Schatten P-norm Research Articles

EXACT LOW-RANK MATRIX RECOVERY VIA NONCONVEX SCHATTEN p-MINIMIZATION

The low-rank matrix recovery (LMR) arises in many fields such as signal and image processing, quantum state tomography, magnetic resonance imaging, system identification and control, and it is generally NP-hard. Recently, Majumdar and Ward [Majumdar, A and RK Ward (2011). An algorithm for sparse MRI reconstruction by Schatten p-norm minimization. Magnetic Resonance Imaging, 29, 408–417]. had successfully applied nonconvex Schatten p-minimization relaxation of LMR in magnetic resonance imaging. In this paper, our main aim is to establish RIP theoretical result for exact LMR via nonconvex Schatten p-minimization. Carefully speaking, letting [Formula: see text] be a linear transformation from ℝm×n into ℝs and r be the rank of recovered matrix X ∈ ℝm×n, and if [Formula: see text] satisfies the RIP condition [Formula: see text] for a given positive integer k ∈ {1, 2, …, m – r}, then r-rank matrix can be exactly recovered. In particular, we obtain a uniform bound on restricted isometry constant [Formula: see text] for any p ∈ (0, 1] for LMR via Schatten p-minimization.

The bounds of restricted isometry constants for low rank matrices recovery

This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 1/√2 or δrA > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ2rA and δrA. Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.

Non-convex algorithm for sparse and low-rank recovery: Application to dynamic MRI reconstruction

In this work we exploit two assumed properties of dynamic MRI in order to reconstruct the images from under-sampled K-space samples. The first property assumes the signal is sparse in the x-f space and the second property assumes the signal is rank-deficient in the x-t space. These assumptions lead to an optimization problem that requires minimizing a combined lp-norm and Schatten-p norm. We propose a novel FOCUSS based approach to solve the optimization problem. Our proposed method is compared with state-of-the-art techniques in dynamic MRI reconstruction. Experimental evaluation carried out on three real datasets shows that for all these datasets, our method yields better reconstruction both in quantitative and qualitative evaluation.

FOCUSS Based Schatten-p Norm Minimization for Real-Time Reconstruction of Dynamic Contrast Enhanced MRI

This work addresses the problem of real-time reconstruction of Dynamic Contrast Enhanced Magnetic Resonance Images (DCE MRI) from partially sampled K-space. The difference image between the current and previous frame is modeled as a rank deficient matrix. It is reconstructed from via nonconvex Schatten-p norm minimization. We develop a FOCUSS (FOCally Under-determined System Solver) based algorithm for solving the said minimization problem. The proposed technique has been compared with state-of-the-art online and offline reconstruction techniques. It yields slightly worse reconstruction results than the offline technique but considerably superior results, both in terms of accuracy and speed compared to the online technique.

Schatten p-norm inequalities related to an extended operator parallelogram law

Let C p be the Schatten p-class for p > 0 . Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A = { A 1 , A 2 , … , A n } and B = { B 1 , B 2 , … , B n } are two sets of operators in, then C 2 ∑ i , j = 1 n ‖ A i - A j ‖ 2 2 + ∑ i , j = 1 n ‖ B i - B j ‖ 2 2 = 2 ∑ i , j = 1 n ‖ A i - B j ‖ 2 2 - 2 ∑ i = 1 n ( A i - B i ) 2 2 . In this paper, we give generalizations of this as pairs of inequalities for Schatten p-norms, which hold for certain values of p and reduce to the equality above for p = 2 . Moreover, we present some related inequalities for three sets of operators.

An algorithm for sparse MRI reconstruction by Schatten p-norm minimization

In recent years, there has been a concerted effort to reduce the MR scan time. Signal processing research aims at reducing the scan time by acquiring less K-space data. The image is reconstructed from the subsampled K-space data by employing compressed sensing (CS)-based reconstruction techniques. In this article, we propose an alternative approach to CS-based reconstruction. The proposed approach exploits the rank deficiency of the MR images to reconstruct the image. This requires minimizing the rank of the image matrix subject to data constraints, which is unfortunately a nondeterministic polynomial time (NP) hard problem. Therefore we propose to replace the NP hard rank minimization problem by its nonconvex surrogate — Schatten p-norm minimization. The same approach can be used for denoising MR images as well.Since there is no algorithm to solve the Schatten p-norm minimization problem, we derive an efficient first-order algorithm. Experiments on MR brain scans show that the reconstruction and denoising accuracy from our method is at par with that of CS-based methods. Our proposed method is considerably faster than CS-based methods.

Schatten p-norm inequalities related to a characterization of inner product spaces

Let $A_1, ... A_n$ be operators acting on a separable complex Hilbert space such that $\sum_{i=1}^n A_i=0$. It is shown that if $A_1, ... A_n$ belong to a Schatten $p$-class, for some $p>0$, then 2^{p/2}n^{p-1} \sum_{i=1}^n \|A_i\|^p_p \leq \sum_{i,j=1}^n\|A_i\pm A_j\|^p_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. Moreover, \sum_{i,j=1}^n\|A_i\pm A_j\|^2_p \leq 2n^{2/p} \sum_{i=1}^n \|A_i\|^2_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch.

An operator equality involving a continuous field of operators and its norm inequalities

Let A be a C∗-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (At)t∈T be a continuous field of operators in A such that the function t↦At is norm continuous on T and the function t↦‖At‖ is integrable. Then the following equality including Bouchner integrals holds(1)∫TAt-∫TAsdP2dP=∫T|At|2dP-∫TAtdP2.This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten p-norm inequalities.

Maps leaving functional values of operator products invariant

Let A be a standard operator algebra on a complex Hilbert space H of dimension greater than 2. By invariants of certain functional values of operator products, we characterize some surjective maps on A . Furthermore, several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as, for example, k-numerical radius ( k ⩾ 1 ) , operator norm, Ky Fan k-norm, Schatten p-norm ( 1 ⩽ p < ∞ ) , and so on.

Mappings that preserve pairs of operators with zero triple Jordan product

Let F be a field and n ⩾ 3 . Suppose S 1 , S 2 ⊆ M n ( F ) contain all rank-one idempotents. The structure of surjections ϕ : S 1 → S 2 satisfying ABA = 0 ⇔ ϕ ( A ) ϕ ( B ) ϕ ( A ) = 0 is determined. Similar results are also obtained for (a) subsets of bounded operators acting on a complex or real Banach space X , (b) the space of Hermitian matrices acting on n-dimensional vectors over a skew-field D , (c) subsets of self-adjoint bounded linear operators acting on an infinite dimensional complex Hilbert space. It is then illustrated that the results can be applied to characterize mappings ϕ on matrices or operators such that F ( ABA ) = F ( ϕ ( A ) ϕ ( B ) ϕ ( A ) ) for all A , B for functions F such as the spectral norm, Schatten p-norm, numerical radius and numerical range, etc.

Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result

We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.

The positive minorant property on matrices

We study the positive minorant property for norms on spaces of matrices. A matrix is said to be a majorant of another if all the entries in the first matrix are greater than or equal to the absolute values of the corresponding entries in the second matrix. For a real number p ≥ 0 the Schatten p-norm of the matrix is the l p -norm of its singular values. The space of n × n matrices with the Schatten p-norm is said to have the positive minorant property if the norm of each nonnegative matrix is greater than or equal to the norm of every nonnegative matrix that it majorizes. It is easy to show that this property holds if p is even. We show that the positive minorant property fails when p < 2( n − 1) and p not even, and provide a simple proof to show the property does hold when p ≥ 2(n − 1)[ (n − 1) 2 ] + 2 .

On the Singular Numbers of Certain Volterra Integral Operators

A criterion for a certain class of integral operators to belong to Schatten-von Neumann symmetric normed ideals is given. In particular, when 2 ≤ p < ∞, it is shown that the Schatten p-norm of such an operator can be estimated by constant multiples of an integral expression which is a natural extension of the well-known formula for the Hilbert-Schmidt norm.

Norm of Schur multiplication for Schatten norm

Let M n denote the algebra of all n×n complex matrices and ||.|| p be the Schatten's p-norm on M n . For each A∈m n , a linear operator S A on Min is defined by S A (X) :=AoX for all X∈M n , where o denotes the Schur product and ||S A || p,q is defined as the operator norm from (M n , ||.|| p ) to (M n , ||.|| q ) for p, q≥1

Facial structures of schattenp-Norms

In this note, we study the facial structure of the closed unit ball Bn of a unitarily invariant norm on n-by-n matrices In particular, we characterize all the extreme points of Bn Furthermore, if the unitarily invariant norm is indeed a Schatten p-norm. then all the proper closed faces of Bn will be described explicitly.

On the Singular Values of a Product of Operators

For compact Hilbert space operators A and B, the singular values of $A^ * B$ are shown to be dominated by those of $\frac{1}{2}(AA^* + BB^* )$.

Lipschitz Continuity of Functions of Operators in the Schatten Classes

We show that the operator-valued map A → f(A) is Lipschitz continuous with respect to the Schatten p-norm if 1 < p < ∞ and f ( u ) = a u + b + ∫ − u ( u − s ) μ ( d s ) , where μ is a signed measure of R of compact support. In the particular case in which f(u) = ∣u∣ this result is false for p = 1 and p = ∞, but results only marginally (logarithmically) weaker do hold in finite dimensions.

Eigenvalue inequalities associated with the cartesian decomposition

Let T=A+iB where A B are Hermitian matrices. We obtain several inequalities relating the lp distance between the eigenvalues of A and those of iB with the Schatten p-norm of T. The majorization results which lead to these inequalities are also used to get simple proofs of some known lower and upper bounds for the determinant of T.

Inequalities for the Schatten $p$-norm V

We prove some inequalities for the Schatten p-norm of operators on a Hubert space. It is shown, among other things, that if A, B, and X are operators such that A + B ^ \X and A + B ^ \X* , then || AX + XB \\pp + || AX* + X*B \\pp for l^p<oo, and max(\\AX + XBl \\AX* + X*B\\) ^ Also, for any three operators A,B, and X, \\\A\X-X\B\\\l+HA*\X-X\B*\\\l£\\AX-XB\\l+\\A*X-XB*

Inequalities for the Schatten p-norm II

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p &lt; ∞), if Σ s1(A)p &lt; ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.