lambda-PSD: Scalable Approximate SNR-Optimised Polynomial Stein Discrepancies

arXiv:2606.26621 · stat.ME, stat.CO, stat.ML · Submitted 2026-06-25 · Read on arXiv

Minh-Long Nguyen, Thanh-Long Vu, Christopher Drovandi, Leah F. South, Trung-Tin Nguyen

stat.ME, stat.CO, stat.ML

Submitted: 2026-06-25

Comments: 15 pages, 5 figures

Code: https://github.com/olivervu25/weightedpsd

License: http://creativecommons.org/licenses/by/4.0/

The gist: Polynomial Stein discrepancies (PSD) provide a scalable alternative to kernel Stein methods for measuring sample quality and goodness-of-fit testing, but their statistical properties remain poorly

Terminology

Abstract

Polynomial Stein discrepancies (PSD) provide a scalable alternative to kernel Stein methods for measuring sample quality and goodness-of-fit testing, but their statistical properties remain poorly understood. We show that increasing polynomial degree primarily amplifies signal without adequately controlling variance, rather than directly optimising the signal-to-noise ratio (SNR). Under suitable assumptions, this might lead to a failure mode in which the SNR squared can provably decay exponentially with polynomial degree. Motivated by this observation, we reformulate Stein discrepancy construction as an explicit SNR squared maximisation problem, yielding a Rayleigh quotient over Stein features. This perspective motivates lambda-PSD, an approximate scalable covariance-aware reweighting scheme defined in a low-dimensional subspace. Under Gaussian settings, we show that lambda-PSD avoids the exponential SNR squared collapse and achieves a stable SNR squared. Empirically, lambda-PSD substantially improves test power while retaining linear-time complexity in the number of samples, highlighting the importance of SNR-aware design for scalable Stein discrepancies.

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