We present numerical methods for computing the Schatten p-norm of positive semi-definite matrices. Our motivation stems from uncertainty quantification and optimal experimental design inverse problems, where defines a measure uncertainty. Computing high-dimensional matrices is computationally expensive. propose matrix-free method to estimate using Monte Carlo estimator derive convergence result...

Abstract. Generalizing Pisier’s idea, we introduce a Hilbertian matrix cross normed space associated with a pair of symmetric normed ideals. When the two ideals coincide, we show that our construction gives an operator space if and only if the ideal is the Schatten class. In general, a pair of symmetric normed ideals that are not necessarily the Schatten class may give rise to an operator space...

This paper presents a patch-wise low-rank based image denoising method with constrained variational model involving local and nonlocal regularization. On one hand, recent patch-wise methods can be represented as a low-rank matrix approximation problem whose convex relaxation usually depends on nuclear norm minimization (NNM). Here, we extend the NNM to the nonconvex schatten p-norm minimization...

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation EF and the conjugate function E∗ of the entanglement function E(ρ) = S(TrAρ). We then consider the conjectured strong superadditivity of the entanglement of formation EF (ρ) ≥ EF (ρI ) + EF (ρII ), where ρI and ρII are the reductions of ρ to the different Hilbert space copies, and pr...

Let H(S) be the Hardy space on the unit sphere S in C, n ≥ 2. Consider the Hankel operator Hf = (1 − P )Mf |H(S), where the symbol function f is allowed to be arbitrary in L(S, dσ). We show that for p > 2n, Hf is in the Schatten class Cp if and only if f − Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if ...