Learning minimal volume uncertainty ellipsoids
Itai Alon, David Arnon, Ami Wiesel
TL;DR
The paper addresses learning minimal-volume uncertainty ellipsoids for parameter estimation under a prescribed coverage level. It shows that, for jointly Gaussian data, the optimal ellipsoid centers at the conditional mean and scales with the conditional covariance, while general distributions require a data-driven approach. To this end, it introduces LMVE, a neural-network framework that outputs ellipsoid parameters, trains via a Lagrangian-like objective, and uses conformal-prediction calibration to guarantee coverage with minimal average volume. The approach blends covariance estimation, nearest-neighbor ideas, and conformal prediction to yield accurate, compact uncertainty regions with low inference cost, demonstrated on four localization datasets and available in open source form. This work advances practical uncertainty quantification for high-stakes estimation problems in data-driven settings.
Abstract
We consider the problem of learning uncertainty regions for parameter estimation problems. The regions are ellipsoids that minimize the average volumes subject to a prescribed coverage probability. As expected, under the assumption of jointly Gaussian data, we prove that the optimal ellipsoid is centered around the conditional mean and shaped as the conditional covariance matrix. In more practical cases, we propose a differentiable optimization approach for approximately computing the optimal ellipsoids using a neural network with proper calibration. Compared to existing methods, our network requires less storage and less computations in inference time, leading to accurate yet smaller ellipsoids. We demonstrate these advantages on four real-world localization datasets.