Convexity properties of the condition number
Abstract
We define in the space of n by m matrices of rank n, n less or equal than m, the condition Riemannian structure as follows: For a given matrix A the tangent space of A is equipped with the Hermitian inner product obtained by multiplying the usual Frobenius inner product by the inverse of the square of the smallest singular value of A denoted sigman(A). When this smallest singular value has multiplicity 1, the function A -> log (sigman(A)(-2)) is a convex function with respect to the condition Riemannian structure that is t -> log (sigman(A(t))(-2)) is convex, in the usual sense for any geodesic A(t). In a more abstract setting, a function alpha defined on a Riemannian manifold (M,<,>) is said to be self-convex when log alpha (gamma(t)) is convex for any geodesic in (M,<,>). Necessary and sufficient conditions for self-convexity are given when alpha is C2. When alpha(x) = d(x,N)(-2) where d(x,N) is the distance from x to a C2 submanifold N of Rj we prove that alpha is self-convex when restricted to the largest open set of points x where there is a unique closest point in N to x. We also show, using this more general notion, that the square of the condition number ||A|||F / sigman(A) is self-convex in projective space and the solution variety.