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Subspace Averaging and its Applications

In: Coherence

Author

Listed:
  • David Ramírez

    (Universidad Carlos III de Madrid)

  • Ignacio Santamaría

    (Universidad de Cantabria)

  • Louis Scharf

    (Colorado State University)

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.

Suggested Citation

  • David Ramírez & Ignacio Santamaría & Louis Scharf, 2022. "Subspace Averaging and its Applications," Springer Books, in: Coherence, chapter 9, pages 259-296, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-13331-2_9
    DOI: 10.1007/978-3-031-13331-2_9
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