Adaptive prediction regions for complex shape data on curved spaces
Adaptive conformal prediction on the Grassmannian
Machine Learning
Summary
Sometimes, data represents complex shapes that live on curved spaces called Grassmann manifolds. The authors developed a method that makes predictions about these shapes while respecting their complex geometry. Their approach adapts to how difficult it is to make a prediction for each case, producing more accurate and reliable prediction zones. They tested this method on simulated data and a tricky airfoil design example, showing it better fits the data’s true shape and uncertainty.
What this means in practice
- •For aerospace engineers: Generate reliable uncertainty estimates when designing airfoil shapes using inverse modeling methods that output shape data on Grassmann manifolds.
- •For computer vision developers: Improve uncertainty quantification in shape regression tasks that involve outputs naturally represented as subspaces or shapes on curved spaces.
Authors
Dimitrios Loukrezis, Zachary J. Grey, Dimitris G. Giovanis
Abstract
We introduce a conformal prediction framework for regression with Grassmann-valued responses. The proposed method constructs prediction regions in the form of metric balls directly on the Grassmann manifold, using basis-invariant geometric distances as non-conformity measures and scaling the corresponding non-conformity scores through a learned prediction difficulty estimator. This yields locally adaptive, input-dependent prediction regions that respect the underlying manifold geometry, while retaining the finite-sample marginal coverage guarantee of split conformal prediction under exchangeability. We evaluate several Grassmannian distance metrics and compare their prediction-region efficiency on synthetic benchmarks and an airfoil inverse shape-design problem. The results highlight the advantages of geometry-aware non-conformity scores, while local adaptation improves empirical coverage uniformity across prediction difficulty levels and, in strongly heteroscedastic settings, can substantially reduce prediction-region size while maintaining the target coverage.