Chordal Averaging on Flag Manifolds and Its Applications
This addresses a mathematical challenge in computer vision for tasks involving nested subspace data, offering a general solution applicable to various matrix spaces.
The paper tackles the problem of computing flag-mean and flag-median on flag manifolds under the chordal metric, presenting a provably-convergent algorithm that transforms the problem into Stiefel-manifold optimization, with applications demonstrated in Grassmann and rotation averaging and principal component analysis.
This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.