Lq Averaging for Symmetric Positive-Definite Matrices

Abstract

We propose a method to find the Lq mean of a set of symmetric positive-definite (SPD) matrices, for 1 ≤ q ≤ 2. Given a set of points, the Lq mean is defined as a point for which the sum of q-th power of distances to all the given points is minimum. The Lq mean, for some value of q, has an advantage of being more robust to outliers than the standard L2 mean. The proposed method uses a Weiszfeld inspired gradient descent approach to compute the update in the descent direction. Thus, the method is very simple to understand and easy to code because it does not required line search or other complex strategy to compute the update direction. We endow a Riemannian structure on the space of SPD matrices, in particular we are interested in the Riemannian structure induced by the Log-Euclidean metric. We give a proof of convergence of the proposed algorithm to the Lq mean, under the Log-Euclidean metric. Although no such proof exists for the affine invariant metric but our experimental results show that the proposed algorithm under the affine invariant metric converges to the Lq mean. Furthermore, our experimental results on synthetic data confirms the fact that the L1 mean is more robust to outliers than the standard L2 mean.