# Probabilistic Linear Solvers for Machine Learning

arXiv: https://arxiv.org/abs/2010.09691 · HTML: https://mlsanity.com/paper/2010.09691/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2020.

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

- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [High-Dimensional Gaussian Process Inference with Derivatives](https://mlsanity.com/paper/2102.07542/) (2021)
- [Computation-Aware Kalman Filtering and Smoothing](https://mlsanity.com/paper/2405.08971/) (2024)
- [ProbNum: Probabilistic Numerics in Python](https://mlsanity.com/paper/2112.02100/) (2021)
- [Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inference](https://mlsanity.com/paper/2411.01036/) (2024)
- [Calibrated Computation-Aware Gaussian Processes](https://mlsanity.com/paper/2410.08796/) (2024)
- [Accelerating Non-Conjugate Gaussian Processes By Trading Off Computation For Uncertainty](https://mlsanity.com/paper/2310.20285/) (2023)
- [Online Conformal Probabilistic Numerics via Adaptive Edge-Cloud Offloading](https://mlsanity.com/paper/2503.14453/) (2025)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)

## Abstract (arXiv)

Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of uncertainty in the field. Unifying earlier work we propose a class of probabilistic linear solvers which jointly infer the matrix, its inverse and the solution from matrix-vector product observations. This class emerges from a fundamental set of desiderata which constrains the space of possible algorithms and recovers the method of conjugate gradients under certain conditions. We demonstrate how to incorporate prior spectral information in order to calibrate uncertainty and experimentally showcase the potential of such solvers for machine learning.