Solving, Tracking and Stopping Streaming Linear Inverse Problems

TL;DR

This paper develops practical residual estimators for streaming linear inverse problem solvers, enabling accurate stopping criteria and improved efficiency in large-scale applications with noisy residual estimates.

Contribution

It introduces a family of residual estimators with uncertainty quantification, enhancing the practical effectiveness of streaming solvers for large-scale inverse problems.

Findings

Residual estimators are accurate and computationally efficient.

Enhanced stopping criteria improve solver reliability.

Methods demonstrated on large-scale linear problems.

Abstract

In large-scale applications including medical imaging, collocation differential equation solvers, and estimation with differential privacy, the underlying linear inverse problem can be reformulated as a streaming problem. In theory, the streaming problem can be effectively solved using memory-efficient, exponentially-converging streaming solvers. In practice, a streaming solver's effectiveness is undermined if it is stopped before, or well-after, the desired accuracy is achieved. In special cases when the underlying linear inverse problem is finite-dimensional, streaming solvers can periodically evaluate the residual norm at a substantial computational cost. When the underlying system is infinite dimensional, streaming solver can only access noisy estimates of the residual. While such noisy estimates are computationally efficient, they are useful only when their accuracy is known. In this work, we rigorously develop a general family of computationally-practical residual estimators and their uncertainty sets for streaming solvers, and we demonstrate the accuracy of our methods on a number of large-scale linear problems. Thus, we further enable the practical use of streaming solvers for important classes of linear inverse problems.