Abstract

The singular values of a matrix AA of size m×nm\times n can be seen as the critical values of the bilinear form ⟨u,Av⟩\langle u,Av\rangle with uu and vv ranging over the unit spheres of Rm\mathbb{R}^m and Rn\mathbb{R}^n, respectively. If uu and vv are further restricted by closed convex cones PP and QQ, respectively, then the criticality conditions are: P∋u⊥(Av−σu)∈P∗P\ni u \perp (Av-\sigma u)\in P^\ast, Q∋v⊥(A⊤u−σv)∈Q∗Q\ni v \perp (A^\top u -\sigma v)\in Q^\ast. This is a coupled system of complementarity problems involving a pair of cones and their dual cones. The parameter σ\sigma is called a singular value of AA relative to (P,Q)(P,Q). The purpose of our work is to study this new concept of singular value. The analysis of such a coupled system is motivated by a number of applications. By way of illustration, we consider a nonnegative Principal Component Analysis problem.

Author information

Contact details are reproduced from the original publication and may be historical.

David Sossa

Instituto de Ciencias de la Ingeniería, Universidad de O'Higgins, Rancagua, Chile

david.sossa@uoh.cl

Keywords

  • Convex cone
  • cone-constrained singular value
  • cone-constrained eigenvalue
  • complementarity problem
  • principal component analysis

Mathematics Subject Classification

15A18, 52A40, 90C26, 90C33

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Suggested citation

A. Seeger, D. Sossa. “Cone-Constrained Singular Value Problems.” Journal of Convex Analysis 30 (2023), No. 4, 1285–1306. https://doi.org/10.68381/jca30057

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Published by Heldermann Verlag, 2023. Rights now held by Banach Press LLC.