Markov matrix perturbations to optimize dynamical and entropy functionals

Giulia Carigi; Manuel Santos Gutierrez; Niccolo Zagli

REVIEW 3 major objections 7 minor 2 cited by

Markov matrix perturbations to optimize dynamical and entropy functionals

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper gives linear algorithms that, for a given Markov matrix, find the perturbation of fixed norm which maximizes or minimizes entropy, Kullback-Leibler divergence, or entropy production, and a numerical recipe to read that matrix…

desk verdict The paper's entropy-production minimizer has a load-bearing algebraic error; the rest is a solid application of existing optimal-response methods, but the main new formula needs fixing before it can be trusted. read the letter →

arxiv 2507.14040 v1 pith:RAA34QXI submitted 2025-07-18 math.DS nlin.CD

Manuel Santos Gutierrez , Niccolo Zagli , Giulia Carigi This is my paper · ORCID

classification math.DSnlin.CD MSC 37A3037M2560J10

keywords MarkovchainperturbationlinearresponsetheoryentropyfunctionalKullback-LeiblerdivergenceproductionUlammethoddriftreconstructiontransferoperators

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

This paper is trying to establish that the problem of finding the external forcing that most changes a system's entropy-like statistics reduces to a linear algebra problem on a Markov matrix. Given a transition matrix $M$ with invariant vector $u$, the authors derive perturbation matrices $P$ that, to first order in the perturbation strength, maximize or minimize entropy, Kullback-Leibler divergence from the unperturbed state, and entropy production, under constraints that preserve the Markov property, the norm, and the sparsity pattern of $M$. Because such Markov matrices arise from Ulam discretizations of transfer operators, the same algorithms apply to discrete maps, stochastic differential equations, chaotic flows, and a periodic-orbit reduced model of turbulence. The paper also proposes a numerical protocol that reads the optimal matrix perturbation back as a vector-field perturbation, using the logarithm of the Markov matrix, so the forcing can be interpreted physically without knowing the underlying equations. If the claims hold, linear response theory becomes a design tool: one computes the forcing that produces a desired response in entropy-related observables instead of only predicting the response to a given forcing.

What carries the argument

The generalized inverse $G=(I_N - M + u\mathbf{1}^\top)^{-1}$ is the core object: it converts any admissible perturbation $P$ into the first-order change $G P u$ of the stationary measure, and every functional considered is a linear or quadratic function of this vector. The constraints $\|P\|_F=1$, $\mathbf{1}^\top P=0$, $P_{ij}=0$ where $M_{ij}=0$, and (for entropy production) $P u=0$ define the admissible set; the Lagrange multiplier solution for entropy production, Eq. (30), is the explicit minimizer. The return path from matrices to vector fields is carried by the ansatz $L=(1/\tau)\log M$ together with the first-moment formula $F(c_i)\approx\sum_j L_{ji}(c_j-c_i)$.

What would settle it

Simulate a system with known drift and diffusion, build Ulam Markov matrices at several box sizes and transition times, reconstruct the drift with Eq. (43), and measure the error against the true drift; if the error does not shrink as the partition refines, the matrix-to-vector-field link fails. The same test can be run on the optimal perturbation: apply the reconstructed forcing to the original SDE and check whether the entropy or KL change matches the value predicted by the linear response formulas.

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Extended reading notes

Core claim

The central claim is that for a mixing Markov matrix $M$, the first-order change in the stationary measure due to a perturbation $\varepsilon P$ is $v_1 = G P u$, with $G=(I_N - M + u\mathbf{1}^\top)^{-1}$, and that entropy, KL divergence, and entropy production are respectively linear, quadratic, and linear functionals of this $v_1$. Consequently, constrained optimization over $P$ with $\|P\|_F=1$, row sums zero, and the sparsity mask of $M$ fixed becomes a solvable linear problem: an explicit Lagrange multiplier formula for entropy, an SVD of a Kronecker-structured matrix for KL divergence, and another explicit formula with a Sherman-Morrison step for entropy production. The same formalism extends to the change in expectation of any observable, so the paper's formulas directly apply to mixing acceleration and to energy maximization in the turbulent-flow model. The paper further claims that the optimal matrix perturbation can be translated into a drift perturbation of the underlying continuous system through $F(c_i)\approx \sum_j L_{ji}(c_j-c_i)$ with $L=(1/\tau)\log M$, and validates this reconstruction on a one-dimensional double-well potential.

Load-bearing premise

The load-bearing premise is that the logarithm of a coarse-grained Ulam Markov matrix, divided by the transition time, faithfully represents the Fokker-Planck generator through the first-moment formula (43), so the reconstructed vector-field perturbation is the true physical forcing; the paper labels this an ansatz and supplies no error analysis.

Editorial extensions

If this is right

For any Ulam-type Markov matrix, the small forcing that maximizes or minimizes the entropy of the stationary distribution is obtained in closed form, so no forced simulations are needed to design the forcing.
The KL maximizer gives a data-driven perturbation that makes the forced statistics maximally distinguishable from the unforced one, the quantity used in predictability or climate-change detection studies.
Minimizing entropy production under the fixed-invariant-measure constraint yields a perturbation that counteracts non-conservative forces, as shown by the reconstructed clockwise field opposing the counterclockwise rotation in the double-well-with-rotation example.
The same entropy-production formula solves other linear objectives, including accelerating mixing while keeping the invariant measure unchanged, because the spectral-gap response has the same structure (as noted in Remark 3.3).
In the periodic-orbit reduced model of turbulence, maximizing kinetic energy translates into a specific repopulation of the periodic orbits, and the opposite-sign perturbation minimizes it, since linearity makes the minimizer $-P$.

Reading between the lines

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

A direct test the paper does not perform is to apply the reconstructed optimal forcing to the full stochastic or chaotic system and compare the measured entropy change with the linear prediction; this would separate the validity of the optimization formulas from the validity of the drift-reconstruction ansatz.
The drift-reconstruction formula uses only first moments of the flux matrix, so any inferred forcing is a drift-type forcing; it cannot recover or optimize the noise amplitude (diffusion) of the SDE, a limitation one should keep in mind in equation-free applications.
Relaxing the sparsity constraint (C3) may open genuinely new transitions; in MCMC or network settings, where there is no continuity constraint, this could give substantially better optima, and the paper's linear framework is a natural starting point for testing that.
Second-order corrections to KL divergence and entropy production would show how large $\varepsilon$ can be before the optimal perturbation must be re-solved; a possible extension is to compare the first-order optimal P with a numerical nonlinear optimization of the exact functionals at moderate $\varepsilon$.

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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

Dear colleague,

Here's my read on arXiv:2507.14040. The paper does three things: optimizes entropy and KL divergence for Markov-chain perturbations (mostly reusing Antown et al. 2018), derives a new entropy-production minimizer with fixed invariant vector, and proposes a heuristic to map matrix perturbations back to vector-field perturbations. The first and third are fine; the second has a real problem in the written derivation.

The entropy and KL parts are straightforward applications of the Antown framework. The paper correctly vectorizes the KL objective, uses SVD with the sparsity constraints, and demonstrates the results on Lanford, double well, and Lorenz. Those experiments look solid and the linear response checks are convincing. I'd also credit the drift reconstruction: it's explicitly called an ansatz, and the 1D validation shows it works reasonably well, spikes and all. That's honest and useful.

The soft spot is Section 3.3. The Lagrange multiplier calculation leading to Eq. (30) is not valid for sparse matrices. In Eq. (34), the double sum over j and i does not collapse to a term gamma^T q independent of k. The coefficient of q_i depends on k through the row support of row k, so the matrix is not rank-one in general. Even if it were, the sign in Eq. (37) is wrong: Eq. (36) gives (Xi - 1 gamma^T)q = alpha, not (Xi + 1 gamma^T)q = alpha. The Sherman-Morrison inversion that produces the explicit P_ij therefore does not follow. The Appendix B comparison doesn't cure this; it just shows that some numerically implemented method reduces entropy production. Possibly the code solves a different, correct system, but the paper as written doesn't derive it.

That's a load-bearing gap, because the entropy-production minimizer is one of the two genuinely new contributions. The drift reconstruction is the other, and it's honestly labeled as heuristic. So the paper's central claim—providing linear optimization algorithms for entropy production—isn't supported by the algebra as presented.

I don't think this is fatal to the whole project. The numerical results in Section 5.3 suggest the method works in practice, and the framework is worth preserving. But the authors need to redo the derivation carefully, either by justifying the rank-one reduction under stated conditions or by giving the correct linear system and its solution. The sign error alone is fixable.

Should we referee it? Yes, I'd send it to a serious referee if I were an editor. The paper is useful, well-motivated, and the numerical work is extensive. The referee will need to check the linear algebra carefully, but that's exactly what referees are for. A major revision with a corrected Section 3.3 could make this a solid contribution.

Reading group? Maybe—the flawed derivation is instructive, and the drift reconstruction is thought-provoking. I wouldn't cite it in my own work until the EP part is fixed.

Referee Report

3 major / 7 minor

Summary. The paper develops a linear-response framework for optimizing three functionals — Shannon entropy, Kullback-Leibler divergence, and entropy production — of the invariant measure or transition matrix of a finite Markov chain, by adding a small perturbation P subject to norm, stochasticity, and sparsity constraints. The authors derive an explicit formula (Eq. 17) for entropy optimization, a singular-value-decomposition approach for KL divergence (Section 3.2), and a Lagrange-multiplier formula (Eq. 30) for entropy-production minimization. They apply the algorithms to Ulam discretizations of the Lanford map, double-well SDEs, the Lorenz 63 system, and a periodic-orbit reduced model of Kolmogorov flow, and propose a numerical protocol to reconstruct vector-field perturbations from matrix perturbations (Eq. 43).

Significance. If the results are correct, the paper provides a useful toolbox for optimal forcing in data-driven dynamical systems, with potential applications in climate science and Markov chain Monte Carlo, extending the line of work by Antown et al. The numerical experiments are rich and the connection between matrix-level and vector-field-level perturbations is appealing. However, the derivation of the entropy-production optimizer contains a load-bearing algebraic gap, and the drift-reconstruction step is heuristic; both need to be addressed before the central claims can be accepted. The paper does not provide machine-checked proofs or publicly available code, but the numerical studies are described in sufficient detail to be reproducible in principle.

major comments (3)

[Section 3.3, Eqs. (34)–(38)] The reduction of Eq. (34) to the rank-one form in Eq. (36) is invalid for sparse Markov matrices. For fixed k, the double sum in Eq. (34) equals Σ_i Γ_{ki} q_i with Γ_{ki} = Σ_{j : (k,j)∈Z, (i,j)∈Z} β_j, which depends on k through the row support {j : M_{kj} > 0}; the replacement of Γ_{ki} by a k-independent γ_i is only justified under a full-support condition that contradicts the sparsity constraint (C3). In addition, Eq. (36) gives α_k + Σ_i γ_i q_i − ξ_k q_k = 0, which implies (Ξ − 1γ^T)q = α, not (Ξ + 1γ^T)q = α as written in Eq. (37); the Sherman–Morrison inversion and the closed-form expression for P_ij in Eq. (30) therefore do not follow from the displayed equations. The numerical comparison in Appendix B does not repair the written algebra unless the implemented code solves a different linear system.
[Section 4.1, Eq. (43)] The drift reconstruction formula (43) is called an ansatz in the text, but it is a central pillar of the paper's claim to provide a numerical link between matrix perturbations and vector-field perturbations. No error analysis is provided, and the only validation is a single one-dimensional double-well example; no dependence on box size N, transition time τ, or noise amplitude is reported, and the text itself acknowledges spiky artifacts. The authors should provide error bounds under suitable hypotheses on the Ulam discretization, or explicitly delimit the regime in which the reconstruction is quantitatively reliable. Without this, the equation-free physical interpretation of the optimized perturbations remains unsupported.
[Appendix B.1, Fig. 11] The claimed equivalence between Method 1 (Eq. (30)) and Method 2 (projection onto the nullspace of the constraint operator) is not established by the presented numerical comparison. The histograms show only that both methods reduce the entropy-production functional relative to the unperturbed matrix; they do not compare the resulting perturbation matrices against the exact optimizer of the constrained problem, and the non-zero relative difference in Fig. 11(B) is attributed to 'numerical issues' without a convergence test as tolerances are varied. Since the main text presents Eq. (30) as the exact solution, this equivalence needs a quantitative check, such as the residual of the Lagrangian stationarity conditions.

minor comments (7)

[Eq. (35a) and Eq. (33)] In the definition of α_k, the term C_{ij} should be C_{kj} (with a fixed row index k); the same index error appears in Eq. (33).
[Eq. (19b)] The indices in the displayed sum are inconsistent: the sum over j uses n instead of N, and the factor P_{kj} should be P_{\ell j} to match the matrix product D^{-1}GPu.
[Section 5.3 and Fig. 5 caption] In the text and caption, the entropy-production curves are both labeled s(M+εPs); the second curve should be s(M+εPr).
[Section 2, Eq. (8)] The definition of G(s) uses s both as the upper limit and as the dummy summation index; it should be G(s) = Σ_{t=0}^s M^t − u1^T.
[Abstract and Section 3.2] The name is spelled 'Kullback-Liebler' throughout; it should be 'Kullback-Leibler'.
[Section 6] The conclusion refers to 'Eq. (60)', but that equation appears in Appendix B and is not part of the main text; the constraints should be referenced by their labels (C1)–(C4).
[Section 3.2] The phrase 'can be writen' contains a typo and should be 'can be written'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the optimization problems are self-contained linear algebra; the one labeled ansatz is openly heuristic and benchmarked, and the self-citations are background only.

full rationale

None of the load-bearing steps reduces to its own inputs. The three optimizations are defined by independent functionals: entropy (Eq. 13), KL divergence (Eq. 18), and entropy production (Eq. 22), and each is expanded in powers of epsilon to obtain a first-order objective (Eqs. 14, 19, 24). The perturbation matrices P are obtained by maximizing or minimizing those objectives subject to constraints (C1)-(C4); no component of the target functional is fitted to the data used to construct the Markov matrix. The entropy optimization explicitly reuses the known algorithm of Antown et al. with observable f_i = log u_i, which is an external, independent result rather than a self-citation. The KL problem is solved by SVD and projection methods taken from that same external reference. The entropy-production problem is derived via Lagrange multipliers in Eqs. (26)-(30), and Appendix B additionally checks both optimization methods against random sparse Markov matrices and against the additive reversibilization; no parameter is fitted to the minimization outcome. The drift reconstruction in Eq. (43) is explicitly labeled an ansatz and is validated against the known drift in Fig. 1, and against the reversal of the rotation in Appendix A; it is an acknowledged heuristic, not a disguised fit or a renamed known result. Self-citations to the authors' earlier works ([31,47,48,49,63]) appear only in the introduction and background discussion of linear response and Ulam-type discretizations, and they are not load-bearing for the derivation of the optimization algorithms. The skeptical algebra critique concerning the collapse from Eq. (34) to Eq. (36) and the sign in Eq. (37) is a correctness risk for the entropy-production minimizer, not a circularity, and therefore does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theoretical algorithms introduce no fitted parameters or new entities; the numerical experiments use user-chosen transition times tau, box counts, and perturbation amplitudes epsilon, which are not fitted to the quantities being optimized. The main structural inputs are the mixing assumption, the first-order linear response ansatz, and the heuristic drift reconstruction.

assumptions (4)

domain assumption The Markov chain M is mixing with a unique strictly positive invariant vector u (Eq. 1).
Required to define the generalized inverse G in Eq. (10) and the linear response v1 = G P u; Prop 2.1 supplies a sufficient condition for the perturbed chain.
domain assumption The perturbation is small enough that the linear response expansion v = u + epsilon v1 + O(epsilon^2) is valid.
All optimizations are first-order in epsilon; the numerical experiments check the linear regime for each chosen epsilon, but the theory assumes it.
domain assumption The Ulam discretization in Eq. (41) represents the Fokker-Planck semigroup, and the matrix logarithm in Eq. (42) yields a valid generator via the spectral mapping theorem.
The paper states rigorous Ulam results are limited to 1D and smooth invariant measures, and explicitly omits a discussion of the transition time tau.
ad hoc to paper The drift reconstruction ansatz, Eq. (43), that [F(c_i)]_k approximately equals the sum over j of L_{ji}(c_j^k - c_i^k).
Introduced heuristically in Section 4.1 with no error bound, and validated only on a one-dimensional double well example.

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Cite this review

Pith. "Pith review of Markov matrix perturbations to optimize dynamical and entropy functionals." pith.science (2026). https://pith.science/paper/RAA34QXI

@misc{pith2026250714040,
  author       = {Pith},
  title        = {Pith review of: Markov matrix perturbations to optimize dynamical and entropy functionals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RAA34QXI}},
  note         = {Machine review of arXiv:2507.14040}
}

read the original abstract

An important problem in applied dynamical systems is to compute the external forcing that provokes the largest response of a desired observable quantity. For this, we investigate the perturbation theory of Markov matrices in connection with linear response theory in statistical physics. We use perturbative expansions to derive linear algorithms to optimize physically relevant quantities such as: entropy, Kullback-Liebler-divergence and entropy production of Markov matrices and their related probability vectors. These optimization algorithms are applied to Markov chain representations of discrete and continuous flows in and out of equilibrium. We consider Markov matrix representations originating from Ulam-type approximations of transfer operators and a reduced order model of a turbulent flow based on unstable periodic orbits theory. We also propose a numerical protocol to recast matrix perturbations into vector field perturbations. The results allow to physically interpret the obtained optimizing perturbations without knowledge of the underlying equations, in a data-driven way.

Figures

Figures reproduced from arXiv: 2507.14040 by the authors.

**Figure 1.** (A) against the the actual drift function F(x) plotted in the black curve. For illustration, we now enquire what is the perturbation P of the matrix in Eq. (45) such that the KLdivergence is maximized. Intuitively, in order to amplify the KL-divergence, one expects that the forcing applied to Eq. (44) should equalize the wells that originally were unbalanced due to the term α. To visualize the forcing, we need Eq. … view at source ↗

**Figure 2.** Perturbations of the Lanford map. Panel (A): in black we show the invariant vector of the matrix in Eq. (47), in blue and red the perturbed invariant vectors— see Eq. (48)— of the two perturbation problems considered. Panel (B): Case of maximizing the entropy functional. In black we show the unperturbed invariant vector, in gray scales, the successive transient linear responses— for the times indicated in the legend… view at source ↗

**Figure 3.** Double well potential: KL-divergence maximization. Panel (A): unperturbed invariant vector. Panel (B): perturbation matrix P1 applied to u. Panel (C): projected responses along the x-axis. The black and red curves show the unperturbed and perturbed invariant vectors. The grey curves show the transient linear responses obtained from P1 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: (B) resemble those of [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: A double well potential with rotation. Panel (A): quiver plot of the vector field rotation R defined in Eq. (53). Panel (B): vector field reconstruction associated with Ps. Panel (C): entropy production function of M + εPs (blue) and M + εPr (red) vs. ε. In this case, …

**Figure 6.** Figure 6: Entropy maximization in the L63 system. Panel (A): perturbed invariant vector of the matrix M + εP1. Panel (B): linear response for the perturbation matrix P1. Panel (C): marginal distributions for the unperturbed and perturbed invariant vectors. The black line shows t…

**Figure 7.** Figure 7: KL-divergence maximization in the L63 system. Panel (A): unperturbed invariant vector of the matrix M. Panel (B): linear response for the perturbation matrix P2. Panel (C): marginal distributions for the unperturbed and perturbed invariant vectors. The black line shows…

**Figure 8.** Figure 8: Markov Chain description of the turbulent flow. Panel (a): Transition matrix [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Properties of the perturbed system. Panel (a): weights of the system. Top: weights of the unperturbed [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: Reconstructing the rotation. Panels (A) and (B) show the x and y components of the applied rotation R in Eq. (53). Panels (C) and (D) shows the x and y components of formula Eq. (43) applied to Pr defined in Eq. (23). B Projection method for constraints In this append…

**Figure 11.** Figure 11: Comparison between optimization methods. Panel (A): normalized histograms for the relative difference between s M(k) + εP (k) i and s M(k) for both methods shown in the legend. Panel (B): normalized histograms of the relative difference between methods in the…

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