Finding the Subspace Mean or Median to Fit Your Need

Many computer vision algorithms employ subspace models to represent data. Many of these approaches benefit from the ability to create an average or prototype for a set of subspaces. The most popular method in these situations is the Karcher mean, also known as the Riemannian center of mass. The prevalence of the Karcher mean may lead some to assume that it provides the best average in all scenarios. However, other subspace averages that appear less frequently in the literature may be more appropriate for certain tasks. The extrinsic manifold mean, the L2-median, and the flag mean are alternative averages that can be substituted directly for the Karcher mean in many applications. This paper evaluates the characteristics and performance of these four averages on synthetic and real-world data. While the Karcher mean generalizes the Euclidean mean to the Grassman manifold, we show that the extrinsic manifold mean, the L2-median, and the flag mean behave more like medians and are therefore more robust to the presence of outliers among the subspaces being averaged. We also show that while the Karcher mean and L2-median are computed using iterative algorithms, the extrinsic manifold mean and flag mean can be found analytically and are thus orders of magnitude faster in practice. Finally, we show that the flag mean is a generalization of the extrinsic manifold mean that permits subspaces with different numbers of dimensions to be averaged. The result is a "cookbook" that maps algorithm constraints and data properties to the most appropriate subspace mean for a given application.