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.

翻译：本文提出一种新的、可证明收敛的算法，用于在弦度量下计算旗流形上点集的旗均值与旗中位数。旗流形是由旗（即向量空间中随维度递增的嵌套子空间序列）构成的数学空间，其作为施蒂费尔流形与格拉斯曼流形等多种已知矩阵空间的超集，是一类适用于广泛计算机视觉问题的通用对象。为攻克旗流形一阶统计量计算的挑战，我们首先将原问题转化为涉及施蒂费尔流形约束辅助变量的新问题。施蒂费尔流形是正交框架的空间，通过利用施蒂费尔流形优化的数值稳定性与高效性，我们能够有效计算旗均值。系列实验表明，本方法在格拉斯曼平均、旋转平均以及主成分分析中均表现出色。我们在 https://github.com/nmank/FlagAveraging 开源了相关代码。