# Probabilistic Interpretation of Linear Solvers

arXiv: https://arxiv.org/abs/1402.2058 · HTML: https://mlsanity.com/paper/1402.2058/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2014.

## Built on by (most cited first, 12 of 12)

- [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)
- [Accelerating Non-Conjugate Gaussian Processes By Trading Off Computation For Uncertainty](https://mlsanity.com/paper/2310.20285/) (2023)
- [Probabilistic Numeric SMC Sampling for Bayesian Nonlinear System Identification in Continuous Time](https://mlsanity.com/paper/2404.12923/) (2024)
- [A probabilistic Taylor expansion with Gaussian processes](https://mlsanity.com/paper/2102.00877/) (2021)
- [Black Box Probabilistic Numerics](https://mlsanity.com/paper/2106.13718/) (2021)
- [Learning to Solve Related Linear Systems](https://mlsanity.com/paper/2503.17265/) (2025)
- [Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems](https://mlsanity.com/paper/2508.09623/) (2025)
- [Affine Tracing: A New Paradigm for Probabilistic Linear Solvers](https://mlsanity.com/paper/2605.10566/) (2026)

## Abstract (arXiv)

This manuscript proposes a probabilistic framework for algorithms that iteratively solve unconstrained linear problems Bx = b with positive definite B for x. The goal is to replace the point estimates returned by existing methods with a Gaussian posterior belief over the elements of the inverse of B, which can be used to estimate errors. Recent probabilistic interpretations of the secant family of quasi-Newton optimization algorithms are extended. Combined with properties of the conjugate gradient algorithm, this leads to uncertainty-calibrated methods with very limited cost overhead over conjugate gradients, a self-contained novel interpretation of the quasi-Newton and conjugate gradient algorithms, and a foundation for new nonlinear optimization methods.