian semimartingale solves the linear SDE (3.1), whose output $Y_t = L X_t$ is the quantity of interest (for instance a model asset price). Defining time-averaged Gramians $P$ and $Q$, a balancing transformation $T$ diagonalizes both simultaneously, so the singular values $\sigma_k = \sqrt{\mathrm{eig}_k(PQ)}$ order the state components by importance; truncating the small-$\sigma$ components yields the reduced system (3.16). In the Bergomi case with $n = 1365$, reduced dimension $\tilde{n} = 5$ gives relative errors around $10^{-2}$ in implied volatility, $\tilde{n} = 11$ gives $10^{-4}$, and $\tilde{n} = 27$ is exact up to machine precision; in the rough Bergomi case with $n = 3280$, $\tilde{n} = 15$ gives around $10^{-3}$ and $\tilde{n} = 55$ reaches machine precision. The $L^2$ error of the output behaves similarly, so the approximation is strong rather than merely weak.

Load-bearing premise

The load-bearing premise is that the size of the singular values $\sigma_k = \sqrt{\mathrm{eig}_k(PQ)}$ faithfully ranks how much each balanced state component contributes to the output, so truncating the small-$\sigma$ components cannot change the output much; the paper supports this only by a bound on a single mode before truncation, not by a global error estimate for the truncated system.

Editorial extensions

If this is right

Signature-based asset price models can be simulated with dozens of state variables instead of thousands, which makes Monte Carlo pricing and calibration substantially cheaper.
The reduction is not tied to option prices: any linear functional of the truncated signature of a time-extended Brownian semimartingale can be reduced the same way, so path-dependent payoffs, sensitivities, and hedging quantities inherit the speedup.
Because the reduced output matches the full output in $L^2$, the reduced model provides a strong approximation, so it can serve as a proxy in problems requiring pathwise accuracy, such as optimal stopping or non-Markovian control.
In the numerical examples the $\sigma_k$ spectrum drops below machine precision after a modest index, indicating that the high-dimensional signature models studied here have an intrinsic low-rank structure that the balancing procedure exposes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same reduction should apply to signature models driven by more general semimartingales, including jump processes, since the paper's framework explicitly accommodates càdlàg paths; a numerical test on a jump-diffusion calibration would show whether the $\sigma_k$ decay remains as fast.
If the fast $\sigma_k$ decay observed here is generic for signature models calibrated to real data, then balanced truncation could become a standard preprocessing step for signature methods in machine learning, where the same dimension explosion limits the truncation level.
A rigorous global error bound for the truncated balanced system may be within reach by combining the nilpotent structure of the $N_i$ with existing bilinear balanced-truncation estimates; the paper itself only proves a one-mode bound.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper does something genuinely new—it applies balanced truncation to truncated signature SDEs, and the nilpotency of the signature vector fields is exploited to compute the Gramians as finite sums. The derivations in Propositions 3.3–3.5 are standard but correct, the balancing transformation is appropriate, and the numerical evidence is strong: in the tested Bergomi example, reducing 1365 states to 27 gives errors at machine precision, and even 5–11 states give practically useful accuracy. I believe the central methodological idea is sound and worth publishing.

Where the soft spots are:

1. No global a priori error bound. Proposition 3.3 bounds the contribution of a single initial-state direction, but truncation also removes the coupling blocks in (3.15). In deterministic balanced truncation, global bounds come from the fact that the Gramians solve infinite-horizon Lyapunov equations and the Hankel singular values are singular values of a Hankel operator. Here P and Q are finite-time covariance integrals, so that theory doesn't transfer, and the paper provides no replacement. This makes the reduction criterion a heuristic. That's a real gap, but it's a standard limitation in many MOR papers, and the numerics clearly show the heuristic working on these examples.

2. The 'exact up to machine precision' statements are numerical-rank observations, not algebraic facts. In the Bergomi case, σ28 < 1e-8 means the 28th eigenvalue of PQ is below 1e-16, so the effective rank is 27 for that fitted model and time horizon. It's a property of the example, not a theorem. The paper is not deceptive, but a reader should not infer a general rule.

3. The errors are measured against the fitted signature model, not against the Bergomi model itself. Section 4 is fairly explicit about this, but the abstract's phrasing could mislead a casual reader. No code or calibration details are given, so the specific numbers aren't reproducible.

The citation pattern is fair and covers the relevant stochastic MOR and signature literature. The paper is honestly written—the authors explicitly note when reducing to small dimensions leads to true approximation error, and they discuss the choice H=0.3 for rough Bergomi as a compromise.

Who should read this: people working on signature-based models in finance or ML, and anyone interested in MOR for stochastic systems. It deserves a serious referee. My recommendation: send to review. The authors should add a limitation paragraph about the missing error bound and provide code or calibration data.

Referee Report

2 major / 5 minor

Summary. This paper proposes a model-order-reduction (MOR) framework for truncated path-signature models. The truncated signature of a time-extended Brownian motion solves the linear SDE (3.1), and the authors define time-averaged Gramians P and Q in (3.3)-(3.4). They show that these Gramians can be computed exactly by finite sums (3.9) and (3.11) because the nilpotent structure of the signature matrices truncates the Lyapunov series. A balancing transformation (Proposition 3.5) simultaneously diagonalizes P and Q, and the authors truncate the balanced system to obtain a reduced linear SDE (3.16). Numerical experiments for signature models fitted to Bergomi and rough Bergomi data report accurate option-price and L2 approximations with reduced dimensions ~n=27 and ~n=55 instead of n=1365 and n=3280.

Significance. If the reported accuracy is representative, the paper provides a practical method for compressing high-dimensional signature models, which is relevant for finance and for signature-based numerics. The main strengths are the clean exact computation of P and Q via nilpotency, the transparent balancing construction, and the realistic numerical validation with two stochastic-volatility models. The principal weakness is that the truncation criterion is justified only by a per-mode heuristic (Proposition 3.3), not by a global a priori error bound for the reduced SDE, so the 'machine precision' claims in Section 4 are empirical rank observations. This gap is fixable either by adding a bound or by carefully restating the contribution as a numerically validated heuristic.

major comments (2)

[Section 3.3, Eq. (3.16)] The central reduction criterion is not backed by a global a priori error bound. Proposition 3.3 bounds only the output energy contributed by the initial condition along one eigenvector q_k before truncation; it does not bound the error Y - ~Y after deleting the blocks starred in (3.15), because the kept component ~X^1 in (3.16) no longer receives the A_{12} X^2 and N_i^{12} X^2 terms and the discarded component's dynamics are removed. In classical balanced truncation such a bound exists because the Gramians solve algebraic Lyapunov equations; here P and Q are finite-time integrals satisfying the differential Lyapunov equation (3.6), so the standard theory does not transfer. Consequently the statements in Section 4.1 ('~n=27 is exact up to machine precision') and Section 4.2 ('~n=55 is an exact model') are numerical-rank observations (sigma_28 < 10^-8, sigma_56 ~ 0) rather than consequences of a theorem. I recommend either adding an error bound for the truncation or explicitly presenting the method as a heuristic and rewriting the 'exact' claims as numerically exact to working precision.
[Section 3.3, Proposition 3.5] The balancing transformation T = Sigma^{1/2} U^T L_P^{-1} is defined via a Cholesky factorization P = L_P L_P^T and therefore requires P positive definite; the simultaneous diagonalization claim also assumes Q positive definite. The numerical examples in Section 4 do not report whether these conditions hold, and Figures 1 and 5 show that many sigma_k are at or below machine precision, implying that PQ is numerically singular. The paper needs to explain how T is constructed (or how the reduction is performed) when P or Q is singular, and to state which parts of the theory remain valid in that case.

minor comments (5)

[Box 1, Section 2.7] The loop bound in Box 1 appears to be a typographical error: the number of basis elements up to level m-1 is (d^m - 1)/(d - 1), not (dm - 1)/(d - 1) as printed.
[Section 3.2, Eqs. (3.9) and (3.11)] The output matrix L and the Lyapunov operator L are both denoted by the same symbol in consecutive sections; this makes equations such as L^j(zz^T) ambiguous. Please use a different calligraphic or script letter for one of them.
[General] The manuscript contains numerous typographical and OCR artifacts (e.g., 'Thispaperpresentsmethods', 'Outline Westartbyproviding', 'we want so approximate', 'th minor relevance') that should be corrected in a careful revision.
[References] The entry [BHR24] contains the placeholder 'XXXX' for the publisher or volume; please complete it before publication.
[Section 4] The fitting procedure for the signature model functional ell and the Monte Carlo parameters behind the L2 errors are not fully specified; adding these details would strengthen reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Gramian-based truncation criterion is computed from the model's own matrices and the reduced-model errors are measured against the full model independently.

full rationale

The paper's central reduction mechanism is self-contained and not circular. The balancing Gramians P and Q are defined directly from the signature model's data: P = ∫_0^T E[X_u X_u^T] du and Q = ∫_0^T E[Φ(u,0)^T L^T L Φ(u,0)] du, with explicit computations via the finite nilpotent sums (3.8)–(3.11). The reduction criterion, namely the spectrum of PQ, is therefore a function of the model matrices A, N_i, L, z, and the horizon T, not of the output data used to validate the reduced model. The reduced system (3.16) is obtained by the balancing transformation (3.14) and truncation, and the reported errors compare this reduced system against the full signature model output Y_t = L X_t, which is an independent benchmark. The upstream calibration of the linear functional ℓ is presented as a separate, preceding modeling step (Section 2.6), and the paper explicitly assumes a calibrated signature model rather than presenting calibration as a prediction of the MOR framework. No load-bearing result is imported solely from self-citations: the universal approximation theorem is proved in the paper via Stone–Weierstrass, and the MOR constructions are standard balanced truncation adapted to the linear SDE structure. The absence of a global a priori error bound for the truncation heuristic—Proposition 3.3 bounds only per-mode contributions—is a rigor or correctness concern, not a circularity, because the numerical claims are empirical validations and the truncation criterion is not fitted to the errors it is used to predict. The statements about machine-precision exactness for certain reduced dimensions are numerical-rank observations based on computed eigenvalues σ_k; this is a limitation of the strength of the claim, not a circular use of the target result.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The MOR derivation itself introduces no free parameters and no invented entities. The numerical demonstration relies on user choices (truncation level m, reduced dimension tilde_n) and on a calibrated signature functional from [CGSF22], none of which are part of the reduction derivation. The key unproven premise is that balanced truncation yields a good global approximation of the output without an error bound.

free parameters (3)

reduced dimension tilde_n = 27 (Bergomi), 55 (rough Bergomi) in the exact cases; 5, 11, 15 for approximate cases
Chosen by the user from the decay of sigma_k; the paper gives a criterion (sigma_{tilde_n+1} small). The numerical illustrations of the central claim depend on this choice, but the method itself accepts any tilde_n.
truncation level m = 5 for Bergomi (d=4), 7 for rough Bergomi (d=3)
Modeling choice for the signature approximation. The state dimension n = (d^{m+1}-1)/(d-1) depends on it. It is an input to the reduction, not fitted by it.
signature model functional ell = not reported numerically
The linear functional defining the asset price model is calibrated to the Bergomi model following [CGSF22]. The calibration details and resulting coefficients are not given, so the numerical experiments are not fully reproducible from the paper alone.

assumptions (4)

domain assumption The driving process X_t is a time-extended Brownian motion (Assumption 3.1), so the signature SDE has constant linear coefficients in Itô form and the Gramian computations apply.
Section 3, Assumption 3.1. This restricts the method to Brownian drivers or semimartingales with quadratic variation included (Remark 3.2). The numerical examples satisfy it, but the abstract's general phrasing is broader.
domain assumption P and Q are positive definite, so the balancing transformation T = Sigma^{1/2} U^T L_P^{-1} is well defined.
Proposition 3.5. If P or Q is singular, generalized balancing is needed; the paper does not treat this case.
ad hoc to paper Truncating balanced coordinates with small sigma_k yields a good approximation of the output.
Section 3.3, after (3.16). Proposition 3.3 gives a one-mode bound, but no error bound for the truncated block is proven. This heuristic is load-bearing for the numerical claims.
domain assumption The signature model S_t^ell = <ell, X_{0,t}^{<=m}> is a good approximation of the target financial model (Bergomi or rough Bergomi), so reducing it is meaningful.
Section 2.6 and Section 4. This is inherited from [CGSF22] and not demonstrated here.

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Cite this review

Pith. "Pith review of Dimension reduction for path signatures." pith.science (2026). https://pith.science/paper/G7TETBTM

@misc{pith2026241214723,
  author       = {Pith},
  title        = {Pith review of: Dimension reduction for path signatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G7TETBTM}},
  note         = {Machine review of arXiv:2412.14723}
}

read the original abstract

This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and their applications in areas such as stochastic differential equations (SDEs) and financial modeling are pointed out. In particular, exploiting the rough paths view, solutions of SDEs continuously depend on the lift of the driver. Such continuous mappings can be approximated using (truncated) signatures, which are solutions of high-dimensional linear systems. In order to lower the complexity of these models, this paper presents methods for reducing the order of high-dimensional truncated signature models while retaining essential characteristics. The derivation of reduced models and the universal approximation property of (truncated) signatures are treated in detail. Numerical examples, including applications to the (rough) Bergomi model in financial markets, illustrate the proposed reduction techniques and highlight their effectiveness.

Figures

Figures reproduced from arXiv: 2412.14723 by the authors.

**Figure 2.** L 2 -error between output of the signature model of (4.1) (n = 1365) and reduced system output for ne = 1, . . . , 26. decay in k and notice that σ28 < 10−8 which is below the machine precision. Consequently, (3.1) has a high reduction potential and allows for an exact approximation in case of choosing ne = 27. The corresponding reduction errors 19 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Implied volatilities of (4.1) for T = 1/12, 1/2, 1 and strike prices KT = (0.8 + j · 0.02) √ T with j = 0, 1, . . . , 20. error, the quality of the reduced signature system is tested in a finance context. First, we determine the fair price E h max{S (ℓ) T − KT , 0} i of a European option, where the values of the strike price KT = (0.8 + j · 0.02) √ T (j = 0, 1, . . . , 20) are chosen depending on the maturity T. The… view at source ↗

**Figure 4.** Relative error between volatilities of signature model associated to [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: L 2 -error between output of the signature model of (4.2) (n = 3280) and reduced system output for ne = 1, . . . , 54. Moreover, we are interested in the quality of the signature approximation (3.16) when IV are aimed to be reproduced. First of all, let us note the tru…

**Figure 7.** Figure 7: Implied volatilities of (4.2) for T = 1/12, 1/2, 1 and strike prices KT = (0.8 + j · 0.02) √ T with j = 0, 1, . . . , 20. these values by the IV of reduced system for ne = 55 and, as expected, we obtain an error that can be fully neglected, see [PITH_FULL_IMAGE:figure…

**Figure 8.** Figure 8: Relative error between volatilities of signature model associated to [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 3 canonical work pages

[1]

Novelty Detection on Radio Astronomy Data using Signatures

[Arr+24] P. Arrubarrena, M. Lemercier, B. Nikolic, T. Lyons, and T. Cass. “Novelty Detection on Radio Astronomy Data using Signatures”. In: arXiv preprint arXiv:2402.14892(2024). [Bay+23a] C. Bayer, P. K. Friz, M. Fukasawa, J. Gatheral, A. Jacquier, and M. Rosenbaum.Rough volatility. SIAM,

work page Pith review arXiv 2024
[274]

Low-dimensional approxi- mations of high-dimensional asset price models

[RBG21] M. Redmann, C. Bayer, and P. Goyal. “Low-dimensional approxi- mations of high-dimensional asset price models”. In:SIAM J. Fi- nancial Math.12.1 (2021), pp. 1–28. [Red22] M. Redmann. “Solving high-dimensional optimal stopping prob- lems using optimization based model order reduction”. In:Applied Mathematical Finance29.2 (2022), pp. 110–140. [RJ22] ...

work page 2021
[2010]

Model reduction in limited time and frequency intervals

[GJ90] W. Gawronski and J. Juang. “Model reduction in limited time and frequency intervals”. In:Int. J. Syst. Sci.21.2 (1990), pp. 349–376. doi: 10.1080/00207729008910366. 24 [Guy22] J. Guyon. “The VIX future in Bergomi models: Fast approxima- tion formulas and joint calibration with S&P 500 skew”. In:SIAM Journal on Financial Mathematics13.4 (2022), pp. ...

arXiv 1990
[2015]

Primal and dual optimal stopping with signatures

[BFG16] C. Bayer, P. K. Friz, and J. Gatheral. “Pricing under rough volatil- ity”. In:Quantitative Finance16.6 (2016), pp. 887–904. 23 [BFZ24] E. Bayraktar, Q. Feng, and Z. Zhang. “Deep Signature Algorithm for Multidimensional Path-Dependent Options”. In:SIAM Journal on Financial Mathematics15.1 (2024), pp. 194–214.doi: 10.1137/ 23M1571563. [BH19] S. Beck...

work page Pith review arXiv 2016
[2023]

Optimal stop- ping with signatures

[Bay+23b] C. Bayer, P. Hager, S. Riedel, and J. Schoenmakers. “Optimal stop- ping with signatures”. In:The Annals of Applied Probability33.1 (2023), pp. 238–273. [BD11] P. Benner and T. Damm. “Lyapunov equations, energy function- als, and model order reduction of bilinear and stochastic systems.” English. In:SIAM J. Control Optim.49.2 (2011), pp. 686–711....

doi:10.1137/09075041x 2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Novelty Detection on Radio Astronomy Data using Signatures

[Arr+24] P. Arrubarrena, M. Lemercier, B. Nikolic, T. Lyons, and T. Cass. “Novelty Detection on Radio Astronomy Data using Signatures”. In: arXiv preprint arXiv:2402.14892(2024). [Bay+23a] C. Bayer, P. K. Friz, M. Fukasawa, J. Gatheral, A. Jacquier, and M. Rosenbaum.Rough volatility. SIAM,

work page Pith review arXiv 2024

[2] [274]

Low-dimensional approxi- mations of high-dimensional asset price models

[RBG21] M. Redmann, C. Bayer, and P. Goyal. “Low-dimensional approxi- mations of high-dimensional asset price models”. In:SIAM J. Fi- nancial Math.12.1 (2021), pp. 1–28. [Red22] M. Redmann. “Solving high-dimensional optimal stopping prob- lems using optimization based model order reduction”. In:Applied Mathematical Finance29.2 (2022), pp. 110–140. [RJ22] ...

work page 2021

[3] [2010]

Model reduction in limited time and frequency intervals

[GJ90] W. Gawronski and J. Juang. “Model reduction in limited time and frequency intervals”. In:Int. J. Syst. Sci.21.2 (1990), pp. 349–376. doi: 10.1080/00207729008910366. 24 [Guy22] J. Guyon. “The VIX future in Bergomi models: Fast approxima- tion formulas and joint calibration with S&P 500 skew”. In:SIAM Journal on Financial Mathematics13.4 (2022), pp. ...

arXiv 1990

[4] [2015]

Primal and dual optimal stopping with signatures

[BFG16] C. Bayer, P. K. Friz, and J. Gatheral. “Pricing under rough volatil- ity”. In:Quantitative Finance16.6 (2016), pp. 887–904. 23 [BFZ24] E. Bayraktar, Q. Feng, and Z. Zhang. “Deep Signature Algorithm for Multidimensional Path-Dependent Options”. In:SIAM Journal on Financial Mathematics15.1 (2024), pp. 194–214.doi: 10.1137/ 23M1571563. [BH19] S. Beck...

work page Pith review arXiv 2016

[5] [2023]

Optimal stop- ping with signatures

[Bay+23b] C. Bayer, P. Hager, S. Riedel, and J. Schoenmakers. “Optimal stop- ping with signatures”. In:The Annals of Applied Probability33.1 (2023), pp. 238–273. [BD11] P. Benner and T. Damm. “Lyapunov equations, energy function- als, and model order reduction of bilinear and stochastic systems.” English. In:SIAM J. Control Optim.49.2 (2011), pp. 686–711....

doi:10.1137/09075041x 2023