A Kernelized Operator Approach to Nonlinear Data-Enabled Predictive Control

Mircea Lazar; Siep Weiland; Thomas de Jong

REVIEW 3 major objections 6 minor 23 references

A Kernelized Operator Approach to Nonlinear Data-Enabled Predictive Control

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper turns nonlinear DeePC into a low-dimensional one-shot optimization by parameterizing the system with a product kernel, cutting per-action computation by about a factor of one hundred.

desk verdict A useful Kronecker-product computational trick buried under an overstated equivalence claim and an unfair speed comparison. read the letter →

arxiv 2501.17500 v1 pith:EXQJE7ET submitted 2025-01-29 math.OC

Thomas de Jong , Siep Weiland , Mircea Lazar This is my paper · ORCID

classification math.OC MSC 93C1093B30

keywords data-enabledpredictivecontrolDeePCreproducingkernelHilbertspaceproductkernelizedoperatordata-drivenuniversallearning

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

This paper proposes a kernelized operator formulation of data-enabled predictive control (DeePC) for nonlinear systems. Instead of using a kernel on the stacked space of initial states and future inputs, it learns the input-to-output map as an operator in a product reproducing kernel Hilbert space, with a product kernel that factors into a state kernel and an input kernel. The central claim is that this product structure shrinks the online optimization: the equality constraints drop from the product of the two dataset sizes to the number of input trajectories, and the decision variables are projected into the output space. In the paper's Van der Pol example, each control action takes about 0.33 seconds instead of 33.87 seconds for the same data, and datasets with 10,000 trajectories become usable online. If the claim holds, nonlinear DeePC can exploit much larger datasets and reach performance comparable to model-based nonlinear MPC without an explicit model.

What carries the argument

The central object is the product kernel $k_\otimes((u_1,x_1),(u_2,x_2))=k_u(u_1,u_2)k_x(x_1,x_2)$ on the product space of input trajectories and initial states, which generates a product reproducing kernel Hilbert space and a Gram matrix $K_\otimes=K_u\otimes K_x$ whose inverse factors as $K_u^{-1}\otimes K_x^{-1}$. The load-bearing identity is the reduction $\Omega(x)=\big(I_{T_u}\otimes k_x^\top(x)/\|k_x(x)\|_2^2\big)K_\otimes$, which replaces the $T_xT_u$-dimensional kernel equation by a $T_u$-dimensional one, and the companion projection $\hat g=Y^\dagger\tilde g$ that moves decision variables into the $pN$-dimensional output space. This machinery is what decouples the input and state data, cuts the optimization size, and makes the Gram inversion and construction much cheaper.

What would settle it

Take a nonlinear benchmark with a non-smooth or discontinuous map (for example a relay or dead-zone nonlinearity), generate data with Algorithm 2, and check whether the multi-step prediction error (18) keeps decreasing as $T_x$ and $T_u$ grow; if the error plateaus above the measurement noise, the universal-approximation premise is violated for that system.

Watch

Extended reading notes

Core claim

The paper claims that a nonlinear discrete-time system's behavior can be parameterized as $G(u)(x)=\Theta^* k_\otimes(u,x)$ in the product reproducing kernel Hilbert space built from the kernel $k_\otimes((u_1,x_1),(u_2,x_2))=k_u(u_1,u_2)k_x(x_1,x_2)$, with Gram matrix $K_\otimes=K_u\otimes K_x$. It then shows that the DeePC prediction equations $K_\otimes g=k_\otimes(u,x)$, $Yg=y$ are equivalent to the reduced system $\Omega(x)g=k_u(u)$, $Yg=y$ with $\Omega(x)=\big(I_{T_u}\otimes k_x^\top(x)/\|k_x(x)\|_2^2\big)K_\otimes$, and that projecting $g$ through $g=Y^\dagger\tilde g$ into the $pN$-dimensional output space yields a one-shot efficient formulation (Problem 4.4). The equivalence is established by Lemma 4.3 under the condition that the output data annihilate the null space of $\Omega(x)$. The numerical section reports per-control-action computation times of $0.3319\,\mathrm{s}$ versus $33.8668\,\mathrm{s}$ for the stacked-kernel formulation at $T=400$, and shows that $T=10000$ datasets are tractable with the product formulation while the stacked formulation is not.

Load-bearing premise

Everything rests on the inherited premise that the product reproducing kernel Hilbert space is dense enough to represent the true system's input-to-output operator on the relevant domain; if the real system is not well approximated there, the predictor is biased no matter how the optimization is sped up.

Editorial extensions

If this is right

For the same dataset of 400 trajectories, the efficient product-kernel DeePC computes a control action in 0.3319 seconds whereas the stacked-kernel formulation needs 33.8668 seconds, a speedup of roughly two orders of magnitude.
The product formulation remains solvable online for 10,000 trajectories (about 1.35 seconds per action), while the stacked formulation cannot handle that dataset online, so users can train on 25 times more data.
With 10,000 trajectories, the product-kernel controller's mean tracking error drops from 0.0994 to 0.0835 and its mean prediction error from 0.0216 to 0.0157, and the paper shows tracking performance comparable to nonlinear MPC.
Under the conditions of Lemma 4.3, the reduced model predicts the same outputs as the original product-kernel model, so the computational savings do not change the predicted trajectories.

Reading between the lines

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

Because the state and input data enter through separate kernels, the same library of input sequences can be reused with many different initial-condition sets; one could update $K_x$ and the corresponding part of $K_\otimes$ without recomputing the whole pipeline when adding new operating regions.
The projection $g=Y^\dagger\tilde g$ buys speed at the cost of limiting the decision space to the row space of the output data, so systems with many outputs or poor output richness may need feature selection in the output space to keep enough flexibility.
The k-means choice of initial conditions suggests a testable extension: distributing initial conditions along unstable or strongly nonlinear regions could improve the operator approximation for systems that are not globally stable, since the density theorem does not say where data must be sampled.

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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

Start with the punchline: the paper has a genuinely useful computational observation. The product kernel structure means the Gram matrix is a Kronecker product, so inversion and construction cost polynomial in Tx and Tu separately rather than in the product TxTu. That idea is sound, and the k-means data-generation scheme is a reasonable practical addition. The problem is that the "efficient formulation" presented as the main contribution is not an equivalent reduction of the original KerODeePC problem, and the speed comparison is between two different optimization problems.

The algebra in Lemma 4.1 is correct as a manipulation, but the word "equivalently" is doing too much work. Multiplying K⊗g = ku(u)⊗kx(x) by the pseudo-inverse of (I⊗kx(x)) gives a necessary condition, not a sufficient one. The extra solutions are exactly the nullspace of Ω(x), and Lemma 4.3 states the right condition for model equivalence: Y must annihilate that nullspace. The paper never checks that condition. In the T=400 experiment, the nullspace has dimension 380 inside R^400 and Y has only 10 rows, so generically the condition is false. Then Problem 4.4 goes further and restricts ĝ to lie in the row space of Y via ĝ=Y†g̃, with no argument that the relevant solutions live there. The only formal bridge back to the original problem is the λ→∞ limit, and the paper does not report the finite λ used in the experiments. So the 100x speedup is real, but it is the speedup of a relaxed controller, not of solving Problem 3.2 faster.

The second soft spot is the foundation: the universality of the product RKHS is imported from the same group's preprint [18], and the approximation error is not quantified here. That is not fatal on its own, but it means the prediction model's bias is unquantified.

On the positive side, the writing is clear, the Kronecker trick is a legitimate contribution, and the empirical trends—more data helps, prediction error comparable to the stacked kernel baseline—look plausible. But the comparison is unfair: the stacked baseline solves a much larger problem. A revision should either verify the nullspace condition, or explicitly present Problem 4.4 as a heuristic relaxation and benchmark it against a same-problem baseline, not against a different formulation.

My recommendation: send it to peer review, but expect major revision. The core idea is worth keeping; the equivalence claims need to be scaled back and the experiments need to be redone or reframed.

Referee Report

3 major / 6 minor

Summary. This paper proposes a kernelized operator approach to nonlinear data-enabled predictive control. It models the input-output behavior of a nonlinear system as a product reproducing kernel Hilbert space operator G(u)(x) = Θ*k⊗(u,x), building on a result from a companion preprint. The KerODeePC problem (Problem 3.2) enforces K⊗g = k⊗(u,x) and Yg = y. To reduce computation, Lemma 4.1 rewrites the first constraint into a lower-dimensional equality Ω(x)g = ku(u), and Lemma 4.3 gives a condition under which the reduced system is output-equivalent. Problem 4.4 further projects the decision variable onto the output space, resulting in an efficient formulation. Numerical experiments on a Van der Pol oscillator show order-of-magnitude speedups relative to a stacked kernel DeePC and comparable performance to NMPC when more data are used.

Significance. The structural idea of factorizing the kernel into input and state parts to reduce the equality constraints from TxTu to Tu is original and potentially valuable. The numerical results are impressive: a reduction from 33.87s to 0.33s per control action at T=400 and the ability to handle T=10000 online. The paper also contributes a systematic data-generation procedure. However, the value of the contribution depends critically on whether Problem 4.4 is a faithful and controllable approximation of Problem 3.2; the current manuscript does not establish this. If the equivalence issues are resolved, the method could be a practical tool for large-scale kernel-based predictive control.

major comments (3)

[Section IV, Lemma 4.1 and Lemma 4.3] Lemma 4.1's claim that (11) is 'equivalently re-written' from (7) is too strong. The left-multiplication by the pseudo-inverse (I_Tu ⊗ kx(x))† is not invertible: from (11a) one can only conclude that P_{range(I⊗kx)} K⊗ g = (I⊗kx)ku(u), not that K⊗g = k⊗(u,x). The reduced system (11) therefore admits extra solutions, namely all g ∈ K⊗^{-1}(range(I⊗kx))^⊥. Lemma 4.3 correctly identifies the necessary and sufficient condition for output equivalence (Y annihilates N(Ω(x))), but this condition is not verified anywhere in the manuscript and is generically false for the reported data dimensions. For the T=400 example with Tx=20 and Tu=20, dim N(Ω(x)) = Tu(Tx-1) = 380 while Y ∈ R^{10×400}; the requirement that a 10-row matrix annihilate a 380-dimensional subspace is a restrictive condition that should be checked numerically. The paper neither checks it nor discusses its plausibility, so the claim that Problem 4.4 preserves the original operator model is not supported.
[Section IV, Problem 4.4 and the paragraph after (15)] Problem 4.4 is not an equivalent reformulation of Problem 3.2. It replaces the decision variable ĝ by Y†˜g, which restricts the search to row(Y), and it changes the regularization term from ||g||² to ||˜g||². Moreover, the constraint (15b) is only a relaxation of the requirement that ĝ ∈ N(Ω(x)); it enforces that Ω(x)Y†˜g=0, i.e., that the output-space variable lies in Y N(Ω(x)). The only formal connection to Problem 3.2 established in the text is the λ→∞ limit, where ˜g→0 and y tends to the unregularized operator predictor. Since the finite value of λ used in the experiments is not reported, the speedups in Table I cannot be attributed to solving the original KerODeePC problem; they are for a modified controller. The manuscript should either provide a bound on the suboptimality/approximation error for finite λ, or explicitly present Problem 4.4 as a relaxation and study its closed-loop behavior as a function of λ.
[Section II, eq. (5)-(7); Section III-A] The paper's central modeling assumption is that the product RKHS H(k⊗, X×U) is universal in the space of nonlinear system operators and that the minimizer of (9) is a universal approximator. This theorem is imported from the companion preprint [18] and is neither stated with its assumptions nor proven in the present manuscript. The paper also does not verify the conditions of that theorem for the Van der Pol oscillator and the data generated by Algorithms 1 and 2. If the universality result does not apply, the prediction model (10) is biased and the claimed nonlinear generalization is unsupported. At minimum, the authors should state the theorem precisely (including the required assumptions on the system, the kernel, and the data) and indicate how their data-generation procedure meets those assumptions. Relying on an unpublished self-cited preprint for such a load-bearing step is risky for the reader.

minor comments (6)

[Section I, Notation] The definition of 1n is garbled: 'A square n × n matrix where all elements of 1/n is denoted by 1n' should read 'A square n × n matrix whose elements are all equal to 1/n is denoted by (1/n)1_n1_n^T' or similar, and the vector 1_n should be defined separately.
[Section II, eq. (3) and (4)] Equations (3) and (4) use k(z1, z1) on the left-hand side; they should be k(z1, z2).
[Definition 2.2] In Definition 2.2, item (i) says 'k(z, ·) ∈ H(k, X)' but the space should be H(k, Z); likewise item (ii) should refer to H(k, Z).
[End of Section IV] The sentence 'when λ → ∞, we have that ˜gk → 0 and thus ˆgk → 0 by Lemma 4.1' incorrectly invokes Lemma 4.1; the implication follows from the definition ˆg = Y†˜g, not from Lemma 4.1.
[Problem 4.4, eq. (15c)] Constraint (15c) contains an extra closing parenthesis after 'u[0,N−1](k)))'; it should read 'YΩ†(xk)ku(u[0,N−1](k)) + ˜gk = y[1,N](k)'.
[Section V, Table I] The entry 'X' for the stacked Gram inversion at T=10000 could be clearer: state explicitly that the stacked formulation is not solvable online at that size, and give the corresponding product-kernel time in the same row.

Circularity Check

1 steps flagged · score 4.0 of 10

The operator-representation premise is imported from a same-author preprint; the computational reformulation is derived independently.

uniqueness imported from authors [Section I (Introduction) and Section II, eqs. (5)-(7)]
"Therein, it was proven that for positive definite kernels [19] the resulting product RKHS is dense in the space of nonlinear systems operators and that its unique minimizer is a universal approximator. ... As pointed out in [18], through the prism of Willems' fundamental lemma, we consider the following system of equations: K⊗g = k⊗(u,x), Yg = y."

The prediction mechanism (7) is not derived in this paper; it is lifted from [18], a preprint by co-author M. Lazar. The universal-approximation and uniqueness theorem from [18] is load-bearing: without it, the product RKHS representation K⊗g = k⊗(u,x), Yg = y has no more justification than any arbitrary kernel interpolation choice. The paper provides no proof, numerical verification, or independent external reference for this theorem; the only support is the same-author citation. The efficient reformulation in Lemma 4.1 and Problem 4.4 is developed in this paper and is independent, so the circularity is confined to the foundational operator-representation premise.

full rationale

The paper's central computational contribution, the Kronecker-product reduction and the efficient KerODeePC formulation, is independently derived and benchmarked against a stacked-kernel baseline and NMPC, so the speed claim is not circular. The foundational premise, however, is the product-RKHS universality result from [18], a same-author preprint; the paper merely cites it and builds equations (5)-(7) on it without re-deriving or independently verifying it. This is a load-bearing self-citation, but it does not by itself make the computational derivation circular. There is also an omitted-proof risk: Lemma 4.1 proves only the forward direction of the claimed equivalence, and Lemma 4.3's condition Y g-hat = 0 on the null space of Omega(x) is never verified, while the finite lambda used in experiments is not reported; these are correctness concerns rather than circularity. Overall, the result is partially dependent on an imported same-author theorem, but the main algorithmic development stands on its own, so a moderate score of 4 is appropriate.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method relies on a self-cited universality result, on positive definiteness of Gram matrices, and on a full-row-rank output matrix. The kernel widths and the unspecified cost weights and regularization are hand-chosen parameters that affect the numerical results. No new physical entities are introduced.

free parameters (3)

Gaussian kernel widths = sigma_u = 50, sigma_x = 3
Chosen by hand for the Van der Pol example; no tuning rule or cross-validation is given. The approximation quality and the conditioning of the Gram matrix depend on these values.
Regularization weight lambda = not specified numerically
Appears in the objective of Problems 3.2 and 4.4; the paper states lambda > 0 but does not report the value used in experiments, making the exact controller unreproducible.
Cost matrices P, Q, R = not specified
Assumed positive definite, but no numerical values are given for the tracking experiments.

assumptions (5)

ad hoc to paper Product RKHS H(k⊗, X × U) is dense in the space of nonlinear system operators, and the unique minimizer of (9) is a universal approximator.
Imported from [18], a self-cited preprint by the same author group; the present paper does not prove or test the conditions under which this holds for systems of form (1).
domain assumption K⊗ is positive definite for the collected data (Lemma 4.3 uses K⊗ ≻ 0).
Justified only by the use of universal kernels and distinct data points; not proven for the specific data generation scheme.
domain assumption The output data matrix Y has full row rank (Np ≤ TxTu), needed for the pseudo-inverse Y† in Problem 4.4.
Stated as necessary in Section IV; no discussion of how to guarantee or verify it.
domain assumption System (1) is deterministic with measured states and unknown f and h.
Standard DeePC setup; no noise model is considered, so the method's robustness to noise is not addressed.
domain assumption Assumption 3.3: X and U are polytopes containing the equilibrium points or references of interest in their interior.
Used to justify quasi-random initialization of centroids in Algorithm 1.

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Cite this review

Pith. "Pith review of A Kernelized Operator Approach to Nonlinear Data-Enabled Predictive Control." pith.science (2026). https://pith.science/paper/EXQJE7ET

@misc{pith2026250117500,
  author       = {Pith},
  title        = {Pith review of: A Kernelized Operator Approach to Nonlinear Data-Enabled Predictive Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EXQJE7ET}},
  note         = {Machine review of arXiv:2501.17500}
}

read the original abstract

This paper considers the design of nonlinear data-enabled predictive control (DeePC) using kernel functions. Compared with existing methods that use kernels to parameterize multi-step predictors for nonlinear DeePC, we adopt a novel, operator-based approach. More specifically, we employ a universal product kernel parameterization of nonlinear systems operators as a prediction mechanism for nonlinear DeePC. We show that by using a product reproducing kernel Hilbert space (RKHS) to learn the system trajectories, big data sets can be handled effectively to construct the corresponding product Gram matrix. Moreover, we show that the structure of the adopted product RKHS representation allows for a computationally efficient DeePC formulation. Compared to existing methods, our approach achieves substantially faster computation times for the same data size. This allows for the use of much larger data sets and enhanced control performance.

Figures

Figures reproduced from arXiv: 2501.17500 by the authors.

**Figure 2.** Comparison of N = 10 step-ahead predictions (colored lines) from the stacked/product kernel prediction models against the true system trajectory (black line) for various initial conditions. reference signal, see Fig.3. The tracking performance of the product kernel formulations is on average slightly better than the performance of the stacked kernel formulation as shown in Table II. However, in Fig.3 a higher oversh… view at source ↗

**Figure 3.** Reference tracking for product kernel, T = 400 (—), stacked kernel, T = 400 (—), product kernel, T = 10000 (—) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 1.** Initial conditions for Tx = 20 and Tx = 200 generated using Algorithm 1. 0 50 100 150 200 250 300 -1 0 1 2 0 50 100 150 200 250 300 -10 1 2 0 50 100 150 200 250 300 -1 0 1 2 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Reference graph

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Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [18]

A universal reproducing kernel Hilbert spac e for learning nonlinear systems operators,

M. Lazar, “A universal reproducing kernel Hilbert spac e for learning nonlinear systems operators,” arXiv, 2024, presented at the SysDo 2024 Symposium, October, Stuttgart, Germany, to appear in S pringer Nature (original manuscript submitted in April 2024). [Onl ine]. Available: https://arxiv.org/abs/2412.18360

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work page 2005

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J. Coulson, J. Lygeros, and F. D¨ orﬂer, “Data-enabled pr edictive control: In the shallows of the deepc,” in 2019 18th European Control Conference (ECC) . IEEE, 2019, pp. 307–312

work page 2019

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F. Fiedler and S. Lucia, “On the relationship between dat a-enabled predictive control and subspace predictive control,” in 2021 European Control Conference (ECC) . IEEE, 2021, pp. 222–229

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M. Lazar, “Basis-functions nonlinear data-enabled pre dictive control: Consistent and computationally efﬁcient formulations,” i n 2024 Euro- pean Control Conference (ECC) . IEEE, 2024, pp. 888–893

work page 2024

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Y . Lian, R. Wang, and C. N. Jones, “Koopman based data-dri ven predictive control,” arXiv preprint arXiv:2102.05122 , 2021

arXiv 2021

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work page 2024

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Koopman Data-Driven Predictive Control with Robust Stability and Recursive Feasibility Guarantees

T. de Jong, V . Breschi, M. Schoukens, and M. Lazar, “Koopm an data- driven predictive control with robust stability and recurs ive feasibility guarantees,” arXiv preprint arXiv:2405.01292 , 2024

work page Pith review arXiv 2024