Fast determinantal sampling on general spaces and diffusion geometry

2026-07-09 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research introduces a novel class of Determinantal Point Processes (DPPs) designed for efficient sampling on general spaces, moving beyond traditional Euclidean settings. The core innovation lies in constructing spectral kernels from eigenfunctions of Laplacian and Markov diffusion operators, applicable to Riemannian manifolds and weighted networks. The study establishes explicit rate guarantees for determinantal sampling, demonstrating that accuracy depends on the intrinsic dimensionality ($d_{\text{int}}$ or $m$) of the underlying space, rather than the often much larger ambient dimension. Specifically, the method achieves variance reduction guarantees of (sample size)$^{-\frac{1}{2}-\frac{1}{2d_{\text{int}}}}$ for accuracy, matching known rates on comparable Euclidean spaces. This significantly improves upon classical i.i.d. samplers, which typically offer $O(n^{-1/2})$ accuracy. The approach integrates concepts from Weyl's Law, Markov diffusions, and pseudodifferential operators, with practical applications in Monte Carlo integration on manifolds and minibatch sampling for low-dimensional data structures like k-nearest neighbor graphs.

Key takeaway

For Machine Learning Engineers optimizing sampling for high-dimensional datasets residing on low-dimensional manifolds, you should consider implementing spectral Determinantal Point Processes (DPPs). This approach offers provable variance reduction, achieving accuracy rates of $n^{-1/2-1/(2m)}$ (where $m$ is intrinsic dimension) compared to $n^{-1/2}$ for i.i.d. sampling. This significantly improves minibatch efficiency and Monte Carlo integration accuracy, particularly when data geometry is complex or non-Euclidean.

Key insights

Spectral DPPs adapt to intrinsic data geometry via Laplacian eigenfunctions, yielding superior sampling rates over i.i.d. methods.

Principles

• DPPs promote diverse, repulsive sample configurations.
• Variance reduction improves sampling accuracy beyond $O(n^{-1/2})$ rates.
• Intrinsic dimensionality governs sampling rates in non-Euclidean domains.

Method

Construct spectral DPPs from Laplacian-type operator eigenfunctions on general spaces (manifolds, graphs). Bound variance using eigenvalue distribution and Weyl-type laws.

In practice

• Apply spectral DPPs for Monte Carlo integration on manifolds.
• Use graph Laplacian eigenvectors for minibatch sampling on k-NN graphs.
• Achieve $n^{-1/2-1/(2m)}$ accuracy for low-dimensional data.

Topics

• Determinantal Point Processes
• Spectral Kernels
• Diffusion Geometry
• Minibatch Sampling
• Monte Carlo Integration
• Intrinsic Dimensionality

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.