Perturbation-Induced Linearization (PIL)

• Perturbation-Induced Linearization (PIL) is a method that converts nonlinear systems into linear representations through explicit perturbation and reconstruction.
• It enables global conjugacy in dynamical systems, finite-dimensional operator construction, and simplifies complex processes in numerical and machine learning contexts.
• PIL serves as a unified design pattern across domains, emphasizing controllability and structured perturbation to achieve reliable linear surrogates.

Perturbation-Induced Linearization (PIL) denotes a family of constructions in which perturbations are introduced, analyzed, or inverted so that a nonlinear system, a nonautonomous evolution, a training dynamics, or an algorithmically difficult numerical procedure can be represented through a linear system, a linearized dynamics, or a linear surrogate. The cited literature uses the label both explicitly and as a precise conceptual framing. In these works, PIL appears in global nonautonomous topological conjugacy for quasilinear perturbations, multiscale linearization of nonautonomous systems, perturbation-based construction of finite-dimensional linear operators, post-hoc uncertainty estimation from linearized training dynamics, generation of unlearnable examples with linear surrogates, perturbation-induced static pivoting for GPU solvers, and recovery of polynomial perturbations from perturbed linearization pencils (Castañeda et al., 2024, Backes et al., 2022, Avillez et al., 2024, Zhang et al., 22 May 2025, Liu et al., 27 Jan 2026, Chevalier et al., 2023, Dmytryshyn, 2020).

1. Scope and recurrent structure

The recurrent structure is a perturbation step followed by a linear representation step. In nonautonomous dynamical systems, one starts from a linear system and a nonlinear or quasilinear perturbation and seeks a topological conjugacy between them. In operator-construction settings, one expands a nonlinear ordinary differential equation in a small parameter and then extends the configuration space so that the expanded dynamics become exactly linear in a polynomial basis. In modern machine learning, one linearizes training dynamics in the lazy / NTK regime, introduces a hypothetical perturbation, and uses the induced fluctuation to define uncertainty. In data protection, one constructs perturbations with a bias-free linear classifier so that downstream deep models rely on linearly separable shortcut patterns. In numerical linear algebra, one perturbs a matrix so that static pivoting becomes possible and then linearly combines perturbed solves to recover the unperturbed solution. In matrix-polynomial analysis, one starts from a perturbation of a linearization pencil and recovers the corresponding perturbation of the underlying polynomial (Castañeda et al., 2024, Avillez et al., 2024, Zhang et al., 22 May 2025, Liu et al., 27 Jan 2026, Chevalier et al., 2023, Dmytryshyn, 2020).

Domain	Linear object constructed or used	Representative paper
Nonautonomous ODEs	Topological conjugacy to the linear part	(Castañeda et al., 2024)
One-sided multiscale dynamics	Topological conjugacy or quasi-conjugacy	(Backes et al., 2022)
Perturbed ODEs with small parameter	Finite-dimensional linear operator $\frac{d\boldsymbol{v}}{d\tau} = M \boldsymbol{v}$	(Avillez et al., 2024)
Test-time uncertainty estimation	Linearized NTK training dynamics	(Zhang et al., 22 May 2025)
Unlearnable examples	Bias-free linear surrogate classifier	(Liu et al., 27 Jan 2026)
GPU linear solvers	Static-pivotable perturbed linear systems	(Chevalier et al., 2023)
Matrix polynomials	Companion linearization of a perturbed polynomial	(Dmytryshyn, 2020)

These uses are technically distinct. The cited works suggest that PIL is better understood as a perturbation-centric design pattern than as a single domain-independent theorem.

2. Global linearization in nonautonomous dynamical systems

A central PIL meaning in dynamical systems is the construction of a global and nonautonomous topological conjugacy between

$$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$

with $t\in\mathbb{R}_0^+$, where the linear system has a nonuniform exponential dichotomy and nonuniform bounded growth, and the perturbation is globally Lipschitz in space with exponential control in time. The conjugacy is global-in-space and global-in-time-forward, with homeomorphisms $H_t$ and $G_t=H_t^{-1}$ satisfying conjugacy of flows and properness at infinity. The explicit constructions are perturbations of the identity,

$$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$

where $w^*$ is given by a Green-operator integral and $z^*$ is the unique fixed point of an operator $\Gamma_{(\tau,\xi)}$ on a weighted Banach space $\mathcal A$ (Castañeda et al., 2024).

The distinctive feature is parameter mixing. The linear system carries dichotomy parameters $$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$0 and bounded-growth parameters $$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$1; the perturbation carries $$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$2. Existence of topological conjugacy is obtained under

$$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$3

$$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$4

and

$$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$5

An additional condition,

$$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$6

yields differentiability of $$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$7, with

$$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$8

For $$\dot{x} = A(t)x \quad\text{and}\quad \dot{y} = A(t)y + f(t,y),$$9-diffeomorphism, the paper introduces increasing sequences $t\in\mathbb{R}_0^+$0 and $t\in\mathbb{R}_0^+$1; by increasing them, the intervals $t\in\mathbb{R}_0^+$2 where $t\in\mathbb{R}_0^+$3 is a $t\in\mathbb{R}_0^+$4-diffeomorphism increase, which is the “broader smoothness interval” stated in the abstract (Castañeda et al., 2024).

A second dynamical-systems formulation removes asymptotic hypotheses such as exponential dichotomy altogether. For one-sided discrete and continuous dynamics on Banach spaces, the phase space is decomposed into finitely many mutually complementary directions,

$t\in\mathbb{R}_0^+$5

with $t\in\mathbb{R}_0^+$6. Direction-dependent boundedness and Lipschitz conditions are imposed only along $t\in\mathbb{R}_0^+$7, together with the global smallness condition

$t\in\mathbb{R}_0^+$8

The resulting theorems yield topological conjugacy when $t\in\mathbb{R}_0^+$9 or when the perturbation vanishes on $H_t$0, and quasi-conjugacy otherwise. The multiscale feature is that each direction has its own control functions and smallness parameter; no exponential dichotomy, hyperbolicity, spectral gap, or prescribed asymptotic behavior is assumed (Backes et al., 2022).

3. Perturbation expansions and linear operators in extended spaces

Another PIL lineage constructs finite-dimensional linear operators that approximately represent nonlinear systems of perturbed ordinary differential equations. The starting point is

$H_t$1

together with a perturbation expansion

$H_t$2

When oscillatory behavior produces secular terms, the methodology applies the Lindstedt-Poincaré expansion, introducing

$H_t$3

and choosing the $H_t$4 to remove secular terms. The expanded equations are then written in polynomial basis functions, and the configuration space is extended so that the dynamics become exactly linear in the lifted variables: $H_t$5 The paper proves finite representability under structural conditions that prevent indefinite growth of monomial degrees under differentiation, and it gives pseudo-code for constructing the basis and the constant matrix $H_t$6 (Avillez et al., 2024).

The Duffing oscillator

$H_t$7

is the basic example. A second-order Lindstedt-Poincaré construction yields a $H_t$8 operator, while a simpler power expansion without frequency correction yields a $H_t$9 operator. For the J2 problem, the method uses an osculating formulation and low-eccentricity variables, derives polynomial perturbation equations, and constructs a $G_t=H_t^{-1}$0 operator without frequency control and a $G_t=H_t^{-1}$1 operator with full Lindstedt-Poincaré control. For J2 with atmospheric drag, a constant-frequency formulation yields a $G_t=H_t^{-1}$2 operator, and a modified Lindstedt-Poincaré method with dynamically adapting frequency yields a $G_t=H_t^{-1}$3 operator. The same development also provides conditions on the osculating Keplerian elements that produce low-eccentricity frozen orbits (Avillez et al., 2024).

A different operator-theoretic PIL problem appears for matrix polynomials. Let

$G_t=H_t^{-1}$4

The first companion linearization $G_t=H_t^{-1}$5 is perturbed to $G_t=H_t^{-1}$6, where $G_t=H_t^{-1}$7 is a full perturbation of the pencil. The paper shows that, if

$G_t=H_t^{-1}$8

then there exist nonsingular $G_t=H_t^{-1}$9 and a polynomial perturbation $$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$0 such that

$$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$1

Moreover,

$$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$2

The recovery is constructive: an iterative algorithm repeatedly solves a coupled Sylvester system for $$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$3, applies $$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$4, and drives the unstructured part of the pencil perturbation to zero while accumulating the structured coefficient perturbation of the polynomial (Dmytryshyn, 2020).

4. Machine-learning formulations: uncertainty and unlearnability

In test-time uncertainty estimation, the paper “TULiP: Test-time Uncertainty Estimation via Linearization and Weight Perturbation” does not use the phrase “Perturbation-Induced Linearization (PIL)” explicitly, but it is organized around the same pattern. The network is linearized around $$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$5,

$$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$6

and training is modeled by linearized NTK dynamics,

$$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$7

A hypothetical perturbation is injected at time $$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$8,

$$G(t,\eta):=\eta+w^{*}(t;(t,\eta)), \qquad H(t,\xi):=\xi+z^{*}(t;(t,\xi)),$$9

and the uncertainty at a test point $w^*$0 is the training fluctuation

$w^*$1

The resulting post-hoc method estimates the bound through weight perturbations, finite differences, Hutchinson-style approximations of $w^*$2, and a surrogate ensemble called the Surrogate Posterior Envelope. The final classifier score is the Shannon entropy of the averaged predictive distribution. The paper reports state-of-the-art performance, particularly for near-distribution samples; on Table 1, CIFAR-10 near-OOD AUROC improves from 88.12 to 89.36, ImageNet-1K near-OOD AUROC from 76.55 to 78.38, and ImageNet-1K far-OOD AUROC from 86.21 to 88.85. The test-time cost is $w^*$3 forward passes; with $w^*$4, the measured wall-clock overhead is about $w^*$5 compared to EBO (Zhang et al., 22 May 2025).

The term PIL is used explicitly in “Perturbation-Induced Linearization: Constructing Unlearnable Data with Solely Linear Classifiers.” Here the task is defensive generation of unlearnable examples. The perturbed dataset is

$w^*$6

and the only surrogate is the bias-free linear classifier

$w^*$7

The perturbation jointly enforces semantic obfuscation and shortcut learning through

$w^*$8

with $w^*$9, perturbation step size $z^*$0, and $z^*$1. The mechanism advanced by the paper is that the perturbations force a downstream DNN to behave more linearly around the training data, relying on linearly separable perturbation patterns rather than nonlinear semantic features. On CIFAR-10, ResNet-18 accuracy falls from 92.11% on clean data to 12.77% with PIL; on CIFAR-100, DenseNet-121 falls from 75.22% to 1.23%; on ImageNet-100, DenseNet-121 falls from 76.98% to 3.14%. The measured perturbation-generation time on CIFAR-10 is 40.53 seconds, compared with 1.65k seconds for EM, 40.14k seconds for TAP, 26.78k seconds for SEP, and 54.46k seconds for REM. The paper further supports the “linearization hypothesis” through larger FGSM accuracy drops, changed Jacobian singular values, near-orthogonality between clean and UE gradients, and the partial-perturbation relation

$z^*$2

under Assumption 1 (Gradient Orthogonality) (Liu et al., 27 Jan 2026).

5. Perturbation-induced static pivoting in GPU linear solvers

A distinct numerical-linear-algebra use of PIL appears in GPU-based sparse direct solving, where dynamic numerical pivoting is a bottleneck. The proposed method solves

$z^*$3

by replacing one difficult, pivot-dependent factorization with a family of perturbed, pivot-free systems

$z^*$4

The averaged perturbation solutions are

$z^*$5

and these are linearly combined through a Vandermonde system to recover the unperturbed solution. With $z^*$6 total solves, the approximation error satisfies

$z^*$7

The method is therefore a perturbation-induced static pivoting scheme: a data-dependent pivoting process is replaced by a fixed family of statically factorizable systems whose outputs are linearly recombined (Chevalier et al., 2023).

The prototype implementation uses CUBLAS.getrf_batched! and CUBLAS.trsm_batched! on a distributed-slack AC power-flow Jacobian $z^*$8 from the PGLib 300-bus case. A naïve no-pivoting GPU solve yields vectors filled with NaN/Inf because a leading zero is encountered on the diagonal, whereas the perturbation-induced method stabilizes the solve. For $z^*$9, the residual $\Gamma_{(\tau,\xi)}$0 decreases roughly exponentially as the number of perturbed systems increases up to $\Gamma_{(\tau,\xi)}$1, until saturation around $\Gamma_{(\tau,\xi)}$2. Across 25 trials, batched LU factorization and batched triangular solves take essentially constant time for $\Gamma_{(\tau,\xi)}$3, and the linear combination of 10 perturbed solutions takes on the order of $\Gamma_{(\tau,\xi)}$4 (Chevalier et al., 2023).

6. Limitations, distinctions, and unifying themes

Several recurring limitations are explicit. In dynamical-systems PIL, existence of conjugacy is generally easier than smoothness: one paper proves topological conjugacy under mixed parameter inequalities and only $\Gamma_{(\tau,\xi)}$5 smoothness on time intervals under additional conditions, while the multiscale one-sided theory gives topological conjugacy or quasi-conjugacy, not higher regularity (Castañeda et al., 2024, Backes et al., 2022). In perturbation-based operator construction, the methodology requires a small parameter, an integrable unperturbed problem, and polynomial structure after expansion; operator dimension grows rapidly with perturbation order and system dimension (Avillez et al., 2024). In TULiP, the theoretical guarantees are tied to the lazy/NTK regime, smooth loss, bounded gradients, near-zero training loss, and a closeness assumption for test points (Zhang et al., 22 May 2025). In unlearnable-data PIL, protection requires labels, and partial perturbation does not prevent learning from whatever clean data remain (Liu et al., 27 Jan 2026). In perturbation-induced static pivoting, the choice of perturbation matrix $\Gamma_{(\tau,\xi)}$6 and sparse batched implementations remain open problems (Chevalier et al., 2023). In the matrix-polynomial setting, the algorithm is presented for the first companion linearization, although the paper states that it can be generalized to broader classes of linearizations (Dmytryshyn, 2020).

Several common misconceptions are therefore excluded by the cited literature. PIL is not restricted to one application area; it is not synonymous with first-order Taylor linearization; and it does not always use deep surrogates. A plausible synthesis is that PIL is best characterized by three ingredients: a perturbation model, a linear or linearized surrogate structure, and an explicit reconstruction or control mechanism. In global nonautonomous conjugacy this mechanism is the Green-function fixed-point construction; in polynomial and operator settings it is an extended finite basis; in test-time uncertainty it is the linearized training fluctuation propagated to inference time; in unlearnable data it is a linear shortcut enforced during training; in GPU solvers it is a Vandermonde recombination of perturbed solves; and in matrix-polynomial analysis it is strict equivalence between a perturbed pencil and the linearization of a perturbed polynomial (Castañeda et al., 2024, Avillez et al., 2024, Zhang et al., 22 May 2025, Liu et al., 27 Jan 2026, Chevalier et al., 2023, Dmytryshyn, 2020).

Taken together, these works suggest that PIL is a broad perturbation-centric methodology for obtaining tractable linear descriptions of otherwise nonlinear, nonuniform, or implementation-constrained problems. What changes from domain to domain is not the perturbation-plus-linearization motif, but the object being linearized, the admissible perturbation class, and the criterion by which the induced linear representation is judged: topological conjugacy, $\Gamma_{(\tau,\xi)}$7 smoothness, operator closure, uncertainty calibration, unlearnability, numerical stability, or backward-error interpretability.