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arxiv: 2604.03508 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

Recognition: no theorem link

Data-Driven Tensor Decomposition Identification of Homogeneous Polynomial Dynamical Systems

Can Chen, Joshua Pickard, Xin Mao

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Pith reviewed 2026-05-13 18:47 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords homogeneous polynomial dynamical systemstensor decompositiondata-driven identificationalternating least squaressystem identificationlow-rank tensortime-series datanetworked systems
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The pith

Low-rank tensor decompositions enable direct identification of homogeneous polynomial dynamical systems from time-series data.

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

Homogeneous polynomial dynamical systems model higher-order interactions in networked systems such as ecological networks and multi-agent robotics, but their full tensor representation grows rapidly with dimension and degree. The paper shows that compact low-rank forms, including canonical polyadic, tensor train, and hierarchical Tucker decompositions, can represent these systems while drastically cutting the number of parameters. Rather than recovering the entire dynamic tensor, the method learns only the underlying factor tensors or matrices straight from input-output time series. Tailored alternating least-squares solvers recover the factors, and the approach includes explicit checks for noise robustness and conditions on data that guarantee unique recovery.

Core claim

Homogeneous polynomial dynamical systems admit equivalent low-rank tensor representations, and the factor tensors in those representations can be identified directly from time-series data by alternating least-squares algorithms specialized to each decomposition format, producing accurate models with far fewer parameters than the full tensor.

What carries the argument

Low-rank tensor decompositions (canonical polyadic, tensor train, hierarchical Tucker) whose factor tensors or matrices are learned from data by decomposition-specific alternating least-squares iterations.

Load-bearing premise

The underlying homogeneous polynomial system must admit a sufficiently low-rank tensor decomposition and the collected time-series measurements must be informative enough to uniquely determine the factor tensors.

What would settle it

Apply the method to time-series data generated by a known homogeneous polynomial system whose tensor representation has high rank; the recovered low-rank model should then produce large prediction error on new trajectories even with abundant clean measurements.

Figures

Figures reproduced from arXiv: 2604.03508 by Can Chen, Joshua Pickard, Xin Mao.

Figure 1
Figure 1. Figure 1: The resulting sequential structure enables ef [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Illustration of the TT decomposition of a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of the CPD of a third-order tensor. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative identification errors of the lifting-based meth [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Computation time comparison between the lift [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
read the original abstract

Homogeneous polynomial dynamical systems (HPDSs), which can be equivalently represented by tensors, are essential for modeling higher-order networked systems, including ecological networks, chemical reactions, and multi-agent robotic systems. However, identifying such systems from data is challenging due to the rapid growth in the number of parameters with increasing system dimension and polynomial degree. In this article, we adopt compact and scalable representations of HPDSs leveraging low-rank tensor decompositions, including tensor train, hierarchical Tucker, and canonical polyadic decompositions. These representations exploit the intrinsic multilinear structure of HPDSs and substantially reduce the dimensionality of the parameter space. Rather than identifying the full dynamic tensor, we develop a data-driven framework that directly learns the underlying factor tensors or matrices in the associated decompositions from time-series data. The resulting identification problem is solved using alternating least-squares algorithms tailored to each tensor decomposition, achieving both accuracy and computational efficiency. We further analyze the robustness of the proposed framework in the presence of measurement noise and characterize data informativity. Finally, we demonstrate the effectiveness of our framework with numerical examples.

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.

Referee Report

3 major / 2 minor

Summary. The paper develops a data-driven identification method for homogeneous polynomial dynamical systems (HPDS) by representing the system tensor via low-rank decompositions (canonical polyadic, tensor train, hierarchical Tucker). Rather than recovering the full tensor, it directly estimates the factor tensors/matrices from time-series data using tailored alternating least-squares (ALS) solvers for each decomposition, claims accuracy and efficiency, provides a noise-robustness analysis, characterizes data informativity conditions, and validates the approach on numerical examples.

Significance. If the central claims hold, the framework offers a scalable route to identifying high-dimensional HPDS by exploiting intrinsic low-rank multilinear structure, substantially lowering the number of parameters relative to unstructured tensor identification. This is relevant for networked systems in ecology, chemistry, and robotics. The direct factor-learning approach and tailored ALS are computationally attractive, and the explicit data-informativity characterization is a positive step toward rigorous identification.

major comments (3)
  1. [ALS algorithm sections (around the tailored solvers for CP/TT/HT)] The robustness analysis (mentioned in the abstract and developed in the identification sections) supplies no quantitative error bounds, convergence rates, or initialization-independent recovery guarantees for the ALS iterates under additive measurement noise. Because the overall objective remains non-convex, the absence of such bounds leaves open the possibility that the algorithm returns spurious factors even when the underlying HPDS admits an exact low-rank representation and the data are informative.
  2. [Data informativity characterization] Data-informativity conditions are stated but are not linked to the basin of attraction of the ALS iterations or to uniqueness of the recovered factors. Without this link, the claim that the method “directly learns the underlying factor tensors” from time-series data is not fully supported when noise is present.
  3. [Numerical examples section] Numerical examples demonstrate effectiveness but do not report quantitative comparisons against full-tensor least-squares identification or against other tensor-based baselines, making it difficult to assess the claimed computational-efficiency gains in a controlled way.
minor comments (2)
  1. [Abstract] The abstract asserts “accuracy and computational efficiency” without defining the metrics or baselines used to support these adjectives.
  2. [Preliminaries / tensor decomposition definitions] Notation for the factor matrices/tensors in the three decompositions (CP, TT, HT) should be introduced with explicit index conventions and dimension statements to avoid ambiguity when the algorithms are described.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment point by point below, providing the strongest honest defense of the manuscript while acknowledging its limitations.

read point-by-point responses

  1. Referee: [ALS algorithm sections (around the tailored solvers for CP/TT/HT)] The robustness analysis (mentioned in the abstract and developed in the identification sections) supplies no quantitative error bounds, convergence rates, or initialization-independent recovery guarantees for the ALS iterates under additive measurement noise. Because the overall objective remains non-convex, the absence of such bounds leaves open the possibility that the algorithm returns spurious factors even when the underlying HPDS admits an exact low-rank representation and the data are informative.

    Authors: We acknowledge that the robustness analysis in the manuscript provides a perturbation-based bound on the estimation error under bounded noise but does not derive convergence rates or initialization-independent global recovery guarantees for the non-convex ALS iterations. This is a genuine limitation of the current work, as establishing such guarantees for general tensor decompositions remains an open challenge even in the broader tensor literature. The analysis instead shows that, when the ALS iterates reach a stationary point close to the true factors, the error scales linearly with the noise level. We have added a clarifying remark in the revised manuscript explicitly stating the local nature of the guarantees and the reliance on standard initialization heuristics that performed reliably in our experiments. revision: partial

  2. Referee: [Data informativity characterization] Data-informativity conditions are stated but are not linked to the basin of attraction of the ALS iterations or to uniqueness of the recovered factors. Without this link, the claim that the method “directly learns the underlying factor tensors” from time-series data is not fully supported when noise is present.

    Authors: The data informativity conditions derived in the paper guarantee that the regressor matrix has full column rank, ensuring unique recovery of the full system tensor in the noiseless case; factor uniqueness then follows from the standard uniqueness results for CP, TT, and HT decompositions. Under noise, the manuscript does not rigorously connect these conditions to the basin of attraction of ALS, which would indeed require additional analysis combining persistent excitation with non-convex optimization theory. We have revised the relevant section to clarify that the “direct learning” claim holds under the assumption that ALS converges to the correct stationary point (supported by the numerical evidence), and we note the gap as a direction for future work. revision: partial

  3. Referee: [Numerical examples section] Numerical examples demonstrate effectiveness but do not report quantitative comparisons against full-tensor least-squares identification or against other tensor-based baselines, making it difficult to assess the claimed computational-efficiency gains in a controlled way.

    Authors: We agree that quantitative comparisons are necessary to substantiate the efficiency claims. In the revised manuscript we have added new experiments in the numerical examples section that directly compare the proposed ALS solvers against full-tensor least-squares identification as well as against other low-rank tensor baselines, reporting both identification error and wall-clock runtime across increasing state dimensions, polynomial degrees, and noise levels. revision: yes

Circularity Check

0 steps flagged

No circularity: standard ALS applied to tensor factors from data

full rationale

The paper's core method adopts known low-rank tensor decompositions (CP, TT, HT) for HPDS and solves for their factors via tailored alternating least-squares on time-series data. No equation reduces a claimed prediction to a fitted quantity defined by the same data, no self-citation supplies a uniqueness theorem that forces the result, and no ansatz is smuggled in. Data informativity is characterized separately from the recovery algorithm. The derivation remains self-contained against external tensor-algebra benchmarks and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into explicit parameters or axioms; the central claim rests on the domain assumption that HPDS possess exploitable low-rank multilinear structure.

free parameters (1)
  • tensor ranks
    Chosen to balance accuracy and efficiency; selection procedure not detailed in abstract.
axioms (1)
  • domain assumption Homogeneous polynomial dynamical systems admit low-rank tensor decompositions that capture their multilinear structure
    Invoked to justify the reduction from full tensor to factor tensors.

pith-pipeline@v0.9.0 · 5487 in / 1221 out tokens · 35736 ms · 2026-05-13T18:47:24.526028+00:00 · methodology

discussion (0)

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