Cone-Constrained Singular Value Problems

The singular values of a matrix

of size

m\times n m × n

can be seen as the critical values of the bilinear form

\langle u,Av\rangle ⟨ u , A v ⟩

with

u u

and

v v

ranging over the unit spheres of

\mathbb{R}^m R m

and

\mathbb{R}^n R n

, respectively. If

u u

and

v v

are further restricted by closed convex cones

P P

and

Q Q

, respectively, then the criticality conditions are:

P\ni u \perp (Av-\sigma u)\in P^\ast P ∋ u ⊥ ( A v − σ u ) ∈ P ∗

,

Q\ni v \perp (A^\top u -\sigma v)\in Q^\ast Q ∋ v ⊥ ( A ⊤ u − σ v ) ∈ Q ∗

. This is a coupled system of complementarity problems involving a pair of cones and their dual cones. The parameter

\sigma σ

is called a singular value of

A A

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.