Probabilistic linear algebra methods assign matrix-valued probability measures to unknown matrix quantities; for example to elements of the positive definite cone.

Philipp Hennig (Project Leader),
Simon Bartels

Linear algebra methods form the basis for the majority of numerical computations. Because they perform a positively elementary task, they have to satisfy strict requirements on computational efficiency and numerical robustness.

Our work has added to a growing understanding that many, widely used, linear solvers can be interpreted as performing probabilistic inference on the elements of a matrix or a vector from observations linear projections of this latent object. In particular, this is true for such foundational algorithms as the method of conjugate gradients and other iterative algorithms in the *conjugate directions* and *projection method* classes.

Our ongoing research effort focusses on ways to use these insights in the large-scale linear algebra problems encountered in machine learning. There, the most frequent linear algebra task is the least-squares problem of solving

$$ Kx = b $$

where $K$ is a very large symmetric positive definite kernel Gram matrix, i.e. a matrix arising from computing pairwise terms over a dataset. An intriguing question is how available knowledge about the structure or generative process underlying the dataset can be translated into helpful prior information for the linear solver.

2 results

**Probabilistic Approximate Least-Squares**
*Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS 2016)*, 51, pages: 676-684, JMLR Workshop and Conference Proceedings, (Editors: Gretton, A. and Robert, C. C. ), 2016 (conference)

**Probabilistic Interpretation of Linear Solvers**
*SIAM Journal on Optimization*, 25(1):234-260, 2015 (article)