arxiv: 2605.14351 · v1 · submitted 2026-05-14 · 📡 eess.SY · cs.LG· cs.SY

Recognition: no theorem link

Randomized Atomic Feature Models for Physics-Informed Identification of Dynamic Systems

Rajiv Singh , Mario Sznaier , Lennart Ljung

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:38 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SY

keywords system identificationrandom atomic featuresstable polesconvex optimizationimpulse responsekernel methodsstability constraintsdisk Bochner representation

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The pith

Impulse responses are recovered as random superpositions of damped exponentials from poles sampled inside a disk using convex optimization with physical constraints.

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

The paper presents a framework for system identification that represents impulse responses as random linear combinations of stable atoms, specifically damped complex exponentials whose poles are drawn inside a user-specified disk. Identification is formulated as a convex regularized least-squares problem that admits optional linear, second-order-cone, and KYP constraints to enforce stability, passivity, and other engineering priors. The approach extends random Fourier and Laplace features to the nonstationary damped regime while preserving modal interpretability and finite-dimensional scalability. Analytically, positive measures on stable poles are shown to induce positive-definite kernels with a controlled radius-dependent shift defect, and the converse characterization links arbitrary kernels to subnormal shifts. A sympathetic reader would value this for recovering physically consistent models from limited or poorly excited data.

Core claim

Impulse responses are represented as random superpositions of stable atoms, namely damped complex exponentials associated with poles sampled inside a prescribed disk. Identification is cast as a convex regularized least-squares problem with optional linear, second-order-cone, and KYP constraints. The main analytic point is an operator-theoretic Disk-Bochner viewpoint: positive measures over stable poles generate positive-definite kernels with a radius-dependent shift defect, while a converse scalar disk moment representation for an arbitrary kernel is characterized by subnormality of the canonical shift. This yields an RKHS-to-l1 embedding, a valid finite atomic gauge from sampled poles, and

What carries the argument

Randomized atomic features consisting of damped complex exponentials with poles inside a disk, which induce a finite atomic gauge and enable an RKHS-to-l1 embedding for sparse recovery under restricted-eigenvalue conditions on the disk-Vandermonde matrix.

If this is right

The framework directly encodes stability margins, modal localization, DC-gain bounds, monotonicity, passivity, relative degree, settling-time targets, and time/frequency-domain error bounds.
Physically meaningful priors compensate for poor excitation and improve constrained impulse-response recovery in under-informative data settings.
Normalized transfer-function problems connect to Nevanlinna-Pick interpolation and LFT set-membership.
Random-feature convergence and sparse-recovery guarantees hold conditionally on the restricted-eigenvalue properties of the realized design matrix.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same randomization idea could be applied to other regions of the complex plane to handle marginally stable or unstable dynamics without altering the convex formulation.
Hybrid combinations with neural-network feature extractors could extend the modal interpretability to high-dimensional nonlinear identification tasks.
Numerical validation on multi-input multi-output industrial datasets would test whether the restricted-eigenvalue guarantees remain practical at scale.

Load-bearing premise

Random sampling of poles inside a prescribed disk produces a sufficiently rich and stable atomic dictionary whose finite realizations satisfy the restricted-eigenvalue properties needed for the stated sparse-recovery guarantees.

What would settle it

A concrete counterexample in which the realized disk-Vandermonde matrix violates the restricted eigenvalue condition for the chosen sampling distribution, causing the convex program to return unstable or non-unique models even when physical constraints are enforced.

Figures

Figures reproduced from arXiv: 2605.14351 by Lennart Ljung, Mario Sznaier, Rajiv Singh.

**Figure 2.** Figure 2: Data and bound prior. Left: training input/output [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: The example is deliberately small and illustrative; it is not intended as a statistical benchmark or as evidence of guaranteed pole recovery. It nevertheless highlights an important point. RAF is not merely a randomized approximation to a kernel estimator. It is a modeling language for combining modal dictionaries with convex side information. When data are rich, the method behaves like a scalable sparse m… view at source ↗

**Figure 3.** Figure 3: Fitting results. Top row: stability prior only. Middle row: pole-sector constraints plus sparsity. Bottom row: RAF [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

read the original abstract

We present a physics-informed framework for system identification based on randomized stable atomic features. Impulse responses are represented as random superpositions of stable atoms, namely damped complex exponentials associated with poles sampled inside a prescribed disk. Identification is then cast as a convex regularized least-squares problem with optional linear, second-order-cone, and KYP constraints. The approach generalizes random Fourier and random Laplace features to the damped, nonstationary regime relevant to engineering systems while retaining modal interpretability and scalable finite-dimensional computation. The main analytic point is an operator-theoretic Disk-Bochner viewpoint: positive measures over stable poles generate positive-definite kernels with a radius-dependent shift defect, while a converse scalar disk moment representation for an arbitrary kernel is characterized by subnormality of the canonical shift. We prove this statement, establish an RKHS-to-l1 embedding, show that sampled poles induce a valid finite atomic gauge, discuss random-feature convergence, and state sparse-recovery guarantees conditionally on the restricted-eigenvalue properties of the realized disk-Vandermonde or input-output design matrix. We also connect the normalized transfer function problem to Nevanlinna-Pick interpolation and LFT set-membership. The framework directly encodes stability margins, modal localization, DC-gain bounds, monotonicity, passivity, relative degree, settling-time targets, and time/frequency-domain error bounds. Numerical comparisons illustrate how physically meaningful priors can compensate for poor excitation and improve constrained impulse-response recovery in an under-informative data setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's Disk-Bochner operator view for randomized stable atoms is a genuine extension of random features to damped systems, but the sparse-recovery guarantees remain conditional on unproven restricted-eigenvalue properties of the disk-Vandermonde matrices.

read the letter

The paper introduces a Disk-Bochner representation that connects positive measures over stable poles to positive-definite kernels with a radius-dependent shift defect, then uses this to build randomized atomic features for system identification. Poles are sampled inside a disk to form a dictionary of damped exponentials, and identification becomes a convex regularized least-squares problem that can incorporate linear, SOC, or KYP constraints for stability, passivity, or settling-time priors. This generalizes random Fourier and Laplace features to the nonstationary damped regime while keeping modal interpretability and finite-dimensional computation. The authors also link the normalized transfer-function case to Nevanlinna-Pick interpolation and LFT set-membership. The operator-theoretic part, including the proof that such measures generate valid kernels and the characterization via subnormality of the shift, is the clearest new piece. For engineers who need to embed physics priors into convex identification with limited data, the framework offers a practical route that standard random-feature methods do not directly supply. The soft spot is the recovery side. Sparse-recovery guarantees are stated only conditionally on the realized disk-Vandermonde or input-output matrix satisfying restricted-eigenvalue properties. No concentration inequalities, explicit sampling-radius conditions, or lower bounds are derived to ensure these properties hold with high probability when poles are drawn uniformly inside the disk. In the under-informative regimes the numerics target, this leaves the claims formal rather than operational. The abstract mentions numerical comparisons showing gains from the priors, but without details on effect sizes or baselines it is difficult to judge how much the method actually improves recovery. This work is aimed at control and system-identification researchers who already use convex optimization and want to add modal or stability constraints without losing tractability. A reader focused on data-driven modeling with engineering priors would get value from trying the atomic-feature construction, even if the probabilistic guarantees need more work. I would send it to peer review. The kernel and operator results look substantive enough to merit referee attention, and the conditional recovery claims can be tightened or qualified during revision.

Referee Report

1 major / 1 minor

Summary. The paper proposes a physics-informed system identification framework that represents impulse responses as random superpositions of stable atoms (damped complex exponentials with poles sampled inside a prescribed disk). Identification is cast as convex regularized least-squares with optional linear, SOC, and KYP constraints. The central analytic contribution is an operator-theoretic Disk-Bochner representation: positive measures over stable poles induce positive-definite kernels with a radius-dependent shift defect, with a converse characterization via subnormality of the canonical shift; the work also establishes an RKHS-to-l1 embedding, shows that sampled poles induce a valid finite atomic gauge, discusses random-feature convergence, and states sparse-recovery guarantees conditionally on restricted-eigenvalue properties of the realized disk-Vandermonde or input-output matrix. Connections to Nevanlinna-Pick interpolation and LFT set-membership are drawn, and the approach is illustrated numerically for incorporating stability margins and other priors in under-informative regimes.

Significance. If the conditional sparse-recovery claims can be strengthened with explicit probabilistic control on the random dictionary, the framework would offer a flexible, interpretable generalization of random Fourier/Laplace features to damped nonstationary systems while directly encoding engineering priors such as stability, passivity, and settling time. The Disk-Bochner viewpoint and RKHS-to-l1 embedding constitute a clean operator-theoretic foundation that could support further modal analysis and set-membership methods.

major comments (1)

[Main analytic contribution (Disk-Bochner viewpoint, RKHS-to-l1 embedding, and sparse-recovery guarantees)] The sparse-recovery guarantees (stated in the main analytic contribution following the Disk-Bochner representation) are conditional on the realized disk-Vandermonde or input-output design matrix satisfying restricted-eigenvalue properties. No concentration inequalities, lower bounds on the minimal eigenvalue, or explicit conditions on disk radius and number of random poles are derived to ensure these properties hold with high probability under uniform sampling inside the disk. This leaves the finite atomic gauge and convex recovery claims formal rather than operational for the under-informative data regimes highlighted in the numerical comparisons.

minor comments (1)

[Abstract and analytic sections] The abstract asserts proofs of the Disk-Bochner statement, RKHS-to-l1 embedding, and random-feature convergence, but the main text would benefit from expanded step-by-step derivations (particularly for the subnormality characterization) to allow independent verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our paper. We provide a point-by-point response to the major comment below.

read point-by-point responses

Referee: The sparse-recovery guarantees (stated in the main analytic contribution following the Disk-Bochner representation) are conditional on the realized disk-Vandermonde or input-output design matrix satisfying restricted-eigenvalue properties. No concentration inequalities, lower bounds on the minimal eigenvalue, or explicit conditions on disk radius and number of random poles are derived to ensure these properties hold with high probability under uniform sampling inside the disk. This leaves the finite atomic gauge and convex recovery claims formal rather than operational for the under-informative data regimes highlighted in the numerical comparisons.

Authors: We acknowledge that the sparse-recovery guarantees are indeed conditional on the restricted-eigenvalue (RE) properties of the realized design matrix, and we do not provide new concentration inequalities or explicit probabilistic bounds in the manuscript. This is because the main focus of our work is on the operator-theoretic foundations, including the Disk-Bochner representation for positive measures over stable poles and the converse characterization via subnormality of the shift operator, as well as the RKHS-to-l1 embedding. The finite atomic gauge and recovery claims are presented in this conditional form, consistent with many results in random feature methods and structured compressive sensing where the probabilistic analysis of the dictionary is treated separately. For the under-informative regimes in our numerics, the constraints (stability, passivity, etc.) provide the practical regularization. In response to this comment, we will revise the manuscript to include additional discussion on the conditions under which the RE property is expected to hold, based on existing literature on random matrices with poles in the disk, and note that full probabilistic guarantees remain an open direction for future work. Thus, we make a partial revision by adding clarifying text without deriving new bounds. revision: partial

Circularity Check

0 steps flagged

No circularity; operator-theoretic claims proven independently and recovery guarantees stated conditionally

full rationale

The paper proves its central Disk-Bochner representation and RKHS-to-l1 embedding directly, states sparse-recovery results only conditionally on the realized matrix satisfying restricted-eigenvalue properties (without deriving those properties from the same data or by construction), and encodes physics priors as explicit constraints rather than tautological fits. No derivation step reduces to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on the assumption that positive measures on stable poles inside a disk yield valid kernels and that finite random samples produce design matrices with adequate restricted-eigenvalue properties; no explicit free parameters or invented entities are named in the abstract.

free parameters (2)

disk radius
Prescribed radius that defines the stability region for pole sampling; chosen by the user to match desired stability margins.
regularization weight
Scalar multiplier on the atomic-norm or sparsity term in the convex program; must be selected for each data set.

axioms (2)

domain assumption Positive measures over stable poles generate positive-definite kernels with a radius-dependent shift defect
Invoked as the central operator-theoretic foundation for the feature map.
domain assumption Sampled poles induce a valid finite atomic gauge
Required for the finite-dimensional convex formulation to be well-posed.

pith-pipeline@v0.9.0 · 5577 in / 1416 out tokens · 30249 ms · 2026-05-15T02:38:40.486298+00:00 · methodology

discussion (0)

Reference graph

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