arxiv: 2605.05901 · v1 · submitted 2026-05-07 · 💻 cs.CE

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Duplicate-Aware Shift-and-Lift Carleman Linearization:Structure, Complexity, and Comparative Evaluation

Takaki Akiba , Youhi Morii

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Pith reviewed 2026-05-08 03:57 UTC · model grok-4.3

classification 💻 cs.CE

keywords Carleman linearizationshift-and-liftduplicate coalescingmoving centernonlinear approximationmonomial basisJacobian comparison

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The pith

Shift-and-lift Carleman linearization with duplicate coalescing removes repeated monomials to cut overhead while keeping truncated dynamics intact.

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

The paper aims to make Carleman linearization practical at higher orders and dimensions by exposing and removing duplicate monomial terms during assembly. It does this through a shift-and-lift structure that allows coefficient coalescing, then pairs the structure with moving local centers so the truncation stays valid along system trajectories. A reader would care because standard Carleman embeddings grow rapidly with duplicates, limiting their use on nonlinear models, whereas this workflow reduces indexing and write costs without changing the underlying affine approximation. The authors test the approach on bilinear and logistic benchmarks and compare it directly to Jacobian linearization using error, step-size, and accuracy-per-cost metrics.

Core claim

The resulting workflow combines symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing to preserve truncated affine dynamics while reducing index-resolution overhead and write-path irregularity. Paired with moving-center expansion that jointly updates shift and lift, the method yields regime-dependent accuracy gains over Jacobian linearization on the bilinear driver and logistic interaction benchmarks, with both approaches converging under refinement.

What carries the argument

Duplicate-aware coefficient coalescing inside the shift-and-lift monomial basis, performed via symmetry reduction and sparse triplet assembly.

If this is right

Higher truncation orders become feasible because monomial count no longer grows with full duplication.
The assembled linear operator exactly matches the truncated nonlinear dynamics after coalescing.
Preprocessing and index-resolution costs scale better with state dimension and order.
Accuracy per computational cost improves relative to Jacobian linearization in some operating regimes.
Truncation error sources can be isolated separately from duplication artifacts.

Where Pith is reading between the lines

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

The same coalescing step could be reused inside repeated linearizations along a control trajectory without full re-assembly each time.
The symmetry-reduced bases might combine with other lifted representations such as Koopman operators to further compress high-dimensional models.
Testing on systems with known closed-form solutions would directly quantify how much of the observed error reduction comes from the moving center versus the duplicate removal.

Load-bearing premise

Jointly updating shift and lift around moving centers does not create new convergence problems or excessive recomputation for higher-order terms.

What would settle it

Measure the number of unique monomials after coalescing versus without coalescing at fixed truncation order on the bilinear benchmark; if the count does not drop substantially while trajectory error stays the same, the overhead-reduction claim fails.

Figures

Figures reproduced from arXiv: 2605.05901 by Takaki Akiba, Youhi Morii.

**Figure 1.** Figure 1: Trajectory comparison for Demo 1 (state x): reference solution versus Jacobian and shift-and-lift approximations view at source ↗

**Figure 2.** Figure 2: Error scaling in Demo 1 versus number of time steps. Both methods converge with view at source ↗

**Figure 3.** Figure 3: Trajectory comparison for Demo 2 (logistic interaction): reference, Jacobian, and shift view at source ↗

**Figure 4.** Figure 4: Error scaling in Demo 2 versus number of time steps for Jacobian and lifted models with view at source ↗

read the original abstract

The primary objective of this study is to remove duplicated monomial contributions that proliferate in Carleman linearization as state dimension and truncation order increase. To do so, we adopt a shift-and-lift architecture, since it exposes repeated exponent targets and allows duplicate-aware coefficient coalescing during lifted-operator assembly. This architecture also makes high-order truncation practical, but that regime intensifies local convergence and closure sensitivity for higher-order nonlinearities. We therefore pair shift-and-lift with a moving-center expansion so that shift and lift are updated jointly around evolving local centers, improving validity of the truncated model along the trajectory. The resulting workflow combines symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing to preserve truncated affine dynamics while reducing index-resolution overhead and write-path irregularity. We analyze variable growth, preprocessing complexity, and truncation-induced error mechanisms, and we compare against Jacobian linearization through fixed-step error, admissible step size, and cost-at-target-accuracy criteria. Two benchmarks (bilinear driver and logistic interaction) show convergence under refinement for both approaches, with regime-dependent accuracy gains for the proposed method rather than universal superiority.

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 gives a practical duplicate-coalescing tweak to shift-and-lift Carleman linearization plus moving centers, but the moving-center cost and convergence details stay thin.

read the letter

I read the paper on duplicate-aware shift-and-lift Carleman linearization. The main point is a workflow that cuts repeated monomial terms during lifted-operator assembly by using symmetry-reduced bases, packed exponent keys, and sparse triplet coalescing, then pairs it with joint moving-center updates to keep the truncated affine model valid longer along trajectories. They analyze growth in variables and preprocessing cost, and they compare the approach to plain Jacobian linearization on error, step size, and cost-at-accuracy metrics. Two benchmarks (bilinear driver and logistic interaction) are shown to converge under refinement, with the new method winning in some regimes but not all. That combination of indexing tricks and the moving-center idea is the concrete new piece; it is not just a restatement of earlier Carleman work. The structure is explained clearly enough that someone already working with lifted linearizations could implement the coalescing step without much guesswork. The benchmarks at least confirm the method does not break basic convergence. The soft spots sit in the validation and the moving-center claim. The abstract states convergence and regime-dependent gains but supplies no numbers, error bars, or tables, so the size of any advantage is hard to judge. More critically, the moving-center update is offered to fix local convergence problems at high order, yet the description gives no bound on update frequency, no accounting for re-assembly cost when the center shifts, and no check that the symmetry-reduced basis and coalescing still preserve the dynamics after repeated recentering. If center motion forces frequent rebuilds, the claimed reduction in index overhead could be lost, especially where the paper already flags sensitivity for higher-order nonlinearities. The stress-test concern about unanalyzed recomputation therefore lands. This is for computational engineers or control people who already use Carleman methods and want to push truncation order without the state space exploding. A reader who needs concrete indexing or coalescing recipes would get something usable from it. The work is clear enough on its own terms to deserve a serious referee rather than a desk reject; the core engineering change is well-motivated and the benchmarks provide a minimal sanity check, even if more quantitative results and cost analysis would be needed before acceptance.

Referee Report

1 major / 0 minor

Summary. The paper proposes a duplicate-aware shift-and-lift Carleman linearization that removes proliferating duplicate monomial contributions via symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing during lifted-operator assembly. It pairs this architecture with a moving-center expansion that jointly updates shift and lift around evolving local centers to improve validity of the truncated affine dynamics along trajectories, particularly under high-order truncation. Complexity analysis covers variable growth and preprocessing, while comparative evaluation against Jacobian linearization uses fixed-step error, admissible step size, and cost-at-target-accuracy on two benchmarks (bilinear driver and logistic interaction), reporting convergence under refinement with regime-dependent gains rather than universal superiority.

Significance. If the structural optimizations and moving-center updates preserve the claimed efficiency without offsetting recomputation costs, the work would advance practical high-order Carleman approximations for nonlinear systems in computational engineering by reducing index-resolution overhead and write-path irregularity. The explicit analysis of truncation-induced error mechanisms and the benchmark comparisons provide a useful framework for assessing when such methods outperform standard linearization, with potential applicability to control and simulation tasks.

major comments (1)

[moving-center expansion and workflow] The moving-center expansion (abstract and workflow description): the central claim that jointly updating shift and lift around evolving centers improves validity of the truncated model without new local convergence problems or excessive recomputation is load-bearing, yet the manuscript provides no explicit bound on center-update frequency, no analysis of interactions with the symmetry-reduced monomial basis or sparse triplet coalescing, and no demonstration that truncated affine dynamics remain preserved under repeated recentering. This directly affects whether the reported reductions in index-resolution overhead hold in the high-order regimes where sensitivity is already noted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The identification of the need for stronger theoretical grounding of the moving-center expansion is helpful, and we address this major comment below by outlining the revisions we will make.

read point-by-point responses

Referee: [moving-center expansion and workflow] The moving-center expansion (abstract and workflow description): the central claim that jointly updating shift and lift around evolving centers improves validity of the truncated model without new local convergence problems or excessive recomputation is load-bearing, yet the manuscript provides no explicit bound on center-update frequency, no analysis of interactions with the symmetry-reduced monomial basis or sparse triplet coalescing, and no demonstration that truncated affine dynamics remain preserved under repeated recentering. This directly affects whether the reported reductions in index-resolution overhead hold in the high-order regimes where sensitivity is already noted.

Authors: We agree that the current manuscript provides only a high-level description and empirical motivation for the moving-center expansion without the requested formal elements. In the revised version we will add: (1) an explicit bound on center-update frequency obtained from the local truncation remainder of the Carleman series together with a Lipschitz estimate on the nonlinearity along the trajectory, guaranteeing that recentering is triggered only when the expansion point would otherwise cause the affine approximation to exceed a user-specified error tolerance; (2) a short analysis showing that the symmetry-reduced monomial basis and packed exponent-key indexing remain invariant under recentering (the origin shift simply translates the exponent vectors while preserving their symmetry orbits, so the sparse triplet coalescing step can be performed incrementally on the updated keys without full re-indexing); (3) a concise invariance argument establishing that the truncated affine dynamics are preserved because the joint shift-and-lift operator commutes with the recentering transformation in the reduced basis. These additions will be accompanied by an extended numerical study on the bilinear and logistic benchmarks that quantifies the amortized preprocessing cost and confirms that recomputation does not offset the index-resolution savings even at high truncation orders. The revisions therefore directly substantiate that the reported overhead reductions continue to hold in the regimes of interest. revision: yes

Circularity Check

0 steps flagged

No circularity: structural algorithmic workflow with independent error comparisons

full rationale

The paper presents a shift-and-lift Carleman linearization augmented by moving-center expansion, symmetry-reduced monomial bases, packed exponent-key indexing, and sparse triplet coalescing. These are described as direct structural modifications to operator assembly that preserve truncated affine dynamics while reducing overhead. No equations, fitted parameters, or predictions are shown that reduce by construction to the inputs (e.g., no parameter fitted on one data subset then relabeled as a prediction on a related quantity). Comparisons to Jacobian linearization rely on external metrics such as fixed-step error, admissible step size, and cost-at-target-accuracy on two benchmarks, which are independent of the method's internal definitions. No self-citation chains or uniqueness theorems are invoked as load-bearing premises. The derivation chain is therefore self-contained as an engineering description rather than a tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard mathematical framework of Carleman linearization and the assumption that monomial duplication is the dominant overhead source; no new entities or fitted constants are introduced in the abstract.

axioms (1)

domain assumption Nonlinear systems can be represented as infinite-dimensional linear systems via monomial lifting, with truncation error controlled by order.
The paper builds directly on Carleman linearization without re-deriving or questioning this foundation.

pith-pipeline@v0.9.0 · 5501 in / 1190 out tokens · 82715 ms · 2026-05-08T03:57:36.256342+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 2 canonical work pages

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