Exploring quasi-geodesics on Stiefel manifolds in order to smooth interpolate between domains

Abstract

Manifold-based algorithms are receiving increasing attention in computer vision and pattern recognition. Geodesic curves in the Grasmann manifold have proven to be very useful in modeling domain shift between a source and target domain, represented as subspaces. To obtain an invariant domain representation, the data is projected into a set of subspaces along the geodesic. In contrast to previous works that mainly explore intermediate subspaces along geodesics, in this paper we propose an alternative approach to address multiple source domain adaptation, by taking advantage of smooth interpolating curves on the Stiefel manifold to walk along a set of multiple domains. This aspect is particularly interesting in temporally or dynamically evolving events that are represented by discrete subsets of the data. To generate such curves, we apply a recent technique based on successive quasi-geodesic interpolation on the Stiefel manifold, that results from a modification of the Casteljau algorithm. To evaluate the usefulness of these smooth interpolating curves in pattern recognition problems, several experiments were conducted. We show the advantage of using such curves in multi-source unsupervised domain adaptation problems and in object recognition problems across dynamically evolving datasets.