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arxiv: 2605.21885 · v1 · pith:AJCTMFCRnew · submitted 2026-05-21 · 🧮 math.OC · cs.NA· math.NA

Proximal Gradient-based Low Rank Tensor Decomposition for State Dependent Riccati Equation

Jiahua Jiang , Carmeliza Navasca This is my paper

Pith reviewed 2026-05-22 05:13 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords optimal controlstate-dependent Riccati equationtensor decompositionlow-rank approximationproximal gradientmodel reductionCP decompositionPDE control
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The pith

Low-rank CP tensor decomposition reduces state-dependent Riccati equations for large control systems.

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

The paper develops a dimensionality reduction technique for optimal control problems that arise when partial differential equations are discretized into large-scale systems. Snapshot data from the full model is organized as a tensor and approximated by a low-rank canonical polyadic decomposition found via proximal gradient methods combined with sparse optimization. The resulting reduced basis produces a smaller state-dependent Riccati equation whose solution yields an approximate optimal control for the original system. Readers would care because solving the full-order Riccati equation directly is usually impossible at these scales, while the reduced version becomes numerically tractable.

Core claim

By computing basis elements from the canonical polyadic decomposition of snapshot tensors taken from large discrete control systems, using proximal gradient methods and flexible hybrid sparse optimization to obtain the low-rank factors, the reduced optimal control problem leads to reduced state-dependent Riccati equations that can be solved efficiently.

What carries the argument

Proximal gradient optimization to recover low-rank CP factors from snapshot tensors, supplying a reduced basis that preserves dynamics for the state-dependent Riccati equation.

If this is right

  • Optimal control becomes feasible for high-dimensional systems obtained from PDE discretizations.
  • The reduced state-dependent Riccati equation can be solved with standard dense or sparse Riccati solvers.
  • The same snapshot tensors can be reused across multiple control problems sharing the same dynamics.
  • Hybrid sparse and proximal methods allow flexible control over the rank and sparsity of the basis.

Where Pith is reading between the lines

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

  • If the low-rank structure is reliable, the same snapshot tensors could support online model adaptation when the underlying PDE changes slowly.
  • The approach may combine naturally with existing projection-based reduction methods such as proper orthogonal decomposition to further improve accuracy.
  • Real-time feedback control on embedded hardware becomes plausible once the reduced Riccati solve fits within tight time budgets.

Load-bearing premise

Snapshot tensors from the large discrete control systems admit accurate low-rank CP approximations that preserve the dynamics required for the optimal control solution.

What would settle it

Run the method on a concrete large-scale PDE control example, solve both the reduced and full-order state-dependent Riccati equations, and check whether the resulting feedback controls produce materially different closed-loop performance.

Figures

Figures reproduced from arXiv: 2605.21885 by Carmeliza Navasca, Jiahua Jiang.

Figure 2
Figure 2. Figure 2: Feedback control [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: State with Feedback Control [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: State with Feedback Control [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We address the optimal control problems arising from partial differential equations with large discrete dimensional control systems. To obtain reduced order models, we find basis elements from the canonical polyadic (CP) decomposition. Tensor datasets are from snapshots of the large models. Our method to reduce the control system is to use dimensionality reduction approaches through sparse optimization and flexible hybrid methods is to obtain low rank CP tensor basis elements. The reduced optimal control problem leads to reduced state-dependent Riccati Equations which can be solved efficiently.

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 manuscript proposes a proximal gradient-based approach to compute low-rank canonical polyadic (CP) decompositions of snapshot tensors obtained from large-scale discrete control systems. These factors supply reduced basis elements that yield a reduced-order optimal control problem whose state-dependent Riccati equation (SDRE) can be solved efficiently.

Significance. If the central claims are substantiated, the work would supply a tensor-based model-reduction route for high-dimensional SDRE problems that combines proximal-gradient sparsity with hybrid dimensionality reduction. The approach could be useful for PDE-constrained control where direct solution of the full SDRE is prohibitive, provided the low-rank truncation preserves the necessary quadratic structure and feedback properties.

major comments (3)
  1. [Theoretical development (around the reduced SDRE derivation)] No section derives an a-priori perturbation bound showing that the residual of the proximal-gradient CP approximation induces a controlled error in the solution of the reduced SDRE or in the resulting feedback law. This bound is load-bearing for the claim that the reduced SDRE remains close to the original in the control sense.
  2. [Algorithm description and convergence statement] The manuscript does not demonstrate that the proximal step respects the algebraic structure (state-dependent coefficients and quadratic form) required for the reduced Riccati operator to stay well-posed; convergence of the decomposition is stated but not linked to preservation of the dynamics.
  3. [Numerical results section] Numerical examples, if present, lack quantitative comparison of the closed-loop performance (e.g., cost or stability margin) between the full and reduced SDRE solutions, leaving the efficiency claim without direct evidence.
minor comments (2)
  1. [Preliminaries] Notation for the CP factors and the mapping from tensor snapshots to the reduced state space is introduced without a clear diagram or table summarizing the dimensions.
  2. [Abstract] The abstract refers to 'flexible hybrid methods' without a forward reference to the specific combination of proximal gradient and other techniques used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We find the comments helpful and will incorporate revisions to strengthen the theoretical and numerical aspects of the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Theoretical development (around the reduced SDRE derivation)] No section derives an a-priori perturbation bound showing that the residual of the proximal-gradient CP approximation induces a controlled error in the solution of the reduced SDRE or in the resulting feedback law. This bound is load-bearing for the claim that the reduced SDRE remains close to the original in the control sense.

    Authors: We agree that an explicit a-priori perturbation bound would strengthen the theoretical justification for the control performance of the reduced SDRE. The current manuscript emphasizes the algorithmic construction via proximal gradient CP decomposition on snapshot tensors and provides empirical evidence of accuracy. In the revision we will add a dedicated subsection that derives a first-order perturbation bound relating the CP residual norm to the error in the reduced Riccati solution and the resulting feedback operator, under standard Lipschitz assumptions on the state-dependent coefficients. revision: yes

  2. Referee: [Algorithm description and convergence statement] The manuscript does not demonstrate that the proximal step respects the algebraic structure (state-dependent coefficients and quadratic form) required for the reduced Riccati operator to stay well-posed; convergence of the decomposition is stated but not linked to preservation of the dynamics.

    Authors: The proximal gradient iterations are performed on the tensor of system snapshots; the resulting low-rank factors define a projection that is applied to the original state-dependent operators, thereby preserving the quadratic structure and state dependence by construction. Convergence of the proximal scheme controls the tensor approximation error, which in turn bounds the perturbation to the reduced dynamics. We will revise the algorithm section to include an explicit argument linking the proximal convergence guarantee to well-posedness of the reduced Riccati operator. revision: yes

  3. Referee: [Numerical results section] Numerical examples, if present, lack quantitative comparison of the closed-loop performance (e.g., cost or stability margin) between the full and reduced SDRE solutions, leaving the efficiency claim without direct evidence.

    Authors: The existing numerical section reports computational speedup and state-trajectory approximation error. We acknowledge that direct closed-loop metrics would provide stronger evidence. In the revised manuscript we will augment the examples with tables comparing the infinite-horizon cost and the minimum real-part eigenvalue of the closed-loop operator for the full-order and reduced SDRE solutions. revision: yes

Circularity Check

0 steps flagged

No circularity: standard proximal gradient applied to external snapshots yields reduced SDRE without self-referential fitting or load-bearing self-citations.

full rationale

The derivation begins with snapshot tensors collected from the original large-scale discrete control system (external data). Proximal gradient optimization is then used to recover low-rank CP factors as basis elements for dimensionality reduction. These factors produce a reduced optimal control problem whose associated state-dependent Riccati equation is solved directly. No fitted parameter is renamed as a prediction, no equation reduces to its own input by construction, and no uniqueness theorem or ansatz is imported via self-citation. The central claim therefore rests on the empirical fidelity of the CP approximation to the snapshot data rather than on any internal redefinition or self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of useful low-rank structure in snapshot tensors and on the ability of proximal gradient to recover it without destroying control-relevant dynamics.

axioms (1)
  • domain assumption Snapshot data from the large discrete control system sufficiently captures the dynamics needed for optimal control.
    Invoked when the authors form tensor datasets from snapshots to obtain basis elements.

pith-pipeline@v0.9.0 · 5608 in / 1128 out tokens · 64472 ms · 2026-05-22T05:13:46.974086+00:00 · methodology

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Reference graph

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