Subspace Averaging and its Applications

David Ramírez, Ignacio Santamaría and Louis Scharf
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David Ramírez: Universidad Carlos III de Madrid
Ignacio Santamaría: Universidad de Cantabria
Louis Scharf: Colorado State University

Chapter 9 in Coherence, 2022, pp 259-296 from Springer

Abstract: Abstract All distances between subspaces are functions of the principal angles between them and thus can ultimately be interpreted as measures of coherence between pairs of subspaces. In this chapter, we first review the geometry of the Grassmann and Stiefel manifolds, in which q-dimensional subspaces and q-dimensional frames live, respectively. Then, we assign probability distributions to these manifolds. We pay particular attention to the problem of subspace averaging using the projection (a.k.a. chordal) distance. Using this metric, the average of orthogonal projection matrices turns out to be the central quantity that determines, through its eigendecomposition, both the central subspace and its dimension. The dimension is determined by thresholding the eigenvalues of an average of projection matrices, while the corresponding eigenvectors form a basis for the central subspace. We discuss applications of subspace averaging to subspace clustering and to source enumeration in array processing.

Keywords: Grassmann manifold; Stiefel manifold; Subspace averaging; Average projection matrix; Order estimation; Source enumeration; Subspace clustering (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-13331-2_9

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DOI: 10.1007/978-3-031-13331-2_9

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