Tangent-Bundle Maps on the Grassmann Manifold: Application to Empirical Arithmetic Averaging

The present paper elaborates on tangent-bundle maps on the Grassmann manifold, with application to subspace arithmetic averaging. In particular, the present contribution elaborates on the work about retraction/lifting maps devised for the Stiefel manifold in the recently published paper T. Kaneko, S. Fiori and T. Tanaka, “Empirical arithmetic averaging over the compact Stiefel manifold,” IEEE Trans. Signal Process., Vol. 61, No. 4, pp. 883-894, February 2013, and discusses the extension of such maps to the Grassmann manifold. Tangent-bundle maps are devised on the basis of the thin QR matrix decomposition, the polar matrix decomposition and the exponential map. Also, tangent-bundle pseudo-maps based on the matrix Cayley transform are devised. Theoretical and numerical comparisons about the devised tangent-bundle maps are performed in order to get an insight into their relative merits and demerits, with special emphasis to their computational burden. The averaging algorithm based on the thin-QR decomposition maps stands out as it exhibits the best trade off between numerical precision and computational burden. Such algorithm is further compared with two Grassmann averaging algorithms drawn from the scientific literature on an handwritten digits recognition data set. The thin-QR tangent-bundle maps-based algorithm exhibits again numerical features that make it preferable over such algorithms.