Schatten P-norms Research Articles

On the geometry of the Birkhoff polytope II: the Schatten p-norms

On the geometry of the Birkhoff polytope II: the Schatten p-norms

Ulam stability of lamplighters and Thompson groups

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten p-norms. These include lamplighters Γ≀Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\wr \\Lambda $$\\end{document} where Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups F,F′,T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F, F', T$$\\end{document} and V. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.

Frobenius non-stability of nilpotent groups

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

Some generalizations of numerical radii and Schatten p-norms inequalities

Some generalizations of numerical radii and Schatten p-norms inequalities

A coarse-to-fine segmentation frame for polyp segmentation via deep and classification features

Accurate polyp segmentation is of great significance for the diagnosis and treatment of colon cancer. Deep convolution network can extract the common high level features of the target. However, most network models ignore some individual features, so the sample prediction results in complex space are fuzzy and lack of regularity. In this paper, a coarse-to-fine segmentation frame for polyp segmentation via deep and classification features is proposed. Firstly, batch schatten-p norms maximization is introduced into a network model to strengthen the predict map. Then, an automatic two classification mechanism is constructed and the prediction map is classified into two categories: simple and complex samples. Since the CNN prediction maps of simple samples are close to binary images, the prediction maps are not processed. Finally, an active contour model segmentation algorithm for saliency detection of complex samples is proposed. Experiments on Kvasir-SEG, CVC-300, CVC-ClinincDB, CVC-ColonDB and ETIS-LaribPolypDB datasets using multiple models verify the effectiveness of the framework. Code is available at https://doi.org/10.24433/CO.7821162.v1.

Smoothness of Schatten norms and sliding-window matrix streams

Large matrices are often accessed as a row-order stream. We consider the setting where rows are time-sensitive (i.e. they expire), which can be described by the sliding-window row-order model, and provide the first (1+ϵ)-approximation of Schatten p-norms in this setting. Our main technical contribution is a proof that Schatten p-norms in row-order streams are smooth, and thus fit the smooth-histograms technique of Braverman and Ostrovsky (FOCS 2007) for sliding-window streams.

Approximate tensor decompositions: Disappearance of many separations

It is well known that tensor decompositions show separations, that is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here, we show that many of these separations disappear in the approximate case. Specifically, for every approximation error ɛ and norm, we define the approximate rank as the minimum rank of an element in the ɛ-ball with respect to that norm. For positive semidefinite matrices, we show that the separations between rank, purification rank, and separable rank disappear for a large class of Schatten p-norms. For non-negative tensors, we show that the separations between rank, positive semidefinite rank, and non-negative rank disappear for all ℓp-norms with p &amp;gt; 1. For the trace norm (p = 1), we obtain upper bounds that depend on the ambient dimension. We also provide a deterministic algorithm to obtain the approximate decomposition attaining our bounds. Our main tool is an approximate version of the Carathéodory theorem. Our results imply that many separations are not robust under small perturbations of the tensor, with implications in quantum many-body systems and communication complexity.

Compiling basic linear algebra subroutines for quantum computers

Efficiently processing basic linear algebra subroutines is of great importance for a wide range of computational problems. In this paper, we consider techniques to implement matrix functions on a quantum computer. We embed given matrices into 3 times larger Hermitian matrices and assume as input a given set of unitary operators generated by the embedding matrices. With the matrix embedding formula, we give Trotter-based quantum subroutines for elementary matrix operations include addition, multiplication, Kronecker sum, tensor product, Hadamard product, and arbitrary real eigenvalue single-matrix functions. We then discuss the composed matrix functions in terms of the estimation of scalar quantities such as inner products, traces, determinants, and Schatten p-norms with bounded errors. We thus provide a framework for compiling instructions for linear algebraic computations into gate sequences on actual quantum computers. The framework for calculating the matrix functions is more efficient than the best classical counterpart for a set of matrices.

Multiplicative perturbation bounds for multivariate multiple linear regression in Schatten p-norms

Multivariate multiple linear regression (MMLR), which occurs in a number of practical applications, generalizes traditional least squares (multivariate linear regression) to multiple right-hand sides. We extend recent MLR analyses to sketched MMLR in general Schatten p-norms by interpreting the sketched problem as a multiplicative perturbation. Our work represents an extension of Maher's results on Schatten p-norms. We derive expressions for the exact and perturbed solutions in terms of projectors for easy geometric interpretation. We also present a geometric interpretation of the action of the sketching matrix in terms of relevant subspaces. We show that a key term in assessing the accuracy of the sketched MMLR solution can be viewed as a tangent of a largest principal angle between subspaces under some assumptions. Our results enable additional interpretation of the difference between an orthogonal and oblique projector with the same range.

The Operator-Valued Parallelism and Norm-Parallelism in Matrices

Let ℌ be a Hilbert space, and let K(ℌ) be the C*-algebra of compact operators on ℌ. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert K(ℌ)-module by employing the Birkhoff-James orthogonality. Among other things, we present a characterization of transitive relation of the norm-parallelism for elements in a certain Hilbert K(ℌ)-module. We also give some characterizations of the Schatten p-norms and the operator norm-parallelism for matrices.

A Unified Scalable Equivalent Formulation for Schatten Quasi-Norms

The Schatten quasi-norm is an approximation of the rank, which is tighter than the nuclear norm. However, most Schatten quasi-norm minimization (SQNM) algorithms suffer from high computational cost to compute the singular value decomposition (SVD) of large matrices at each iteration. In this paper, we prove that for any p, p1, p2&gt;0 satisfying 1/p=1/p1+1/p2, the Schatten p-(quasi-)norm of any matrix is equivalent to minimizing the product of the Schatten p1-(quasi-)norm and Schatten p2-(quasi-)norm of its two much smaller factor matrices. Then, we present and prove the equivalence between the product and its weighted sum formulations for two cases: p1=p2 and p1≠p2. In particular, when p&gt;1/2, there is an equivalence between the Schatten p-quasi-norm of any matrix and the Schatten 2p-norms of its two factor matrices. We further extend the theoretical results of two factor matrices to the cases of three and more factor matrices, from which we can see that for any 0&lt;p&lt;1, the Schatten p-quasi-norm of any matrix is the minimization of the mean of the Schatten (⌊1/p⌋+1)p-norms of ⌊1/p⌋+1 factor matrices, where ⌊1/p⌋ denotes the largest integer not exceeding 1/p.

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Let Mm,n be the space of m×n real or complex rectangular matrices. Two matrices A,B∈Mm,n are disjoint if A∗B=0n and AB∗=0m. We show that a linear map Φ:Mm,n→Mr,s preserving disjointness exactly when Φ(A)=U⎛⎜⎝A⊗Q1000At⊗Q2000⎞⎟⎠V,∀A∈Mm,n, for some unitary matrices U∈Mr,r and V∈Ms,s, and positive diagonal matrices Q1,Q2, where Q1 or Q2 may be vacuous. The result is used to characterize nonsurjective linear maps between rectangular matrix spaces preserving (zero) JB∗-triple products, the Schatten p-norms or the Ky--Fan k-norms.

On singular values of block circulant matrices

In this paper, singular values of block circulant matrices are investigated. Some results of W. Bani-Domi and F. Kittaneh [3] are extended. Some weak majorization inequalities are established for singular values of matrices transformed by certain linear operators with values in the space of block-circulant matrices. Interpretations are provided for Ky Fan k-norms and Schatten p-norms.

Robust principal component analysis via optimal mean by joint ℓ2,1 and Schatten p-norms minimization

Abstract Since principal component analysis (PCA) is sensitive to corrupted variables or observations that affect its performance and applicability in real scenarios, some convex robust PCA methods have been developed to enhance the robustness of PCA. However, most of them neglect the optimal mean calculation problem. They center the data with the mean calculated by the l2-norm, which is incorrect because the l1-norm objective function is used in the following steps. In this paper, we consider a novel robust PCA method that can pursue and remove outliers, exactly recover a low-rank matrix and calculate the optimal mean. Specifically, we propose an optimization model constituted by a l2,1-norm based loss function and a Schatten p-norm regularization term. The l2,1-norm used in loss function aims to pursue and remove outliers, the Schatten p-norm can suppress the singular values of reconstructed data at smaller p (0

Refinements of Inequalities Related to Landau–Grüss Inequalities for Elementary Operators Acting on Ideals Associated to p-Modified Unitarily Invariant Norms

In this paper we find the explicit form for the equalizing term in the inequality related to Landau–Gruss inequality for elementary operators, which enabled us to refine inequalities related to operator Landau–Gruss norm inequalities for p-modifications of initarily invariant norms. Specially, for Schatten p-norms with \(p\ge 2\) those inequalities have additionally simplified form, which enables, among others, to analyze equality cases in considered operator inequalities. A broader family of mixed norms inequalities related to the operator Landau–Gruss inequality is also presented.

A Determinantal Inequality Involving Partial Traces

AbstractLet A be a density matrix in . Audenaert [J. Math. Phys. 48(2007) 083507] proved an inequality for Schatten p-norms:where Tr1 and Tr2 stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality

Isoperimetric inequalities for Schatten norms of Riesz potentials

In this note we prove that the ball is a maximiser for integer order Schatten p-norms of the Riesz potential operators among all domains of a given measure in Rd. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh–Faber–Krahn and Hong–Krahn–Szegö inequalities.

Norm inequalities involving accretive-dissipative 2 × 2 block matrices

Let T11,T12,T21, and T22 be n×n complex matrices, and let T=(T11T12T21T22) be accretive-dissipative. It is shown that if f is an increasing convex function on [0,∞) such that f(0)=0, then⦀f(|T12|2)+f(|T21⁎|2)⦀≤⦀f(|T|2)⦀ for every unitarily invariant norm ⦀⋅⦀. Moreover, if f is an increasing concave function on [0,∞) such that f(0)=0, then⦀f(|T12|2)+f(|T21⁎|2)⦀≤4⦀f(|T|24)⦀ for every unitarily invariant norm ⦀⋅⦀. Among other inequalities for the Schatten p-norms, it is shown that‖T12‖pp+‖T21‖pp≤2p−1‖T11‖pp/2‖T22‖pp/2 for p≥2.

A note on isoperimetric inequalities for logarithmic potentials

We give an inequality for all Schatten p-norms with p≥2, on a class of bounded domains majorized by a given domain. We partially prove the Ruzhansky–Suragan's conjecture [12], by showing that for any integer p≥3, the square (cube) is a maximizer of Schatten p-norm of the logarithmic (Newtonian) potential among all rectangles (rectangular parallelepipeds) of a given Lebesgue measure.

An arithmetic–geometric mean inequality for products of three matrices

Consider the following noncommutative arithmetic–geometric mean inequality: Given positive-semidefinite matrices A1,…,An, the following holds for each integer m≤n:1nm∑j1,j2,…,jm=1n⦀Aj1Aj2…Ajm⦀≥(n−m)!n!∑j1,j2,…,jm=1all distinctn⦀Aj1Aj2…Ajm⦀, where ⦀⋅⦀ denotes a unitarily invariant norm, including the operator norm and Schatten p-norms as special cases. While this inequality in full generality remains a conjecture, we prove that the inequality holds for products of up to three matrices, m≤3. The proofs for m=1,2 are straightforward; to derive the proof for m=3, we appeal to a variant of the classic Araki–Lieb–Thirring inequality for permutations of matrix products.