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  • Positive Curvature and Hamiltonian Monte Carlo Christof Seiler, Simon Rubinstein-Salzedo, and Susan Holmes

    Department of Statistics, Stanford University

    Goal Running time estimates for sampling from high di- mensional probability distributions using Hamilto- nian Monte Carlo (HMC).

    Hamiltonian Monte Carlo

    Bayesian posterior distribution: π(q | data) = 1Z L(data | q) π(q)

    parameter q ∈ Rd, d� observations, and unknown normalizing constant Z.

    Expectation of f : Rd→ R I := Eπ(f ) =

    ∫ f (q) π(q | data) dq

    Empirical mean: ̂I := 1T


    t=1 Qt, Q1, . . . , QT ∼ π(q | data)

    simulated using HMC for running time T HMC algorithm: Metropolis-Hastings with proposal

    step following Hamiltonian trajectories Error and running time: Bound the error | ̂I − I| as

    a function of the running time T

    Sectional Curvature

    Sectional curvature Sec describes the nature of geodesics


    ✓ 1 � "


    2 Secx(u, v) + O("

    3 + "2�)

    Secx(u, v) > 0

    Positive curvature: geodesics meet Flat and negative curvature: geodesics deviate

    Probability Meets Geometry

    Coarse Ricci curvature κ describes the deviation from independence of subsequent draws from Markov chain Monte Carlo •The larger κ the more independent •The larger κ the smaller T for the same error •We calculate the κ of HMC

    Concentration inequality (JO10) P ( | ̂I − I| ≥ r‖f‖Lip

    ) ≤ 2e−r2/(16V 2(κ,T ))

    Jacobi Metric

    Jacobi metric gh links Hamiltonians to Riemannian geometry

    Hamiltonian function: h = V (q) + K(p)

    Potential energy: V (q) = − log π(q | data) + C

    Kinetic energy: K(p) = 12 〈p, p〉

    Hamiltonian equations: dq

    dt = ∂H ∂p

    and dp dt

    = −∂H ∂q

    Jacobi-Maupertuis Principle:



    p(0) q(0)

    Hamiltonian trajectories q(t) with fixed h are geodesics on Riemannian manifolds X with the Jacobi metric:

    gh(p, p) = 2 (h− V ) 〈p, p〉.

    HMC Meets Geometry

    Reference metric 〈·, ·〉 Conformal scaling of a metric 2 (h− V ) Equivalence table:

    HMC Riemannian geometry posterior distribution scaling of reference metric covariance of proposal reference metric pick from proposal geodesics

    Gaussian Example

    •HMC sampling from a Gaussian in Rd with zero mean and covariance matrix Λ−1

    •Calculate sectional curvature of Jacobi metric • Sec is random because p is random •Calculate coarse Ricci curvature from Sec Our Lemma on positive curvature of HMC:

    P(d2 Sec ≥ K1) ≥ 1−K2e−K3 √ d

    Our Corollary in high dimensions:

    Sec ≥ Tr(Λ) d3

    as d→∞ Our Proposal on coarse Ricci curvature of HMC:

    κ ≥ Tr(Λ)6d2 + O  

    1 d2

      as d→∞

    Numerical simulations: Sample Sec’s uniformly at each step, q(0), . . . , q(T − 1)

    Histogram of sectional curvatures (d = 100)

    Fr eq

    ue nc


    0.00005 0.00015 0.00025 0.00035

    0 20

    00 0

    40 00

    0 60

    00 0

    expectation E(Sec) sample mean

    Histogram of sectional curvatures (d = 10)

    Fr eq

    ue nc


    −3 −2 −1 0 1 2 3 4

    0 20

    00 0

    40 00

    0 60

    00 0

    80 00


    expectation E(Sec) sample mean

    (d = 10)

    (d = 1000)


    0-3 3


    Computational Anatomy

    Conclusion Linking HMC to geodesics of the Jacobi metric en- ables us to analyze HMC through the rich mathe- matical framework of Riemannian geometry.


    (JO10) A. Joulin and Y. Ollivier. Curvature, Concentration and Error Estimates for Markov Chain Monte Carlo. Ann. Probab., 2010.

    (Pin75) O. C. Pin. Curvature and Mechanics. Advances in Math., 1975.


    CS is supported by a postdoctoral fellowship from the Swiss Na- tional Science Foundation and a travel grant from the France- Stanford Center. SRS and SH are supported by NIH grant R01- GM086884.