Certification of kernel-based balanced truncation for nonlinear systems: empirical Hankel spectra, a priori error bounds, and the spectral-gap reducibility conjecture

Bouvrie and Hamzi (SIAM J. Control Optim., 2017) proposed a data-driven model reduction method for nonlinear control systems that embeds trajectory data into a reproducing kernel Hilbert space (RKHS), performs linear balanced truncation implicitly there using empirical kernel Gram matrices, and learns a closed low-order model. The certification of this method is open: nothing rigorous is known about what the empirical Hankel singular values of the Hankel kernel matrix say about the true nonlinear Hankel operator; no a priori input-output error bound analogous to the classical linear balanced truncation bound exists; and the authors' conjecture that a nonlinear system is model reducible if there exists an RKHS whose Gramians exhibit a spectral gap is unproven and unformalized. This problem asks for the missing foundations as one coherent objective: an operator-level characterization of the empirical kernel Hankel spectrum, an end-to-end a priori error analysis covering feature-space truncation, preimage and Jacobian approximations, RKHS regression, and finite sampling, and a formalization and resolution of the spectral-gap reducibility conjecture with verifiable conditions. The 2017-2026 follow-up literature develops algorithms along adjacent routes but explicitly leaves all three questions open.

Background

Model reduction seeks a low-dimensional surrogate of a dynamical system that reproduces its input-output behavior. For a linear time-invariant system $\dot x = Fx + Gu$, $y = Hx$, Moore's balancing (1981) simultaneously diagonalizes the controllability and observability Gramians -- positive semidefinite matrices measuring how strongly inputs drive each state and how strongly each state influences the output -- and orders the states by their Hankel singular values, the singular values of the Hankel operator that maps an input signal's past to the corresponding future output. Deleting balanced states with small Hankel singular values is balanced truncation, and it is certified: the truncation error in Hankel norm equals the first neglected singular value, and the output error obeys Glover's a priori bound

$$\|y - y_r\|_2 \le 2\left(\sum_{j>k} \sigma_j\right)\|u\|_2,$$

where the norms are taken on the input and output signals and $y_r$ is the output of the order-$k$ reduced system. For nonlinear systems no certified analog exists. Scherpen (1993) extended balancing through controllability and observability energy functions defined by nonlinear optimal-control problems, and Gray and Scherpen (1998) developed the associated theory of Hankel operators for nonlinear systems; but the energy functions solve Hamilton-Jacobi-type partial differential equations that are intractable in high dimension, and no analog of the linear error bound is known. Data-driven approximations include the snapshot-based subspace approach of Lall, Marsden and Glavaski (2002) and empirical Gramian methods.

Bouvrie and Hamzi (2017) introduced a kernel-based route. A Mercer kernel -- a continuous, symmetric, positive-definite function of two states, written $k(x, y) = \langle \Phi(x), \Phi(y)\rangle$ through its feature map $\Phi$ into a reproducing kernel Hilbert space (RKHS; a Hilbert space of functions in which point evaluation is a continuous linear functional) -- lets linear computations in the feature space represent nonlinear functions of the state. From trajectory samples of controllability experiments ($x_\nu$) and observability experiments ($d_\mu$), the method forms kernel Gram matrices $K_c$ and $K_o$ and the Hankel kernel matrix $(K_{o,c})_{\mu\nu} = k(d_\mu, x_\nu)$; the square roots of the eigenvalues of $K_{o,c} K_{o,c}^{\top}$ are interpreted as empirical Hankel singular values of the system lifted into the RKHS. The paper proves that a spectral gap $\sigma_q \gg \sigma_{q+1}$ in this spectrum licenses a nonlinear reduction map $\Pi$ built from the leading eigenvectors, after which the reduced dynamics and output map are estimated by RKHS regression and simulated through a preimage -- an approximate inverse of $\Phi$, which is generally non-surjective, so the exact reduced system does not literally exist. The method is validated only empirically, on two- and seven-dimensional examples.

The authors leave the method's foundations open in three linked respects: (i) the relation between the empirical Hankel singular values and eigenvectors of $K_{o,c} K_{o,c}^{\top}$ and the singular values and eigenfunctions of the true nonlinear Hankel operator is stated to be an open problem; (ii) no end-to-end error analysis exists covering feature-space truncation, the preimage and Jacobian approximations (the pseudo-inverse $\Pi^\dagger$ by which feature-space projected points are pulled back to the state space, and the derivatives of the learned maps), RKHS regression error, and finite-sample effects -- the paper notes that formally extending the linear bound to this setting is tempting but unwarranted; and (iii) the conclusion conjectures that a nonlinear system is model reducible if there exists an RKHS in whose Gramians a spectral gap appears, without formal definitions of reducibility or of the gap. These three gaps, which are what would give the method the certification that linear balanced truncation possesses, define the present problem.

Problem Statement

Consider the setting of the cited work: a finite-dimensional nonlinear control system $\dot x = f(x, u)$, $y = h(x)$ (or its discrete-time analog), its nonlinear Hankel operator $\mathcal{H}$ mapping past input histories to future outputs, a Mercer kernel $k$ with RKHS feature map $\Phi$, and trajectory samples from controllability and observability experiments, from which the kernel Gram matrices $K_c$, $K_o$ and the Hankel kernel matrix $(K_{o,c})_{\mu\nu} = k(d_\mu, x_\nu)$ are formed. The problem is to establish -- or disprove -- the certification theory for kernel-based nonlinear balanced truncation, by answering three linked questions about these objects.

(1) Spectral correspondence. Characterize the relationship between the empirical Hankel singular values and eigenvectors of $K_{o,c} K_{o,c}^{\top}$ and the singular values and eigenfunctions of the nonlinear Hankel operator $\mathcal{H}$: under which assumptions on the system class, the kernel/RKHS, and the sampling design do the former approximate the latter (and at what rate, including finite-sample effects), and what exactly do the empirical values measure when they do not?

(2) A priori error bounds. Derive an a priori bound on the input-output error between the original system and the kernel-reduced model in the spirit of the linear bound $\|y - y_r\| \le 2\left(\sum_{j>q} \sigma_j\right)\|u\|$ of the cited work, accounting for the error sources it identifies -- truncation in feature space, the preimage and Jacobian approximations, RKHS regression error, and finite-sample effects -- or prove that no such bound can hold in general, by an explicit counterexample.

(3) Reducibility conjecture. Formalize model reducibility (the existence of a lower-order model reproducing the input-output behavior to a specified accuracy) and the spectral-gap condition on the Gramians in this kernel setting, then settle the authors' conjecture that a nonlinear system is model reducible if there exists an RKHS in which the Gramians exhibit a spectral gap, and derive verifiable conditions on the system and the data under which such an RKHS and gap can be found or certified.

The three questions form one objective -- what the data-driven kernel spectra of a nonlinear system provably say about its Hankel behavior and its reducibility.

References

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