Both sparse coding and rank minimization have led to great successes in various image processing tasks. Though the underlying principles of these two approaches are similar, no theory is available to demonstrate the correspondence. In this paper, starting by designing an adaptive dictionary for each group of image patches, we analyze the sparsity of image patches in each group using the rank mi...

A countable discrete group is said to be Frobenius stable if every function from the unitary matrices that “almost multiplicative” in norm “close” a representation norm. The purpose of this paper show finitely generated nilpotent groups are not virtually cyclic stable. Our argument proves same result for other unnormalized Schatten p-norms with 1<p≤∞.

Given an n×d matrix A, its Schatten-p norm, p ≥ 1, is defined as ‖A‖p = (∑rank(A) i=1 σi(A) p )1/p , where σi(A) is the i-th largest singular value of A. These norms have been studied in functional analysis in the context of non-commutative `p-spaces, and recently in data stream and linear sketching models of computation. Basic questions on the relations between these norms, such as their embed...

We apply the complex interpolation method to prove that, given two spacesB p0,a;s0 , B (n) p1,b;s1 of n-tuples of operators in the p-Schatten class of a Hilbert space H , endowed with weighted norms associated to positive and invertible operators a and b of B(H) then, the curve of interpolation (B p0,a;s0 , B (n) p1,b;s1 )[t] of the pair is given by the space of n-tuples of operators in the pt-...

Let ‖ · ‖ be a unitary similarity invariant norm on the set Mn of n× n complex matrices. A description is obtained for surjective maps φ on Mn satisfying ‖AB− BA‖ = ‖φ(A)φ(B)−φ(B)φ(A)‖ for all A,B ∈ Mn . The general theorem covers the special cases when the norm is one of the Schatten p -norms, the Ky-Fan k -norms, or the k -numerical radii. Mathematics subject classification (2000): Primary 15...

In this paper, we investigate the known operator inequalities for p-Schatten norm and obtain some refinements of these when parameters taking values in different regions. Let A1,...,An, B1,..., Bn ? Bp(H) such that ?ni,j=1Ai*Bj=0. Then p 2, 21/p-?/4n3/p-?/4-1/2(?ni=1 ||Ai||4/?p + ?ni=1 ||Bi||4/?p )?/4 n2/p-1/2||?ni=1 |Ai|2 |Bi|2||1/2 p/2 n2/p-2/?(?ni,j=1 ||Ai Bj||?p)1/?. For 0 &lt; &gt; are rev...