Sampling and Estimation on Manifolds using the Langevin Diffusion
Authors: Karthik Bharath, Alexander Lewis, Akash Sharma, Michael V. Tretyakov
JMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Numerical illustrations with distributions, log-concave and otherwise, on the manifolds of positive and negative curvature elucidate on the derived bounds and demonstrate practical utility of the sampling algorithm. We carry out numerical experiments to verify the theoretical results from Section 4 for compact manifolds, and demonstrate utility of the proposed algorithm for both compact and non-compact M. Section 6.1 considers sampling using retractions from the von-Mises distribution on the two-dimensional unit sphere S2. ... Section 6.2 considers two distributions on the non-compact manifold of symmetric positive definite matrices: the Riemannian-Gaussian distribution with convex φ, and a distribution with non-convex potential. Versatility of the proposed algorithm to handle both convex and non-convex potential is demonstrated. |
| Researcher Affiliation | Academia | Karthik Bharath EMAIL School of Mathematical Sciences University of Nottingham Nottingham, UK, Alexander Lewis EMAIL Institut f ur Mathematische Stochastik Georg-August-Universit at G ottingen 37077 G ottingen, Germany, Akash Sharma EMAIL Department of Mathematical Sciences Chalmers University of Technology and the University of Gothenburg 41296 Gothenburg, Sweden, Michael V. Tretyakov EMAIL School of Mathematical Sciences University of Nottingham Nottingham, UK |
| Pseudocode | Yes | Algorithm 1 Sample from a von-Mises Fisher distribution on S2 using a retraction. ... Algorithm 2 Algorithm to sample from Riemannian-Gaussian distribution on P3 with O = I3, σ = 1/2 and X0 = hvec 1((2, 4, 2, 1, 1, 0) ). ... Algorithm 3 Algorithm to sample from double-well potential distribution on P3 with O = Im, X0 = hvec 1((2, 4, 2, 1, 1, 0) ). |
| Open Source Code | No | The paper does not provide explicit statements about the release of source code for the methodology described, nor does it include a link to a code repository. |
| Open Datasets | No | The paper focuses on sampling from defined mathematical distributions (e.g., von-Mises Fisher distribution, Riemannian-Gaussian distribution, distribution with non-convex potential) rather than analyzing pre-existing experimental datasets. Therefore, no external datasets are used or made available. |
| Dataset Splits | No | The paper does not use any experimental datasets; it focuses on sampling from defined mathematical distributions. Therefore, the concept of dataset splits is not applicable. |
| Hardware Specification | Yes | Simulations were performed in R using a parallel architecture on 30 cores, using the packages purrr and furrr on a Supermicro 620U Linux RHEL8.8 server with 48 Intel Xeon (Ice Lake class) CPUs. A further experiment was implemented in Julia using GPU kernels and run on an NVIDIA A100 GPU. |
| Software Dependencies | No | The paper mentions software like R (with packages purrr and furrr), Julia, and Mathematica, but does not provide specific version numbers for any of these software components or their libraries. |
| Experiment Setup | Yes | The parameters are λ = 1, X0 = (r0, θ0) = (π/4, π/4) and T = 5. (Table 1 caption); The parameters are O = I3, σ = 1/2, X0 = hvec 1((2, 4, 2, 1, 1, 0) ) and T = 10. (Table 3 caption); The parameters are O = Im, X0 = hvec 1((2, 4, 2, 1, 1, 0) ) and T = 5. (Table 4 caption). It also specifies "L" (number of independent realisations) and "h" (time step) values in the tables. |