caa a22 4500 445331577 CHVBK 20180317142739.0 cr unu---uuuuu 170323e20110601xx s 000 0 eng 10.1007/s10208-011-9086-4 doi (NATIONALLICENCE)springer-10.1007/s10208-011-9086-4 On the Existence of Optimal Unions of Subspaces forDataModelingandClustering [Elektronische Daten] [Akram Aldroubi, Romain Tessera] Given a set of vectors F={f 1, ,f m} in a Hilbert space $\mathcal {H}$, and given a family $\mathcal {C}$ of closed subspaces of $\mathcal {H}$, the subspace clustering problem consists in finding a union of subspaces in $\mathcal {C}$ that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces $\mathcal {C}$ for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set $\mathcal {C}^{+}$. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families $\mathcal {C}$ of unitary representations of Abelian groups is derived. SFoCM, 2011 Subspace clustering nationallicence Hybrid linear modeling nationallicence Data modeling nationallicence Union of subspaces nationallicence Aldroubi Akram Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, 37240, Nashville, TN, USA aut Tessera Romain UMPA, ENS Lyon, 146 Allée d'Italie, 69364, Lyon Cedex, France aut Foundations of Computational Mathematics Springer-Verlag 11/3(2011-06-01), 363-379 1615-3375 11:3<363 2011 11 10208 https://doi.org/10.1007/s10208-011-9086-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10208-011-9086-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Aldroubi Akram Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, 37240, Nashville, TN, USA aut NATIONALLICENCE 700 1- Tessera Romain UMPA, ENS Lyon, 146 Allée d'Italie, 69364, Lyon Cedex, France aut NATIONALLICENCE 773 0- Foundations of Computational Mathematics Springer-Verlag 11/3(2011-06-01), 363-379 1615-3375 11:3<363 2011 11 10208 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer