Solving Implicit Inverse Problems with Homotopy-Based Regularization Path

Davide Parodi; Federico Benvenuto; Michele Piana; Sara Garbarino

REVIEW 3 major objections 5 minor 33 references

Solving Implicit Inverse Problems with Homotopy-Based Regularization Path

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes a homotopy-based regularization path that solves implicitly constrained inverse problems by tracing decreasing-α regularized minimizers with warm starts, and shows it recovers sparse latent ODE dynamics from noisy data.

desk verdict A clean, well-written assembly of known tools for implicit inverse problems, but the headline recovery numbers are oracle-selected and the paper still needs a non-oracle parameter choice rule and baselines. read the letter →

arxiv 2505.19608 v1 pith:Y7SH4WHI submitted 2025-05-26 math.NA cs.NAmath.OC

Davide Parodi , Federico Benvenuto , Sara Garbarino , Michele Piana This is my paper

classification math.NAcs.NAmath.OC MSC 65J2249M1565K10

keywords implicitinverseproblemshomotopymethodadjointstateregularizationpathdynamicalsystemsODElearningsparserecoverysemi-convergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that a homotopy-based regularization path, which solves a sequence of regularized constrained minimizations with decreasing α and warm starts, gives a stable and practically usable way to solve implicit inverse problems. Implicit inverse problems ask for a nonlinear functional acting on an unknown function, given only noisy samples of that function, and they are typically ill-posed with multiple solutions; the paper's claim is that tracing the path makes the non-unique problem tractable. The concrete demonstration is in latent dynamics discovery, where the paper shows that a sparse cosine-basis representation of an ODE's right-hand side can be recovered from noisy time series, with relative parameter errors between roughly 0.01 and 0.12 depending on the sparsity of the ground truth. If the claim holds, the method is a general template for equation learning, optimal control, and data assimilation problems framed as implicit inverse problems.

What carries the argument

The central object is the regularization path (19) generated by the outer loop, together with the inner-loop Lagrangian $L_{d,\alpha}(u,m,\lambda)=h_d(u)+\alpha g(m)-\langle\lambda,F(u,m)\rangle$ and its stationarity conditions from Proposition 1. The adjoint state method computes the gradient of the data term with respect to the parameters by solving the adjoint linear system (16); Newton–Raphson solves the forward nonlinear equation $F(u,m)=0$; Landweber iteration solves the adjoint system; and a hard-thresholding operator (26) enforces sparsity on the parameter vector after each update. Homotopy continuation is what carries the method: warm-starting each $\alpha_{l+1}$ problem from the solution at $\alpha_l$ keeps the iterates in the basin of attraction of a desired local solution and makes the whole path a continuous deformation from a strongly regularized, unique problem to the ill-posed one.

What would settle it

Take a synthetic ODE whose right-hand side includes a term outside the cosine library, for instance $f(u)=e^u$ or $f(u)=u^2$, run the same homotopy, and compare recovered coefficients and solution error with the in-library cases; a sharp rise in relative error as the basis mismatch grows would show that the library assumption, not the homotopy mechanism, is what carries the recovery result.

Watch

Extended reading notes

Core claim

The central claim is that the solution of the discretized regularized problem (8), computed for a decreasing sequence of regularization parameters $\alpha_0>\alpha_1>\dots>\alpha_L>0$ with each problem warm-started from the previous one, traces a regularization path that approaches a solution of the unregularized problem (9) while remaining numerically stable. Under this scheme, the adjoint state method gives the gradient of the data term with respect to parameters, and Newton–Raphson solves the nonlinear forward constraint $F(u,m)=0$ at each inner iteration. In the numerical study on latent dynamics discovery with the periodic basis $\Phi(u)=(\cos u,\dots,\cos 6u)$, the paper reports that the method recovers the ground truth coefficients: for the sparse vector $m_1=(1,-1,0,0,0,0)$ the relative parameter error stays around $0.01$–$0.02$ across noise levels, while for the denser $m_2=(-1.5,1.5,-1.5,1,-1,0)$ it is around $0.11$–$0.12$, with solution errors below $0.05$ in most settings. The paper also documents semi-convergence: the best parameter reconstruction is obtained at an intermediate $\alpha$, and continuing the path toward $\alpha\to0$ reconstructs noise instead of signal.

Load-bearing premise

The method assumes the true dynamics lie exactly in the prescribed finite cosine basis, with no model error, and that the forward ODE solver reliably finds the correct solution branch of the Cauchy problem for every parameter vector on the path.

Editorial extensions

If this is right

Because the method only requires the forward map and its Jacobians, the same homotopy template applies to any implicit inverse problem with a parametric nonlinear constraint, not just ODE learning.
For sparse ground truth, the recovered parameter errors are about an order of magnitude smaller than for a denser ground truth, so the method is most reliable when the true dynamics are governed by few active terms.
The semi-convergence documented in the paper implies the path should not be followed to $\alpha \to 0$ blindly; the optimal reconstruction sits at an intermediate regularization level, giving an early-stopping target.
The regularization path itself acts as a stability diagnostic: smooth paths with nearby solutions across adjacent $\alpha_l$ indicate that the problem is well-behaved, while irregular paths flag ill-posedness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the same homotopy could be used to prune the basis library online, removing coefficients that stay at zero across several $\alpha_l$ before the next stage, which would directly address the paper's stated limitation on basis design.
Beyond the paper: the semi-convergence curve suggests an automated stopping rule based on where the data-loss path flattens or where the parameter path starts to grow, analogous to the L-curve criterion for linear inverse problems.
Beyond the paper: a direct robustness test would be to add a small non-cosine term to the true dynamics; if the recovered coefficients change discontinuously along the path, basis mismatch is the controlling failure mode.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a well-written assembly of standard pieces — homotopy continuation, adjoint-state gradients, smooth L1 with hard thresholding — specialized to implicit inverse problems and ODE learning. The contribution is the combination, not a new principle, and that is fine. The math is coherent; Proposition 1 is textbook Lagrange/adjoint material, and the algorithm is described clearly enough to reimplement, except for the details noted below.

What I like: the paper is honest about semi-convergence and actually shows it (Figure 4), and it states its limitations in the conclusion (basis library design, manual derivatives). The synthetic experiments cover two ground truths and three noise levels, and the regularization paths look smooth. For the sparse case the parameter errors are low. That is real evidence that the machinery can work on nice no-model-error problems.

The soft spots are concentrated in the numerical claim. The reported errors are computed at the oracle point alpha*_q from Eq. (30), which minimizes parameter error against the true m. The paper explicitly demonstrates that error increases after that point, so Algorithm 1 returns a path, not a solution. Without a data-driven rule to pick alpha, the phrase 'recovers the important terms' is only true if someone hands you the ground truth. That is the load-bearing gap. The stress-test note is right on this; I do not see a way around it. Second, there is no baseline: no SINDy, no standard gradient matching, no fixed-alpha optimization without homotopy, so we cannot tell whether the continuation and warm starts add anything. Third, implementation details needed for independent reproduction are missing: how du/dt is discretized in Eq. (24), the value of epsilon in Eq. (25), initialization m^(0,0), NR and Landweber tolerances, and the early-stopping criterion. No code. Fourth, all tests assume the true dynamics lie exactly in the cosine basis (27), so there is zero model error; the paper never tests basis mismatch or branch jumping in the forward solver. That last point is not fatal, but it means the demonstrated scope is one clean, one-dimensional periodic case.

I would send this to peer review. The derivation is sound, the assembled method is plausible, and the limitations are acknowledged, but acceptance should require a non-oracle parameter-choice rule, at least one baseline, and enough implementation detail to reproduce. It is a useful paper for practitioners in equation learning and data assimilation, not a landmark.

Referee Report

3 major / 5 minor

Summary. The paper proposes a homotopy-based continuation method for implicit inverse problems, in which parameters m of a parametric constraint F(u,m)=0 are inferred from noisy observations d of u. For a fixed regularization parameter α, the method minimizes the regularized objective k_{d,α}(u_m,m) by gradient descent using adjoint-state gradients, with the forward constraint solved by Newton-Raphson and the adjoint equation by Landweber iteration. The outer loop decreases α over a logarithmic grid and warm-starts each inner optimization from the previous α. The method is specialized to latent ODE discovery with a periodic cosine basis and is tested on two synthetic one-dimensional systems (m1 sparse, m2 less sparse) under three noise levels. The paper reports relative parameter errors between 0.01 and 0.12 and solution errors between 0.01 and 0.05 for an oracle-selected α* on each trial, and it demonstrates semi-convergence of the parameter error along the path.

Significance. If the central claim were established, the proposed framework would be a useful contribution to implicit inverse problems and equation learning: the adjoint-state formulation avoids differentiating through the forward solver, the homotopy continuation is natural for ill-posed nonlinear problems, and the explicit reporting of semi-convergence is honest and informative. The variational derivation in Proposition 1 is standard and internally consistent, and the algorithmic structure is clearly laid out. However, the numerical evidence as reported does not yet demonstrate a practical recovery method, because the reported errors are selected in hindsight using the ground-truth parameters. The missing implementation details also prevent independent verification. The contribution is therefore promising but needs substantial revision of the numerical evaluation before the recovery claims can be accepted.

major comments (3)

[§4.1, Eq. (30)] Table 1 and the violin plots in Figure 2 report errors of m*_q selected by α*_q = argmin_l ||m_q^{(l)} − m|| / ||m||, which uses the ground-truth m. Since m is unknown in an inverse problem, this is an oracle selection, not a practical parameter-choice rule. Figure 4 shows that the parameter error is non-monotone along the path: it decreases, reaches a minimum, and then increases as α becomes small while the data loss continues to decrease. Without a data-driven stopping or selection rule, the claim that the method 'recovers the important terms' is not demonstrated for any user of the algorithm. I ask the authors to either add a practical parameter-choice rule (e.g., discrepancy principle, L-curve, or cross-validation) and re-report the errors for the automatically selected α, or explicitly restrict the conclusions to properties of the oracle-selected path.
[§4.1, Algorithm 1, Eqs. (24)–(26)] Several implementation details that are load-bearing for the numerical results are missing: the discretization used for the time derivative ˙u in Eq. (24); the value of ϵ in the smooth L1 approximation (25); the tolerances and stopping criteria for the inexact Newton-Raphson and Landweber solves; the initial parameter vector m^(0,0); the number of inner iterations N_iter and how early stopping is triggered; and whether the hard threshold (26) is applied before or after the gradient update. Moreover, the text after Eq. (26) says the threshold is implemented at 'line 7' of Algorithm 1, but line 7 of the pseudocode is 'Compute ∇mLd,α' and no thresholding appears anywhere in Algorithm 1. Without these details the experiments cannot be reproduced, and the specific effect of the hard-thresholding step on the reported errors is unclear.
[§4 and §5] The numerical validation is limited to the no-model-error setting in which the true dynamics lie exactly in the candidate basis (27). The paper states this basis assumption but never tests robustness to basis mismatch, and the forward ODE (23) is assumed to have a unique solution that the Newton-Raphson solver can reliably find for every parameter vector on the path. Since the abstract and introduction frame the method for general implicit inverse problems, the current evidence covers only one favorable synthetic class. A test with an out-of-basis target or with a forward map that has multiple branches would clarify the scope and limitations of the method.

minor comments (5)

[§2] After equation (4), the text contains an incomplete citation placeholder: 'the regularization function [22, 23] [REF!!!!].' This should be completed or removed.
[§1] In the third paragraph of the Introduction, 'the forward model is given by a an autonomous differential equation' contains a duplicated article 'a an'; please correct to 'an autonomous'.
[§5] In the Conclusions, 'the ground truth was scattered' should presumably read 'the ground truth was sparse,' matching the terminology used elsewhere in the paper.
[Algorithm 1, line 10] The line 'Set m(l+1,0) ← m(l) ← m(l,N iter)' contains a double assignment and uses N_iter, which is not defined in the pseudocode or the parameter list; please clarify the intended update and the stopping criterion.
[Table 1] The column headers 'Low', 'Medium', 'High' are not defined in the caption or the text, and the text later refers to 'σ = [0.01, 0.1, 0.2]' without connecting these labels. Please label the columns with the actual σ values.

Circularity Check

1 steps flagged · score 2.0 of 10

The Lagrange/adjoint derivation is self-contained; the only circularity-adjacent element is Eq. (30), where the ground-truth m selects the reported point on each regularization path, making Table 1 a best-case envelope rather than a standalone recovery result.

fitted input called prediction [Section 4.1, Eq. (30) and Table 1]
"For each trial q = 1, ..., n, we define α∗ q = arg min α(l),l=0,...,99 ∥m(l) q − m∥2/∥m∥2 (30) ... We define m∗ q the set of parameters associated with the best regularization parameter α∗ q (30). ... Results obtained are summarized in Table 1."

The error reported in Table 1 is the same quantity minimized in Eq. (30): the relative parameter error against the true m. Because α∗q is chosen using the ground truth, m∗q is, by construction, the closest point on each trial's path to the target. The quoted errors are therefore a lower envelope over the path, not errors achieved by any parameter-choice rule that a user could apply when m is unknown. The paper itself shows semi-convergence (Figure 4), so without Eq. (30) the algorithm returns a path rather than a selected solution. Thus the empirical claim that the method 'recovers the important terms' is an oracle-selected benchmark statement, although it does not make the underlying Lagrange/adjoint derivation equivalent to its input.

full rationale

The mathematical derivation is not circular: Proposition 1 follows directly from the Lagrange multiplier theorem, the adjoint-state equation is derived from the stationarity conditions, and the numerical experiments use external synthetic ground truths generated from known coefficient vectors. The only circularity-adjacent practice is the oracle alpha* selection in Eq. (30), which inflates the reported recovery errors by choosing the best point on each regularization path using the true m; this is an evaluation-protocol weakness, not a circular derivation step. The text also contains a literal '[REF!!!!]' placeholder in Section 2 after the regularization-function citations, indicating an incomplete reference, but that is an editorial defect rather than a circularity. Overall, the method's derivation is self-contained and its central algorithmic claims are independently supported, so the circularity score is low.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard optimization theory and on strong domain assumptions: the true dynamics are in the chosen basis, the forward problem is uniquely solvable, and the noise model is Gaussian. No new physical entities are introduced. Several algorithm hyperparameters and the oracle alpha* selection are underreported or chosen by hand, which affects reproducibility and the strength of the numerical validation.

free parameters (7)

Regularization schedule alpha_l = L=100 log-spaced points from 10^0 to 10^-6
The outer loop depends on this schedule; no adaptive or data-driven rule is given.
Per-trial oracle regularization level alpha*_q = chosen by minimizing relative parameter error against true m (Eq. 30)
Used only in evaluation; makes reported errors best-case along path rather than achievable predictions.
Gradient step tau = 10^-3
Hand-chosen inner-loop step; no convergence or step-size rule is provided.
Smooth L1 smoothing parameter epsilon = not reported
The approximation g(m)=sum sqrt(m_p^2+epsilon^2) requires a specific epsilon; without it the gradient and thresholding behavior cannot be reproduced.
Hard threshold factor = alpha_l/2 in HT (Eq. 26)
The factor 1/2 and its coupling to alpha_l determine which coefficients are zeroed; no justification is given.
Iteration and early stopping counts = Nmax=1000, Rmax=50, Lmax=100, NES=5
All fixed by hand; the early stopping criterion is not specified beyond checking NES loss values.
Initial parameter vector m^(0,0) = not reported
The warm-start path and final solution depend on initialization; the value is absent from the paper.

assumptions (6)

standard math Lagrange multiplier theorem applies to the nonconvex, nonlinearly constrained problem (8)
Used to prove Proposition 1; requires a constraint qualification that is not checked.
domain assumption The solution map m to u_m is well-defined and the forward equation F(u,m)=0 has a unique solution for each m
The Newton-Raphson forward loop in Algorithm 1 assumes a solvable, stable forward problem, while implicit inverse problems can have non-unique solutions.
domain assumption The true dynamics lie in the finite cosine basis Phi(u)=(cos(u),...,cos(6u))
Assumed in Section 4 and Eq. (21); no basis-mismatch or model-error experiments are performed.
domain assumption Measurement noise is additive Gaussian with known standard deviation, justifying squared L2 fidelity
Eq. (3) and Eq. (5); if noise is non-Gaussian, the data term would need to change.
ad hoc to paper Inexact Newton-Raphson and Landweber solves with early stopping still produce valid descent directions for the outer gradient descent
Remarks 3-5 state these systems are never solved to convergence; no formal guarantee is given that inaccurate gradients converge.
ad hoc to paper The discrete time derivative dot u in (24) is computed by a stable, unspecified scheme
Eq. (24) uses discrete derivative values but the paper never states the discretization, which is critical under noise.

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Cite this review

Pith. "Pith review of Solving Implicit Inverse Problems with Homotopy-Based Regularization Path." pith.science (2026). https://pith.science/paper/Y7SH4WHI

@misc{pith2026250519608,
  author       = {Pith},
  title        = {Pith review of: Solving Implicit Inverse Problems with Homotopy-Based Regularization Path},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y7SH4WHI}},
  note         = {Machine review of arXiv:2505.19608}
}

read the original abstract

Implicit inverse problems, in which noisy observations of a physical quantity are used to infer a nonlinear functional applied to an associated function, are inherently ill posed and often exhibit non uniqueness of solutions. Such problems arise in a range of domains, including the identification of systems governed by Ordinary and Partial Differential Equations (ODEs/PDEs), optimal control, and data assimilation. Their solution is complicated by the nonlinear nature of the underlying constraints and the instability introduced by noise. In this paper, we propose a homotopy based optimization method for solving such problems. Beginning with a regularized constrained formulation that includes a sparsity promoting regularization term, we employ a gradient based algorithm in which gradients with respect to the model parameters are efficiently computed using the adjoint state method. Nonlinear constraints are handled through a Newton Raphson procedure. By solving a sequence of problems with decreasing regularization, we trace a solution path that improves stability and enables the exploration of multiple candidate solutions. The method is applied to the latent dynamics discovery problem in simulation, highlighting performance as a function of ground truth sparsity and semi convergence behavior.

Figures

Figures reproduced from arXiv: 2505.19608 by the authors.

**Figure 1.** m1 (first row), m2 (second row) synthetic data (light blue line) and noisy discrete synthetic data (blue dots) with different amount of noise: from left to right σ = [0.01, 0.1, 0.2]. x−axis shows time. y−axis shows the values of the state variable. Results obtained are summarized in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** Violin plots of relative errors on parameters and solutions computed with best regularization parameter α ∗ q for each trial q = 1, ..., n. First row shows results about parameters m. Second row shows results about solutions u. x-axis reports different levels of noise σ = [0.01, 0.1, 0.2]. y-axis reports the relative error [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Regularization paths with level of noise σ = 0.1. First row shows results for m1 (28), second row shows results for m2 (29). The first column reports the regularization paths of the solutions. The colors of the curves transition from blue (l = 0) to red (l = 99). The x−axis reports the time, the y−axis the value of the state variable. The second column reports the regularization paths of the parameters. x−axis shows… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: is obtained by considering a noise level of σ = 0.2, a choice that allows us to operate in a context where regularization is indispensable, while keeping the perturbed data sufficiently close to the exact data. The figure shows a representative example of the semi-conv…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Mosegaard K and Rygaard-Hjalsted C 1999 Inverse problems 15 573

work page 1999

[2] [2]

Nino-Ruiz E D, Ardila C and Capacho R 2018 Soft Computing 22 4819–4832

work page 2018

[3] [3]

Brunton S L, Proctor J L and Kutz J N 2016 Proceedings of the national academy of sciences 113 3932–3937

work page 2016

[4] [4]

Chen Z, Liu Y and Sun H 2021 Nature communications 12 6136

work page 2021

[5] [5]

Heinonen M, Yildiz C, Mannerstr¨ om H, Intosalmi J and L¨ ahdesm¨ aki H 2018 Learning unknown ode models with gaussian processes International conference on machine learning (PMLR) pp 1959–1968

work page 2018

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Lorenzi M and Filippone M 2018 Constraining the dynamics of deep probabilistic models International Conference on Machine Learning (PMLR) pp 3227–3236

work page 2018

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Dondelinger F, Husmeier D, Rogers S and Filippone M 2013 Ode parameter inference using adaptive gradient matching with gaussian processes Artificial intelligence and statistics (PMLR) pp 216–228

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Williams C K and Rasmussen C E 2006 Gaussian processes for machine learning vol 2 (MIT press Cambridge, MA)

work page 2006