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 groups, 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.

翻译：本文提出了一种新的可证明收敛算法，用于在弦度量下计算旗形流形上点集的旗形均值与旗形中位数。旗形流形是由旗构成的数学空间，其中旗是向量空间中维度递增的嵌套子空间序列。该流形是包括施蒂费尔流形和格拉斯曼流形在内的广泛矩阵族的超集，使其成为适用于多种计算机视觉问题的通用数学对象。为攻克一阶旗形统计量计算的难题，我们首先将问题转化为涉及受限于施蒂费尔流形辅助变量的形式。施蒂费尔流形是正交标架空间，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形均值。通过系列实验，我们展示了该方法在格拉斯曼平均、旋转平均以及主成分分析中的优越性能。