Fast determinantal sampling on general spaces and diffusion geometry

2026-07-07

AI Summary

This work extends determinantal point process (DPP) sampling theory to general non-Euclidean metric spaces—such as Riemannian manifolds and weighted graphs—for the first time. By constructing kernel functions based on spectral properties of Laplace and Markov diffusion operators, the authors propose an adaptive DPP sampling framework that automatically adjusts to the intrinsic dimensionality of the data. Leveraging tools from Weyl’s law, Dirichlet forms, and pseudodifferential operator theory, the method achieves a sampling error rate of $O(n^{-1/2 - 1/(2d_{\text{int}})})$ on compact manifolds and k-nearest neighbor graphs, significantly outperforming conventional i.i.d. sampling and matching the optimal rates known in Euclidean settings.

Abstract

Determinantal point processes have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficient minibatches, coresets, and other compact representations of large-scale datasets. In particular, sampling mechanisms based on DPPs are believed to demonstrate better approximation properties compared to classical i.i.d. samplers, even at the scale of the exponent. One of the key strengths of DPP based samplers is that they can be deployed over very general spaces, in contrast to more classical sampling methods beyond i.i.d. which tend to work in very well-structured settings, principally Euclidean spaces. In this work, we establish explicit rate guarantees for determinantal sampling in spaces that extend far beyond known Euclidean setups, focusing on spectral kernels obtained from eigenspaces of naturally associated Laplacian and other Markov diffusion operators. This includes, in particular, Riemannian manifolds and weighted networks. In determinantal sampling from compact Riemannian manifolds, we establish sampling rates that automatically pick up the intrinsic dimensionality $d_{\text{int}}$ of the underlying manifold. In the setting of networks, we investigate DPP-based samplers on the celebrated k-nearest neighbour graphs, as well as weighted random geometric graphs, and demonstrate a similar improved dependence on the intrinsic dimensionality of the data. Overall, our approach achieves guarantees of $\big(\text{sample size}\big)^{-\frac{1}{2}-\frac{1}{2d_{\text{int}}}}$ that match known rates on Euclidean spaces of comparable dimension. In terms of techniques, we connect to the celebrated Weyl's Law for manifold spectra, and leverage tools from the theory of Markov diffusions and Dirichlet forms as well as certain ingredients from the theory of pseudodifferential operators, which could be of independent interest in this area.

Problem

• determinantal point processes
• sampling rates
• intrinsic dimensionality
• Riemannian manifolds
• weighted networks

Innovation

• determinantal point processes
• diffusion geometry
• intrinsic dimensionality
• spectral kernels
• non-Euclidean sampling