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On the unconventional Hug integrator

Christophe Andrieu, J. M. Sanz-Serna

TL;DR

This work provides a rigorous framework for understanding the unconventional Hug integrator as a consistent discretization of a nonstandard constrained dynamical system. By generalizing Hug to manifolds of arbitrary codimension, it derives a continuous-time ODE on an extended phase space and proves key properties such as volume preservation, time-reversibility, and invariants, linking discrete steps to the ODE via explicit tangential/normal decompositions. A notable finding is supraconvergence: despite first-order local consistency, the method achieves second-order global accuracy due to cancellation of stepwise errors, while also revealing dynamics that can fold back on themselves, potentially limiting manifold exploration in low dimensions. Numerical experiments illustrate these phenomena and suggest practical mitigations, such as nonisotropic velocity updates and occasional velocity refreshes, with implications for Hug-based methods beyond sampling contexts. Overall, the paper provides foundational tools to analyze Hug-type integrators and offers insights into their geometric and numerical behavior on manifolds.

Abstract

Hug is a recently proposed iterative mapping used to design efficient updates in Markov chain Monte Carlo (MCMC) methods. Hug generates proposals that remain very close to hypersurfaces (level sets) of constant probabilty density. We analyse a generalization of Hug from hypersurfaces to manifolds of arbitrary dimensions, not necessarily arising in a sampling context. The analysis is based on interpreting, in a nonstandard way, Hug as a consistent discretization of a system of differential equations with a rather complicated structure. The proof of convergence of this discretization includes a number of unusual features we explore fully, in particular a supraconvergence property is established, whereby second order of convergence is attained with consistency of the first order. We uncover and discuss an unexpected property of the solutions of the underlying dynamical system that manifest itself by the existence of Hug trajectories that fail to cover the manifold of interest.

On the unconventional Hug integrator

TL;DR

This work provides a rigorous framework for understanding the unconventional Hug integrator as a consistent discretization of a nonstandard constrained dynamical system. By generalizing Hug to manifolds of arbitrary codimension, it derives a continuous-time ODE on an extended phase space and proves key properties such as volume preservation, time-reversibility, and invariants, linking discrete steps to the ODE via explicit tangential/normal decompositions. A notable finding is supraconvergence: despite first-order local consistency, the method achieves second-order global accuracy due to cancellation of stepwise errors, while also revealing dynamics that can fold back on themselves, potentially limiting manifold exploration in low dimensions. Numerical experiments illustrate these phenomena and suggest practical mitigations, such as nonisotropic velocity updates and occasional velocity refreshes, with implications for Hug-based methods beyond sampling contexts. Overall, the paper provides foundational tools to analyze Hug-type integrators and offers insights into their geometric and numerical behavior on manifolds.

Abstract

Hug is a recently proposed iterative mapping used to design efficient updates in Markov chain Monte Carlo (MCMC) methods. Hug generates proposals that remain very close to hypersurfaces (level sets) of constant probabilty density. We analyse a generalization of Hug from hypersurfaces to manifolds of arbitrary dimensions, not necessarily arising in a sampling context. The analysis is based on interpreting, in a nonstandard way, Hug as a consistent discretization of a system of differential equations with a rather complicated structure. The proof of convergence of this discretization includes a number of unusual features we explore fully, in particular a supraconvergence property is established, whereby second order of convergence is attained with consistency of the first order. We uncover and discuss an unexpected property of the solutions of the underlying dynamical system that manifest itself by the existence of Hug trajectories that fail to cover the manifold of interest.

Paper Structure

This paper contains 18 sections, 6 theorems, 62 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Hug algorithm
  3. The algorithm
  4. A generalization of the Hug timestepping mechanism
  5. Properties of the timestepping mechanism
  6. The ODE
  7. The derivative $N^\prime(x)$
  8. The ODE being approximated
  9. Relating the Algorithm and the ODE
  10. Dynamics
  11. Numerical experiments and future work
  12. Proofs
  13. Proof of Proposition \ref{['prop:alg']}
  14. Proof of Lemma \ref{['lemma:nprime']}
  15. Proof of Theorem \ref{['th:odebis']}
  16. ...and 3 more sections

Key Result

Proposition 1

The following properties hold: Furthemore, let eq:alg1--eq:alg3 hold for $k=0,\dots, K-1$. Then:

Figures (8)

  • Figure 1: Computation of the iterate $x_2$ for a bivariate Gaussian. The plot depicts the plane of the variable $x=(x_{(1)},x_{(2)})\in\mathbb R^2$. The solid curve is the contour $\{y :\ell(y) = \ell(x_0)\}$. The dashed curve is (part of) the contour $\{y :\ell(y) = \ell(x_{1/2})\}$. The segments $[x_0,x_{1/2}]$, $[x_{1/2},x_{1}]$, $[x_1,x_{3/2}]$, $[x_{3/2},x_{2}]$ share a common Euclidean length $(\delta/2)\|v_0\|$.
  • Figure 2: Phase plane of the system \ref{['eq:edophi']}--\ref{['eq:edop']}, when $a=1$, $b=4$, $c=\sqrt{2}$. The portrait is $2\pi$-periodic in $\phi$. The thicker lines correspond to the heteroclinic connections between saddle points. Outside the heteroclinic connections trajectories rotate; inside they librate.
  • Figure 3: The continuous arc is the solution of the system \ref{['eq:odebis1']}--\ref{['eq:odebis2']} when $f(x) = -x_{(1)}^2-4x_{(2)}^2$, $x(0)=[1,0]$, and $v(0)= [\sqrt{7/4}, 1/2]^{\top}$ depicted in the $x$ plane for $0\leq t \leq 1.4$. (This corresponds to system \ref{['eq:edophi']}--\ref{['eq:edop']} with $\phi(0) = 0$, $p(0) = 1/2$, $c=\sqrt{2}$.) The solution $x(t)$ first moves anticlockwise starting from $x(0)$ but then folds back on itself, moves clockwise and folds back once more. At the final time, $x(14)$ happens to be close to the initial location $x(0)$. Also depicted is the approximation provided by Algorithm \ref{['eq:alg1']}--\ref{['eq:alg3']}, when $\delta= 0.1$ and $K=14$. The numerical solution mimics the behaviour of the ODE; the numbers $0-14$ identify the stepnumber $k$ of the iterates $x_k$.
  • Figure 4: $K=10000$ integration steps of the dynamics constrained to a $3D$ ellipsoid $(n=3)$, for four initial unit length velocities.
  • Figure 5: Scatter plots $i\mapsto(\|v_{\perp}^{(i)}(0)\|,d_{\rm max}(v_{\perp}(0)):=\max_{0\leq k\leq K}\|x^{(i)}(k)-x(0)\|)$ ($n=3$)
  • ...and 3 more figures

Theorems & Definitions (8)

  • Proposition 1
  • Example 1
  • Lemma 2
  • Example 2
  • Theorem 3
  • Proposition 4
  • Theorem 5
  • Theorem 6