Active Subspaces in Infinite Dimension
This work provides a theoretical extension for dimension reduction in functional data analysis, but it is incremental as it builds on existing active subspace methods.
The authors extended active subspace analysis, a supervised dimension reduction technique, to infinite-dimensional Hilbert spaces, showing that key properties hold and applying it to improve modeling and optimization on test problems.
Active subspace analysis uses the leading eigenspace of the gradient's second moment to conduct supervised dimension reduction. In this article, we extend this methodology to real-valued functionals on Hilbert space. We define an operator which coincides with the active subspace matrix when applied to a Euclidean space. We show that many of the desirable properties of Active Subspace analysis extend directly to the infinite dimensional setting. We also propose a Monte Carlo procedure and discuss its convergence properties. Finally, we deploy this methodology to create visualizations and improve modeling and optimization on complex test problems.