arXiv research

Lower bounds on MALA and HMC for well-conditioned distributions.

We introduce a recent symplectic integration scheme derived for solving physically motivated systems with non-separable Hamiltonians. We show its relevance to Riemannian manifold Hamiltonian Monte Carlo (RMHMC) and provide an alternative to the currently used generalised leapfrog symplectic integrator, which relies on …

WALNUTS improves sampling efficiency and robustness for multi-scale distributions.

HMC with leapfrog integrator mixes faster than MALA under certain smoothness conditions.

Adaptive HMC improves sampling efficiency by optimizing mass matrix.

Hamiltonian Monte Carlo on ReLU networks is inefficient due to large local error.

Model predicts alternating market dominance for two competing firms.

Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo sampling algorithm for drawing samples from smooth probability densities over continuous spaces. We study the variant most widely used in practice, Metropolized HMC with the Störmer-Verlet or leapfrog integrator, and make two primary contribut…

The hybrid Monte Carlo algorithm (HMCA) is applied for Bayesian parameter estimation of the realized stochastic volatility (RSV) model. Using the 2nd order minimum norm integrator (2MNI) for the molecular dynamics (MD) simulation in the HMCA, we find that the 2MNI is more efficient than the conventional leapfrog integr…

A new clustering method improves recovery guarantees by re-embedding data.

Hamiltonian Monte Carlo (HMC) exploits Hamiltonian dynamics to construct efficient proposals for Markov chain Monte Carlo (MCMC). In this paper, we present a generalization of HMC which exploits \textit{non-canonical} Hamiltonian dynamics. We refer to this algorithm as magnetic HMC, since in 3 dimensions a subset of th…

New method improves sampling efficiency for complex distributions.

GIST adapts HMC by tuning parameters based on position and momentum.

We present a novel technique for learning the mass matrices in samplers obtained from discretized dynamics that preserve some energy function. Existing adaptive samplers use Riemannian preconditioning techniques, where the mass matrices are functions of the parameters being sampled. This leads to significant complexiti…

Hamiltonian Monte Carlo (HMC) is a widely deployed method to sample from high-dimensional distributions in Statistics and Machine learning. HMC is known to run very efficiently in practice and its popular second-order "leapfrog" implementation has long been conjectured to run in $d^{1/4}$ gradient evaluations. Here we …

New HMC method uses asymmetrical momentum distributions and improves performance.

MALA mixes optimally in κ√d steps for log-concave sampling.

ATLAS adapts HMC step size and trajectory length for complex geometries.

A new method improves actor-critic RL by integrating HMC, enhancing policy distribution and exploration.

This paper analyzes the convergence of dynamic HMC and NUTS methods.

LAVA values data without needing a specific learning algorithm.

The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.

Geometrically interprets integrability of geodesic flow using web theory.

Proof shows volume equals integral points for certain manifolds.

Integrates rough geometric forms on manifolds.

The paper defines and analyzes set-valued stochastic integrals for Lévy processes.

New integration theory on topological spaces, including fractals.

We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can va…

We define a non-absolutely convergent integration on integral currents of dimension 1 in Euclidean space. This integral is closely related to the Henstock-Kurzweil and Pfeffer Integrals. Using it, we prove a generalized Fundamental Theorem of Calculus on these currents. A detailed presentation of Henstock-Kurzweil Inte…

The paper defines and proves the existence of decompositions of integral varifolds.

Counterexample shows Ito integrand needn't be locally square integrable.

Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.

TQ separates sampling and integration for high-dimensional integrals.

We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…

The article constructs stochastic integration in Riemannian manifolds.

Method finds differential equations for integrable billiard tables.

Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.

Investigates integrable systems with linear periodic integral for e(3) Lie algebra.

We use neural networks as control variates with geometric integration techniques.

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…

This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the pert…

Surveying integrability of Lie algebroids and structures.

This paper provides an existence-and-uniqueness theorem characterizing the stochastic integral with respect to a Wiener process. The integral is represented as a mapping from the space of measurable and adapted pathwise locally integrable processes to the space of continuous adapted processes. It is characterized in te…

We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a "Lagrangian" sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrabl…

Integrable LCK manifolds characterized as Kähler Lie algebras.

The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integ…

New integrable deformations for topological hierarchies from Frobenius manifolds.

Sharp lower bound found for integral varifolds' mean curvature.