Probabilistic Error Analysis for Inner Products

Probabilistic models are proposed for bounding the forward error in the numerically computed inner product (dot product, scalar product) between of two real $n$-vectors. We derive probabilistic perturbation bounds, as well as probabilistic roundoff error bounds for the sequential accumulation of the inner product. These bounds are non-asymptotic, explicit, and make minimal assumptions on perturbations and roundoffs. The perturbations are represented as independent, bounded, zero-mean random variables, and the probabilistic perturbation bound is based on Azuma's inequality. The roundoffs are also represented as bounded, zero-mean random variables. The first probabilistic bound assumes that the roundoffs are independent, while the second one does not. For the latter, we construct a Martingale that mirrors the sequential order of computations. Numerical experiments confirm that our bounds are more informative, often by several orders of magnitude, than traditional deterministic bounds -- even for small vector dimensions~$n$ and very stringent success probabilities. In particular the probabilistic roundoff error bounds are functions of $\sqrt{n}$ rather than~$n$, thus giving a quantitative confirmation of Wilkinson's intuition. The paper concludes with a critical assessment of the probabilistic approach

Probabilistic Linear Solvers: A Unifying View

Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behavior of the conjugate gradient method as a prototypical iterative method. In this work surprisingly general conditions for equivalence of these disparate methods are presented. We also describe connections between probabilistic linear solvers and projection methods for linear systems, providing a probabilistic interpretation of a far more general class of iterative methods. In particular, this provides such an interpretation of the generalised minimum residual method. A probabilistic view of preconditioning is also introduced. These developments unify the literature on probabilistic linear solvers, and provide foundational connections to the literature on iterative solvers for linear systems

Conditioning of Leverage Scores and Computation by QR Decomposition

The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range(A). We show that corresponding leverage scores of two matrices A and A + \Delta A are close in the relative sense, if they have large magnitude and if all principal angles between the column spaces of A and A + \Delta A are small. We also show three classes of bounds that are based on perturbation results of QR decompositions. They demonstrate that relative differences between individual leverage scores strongly depend on the particular type of perturbation \Delta A. The bounds imply that the relative accuracy of an individual leverage score depends on: its magnitude and the two-norm condition of A, if \Delta A is a general perturbation; the two-norm condition number of A, if \Delta A is a perturbation with the same norm-wise row-scaling as A; (to first order) neither condition number nor leverage score magnitude, if \Delta A is a component-wise row-scaled perturbation. Numerical experiments confirm the qualitative and quantitative accuracy of our bounds.Comment: This version has been accepted to SIMAX but has not yet gone through copy editin