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Accelerated Information Gradient flow

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Damping Hamiltonian flows on probability space yields accelerated sampling with exponential convergence.

desk verdict Useful unified framework; the convergence proofs are conditional on unproven regularity, and the F-AIG division-by-zero gap is real but fixable. read the letter →

arxiv 1909.02102 v3 pith:YHNBBEVB submitted 2019-09-04 math.OC cs.ITmath.ITstat.COstat.ML

Yifei Wang , Wuchen Li This is my paper · ORCID
classification math.OCcs.ITmath.ITstat.COstat.ML MSC 49Q2265K1060J6058B20
keywords acceleratedgradientflowinformationgeometryFisher-RaometricWassersteinmean-fieldsamplingparticlevariationalinferenceBayesianadaptiverestart
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 tries to establish that the acceleration trick used in classical accelerated gradient methods carries over to optimization over probability densities, where the objects are full distributions rather than single points. It defines a family of dynamics it calls Accelerated Information Gradient (AIG) flows, obtained by adding a damping term to Hamiltonian flows generated by information metrics. For the Fisher-Rao and Wasserstein metrics, it proves that the value functional converges as $O(e^{-\sqrt{\beta} t})$ when the functional is strongly geodesically convex and as $O(t^{-2})$ when it is merely convex. The practical target is mean-field sampling for Bayesian inverse problems, where these flows become particle dynamical systems. If the framework is right, a single construction covers Fisher-Rao, Wasserstein, Kalman-Wasserstein, and Stein metric samplers and accelerates them in a systematic way.

What carries the argument

The central object is the AIG flow itself: $$\partial_t\rho_t - G(\rho_t)^{-1}\Phi_t = 0, \qquad \partial_t\Phi_t + \alpha_t\Phi_t + \frac12 \frac{\delta}{\delta\rho_t}\int \Phi_t G(\rho_t)^{-1} \Phi_t\, dx + \frac{\delta E}{\delta\rho_t}=0,$$ where $G(\rho)$ is the chosen information metric and $\Phi_t$ is the momentum variable. The damping coefficient $\alpha_t$ is the only free scheduling input: $2\sqrt{\beta}$ for strongly convex energies and $3/t$ for convex ones. The convergence proofs are carried by Lyapunov functions written with the optimal transport map $T_t$ from $\rho_t$ to the target density $\rho^*$; the crucial technical lemma is that the vector field $u_t=\partial_t(T_t^{-1})\circ T_t$ satisfies $\nabla\cdot(\rho_t(u_t-\nabla\Phi_t))=0$, which yields the inner-product identities that make the Lyapunov derivative non-positive.

What would settle it

A concrete way to test the theorem is to solve the Wasserstein AIG flow for a target whose KL divergence is known, starting from a smooth positive density, and check whether $E(\rho_t)$ obeys the predicted $t^{-2}$ decay envelope under $\alpha_t=3/t$; any finite-time loss of smoothness or positivity in a high-resolution PDE simulation would place the solution outside the theorem's scope and expose the missing regularity condition.

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Extended reading notes

Core claim

The paper's central claim is that a single mechanism, damping a Hamiltonian flow on the density manifold, reproduces accelerated-gradient dynamics in probability space. Its Theorem 1 states that for either the Fisher-Rao or the Wasserstein metric, if the energy functional $E(\rho)$ is $\beta$-strongly convex along geodesics then the AIG flow with damping coefficient $\alpha_t=2\sqrt{\beta}$ satisfies $E(\rho_t)\le C_0 e^{-\sqrt{\beta}t}$, and if $E$ is only convex then the flow with $\alpha_t=3/t$ satisfies $E(\rho_t)\le C_0 t^{-2}$. The constants $C_0$ depend only on the initial density. Alongside this, the paper gives particle formulations of the Wasserstein, Kalman-Wasserstein, and Stein AIG flows, a bandwidth selection rule learned from Brownian-motion samples, and an adaptive restart rule that keeps the discrete-time energy decreasing.

Load-bearing premise

The load-bearing premise is that the AIG flow's density solution stays smooth and strictly positive for all future time, so the optimal transport maps and the Fisher-Rao transport formula remain finite; without a global regularity theorem, the convergence proof is established only for such smooth solutions.

Editorial extensions

If this is right

  • A strongly log-concave target can be sampled to accuracy $\varepsilon$ in time $O(\beta^{-1/2}\log(1/\varepsilon))$, compared with $O(\beta^{-1}\log(1/\varepsilon))$ for the unaccelerated gradient flow.
  • The particle systems for W-AIG, KW-AIG, and S-AIG provide concrete deterministic samplers that fit the standard Bayesian inference setting, and the numerical experiments on Bayesian logistic regression and Bayesian neural networks show faster early progress than the corresponding gradient-flow samplers.
  • The adaptive restart rule gives a practical way to keep the discrete iteration inside the regime where the continuous-time theorem applies, by resetting momentum whenever the energy begins to increase.
  • The same damped-Hamiltonian template extends the acceleration construction to at least two more metrics, Kalman-Wasserstein and Stein, for which the paper supplies flow equations and particle updates.

Reading between the lines

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

  • Editorial inference: the proof is written for smooth positive solutions, so the practical acceleration can be expected to survive only while the particle approximation keeps the density regular; the reported stiffness near the boundary suggests that the restart step is not just a trick but a needed safeguard for the unproved regularity.
  • Editorial inference: the Brownian-motion bandwidth selector is derived by matching one step of the particle update to a heat flow, so for targets far from Gaussian the selected bandwidth could be biased; a directly testable variant would match to the local Fokker-Planck flow instead.
  • Editorial inference: if the Lyapunov machinery only needs a transport map and geodesic convexity, then analogous accelerated flows should be derivable for other metrics with well-behaved exponential maps, giving a template for accelerated versions of other mean-field samplers.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces Accelerated Information Gradient (AIG) flows obtained by adding linear damping to Hamiltonian flows on probability density manifolds, for four information metrics: Fisher-Rao, Wasserstein-2, Kalman-Wasserstein, and Stein. The central theoretical result, Theorem 1, asserts that if the objective functional E(ρ) is β-strongly geodesically convex with respect to the Fisher-Rao or Wasserstein metric, then the corresponding AIG flow with α_t = 2√β satisfies E(ρ_t) ≤ C₀e^{-√βt}, and if E is merely geodesically convex, α_t = 3/t gives E(ρ_t) ≤ C′₀t^{-2}. The paper also proposes a particle-level discrete-time algorithm for the Wasserstein AIG flow, including a kernel-bandwidth selection method based on Brownian-motion samples (the BM method) and an adaptive restart technique. Numerical experiments on Bayesian logistic regression and Bayesian neural networks compare the proposed methods with SVGD, WNAG, and WNes.

Significance. If Theorem 1 is valid under its stated hypotheses, the paper provides a useful unifying PDE-level framework for accelerated sampling in probability space and extends prior Wasserstein acceleration results to the Fisher-Rao metric. The Wasserstein convergence proof is detailed and almost self-contained modulo standard optimal transport machinery, and the paper explicitly compares its Lyapunov argument with that of Taghvaei and Mehta [32], explaining how it avoids a technical assumption there. The particle formulations for the Kalman-Wasserstein and Stein AIG flows, together with the BM bandwidth selection and restart heuristics, are potentially useful for practitioners, and the numerical results illustrate competitive performance on realistic problems. However, the theoretical claims are presently conditional on unproven regularity and positivity of the PDE solutions, and the discrete algorithm is not analyzed, so the paper's main theoretical contribution needs to be strengthened before the stated result can be regarded as fully proved.

major comments (3)
  1. [Section 4, Theorem 1 and Appendix C] Theorem 1 asserts convergence rates for 'the solution ρ_t' to (W-AIG) without stating any well-posedness or regularity hypotheses. The proof in Appendix C requires the optimal transport map T_t = ∇Ψ_t to exist with positive-definite ∇T_t and requires T_t to be time-differentiable (Lemma 2), and Lemma 3 performs integrations by parts that assume enough regularity of ρ_t and ∇Φ_t. No global existence, uniqueness, or regularity theorem for the W-AIG PDE is supplied. The theorem should either be restricted to classical solutions on an interval for which these hypotheses are verified, or the missing well-posedness result should be established. As it stands, the claimed rates are proved only in an uncharacterized smooth regime.
  2. [Appendix D, Remark 3 and Lemma 4] The Fisher-Rao proof defines T_t(x) = 2H_t/sin(H_t) · (R_*(x) - R_t(x)cos H_t)/R_t(x), dividing by R_t(x) = sqrt(ρ_t(x)). The paper's own Remark 3 notes that 'it may be problematic if R_t(x)=0 for some x' and only observes that the integrated quantity ∫T_t²ρ_t dx is finite. Lemma 4 then differentiates T_t and uses expressions such as R_* R_t^{-1}; if R_t has a zero, or if the flow does not maintain Sobolev regularity, these differentiations and the subsequent Cauchy estimates are not justified. The proof of the F-AIG convergence rates therefore contains a gap at a load-bearing point, and the assertion in Theorem 1 for F-AIG is not established without an additional argument ruling out zeros of R_t or otherwise regularizing the division.
  3. [Section 5, Algorithm 1] The discrete-time particle algorithm is not connected to Theorem 1. The update rule (8) uses a kernel-density estimate ξ_k for ∇logρ_k, and the restart criterion φ_k<0 defined in (12) is heuristic; neither is shown to approximate the continuous-time W-AIG flow in a way that preserves the O(e^{-√βt}) or O(t^{-2}) rates. Consequently, the numerical experiments in Section 6 do not provide independent support for the PDE convergence claims, and the practical acceleration claims rest on empirical evidence alone. The paper should state explicitly that the discrete algorithm is not covered by Theorem 1.
minor comments (5)
  1. [Abstract and Section 1] The abstract and introduction describe the method as 'MCMC' and 'sampling' algorithms, but the particle implementations are deterministic mean-field dynamics rather than Markov chains with a stationary distribution; the terminology should be clarified to avoid overstating the link to MCMC.
  2. [Section 5.1] The derivation of the BM bandwidth selection method assumes that both particle systems Y_k(h) and Z_k approximate solutions of the heat equation, but the update rule (8) is for the AIG flow, not a Brownian motion; the relationship between the two processes is not rigorously established, so the BM method is only heuristically motivated.
  3. [Appendix C.3] The Euclidean Lyapunov function displayed at the start of C.3 contains stray '‖‖‖‖' symbols, making the formula difficult to read; it should be typeset as (1/2)‖x_t - x_* + t/2 ẋ_t‖².
  4. [Table 2] In the caption and table body, 'WRes' appears to be a typo for 'WNes' (the WNes method of [13]); please correct.
  5. [Remark 1] Remark 1 says [5] prove similar results with a constant damping coefficient; this is correct for the strongly convex case α_t = 2√β, but the sentence could specify that the constant coefficient here is exactly 2√β.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; Theorem 1's convergence rates are derived from the stated geodesic-convexity assumptions via self-contained Lyapunov estimates, with acknowledged prior work used as independent support.

full rationale

The central theoretical claim, Theorem 1, is a conditional statement: if E is β-strongly geodesically convex (or geodesically convex), then the solution to (F-AIG) or (W-AIG) with the indicated damping coefficient satisfies the stated exponential or quadratic rate. The proof supplies explicit Lyapunov functionals and derives monotonicity from the AIG PDEs, the assumed convexity inequalities, and standard optimal-transport or Fisher-Rao geodesic identities. The rate is not baked into the definition of convexity, the Lyapunov function, or the damping coefficient; it follows from the estimates. The Wasserstein part overlaps with [5,32], but the paper explicitly acknowledges this overlap, supplies its own proofs, and fixes a technical assumption in [32] rather than relying on it; this is independent support, not circularity. The self-citations present ([7], [10], [35], [36]) concern the Hamiltonian-flow formulation, the Kalman-Wasserstein metric definition, a restart heuristic, and future work, and none is load-bearing for Theorem 1. Appendix D Remark 3 flags the possibility that R_t(x)=0 makes T_t formally ill-defined, but this is an unproven regularity/positivity gap for smooth-solution regimes, not a circular reduction: it does not make the conclusion equivalent to the assumptions. The BM bandwidth-selection method is a fitted heuristic for approximating ∇logρ in the discrete particle algorithm; it is not used as an input to the PDE convergence proof, and the numerical experiments are empirical validation rather than part of the derivation. No 'prediction' in the paper reduces by construction to a fitted parameter or to a self-citation chain.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central theoretical claim rests on standard optimal transport machinery plus a strong geodesic convexity assumption on the energy; the numerical part rests on heuristic choices of bandwidth and step sizes. The paper introduces no new physical entities, forces, or conserved quantities.

free parameters (3)
  • kernel bandwidth h = selected by MED initially and by the BM method each iteration
    In Algorithms 1-3, h controls the KDE estimate ξ_k of ∇logρ_k; the BM method fits h by minimizing MMD between a score step and a Brownian step, and numerical performance depends on it.
  • step size τ_k = grid-searched per method and dataset (Tables 2 and 5)
    Discrete algorithms require step sizes; the experiments choose them by grid search to ensure best performance, so comparisons are tuned.
  • Kalman-Wasserstein regularization λ = 1 in the toy example
    Regularizes the covariance in C_λ(ρ); no theory guides the choice.
assumptions (4)
  • domain assumption E(ρ) is β-geodesically convex with respect to the chosen metric
    Hypothesis of Theorem 1; not verified for KL divergence in the numerical experiments.
  • standard math Brenier's theorem: existence, uniqueness and smoothness of optimal transport maps
    Appendix C uses T_t=∇Ψ_t with ∇T_t symmetric positive definite to construct the Lyapunov function.
  • ad hoc to paper The solution ρ_t to the AIG PDE remains smooth, positive and absolutely continuous on the time horizon
    The proofs differentiate along the flow and, in the Fisher-Rao case, divide by R_t(x); no well-posedness theorem is given (see Remark 3).
  • domain assumption The KDE with BM-selected bandwidth provides a sufficiently accurate approximation of ∇logρ_k
    Algorithms 1-3 replace the true score by ξ_k; no convergence guarantee is provided for the discretized particle system.

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Cite this review

Pith. "Pith review of Accelerated Information Gradient flow." pith.science (2026). https://pith.science/paper/YHNBBEVB

@misc{pith2026190902102,
  author       = {Pith},
  title        = {Pith review of: Accelerated Information Gradient flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YHNBBEVB}},
  note         = {Machine review of arXiv:1909.02102}
}
read the original abstract

We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo (MCMC) algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein metric and Stein metric. For both Fisher-Rao and Wasserstein-2 metrics, we prove convergence properties of accelerated gradient flows. In implementations, we propose a sampling-efficient discrete-time algorithm for Wasserstein-2, Kalman-Wasserstein and Stein accelerated gradient flows with a restart technique. We also formulate a kernel bandwidth selection method, which learns the gradient of logarithm of density from Brownian-motion samples. Numerical experiments, including Bayesian logistic regression and Bayesian neural network, show the strength of the proposed methods compared with state-of-the-art algorithms.

Figures

Figures reproduced from arXiv: 1909.02102 by the authors.

Figure 1
Figure 1. Results on Bayesian logistic regression, averaged over [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Comparison of different AIG flows on a toy example. [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. The effect of the BM method. Samples are plotted as blue dots. Left to right: MCMC, [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

38 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [32]

    Accelerated flow for probability distributions

    Amirhossein Taghvaei and Prashant G Mehta. Accelerated flow for probability distributions. arXiv preprint arXiv:1901.03317, 2019

    arXiv 1901

  2. [1]

    Natural gradient works efficiently in learning.Neural computation, 10(2):251– 276, 1998

    Shun-Ichi Amari. Natural gradient works efficiently in learning.Neural computation, 10(2):251– 276, 1998

    work page 1998

  3. [2]

    Information geometry and its applications, volume 194

    Shun-ichi Amari. Information geometry and its applications, volume 194. Springer, 2016

    2016

  4. [3]

    Differential geometry in statistical inference

    Shun’ichi Amari, Ole E Barndorff-Nielsen, Robert E Kass, Steffen L Lauritzen, and CR Rao. Differential geometry in statistical inference. IMS, 1987

    work page 1987

  5. [4]

    Langevin Monte Carlo and JKO splitting

    Espen Bernton. Langevin Monte Carlo and JKO splitting. In Conference On Learning Theory, pages 1777–1798, 2018

    work page 2018

  6. [5]

    Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces

    José A Carrillo, Young-Pil Choi, and Oliver Tse. Convergence to equilibrium in Wasserstein distance for damped Euler equations with interaction forces. Communications in Mathematical Physics, 365(1):329–361, 2019

    work page 2019

  7. [6]

    Underdamped Langevin MCMC: A non-asymptotic analysis

    Xiang Cheng, Niladri S Chatterji, Peter L Bartlett, and Michael I Jordan. Underdamped Langevin MCMC: A non-asymptotic analysis. arXiv preprint arXiv:1707.03663, 2017

    arXiv 2017

  8. [7]

    Wasserstein Hamiltonian flows

    Shui-Nee Chow, Wuchen Li, and Haomin Zhou. Wasserstein hamiltonian flows. arXiv preprint arXiv:1903.01088, 2019

    work page Pith review arXiv 1903

Show all 38 references
  1. [8]

    An information-theoretic approach to neural computing

    Gustavo Deco and Dragan Obradovic. An information-theoretic approach to neural computing. Springer Science & Business Media, 2012

    2012

  2. [9]

    On the geometry of stein variational gradient descent

    A Duncan, N Nüsken, and L Szpruch. On the geometry of stein variational gradient descent. arXiv preprint arXiv:1912.00894, 2019

    1912 arXiv

  3. [10]

    Interact- ing Langevin diffusions: Gradient structure and ensemble Kalman sampler

    Alfredo Garbuno-Inigo, Franca Hoffmann, Wuchen Li, and Andrew M Stuart. Interact- ing Langevin diffusions: Gradient structure and ensemble Kalman sampler. arXiv preprint arXiv:1903.08866, 2019

    1903 arXiv

  4. [11]

    Adam: A method for stochastic optimization

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

    2014 arXiv

  5. [12]

    The density manifold and configuration space quantization

    John D Lafferty. The density manifold and configuration space quantization. Transactions of the American Mathematical Society, 305(2):699–741, 1988

    1988

  6. [13]

    Understanding and accelerating particle-based variational inference

    Chang Liu, Jingwei Zhuo, Pengyu Cheng, Ruiyi Zhang, and Jun Zhu. Understanding and accelerating particle-based variational inference. In International Conference on Machine Learning, pages 4082–4092, 2019

    2019

  7. [14]

    Ac- celerated first-order methods on the Wasserstein space for Bayesian inference

    Chang Liu, Jingwei Zhuo, Pengyu Cheng, Ruiyi Zhang, Jun Zhu, and Lawrence Carin. Ac- celerated first-order methods on the Wasserstein space for Bayesian inference. arXiv preprint arXiv:1807.01750, 2018

    2018 arXiv

  8. [15]

    Stein variational gradient descent as gradient flow

    Qiang Liu. Stein variational gradient descent as gradient flow. In I. Guyon, U. V . Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors,Advances in Neural Information Processing Systems 30, pages 3115–3123. Curran Associates, Inc., 2017

    2017

  9. [16]

    Stein variational gradient descent: A general purpose bayesian inference algorithm

    Qiang Liu and Dilin Wang. Stein variational gradient descent: A general purpose bayesian inference algorithm. In Advances in neural information processing systems, pages 2378–2386, 2016

    2016

  10. [17]

    Accelerated first-order methods for geodesically convex optimization on Riemannian manifolds

    Yuanyuan Liu, Fanhua Shang, James Cheng, Hong Cheng, and Licheng Jiao. Accelerated first-order methods for geodesically convex optimization on Riemannian manifolds. InAdvances in Neural Information Processing Systems, pages 4868–4877, 2017. 32

    2017

  11. [18]

    Is there an analog of Nesterov acceleration for MCMC? arXiv preprint arXiv:1902.00996, 2019

    Yi-An Ma, Niladri Chatterji, Xiang Cheng, Nicolas Flammarion, Peter Bartlett, and Michael I Jordan. Is there an analog of Nesterov acceleration for MCMC? arXiv preprint arXiv:1902.00996, 2019

    1902 arXiv

  12. [19]

    Hamiltonian descent methods

    Chris J Maddison, Daniel Paulin, Yee Whye Teh, Brendan O’Donoghue, and Arnaud Doucet. Hamiltonian descent methods. arXiv preprint arXiv:1809.05042, 2018

    2018 arXiv

  13. [20]

    Natural gradient, fitness modelling and model selection: A unifying perspective

    Luigi Malago, Matteo Matteucci, and Giovanni Pistone. Natural gradient, fitness modelling and model selection: A unifying perspective. In 2013 IEEE Congress on Evolutionary Computation, pages 486–493. IEEE, 2013

    2013

  14. [21]

    Optimizing neural networks with kronecker-factored ap- proximate curvature

    James Martens and Roger Grosse. Optimizing neural networks with kronecker-factored ap- proximate curvature. In International conference on machine learning , pages 2408–2417, 2015

    2015

  15. [22]

    A method of solving a convex programming problem with convergence rate O(1/k2)

    Yurii Nesterov. A method of solving a convex programming problem with convergence rate O(1/k2). Soviet Mathematics Doklady, 27(2):372–376, 1983

    1983

  16. [23]

    The geometry of dissipative evolution equations: the porous medium equation

    Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations, 26(1-2):101–174, 2001

    2001

  17. [24]

    Adaptive restart for accelerated gradient schemes

    Brendan O’donoghue and Emmanuel Candes. Adaptive restart for accelerated gradient schemes. Foundations of computational mathematics, 15(3):715–732, 2015

    2015

  18. [25]

    Information theoretic learning

    Jose C Principe, Dongxin Xu, John Fisher, and Simon Haykin. Information theoretic learning. Unsupervised adaptive filtering, 1:265–319, 2000

    2000

  19. [26]

    Variational inference with normalizing flows

    Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015

    2015 arXiv

  20. [27]

    A Geometric Framework for Modeling and Inference using the Nonparametric Fisher–Rao metric

    Abhijoy Saha. A Geometric Framework for Modeling and Inference using the Nonparametric Fisher–Rao metric. PhD thesis, The Ohio State University, 2019

    2019

  21. [28]

    Improvement on some known nonparametric uniformly consistent estimators of derivatives of a density

    Radhey S Singh. Improvement on some known nonparametric uniformly consistent estimators of derivatives of a density. The Annals of Statistics, pages 394–399, 1977

    1977

  22. [29]

    Functional and shape data analysis, volume 475

    Anuj Srivastava and Eric P Klassen. Functional and shape data analysis, volume 475. Springer, 2016

    2016

  23. [30]

    Inverse problems: a Bayesian perspective

    Andrew M Stuart. Inverse problems: a Bayesian perspective. Acta numerica, 19:451–559, 2010

    2010

  24. [31]

    Weijie Su, Stephen Boyd., and Emmanuel J. Candés. A differential equation for modeling Nesterov’s accelerated gradient method: Theory and insights. Journal of Machine Learning Research, 2016

    2016

  25. [33]

    Topics in optimal transportation

    Cédric Villani. Topics in optimal transportation. American Mathematical Soc., 2003

    2003

  26. [34]

    Stein variational gradient descent with matrix-valued kernels

    Dilin Wang, Ziyang Tang, Chandrajit Bajaj, and Qiang Liu. Stein variational gradient descent with matrix-valued kernels. In Advances in neural information processing systems , pages 7834–7844, 2019

    2019

  27. [35]

    The Search direction Correction makes first-order methods faster

    Yifei Wang, Zeyu Jia, and Zaiwen Wen. The Search direction Correction makes first-order methods faster. arXiv preprint arXiv:1905.06507, 2019

    1905 arXiv

  28. [36]

    Information newton’s flow: second-order optimization method in probability space

    Yifei Wang and Wuchen Li. Information newton’s flow: second-order optimization method in probability space. arXiv preprint arXiv:2001.04341, 2020

    2001 arXiv

  29. [37]

    Proximal Langevin Algorithm: Rapid convergence under isoperimetry

    Andre Wibisono. Proximal Langevin Algorithm: Rapid convergence under isoperimetry. arXiv preprint arXiv:1911.01469, 2019

    1911 arXiv

  30. [38]

    Towards Riemannian accelerated gradient methods

    Hongyi Zhang and Suvrit Sra. Towards Riemannian accelerated gradient methods. arXiv preprint arXiv:1806.02812, 2018. 33

    2018 arXiv

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