# Asymptotic Analysis of Sampling Estimators for Randomized Numerical Linear Algebra Algorithms

arXiv: https://arxiv.org/abs/2002.10526 · HTML: https://mlsanity.com/paper/2002.10526/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2020.

## Built on by (most cited first, 10 of 10)

- [Towards a statistical theory of data selection under weak supervision](https://mlsanity.com/paper/2309.14563/) (2023)
- [Recent and Upcoming Developments in Randomized Numerical Linear Algebra for Machine Learning](https://mlsanity.com/paper/2406.11151/) (2024)
- [An optimal transport approach for selecting a representative subsample with application in efficient kernel density estimation](https://mlsanity.com/paper/2206.01182/) (2022)
- [Model-free Subsampling Method Based on Uniform Designs](https://mlsanity.com/paper/2209.03617/) (2022)
- [Distributed Least Squares in Small Space via Sketching and Bias Reduction](https://mlsanity.com/paper/2405.05343/) (2024)
- [Fundamental Bias in Inverting Random Sampling Matrices with Application to Sub-sampled Newton](https://mlsanity.com/paper/2502.13583/) (2025)
- [Optimal Sampling Designs for Multi-dimensional Streaming Time Series with Application to Power Grid Sensor Data](https://mlsanity.com/paper/2303.08242/) (2023)
- [Inference in Randomized Least Squares and PCA via Normality of Quadratic Forms](https://mlsanity.com/paper/2404.00912/) (2024)
- [Optimal Sampling for Generalized Linear Model under Measurement Constraint with Surrogate Variables](https://mlsanity.com/paper/2501.00972/) (2025)
- [Core-elements Subsampling for Alternating Least Squares](https://mlsanity.com/paper/2509.18024/) (2025)

## Abstract (arXiv)

The statistical analysis of Randomized Numerical Linear Algebra (RandNLA) algorithms within the past few years has mostly focused on their performance as point estimators. However, this is insufficient for conducting statistical inference, e.g., constructing confidence intervals and hypothesis testing, since the distribution of the estimator is lacking. In this article, we develop an asymptotic analysis to derive the distribution of RandNLA sampling estimators for the least-squares problem. In particular, we derive the asymptotic distribution of a general sampling estimator with arbitrary sampling probabilities. The analysis is conducted in two complementary settings, i.e., when the objective of interest is to approximate the full sample estimator or is to infer the underlying ground truth model parameters. For each setting, we show that the sampling estimator is asymptotically normally distributed under mild regularity conditions. Moreover, the sampling estimator is asymptotically unbiased in both settings. Based on our asymptotic analysis, we use two criteria, the Asymptotic Mean Squared Error (AMSE) and the Expected Asymptotic Mean Squared Error (EAMSE), to identify optimal sampling probabilities. Several of these optimal sampling probability distributions are new to the literature, e.g., the root leverage sampling estimator and the predictor length sampling estimator. Our theoretical results clarify the role of leverage in the sampling process, and our empirical results demonstrate improvements over existing methods.