Fast determinantal sampling on general spaces and diffusion geometry

2026-07-07 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

This research introduces fast determinantal sampling on general spaces, extending the application of determinantal point processes (DPPs) beyond traditional Euclidean settings. DPPs serve as a kernel-based alternative to independent sampling for creating efficient minibatches and compact dataset representations, offering improved approximation properties. The work establishes explicit rate guarantees for determinantal sampling using spectral kernels derived from Laplacian and Markov diffusion operators, applicable to Riemannian manifolds and weighted networks. Specifically, for compact Riemannian manifolds, the sampling rates inherently capture the intrinsic dimensionality $d_{ ext{int}}$. On k-nearest neighbor graphs and weighted random geometric graphs, the method demonstrates similar improved dependence on intrinsic dimensionality. Overall, the approach achieves guarantees of $( ext{sample size})^{-rac{1}{2}-rac{1}{2d_{ ext{int}}}}$, aligning with known rates in Euclidean spaces of comparable dimension, utilizing tools like Weyl's Law and Markov diffusion theory.

Key takeaway

For Machine Learning Engineers developing sampling strategies for large-scale, non-Euclidean datasets, adopting determinantal point processes (DPPs) can significantly improve approximation properties. You should consider implementing DPP-based samplers, especially for data on Riemannian manifolds or complex networks. This approach achieves better representation quality and sampling rates adapting to your data's intrinsic dimensionality. It offers a robust alternative to traditional i.i.d. sampling.

Key insights

DPPs offer superior sampling rates on general non-Euclidean spaces by leveraging diffusion geometry.

Principles

• DPPs improve approximation over i.i.d. samplers.
• Sampling rates can reflect intrinsic dimensionality.
• Spectral kernels extend sampling to general spaces.

Method

The method connects determinantal sampling to Weyl's Law, Markov diffusions, Dirichlet forms, and pseudodifferential operators, using spectral kernels from Laplacian and Markov diffusion operators.

In practice

• Construct efficient minibatches for large datasets.
• Create compact dataset coresets.
• Apply sampling to Riemannian manifolds or networks.

Topics

• Determinantal Point Processes
• Kernel-based Sampling
• Diffusion Geometry
• Riemannian Manifolds
• Weighted Networks
• Laplacian Operators

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.