Pith. sign in

REVIEW 2 major objections 4 minor 37 references

Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Hellinger–Kantorovich–Boltzmann gradient flow preserves the class of Gaussian measures, and the paper derives the exact finite-dimensional ODEs for their mean, covariance, and mass.

desk verdict Solid reduction of HK-Boltzmann flow to Gaussian ODEs, but Section 5 overstates two results: the slope comparison in Theorem 5.11 is inverted and Theorem 5.8 drops initial-data prefactors. read the letter →

arxiv 2504.20400 v1 pith:5C62KV77 submitted 2025-04-29 math.AP math.PRstat.ML

Matthias Liero , Alexander Mielke , Oliver Tse , Jia-Jie Zhu This is my paper · ORCID
classification math.APmath.PRstat.ML MSC 49Q2235Q49
keywords GaussianmeasuresHellinger-Kantorovichdistancegradient-flowequationrelativeBoltzmannentropyKullback-LeiblerdivergencereducedOnsageroperatorPolyak-Lojasiewiczinequalitylog-concavetarget
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 establishes that, when the target measure is a (scaled) Gaussian, the gradient flow of the relative Boltzmann entropy in the Hellinger–Kantorovich (HK) geometry — the metric that combines spatial transport with mass growth — leaves the class of Gaussian measures invariant. Because of this invariance, the infinite-dimensional flow reduces exactly to a finite-dimensional gradient system on the parameters $(\Sigma,m,\kappa)$, with explicit ODEs for the covariance, mean, and mass. The reduction is carried out through a reduced Onsager operator that is a linear combination of the transport and Hellinger parts, so the additive structure of the HK metric survives in the parameter space. Using this reduced structure, the paper proves exponential convergence to equilibrium for Gaussian targets, with decay rates refined by tracking the eigenvalues of the normalized covariance, and it extends the analysis to strongly log-$\lambda$-concave targets. Together, these results give a tractable family of dynamics for Gaussian variational inference, with explicit convergence rates in the Gaussian-target case and numerical support in non-Gaussian applications.

What carries the argument

The load-bearing object is the reduced Onsager operator on the parameter space $P=\mathbb{R}^{d\times d}_{\mathrm{spd}}\times\mathbb{R}^d\times(0,\infty)$. It is obtained by restricting the full HK Onsager operator $K_{\alpha,\beta}(\rho)\xi=-\alpha\,\mathrm{div}(\rho\nabla\xi)+\beta\rho\xi$ (the object that pairs entropic gradients with velocities in the HK metric) to the Gaussian manifold, using the general reduction formula for Onsager operators, which amounts to a Schur complement after Lagrange-multiplier elimination. The reduction is additive, $K^{\mathrm{red}}_{\alpha,\beta}(p)=\alpha K^{\mathrm{red}}_{\mathrm{tr}}(p)+\beta K^{\mathrm{red}}_{\mathrm{He}}(p)$ as in (3.3), and inserting it into $\dot p=-K^{\mathrm{red}}DE(p)$ yields the explicit ODEs (3.6). The argument also uses the natural parametrization $\rho(x)=\exp(c+b\cdot x-\tfrac12 x\cdot Ax)$, in which the flow equations become polynomial, and the evolution of the normalized covariance $B=\Gamma^{-1/2}\Sigma\Gamma^{-1/2}$, whose eigenvalues obey the scalar equations $\dot b_i=-(2\alpha\langle v_i|\Gamma^{-1}v_i\rangle+\beta b_i)(b_i-1)$ obtained by the standard eigenvalue perturbation formula.

What would settle it

Take $d=1$, target $\pi=\mathcal{N}(0,1)$, covariance $\Sigma=s^2$ with $s<1$, and any mean $m\ne0$. Write $g=\int \nabla V\,G\,dx$. The transport slope is $(1-s^2)^2/s^2+g^2$ and the Hellinger slope is $(1-s^2)^2+s^2g^2$. The inequality used in Theorem 5.11 would require $(1-s^2)^2/s^2+g^2 \le s^2((1-s^2)^2+s^2g^2)$, which simplifies to $(1-s^4)((1-s^2)^2+s^2g^2)\le0$, false for every $s<1$. This one-dimensional check falsifies the slope comparison and hence the stated decay rate.

Watch

Extended reading notes

Core claim

The central claim is that the relative Boltzmann entropy flow in the $(\alpha,\beta)$-HK geometry preserves scaled Gaussians whenever the reference measure is a scaled Gaussian, and that the restricted flow is exactly the gradient flow of the reduced entropy under the reduced Onsager operator $K^{\mathrm{red}}_{\alpha,\beta}(p)=\alpha K^{\mathrm{red}}_{\mathrm{tr}}(p)+\beta K^{\mathrm{red}}_{\mathrm{He}}(p)$. Consequently the ODEs (3.6) fully describe the infinite-dimensional flow on the Gaussian manifold, not merely a projected approximation. On normalized Gaussians the paper shows that global geodesic $\lambda$-convexity holds only in the pure transport case ($\beta=0$ with $\lambda\le\alpha\nu_{\min}(\Gamma^{-1})$), while a sublevel version of semi-convexity holds in general. For Gaussian targets it proves exponential decay of the relative Boltzmann entropy with explicit rates, the sharpest being $\nu_{\mathrm{cov}}=2\alpha\nu_{\min}(\Gamma^{-1})+\beta$ for the covariance part and $\nu_{\mathrm{m}}=2\alpha\nu_{\min}(\Gamma^{-1})+2\beta$ for the mean part. For non-Gaussian targets that are strongly log-$\lambda$-concave, it claims exponential convergence with rate $\lambda(2\alpha+\beta r_E^{-1})$ obtained from a slope comparison between the transport and Hellinger dissipations.

Load-bearing premise

The load-bearing premise for the non-Gaussian decay theorem is the slope comparison $|\partial_{\mathrm{Otto}}E_1|^2 \le \sigma_{\min}|\partial_{\mathrm{SHe}}E_1|^2$, which direct computation seems to reverse to $|\partial_{\mathrm{Otto}}E_1|^2 \le \sigma_{\min}^{-1}|\partial_{\mathrm{SHe}}E_1|^2$, so the stated rate $\lambda(2\alpha+\beta r_E^{-1})$ is unsupported unless that inequality is repaired.

Editorial extensions

If this is right

  • In the pure transport case ($\beta=0$) and the pure Hellinger case ($\alpha=0$) the covariance equation has closed-form solutions, so the HK flow's decay can be compared exactly with its two limiting geometries.
  • The additive form of the reduced Onsager operator means the HK flow interpolates between mass-preserving transport and pure mass growth without introducing coupling terms at the metric level, which is what makes the parameter-space analysis tractable.
  • For Gaussian targets the refined decay rates are independent of the initial energy level, so the long-time behavior is governed only by $\alpha$, $\beta$, and the smallest eigenvalue of $\Gamma^{-1}$.
  • The discrete Algorithm 1 provides a practical Gaussian variational-inference scheme for non-Gaussian targets such as Bayesian logistic regression, alternating transport and Hellinger updates with Monte Carlo estimation; the experiments indicate transport steps dominate far from equilibrium while Hellinger steps converge faster near it.

Reading between the lines

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

  • Beyond the paper, the same reduction should apply to any exponential family whose log-density is quadratic: if the target lies in the family, the HK–Boltzmann flow stays in it, and the finite-dimensional ODEs can be derived by the same algebraic reduction.
  • Beyond the paper, if the slope comparison behind Theorem 5.11 is indeed reversed, the exponential-decay conclusion may survive with a different rate; a natural repair is to work with $\sigma_{\min}^{-1}$ or a different sublevel bound, which would change the rate constant but not the qualitative behavior.
  • Beyond the paper, the explicit covariance formula in Remark 2.3 offers a sharp numerical test of the PL rates: computing the exact covariance at time $t$ and comparing the exponent with $\nu_{\mathrm{cov}}$ would show whether the prefactor in (5.9) is an artifact of the proof.
  • Beyond the paper, a promising extension is a variational (minimizing-movement) discretization of the reduced flow rather than the Euler splitting of Algorithm 1; the additive Onsager structure suggests exact substeps for transport and Hellinger would preserve Gaussianity at every iteration.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the gradient flow of the relative Boltzmann entropy with respect to Gaussian reference measures in Hellinger-Kantorovich (HK) geometry. It proves that the class of scaled Gaussians is invariant under the HK-Boltzmann flow, derives the reduced Onsager operators on the parameter space P=(Sigma,m,kappa), obtains the explicit ODE system (2.4)/(3.6), and shows that this parameter-space system is the exact reduced gradient flow. The paper then analyzes geodesic convexity of the normalized Gaussian system, establishes Polyak-Lojasiewicz-type inequalities and refined eigenvalue-based decay estimates for Gaussian targets, extends the results to strongly log-lambda-concave targets, and reports numerical experiments including Bayesian logistic regression.

Significance. The reduction results are a genuine contribution: the closed-form reduced Onsager operators in Theorem 3.8 and the explicit ODEs in (2.4) give a parameter-free, computationally transparent description of HK-Boltzmann flow on Gaussians. The geodesic-convexity analysis clarifies the loss of global convexity for beta>0 while retaining sublevel semi-convexity, and the PL estimates provide explicit rates in alpha, beta, and Gamma. The numerical experiments complement the theory. However, the non-Gaussian decay theorem in Section 5.3 rests on an inverted slope comparison, so that part of the paper needs correction before the stated results can be accepted.

major comments (2)
  1. [Section 5.3, slope comparison preceding Theorem 5.11] The displayed inequality |\partial_Otto E_1|^2 <= sigma_min |\partial_SHe E_1|^2 is reversed. Writing A=I-Gamma^{-1}Sigma and g=integral nabla V G dx, the two slopes are |\partial_Otto E_1|^2=tr(A^2 Sigma^{-1})+|g|^2 and |\partial_SHe E_1|^2=tr(A^2)+g^T Sigma g. Since Sigma^{-1} <= sigma_min^{-1} I and |g|^2 <= sigma_min^{-1} g^T Sigma g, the correct comparison is |\partial_Otto E_1|^2 <= sigma_min^{-1}|\partial_SHe E_1|^2. The printed claim already fails for d=1, Gamma=1, Sigma=epsilon in (0,1), g=0, where it would assert (1-epsilon)^2/epsilon <= epsilon(1-epsilon)^2. The Gronwall chain immediately below the display therefore does not follow as written: the valid argument gives dE_1/dt <= -(alpha+beta sigma_min)|\partial_Otto E_1|^2, and with (5.12) this yields a rate of the form lambda(2alpha+2beta r_E^{-1}), not lambda(2alpha+beta r_E^{-1}). The qualitative statement of Theorem 5.11 is repairable, but the theorem as stated is not supported by the displayed computation.
  2. [Theorem 5.8 and Eqs. (5.9)-(5.10)] The statement of Theorem 5.8 omits the multiplicative prefactors that are part of the estimates proved in (5.9) and (5.10). The theorem displays E_1(p(t)) <= e^{-nu_cov t}H_cov(Sigma(0)) + e^{-nu_m t}H_m(m(0)), whereas (5.9) and (5.10) contain the constants [min{1,b_min(0)}]^{-beta/nu_cov} and [min{1,b_min(0)}]^{-2beta/nu_m}, which can exceed 1 when b_min(0)<1. As printed the theorem is false; either the prefactors must be included in the display or the statement should explicitly say that the bounds hold up to those constants.
minor comments (4)
  1. [Section 4.2, displayed DE1] The formula DE_1(p)=(2(Gamma^{-1}-Sigma^{-1}), Gamma^{-1}(m-n)) is inconsistent with E_1(p)=H(p|Gamma,n), whose covariance derivative is (1/2)(Gamma^{-1}-Sigma^{-1}); it is also inconsistent with the vector field V(p) displayed immediately below. The downstream formulas in (5.4) appear to use the correct prefactor, so please correct the display.
  2. [Remark 4.1, Eq. (4.4c)] For the Otto part at kappa=1, the sign of the first term in \dot S_s is opposite to Hamilton's equations (4.3); in the scalar case it should be -2 alpha S_s^2, not +2 alpha S_s^2. Since the remark is declared unused this is not load-bearing, but it should be corrected or the remark removed.
  3. [Remark 5.9] The parenthetical 'cf. Remark' is a dangling reference; please supply the intended number or delete the parenthetical.
  4. [Lemma 5.7, Eq. (5.8)] The symbol b_i(t) is used both for the evolving eigenvalue and for the explicit upper/lower bound 1+(b_i(0)-1)e^{-nu_cov t} in the same display; please use a different symbol, such as \bar b_i(t), for the bound.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the reduced gradient system is computed directly from the Onsager operators and Gaussian integrals, and the decay rates depend only on explicit parameters.

full rationale

The paper's central derivation is self-contained. The reduced Onsager operator (3.3) is obtained in Lemma 3.6 by direct Gaussian integration of the quadratic ansatz against KOtto and KHe, not by assuming the target ODE. Theorem 3.8 then verifies that p-dot = -Kred DE reproduces the independently derived ODE system (2.4). The Polyak-Lojasiewicz estimates in Propositions 5.3 and 5.5 are explicit inequalities in alpha, beta, and Gamma, with no fitted parameter and no data-dependent constant that is renamed as a prediction. The self-citations present in the paper are not load-bearing in a circular sense: Theorem 3.2 is quoted from [MaM20] but is a general Schur-complement reduction result, and Lemma 3.10 is quoted from [MiZ25] and states a relation between HK and SHK flows whose assumptions do not include Gaussian invariance or the target decay rates. Both results are prior, external support rather than restatements of the paper's conclusion. The only passage that warrants attention is the inequality in Section 5.3 claiming |∂OttoE1|^2 ≤ sigma_min |∂SHeE1|^2, which appears algebraically reversed and would weaken the stated rate in Theorem 5.11; however, that is a correctness or verification issue, not a circularity, because the claimed rate is an output of the argument rather than an input disguised as a result. Overall, the derivation chain does not reduce to its own inputs, so the circularity score is 1.

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

The paper introduces no new entities and fits no free parameters. It relies on established reduction theory, a self-cited equivalence lemma, the Hellmann-Feynman theorem, and standard geodesic convexity facts.

assumptions (4)
  • domain assumption The Schur complement reduction formula for Onsager gradient systems (Theorem 3.2)
    Taken from [MaM20, Sec. 6.1]; used to derive the reduced Onsager operator in Section 3.2.
  • domain assumption The equivalence between HK-flow and SHK-flow (Lemma 3.10)
    Cited from [MiZ25], a preprint by two of the present authors; used to separate shape and mass evolution in Section 3.4.
  • standard math Hellmann-Feynman theorem for derivatives of eigenvalues
    Used in Section 5.2 to derive equation (5.7) for the eigenvalues of the normalized covariance; assumes differentiable eigenpairs.
  • standard math Geodesic λ-convexity of relative entropy under W2 for log-λ-concave targets
    Used in Section 5.3 via equation (5.12) and [AGS05].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow." pith.science (2026). https://pith.science/paper/5C62KV77

@misc{pith2026250420400,
  author       = {Pith},
  title        = {Pith review of: Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5C62KV77}},
  note         = {Machine review of arXiv:2504.20400}
}
read the original abstract

This study leverages the basic insight that the gradient-flow equation associated with the relative Boltzmann entropy, in relation to a Gaussian reference measure within the Hellinger-Kantorovich (HK) geometry, preserves the class of Gaussian measures. This invariance serves as the foundation for constructing a reduced gradient structure on the parameter space characterizing Gaussian densities. We derive explicit ordinary differential equations that govern the evolution of mean, covariance, and mass under the HK-Boltzmann gradient flow. The reduced structure retains the additive form of the HK metric, facilitating a comprehensive analysis of the dynamics involved. We explore the geodesic convexity of the reduced system, revealing that global convexity is confined to the pure transport scenario, while a variant of sublevel semi-convexity is observed in the general case. Furthermore, we demonstrate exponential convergence to equilibrium through Polyak-Lojasiewicz-type inequalities, applicable both globally and on sublevel sets. By monitoring the evolution of covariance eigenvalues, we refine the decay rates associated with convergence. Additionally, we extend our analysis to non-Gaussian targets exhibiting strong log-lambda-concavity, corroborating our theoretical results with numerical experiments that encompass a Gaussian-target gradient flow and a Bayesian logistic regression application.

Figures

Figures reproduced from arXiv: 2504.20400 by the authors.

Figure 5.1
Figure 5.1. Plot of the auxiliary function H(δ, β; y) in Lemma 5.2. Lemma 5.2 Let J = (J−, J+) ⊂ (0, +∞) with J− < 1 < J+, then h(1, 0; J) = lim y→J+ ζϕ(y) ϕ(y) ∈ [1, 2), h(0, 1; J) = lim y→J− yζϕ(y) ϕ(y) ∈ [0, 2). 21 [PITH_FULL_IMAGE:figures/full_fig_p021_5_1.png] view at source ↗
Figure 6
Figure 6. [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
Figure 6.1
Figure 6.1. (Top) Evolution of the Gaussian probability measure [PITH_FULL_IMAGE:figures/full_fig_p032_6_1.png] view at source ↗
Figures from the paper (3 more)
Figure 6.2
Figure 6.2. Figure 6.2: (Top) Evolution of the KL divergence HB(µk|π) (blue line) for the three dif￾ferent gradient flows of probability measures defined on R 100: the Hellinger-Kantorovich￾Gaussian gradient flow (blue; abbreviated as BWFR (Bures-Wasserstein-Fisher-Rao)), the Fisher-Rao gra…
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p034_6.png]
Figure 6.3
Figure 6.3. Figure 6.3: (Left) Evolution of the Gaussian probability measure [PITH_FULL_IMAGE:figures/full_fig_p034_6_3.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 32 canonical work pages

  1. [1]

    urich. Birkh\

    L. Ambrosio, N. Gigli, and G. Savar \'e . Gradient flows in metric spaces and in the space of probability measures . Lectures in Mathematics ETH Z\"urich. Birkh\"auser Verlag, Basel, 2005

    work page 2005

  2. [2]

    S.-i. Amari . Information geometry. Japanese Journal of Mathematics , 16(1), 1--48, January 2021

    work page 2021

  3. [3]

    Amari and H

    S.-i. Amari and H. Nagaoka . Methods of Information Geometry . American Mathematical Soc., 2000

    work page 2000

  4. [4]

    Bakry and M

    D. Bakry and M. \'E mery . Diffusions hypercontractives. In J. Az \'e ma and M. Yor, editors, S \'e minaire de Probabilit \'e s XIX 1983/84 , volume 1123, pages 177--206. Springer Berlin Heidelberg, Berlin, Heidelberg, 1985

    work page 1983

  5. [5]

    C. Bishop . Pattern recognition and machine learning. Springer google schola , 2, 5--43, 2006

    work page 2006

  6. [6]

    D. M. Blei, A. Kucukelbir, and J. D. McAuliffe . Variational Inference : A Review for Statisticians . Journal of the American Statistical Association , 112(518), 859--877, April 2017

    work page 2017

  7. [7]

    J. A. Carrillo, Y. Chen, D. Z. Huang, J. Huang, and D. Wei . Fisher-rao gradient flow: geodesic convexity and functional inequalities. arXiv preprint arXiv:2407.15693 , 2024

    arXiv 2024

  8. [8]

    Y. Chen, D. Z. Huang, J. Huang, S. Reich, and A. M. Stuart . Gradient Flows for Sampling : Mean-Field Models , Gaussian Approximations and Affine Invariance . Preprint , arXiv:2302.11024, November 2023

    arXiv 2023

Show all 37 references
  1. [9]

    Daneri and G

    S. Daneri and G. Savar\'e . Eulerian calculus for the displacement convexity in the W asserstein distance. SIAM J. Math. Analysis , 40, 1104--1122, 2008

    2008

  2. [10]

    Floerchinger and M

    S. Floerchinger and M. Schr\"ofl . Relative entropy and mutual information in Gaussian statistical field theory. Ann. Henri Poincar\'e , pages 1--87, 2024

    2024

  3. [11]

    Hellmann

    H. Hellmann . Hans Hellmann: Einf \"u hrung in Die Quantenchemie: Mit Biografischen Notizen von Hans Hellmann Jr. Springer-Verlag, 2015

    2015

  4. [12]

    M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul . An Introduction to Variational Methods for Graphical Models . Machine Learning , 37(2), 183--233, November 1999

    1999

  5. [13]

    Karimi, J

    H. Karimi, J. Nutini, and M. Schmidt . Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak- L \ \ ojasiewicz Condition . (arXiv:1608.04636), September 2020

    2020 arXiv

  6. [14]

    Laschos and A

    V. Laschos and A. Mielke . Evolutionary Variational Inequalites on the Hellinger-Kantorovich and the spherical Hellinger-Kantorovich spaces. Submitted , 2023. arXiv:2207.09815v3

    2023 arXiv

  7. [15]

    Lambert, S

    M. Lambert, S. Chewi, F. Bach, S. Bonnabel, and P. Rigollet . Variational inference via Wasserstein gradient flows. Preprint , arXiv:2205.15902, 2022

    2022 arXiv

  8. [16]

    Liero and A

    M. Liero and A. Mielke . Gradient structures and geodesic convexity for reaction-diffusion systems. Phil. Trans. Royal Soc. A , 371(2005), 20120346, 28, 2013

    2005

  9. [17]

    Liero, A

    M. Liero, A. Mielke, and G. Savar\'e . Optimal transport in competition with reaction -- the H ellinger-- K antorovich distance and geodesic curves. SIAM J. Math. Analysis , 48(4), 2869--2911, 2016

    2016

  10. [18]

    Liero, A

    M. Liero, A. Mielke, and G. Savar\'e . Optimal entropy-transport problems and a new H ellinger-- K antorovich distance between positive measures. Invent. math. , 211, 969--1117, 2018

    2018

  11. [19]

    Liero, A

    M. Liero, A. Mielke, and G. Savar\'e . Fine properties of geodesics and geodesic -convexity for the Hellinger--Kantorovich distance. Arch. Rational Mech. Anal. , 247(112), 1--73, 2023

    2023

  12. [20]

    Maas and A

    J. Maas and A. Mielke . Modeling of chemical reaction systems with detailed balance using gradient structures. J. Stat. Physics , 181, 2257--2303, 2020

    2020

  13. [21]

    R. H. Martin, Jr. Nonlinear operators and differential equations in B anach spaces . Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976

    1976

  14. [22]

    R. J. McCann . A convexity principle for interacting gases. Adv. Math. , 128, 153--179, 1997

    1997

  15. [23]

    Mielke and J.-J

    A. Mielke and J.-J. Zhu . Hellinger-Kantorovich gradient flows: Global exponential decay of entropy functionals. arXiv preprint arXiv:2501.17049 , 2025

    2025 arXiv

  16. [24]

    K. Modin . Geometry of matrix decompositions seen through optimal transport and information geometry. J. Geom. Mech. , 9(3), 335--390, 2017

    2017

  17. [25]

    Mielke, G

    A. Mielke, G. Savar\'e, and O. Tse . Sublevel evi and the gradient flows in the spherical hellinger geometry. In preparation , 2025

    2025

  18. [26]

    Opper and C

    M. Opper and C. Archambeau . The variational G aussian approximation revisited. Neural Comput. , 21(3), 786--792, 2009

    2009

  19. [27]

    F. Otto . The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eqns. , 26, 101--174, 2001

    2001

  20. [28]

    Otto and C

    F. Otto and C. Villani . Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. , 173(2), 361--400, 2000

    2000

  21. [29]

    Otto and M

    F. Otto and M. Westdickenberg . Eulerian calculus for the contraction in the W asserstein distance. SIAM J. Math. Analysis , 37, 1227--1255, 2005

    2005

  22. [30]

    Pidstrigach and S

    J. Pidstrigach and S. Reich . Affine-invariant ensemble transform methods for logistic regression. Found. Comput. Math. , 23(2), 675--708, 2023

    2023

  23. [31]

    S. Sarkka . On unscented kalman filtering for state estimation of continuous-time nonlinear systems. IEEE Transactions on Automatic Control , 52(9), 1631--1641, 2007

    2007

  24. [32]

    A. Takatsu . Wasserstein geometry of Gaussian measures. Osaka J. Math. , 48, 1005--1026, 2011

    2011

  25. [33]

    C. Villani . Topics in optimal transportation , volume 58 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2003

    2003

  26. [34]

    M. J. Wainwright and M. I. Jordan . Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning , 1(1 2), 1--305, 2008

    2008

  27. [35]

    Wibisono

    A. Wibisono . Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem. In Proceedings of the 31st Conference On Learning Theory , pages 2093--3027. PMLR, July 2018

    2018

  28. [36]

    A. Zellner . Optimal information processing and bayes's theorem. The American Statistician , 42(4), 278--280, 1988

    1988

  29. [37]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence '...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.