Let Sp be the Schatten-von Neumann ideal of compact operators equipped with the norm Np(·). For an A ∈ Sp (1 < p <∞), the inequality [ ∞ ∑ k=1 |Reλk(A)| ] 1 p + bp [ ∞ ∑ k=1 | Imλk(A)| ] 1 p ≥ Np(AR)− bpNp(AI) (bp = const. > 0) is derived, where λj(A) (j = 1, 2, . . . ) are the eigenvalues of A, AI = (A − A∗)/2i and AR = (A + A∗)/2. The suggested approach is based on some relations between the ...

It is well-known that tensor decompositions show separations, is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here we many of these separations disappear the approximate case. Specifically, for every approximation error $\varepsilon$ and norm, define rank minimum element $\varepsilon$-ball with respect to norm. For positive semide...

For a positive integer n, let Mn be the set of n × n complex matrices. Suppose ‖ · ‖ is the Ky Fan k-norm with 1 ≤ k ≤ mn or the Schatten p-norm with 1 ≤ p ≤ ∞ (p 6= 2) on Mmn, where m,n ≥ 2 are positive integers. It is shown that a linear map φ : Mmn →Mmn satisfying ‖A⊗B‖ = ‖φ(A⊗B)‖ for all A ∈Mm and B ∈Mn if and only if there are unitary U, V ∈ Mmn such that φ has the form A ⊗ B 7→ U(φ1(A) ⊗ ...

This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients {〈k,Φm,n〉} of certain frame expansions of the kernel k of an integral operator are in l, then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn-Nirenberg symbol of a pseudodifferential operator lies in a parti...

We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, X states. An expression for the trace distance coherence for all pure states and a semi definite program for arbitrary states is provided. We also explore the relation between l1-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it...

The affine rank minimization problem is to minimize the rank of a matrix under linear constraints. It has many applications in various areas such as statistics, control, system identification and machine learning. Unlike the literatures which use the nuclear norm or the general Schatten q (0 < q < 1) quasi-norm to approximate the rank of a matrix, in this paper we use the Schatten 1/2 quasi-nor...

Abstract Low-rank matrix completion is a hot topic in the field of machine learning. It widely used image processing, recommendation systems and subspace clustering. However, traditional method uses nuclear norm to approximate rank function, which leads only suboptimal solution. Inspired by closed-form formulation $$L_{2/3}$$ ...