# λ-PSD: Scalable Approximate SNR-Optimised Polynomial Stein Discrepancies

arXiv: https://arxiv.org/abs/2606.26621 · HTML: https://mlsanity.com/paper/2606.26621/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [A Linear-Time Kernel Goodness-of-Fit Test](https://mlsanity.com/paper/1705.07673/) (2017)
- [Random Feature Stein Discrepancies](https://mlsanity.com/paper/1806.07788/) (2018)
- [Black-box Importance Sampling](https://mlsanity.com/paper/1610.05247/) (2016)
- [Semi-Exact Control Functionals From Sard's Method](https://mlsanity.com/paper/2002.00033/) (2020)
- [Sparse Generalized Eigenvalue Problem: Optimal Statistical Rates via Truncated Rayleigh Flow](https://mlsanity.com/paper/1604.08697/) (2016)
- [Neural Stein critics with staged L2-regularization](https://mlsanity.com/paper/2207.03406/) (2022)
- [The Polynomial Stein Discrepancy for Assessing Moment Convergence](https://mlsanity.com/paper/2412.05135/) (2024)

## Abstract (arXiv)

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 SNR2 can provably decay exponentially with polynomial degree. Motivated by this observation, we reformulate Stein discrepancy construction as an explicit SNR2 maximisation problem, yielding a Rayleigh quotient over Stein features. This perspective motivates λ-PSD, an approximate scalable covariance-aware reweighting scheme defined in a low-dimensional subspace. Under Gaussian settings, we show that λ-PSD avoids the exponential SNR2 collapse and achieves a stable SNR2. Empirically, λ-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.