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Recursive Frechet Mean Computation on the Grassmannian and its Applications to Computer Vision

In the past decade, Grassmann manifolds (Grassmannian) have been commonly used in mathematical formulations of many Computer Vision tasks. Averaging points on a Grassmann manifold is a very common operation in many applications including but not limited to, tracking, action recognition, video-face recognition, face recognition, etc. Computing the intrinsic/Frechet mean (FM) of a set of points on the Grassmann can be cast as finding the global optimum (if it exists) of the sum of squared geodesic distances cost function. A common approach to solve this problem involves the use of the gradient descent method. An alternative way to compute the FM is to develop a recursive/inductive definition that does not involve optimizing the aforementioned cost function. In this paper, we propose one such computationally efficient algorithm called the it Grassmann inductive Frechet mean estimator (GiFME). In developing the recursive solution to find the FM of the given set of points, GiFME exploits the fact that there is a closed form solution to find the FM of two points on the Grassmann. In the limit as the number of samples tends to infinity, we prove that GiFME converges to the FM (this is called the weak consistency result on the Grassmann manifold). Further, for the finite sample case, in the limit as the number of sample paths (trials) goes to infinity, we show that GiFME converges to the finite sample FM. Moreover, we present a bound on the geodesic distance between the estimate from GiFME and the true FM. We present several experiments on synthetic and real data sets to demonstrate the performance of GiFME in comparison to the gradient descent based (batch mode) technique. Our goal in these applications is to demonstrate the computational advantage and achieve comparable accuracy to the state-of-the-art.

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