Abstract
We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate ell ^p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different ell ^p-norms of subsets of entries.
Highlights
Let A be an m × n matrix and consider the matrix norm A β →α = max x =0Ax α, xβ where · α and · β are vector norms
In this work we consider the case of a matrix norm defined in terms of arbitrary vector norms · α and · β and we prove Theorem 4 below, which is a new version of Theorem 1, holding for general vector norms, provided that suitable and mild differentiability and monotonicity conditions are satisfied
On top of being a classical problem in numerical analysis, computing the norm of a matrix A β→α is a problem that appears in many recent applications in data mining and optimiza
Summary
Ax α , xβ where · α and · β are vector norms. Computing A β→α is a classical problem in computational mathematics, as norms of this kind arise naturally in many situations, such as approximation theory, estimation of matrix condition numbers and approximation of relative residuals [26]. For example, the computation of A q→p is well known to be NP-hard for p > q, we show that for a non-trivial class of nonnegative matrices the power method converges to A q→p in polynomial time even for p sensibly larger than q To our knowledge this is the first global optimality result for this problem that does not require the condition p ≤ q. Further main result of this work addresses this issue for the particular case of norms of the type (1) For this family of norms we provide an explicit convergence bound and an explicit formula for the power iterator for the computation of the corresponding matrix norm A β→α.
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