Convex Set Of Matrices Research Articles

Some points of view on Grothendieck's inequalities

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constantkGC.

Enabling Quasi-Static Reconfigurable Networks With Robust Topology Engineering

Many optical circuit switched data center networks (DCN) have been proposed in the last decade to attain higher capacity and topology reconfigurability, though commercial adoption of these architectures have been minimal. One major challenge these architectures face is the difficulty of handling uncertain traffic demands using commercial optical circuit switches (OCS) with high switching latency. Prior works have generally focused on developing fast-switching OCS prototypes to quickly react to traffic variations through frequent reconfigurations. This approach, however, adds tremendous complexity overhead to the control plane, and raises the barrier for commercial adoption of optical circuit switched data center networks. We propose, a robust topology and routing optimization framework for reconfigurable optical circuit switched data centers. co-optimizes topology and routing based on a convex set of traffic matrices, and offers strict throughput guarantees for any future traffic matrices bounded by the convex set. For the bursty traffic demands that are unbounded by the convex set, we employ a desensitization technique to reduce performance hit. This enables to generate topology and routing solutions capable of handling unexpected traffic changes without relying on frequent topology reconfigurations. Our extensive evaluations based on Facebook’s production DCN traces show that, even with daily reconfigurations which could be realized by current commercial MEMS-based OCSs from Calient Technologies, achieves about 20% lower max link utilization, and about 32% lower average hop count compared to cost-equivalent static topologies. Our work shows that adoption of reconfigurable topologies in commercial DCNs is feasible even without fast OCSs.

The Siegel-Klein Disk: Hilbert Geometry of the Siegel Disk Domain.

We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.

The extreme points of centrosymmetric transportation polytopes

Denote by Uπ(R,S) the convex set of nonnegative centrosymmetric matrices with given row sum vector R and column sum vector S, and denote by U≤π(R,S) the convex set of nonnegative centrosymmetric matrices with the row sum vector componentwisely dominated by R and the column sum vector componentwisely dominated by S respectively. We characterize all extreme points of Uπ(R,S) and U≤π(R,S). In addition, we show that the extreme points of Ωnπ, the polytope of all n×n centrosymmetric doubly stochastic matrices, and the extreme points of ωnπ, the polytope of all n×n centrosymmetric doubly substochastic matrices, can be obtained by letting R=S=(1,1,…,1) in Uπ(R,S) and U≤π(R,S) respectively.

Centrosymmetric stochastic matrices

We consider the convex set of stochastic matrices and the convex set of centrosymmetric stochastic matrices (stochastic matrices that are symmetric under rotation by 180). For , we demonstrate a Birkhoff theorem for its extreme points and create a basis from certain -matrices. For , we characterize its extreme points and create bases, whose construction depends on the parity of m, using our basis construction for stochastic matrices. For each of and , we further characterize their extreme points in terms of their associated bipartite graphs, we discuss a graph parameter called the fill and compute it for the various basis elements, and we examine the number of vertices of the faces of these sets. We provide examples illustrating the results throughout.

Quasi-Newton algorithm for optimal approximate linear regression design: Optimization in matrix space

Given a linear regression model and an experimental region for the independent variable, the problem of finding an optimal approximate design calls for minimizing a convex optimality criterion over a convex set of information matrices of feasible approximate designs. For numerical solution pure gradient methods are often used by design theorists, as vertex direction, vertex exchange, multiplicative algorithms, or combinations hereof. These methods have two major deficiencies: a slow convergence rate after a quick but rough approximation to the optimum, and often a large support of the obtained nearly optimal design. The latter feature is related to the fact that the methods optimize in the space of design measures which is usually of high or even infinite dimension, whereas the dimension of the information matrices is often small or moderate. For such situations a quasi-Newton method is revisited which was originally established by Gaffke & Heiligers (1996). In the present paper new possibilities of its application are demonstrated. The algorithm optimizes in matrix space. It shows a good global and an excellent local convergence behavior resulting in an accurate approximation of the optimum. A crucial subroutine solves convex quadratic minimization over the set of information matrices via repeated linear minimization over that set, providing thus the quasi-Newton step of the algorithm. This may also be of interest as a tool for computing an approximate design from a given information matrix and such that the support size of the design keeps Carathéodory’s bound. Illustrations are given for D- and I-optimality in particular multivariate random coefficient regression models and for T-optimal discriminating design in univariate polynomial models. Moreover, the behavior of the algorithm is tested for cases of larger dimensions: D- and I-optimal design for a third order polynomial model in several variables on a discretized cube.

Some geometric interpretations of quantum fidelity

We consider quantum fidelity between two states ρ and σ, where we fix ρ and allow σ to be sent through a quantum channel. We determine the minimal fidelity where one minimizes over (a) all unital channels, (b) all mixed unitary channels, and (c) arbitrary channels. We derive results involving the minimal eigenvalue of ρ, which we can interpret as a convex combination coefficient. As a consequence, we give a new geometric interpretation of the minimal fidelity with respect to the closed, convex set of density matrices and with respect to the closed, convex set of quantum channels. We further investigate the geometric nature of fidelity by considering density matrices arising as normalized projections onto subspaces; in this way, fidelity can be viewed as a geometric measure of distance between two spaces. We give a connection between fidelity and the canonical (principal) angles between the subspaces.

Geometric and asymptotic properties associated with linear switched systems

Consider a continuous-time linear switched system on Rn associated with a compact convex set of matrices. When it is irreducible and its largest Lyapunov exponent is zero there always exists a Barabanov norm associated with the system. This paper deals with two types of issues: (a) properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; (b) asymptotic behavior of the extremal solutions of the linear switched system. Regarding Issue (a), we provide partial answers and propose four related open problems. As for Issue (b), we establish, when n=3, a Poincaré–Bendixson theorem under a regularity assumption on the set of matrices. We then revisit a noteworthy result of N.E. Barabanov describing the asymptotic behavior of linear switched system on R3 associated with a pair of Hurwitz matrices {A,A+bcT}. After pointing out a gap in Barabanov's proof we partially recover his result by alternative arguments.

Neural network segmented CD algorithm-based PET liver image reconstruction

In this paper, reconstruction of the Positron Emission Tomography (PET) images, a CD algorithm was instigated with NN based image segmentation techniques called Neural Network Segmentation based Coordinate Descent–Weighted Least Square (NNCD–WLS). Thus, NNCD–WLS of the function is not quadratic, but natural. The iterative algorithm achieve a fashion equivalent to an analytic derivation of the Maximum Likelihood–Expectation Maximisation (ML–EM) algorithm, which gives a different minimisation process between two convex sets of matrices. Conversely the distance metric is quite distinct, and more intricate to analyse. This algorithm is similar type, shares many properties acquainted with the ML–EM algorithm. Unlike WLS algorithm, NNCD–WLS method minimises the WLS objective function. The NNCD–WLS algorithm instigates via NN based segmentation process in image reconstruction. Image quality parameter of the PSNR value, NNCD–WLS algorithm and the denoising algorithm is compared. The PET input image is reconstructed and simulated in the MATLAB/Simulink package.

Volumes of conditioned bipartite state spaces

We analyze the metric properties of conditioned quantum state spaces These spaces are the convex sets of density matrices that, when partially traced over m degrees of freedom, respectively yield the given n × n density matrix η. For the case n = 2, the volume of equipped with the Hilbert–Schmidt measure can be conjectured to be a simple polynomial of the radius of η in the Bloch-ball. Remarkably, for we find numerically that the probability to find a separable state in is independent of η (except for η pure). For , the same holds for , the probability to find a state with a positive partial transpose in . These results are proven analytically for the case of the family of 4 × 4 X-states, and thoroughly numerically investigated for the general case. The important implications of these findings for the clarification of open problems in quantum theory are pointed out and discussed.

Coalgebraic Quantum Computation

Coalgebras generalize various kinds of dynamical systems occuring in mathematics and computer science. Examples of systems that can be modeled as coalgebras include automata and Markov chains. We will present a coalgebraic representation of systems occuring in the field of quantum computation, using convex sets of density matrices as state spaces. This will allow us to derive a method to convert quantum mechanical systems into simpler probabilistic systems with the same probabilistic behaviour.

Controlling singular values with semidefinite programming

Controlling the singular values of n -dimensional matrices is often required in geometric algorithms in graphics and engineering. This paper introduces a convex framework for problems that involve singular values. Specifically, it enables the optimization of functionals and constraints expressed in terms of the extremal singular values of matrices. Towards this end, we introduce a family of convex sets of matrices whose singular values are bounded. These sets are formulated using Linear Matrix Inequalities (LMI), allowing optimization with standard convex Semidefinite Programming (SDP) solvers. We further show that these sets are optimal, in the sense that there exist no larger convex sets that bound singular values. A number of geometry processing problems are naturally described in terms of singular values. We employ the proposed framework to optimize and improve upon standard approaches. We experiment with this new framework in several applications: volumetric mesh deformations, extremal quasi-conformal mappings in three dimensions, non-rigid shape registration and averaging of rotations. We show that in all applications the proposed approach leads to algorithms that compare favorably to state-of-art algorithms.

Results and open problems in matchings in regular graphs

This survey paper deals with upper and lower bounds on the number of k-matchings in regular graphs on N vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its variants. We then discuss non-bipartite graphs. Little is known beyond the recent proof of the Lovasz-Plummer conjecture on the exponential growth of perfect matchings in cubic bridgeless graphs. We discuss the problem of the minimum of haffnians on the convex set of matrices, whose extreme points are the adjacency matrices of subgraphs of the complete graph corresponding to perfect matchings. We also consider infinite regular graphs. The analog of k-matching is the p-monomer entropy, where p ∈ [0, 1] is the density of the number of matchings. AMS subject classifications. 05C30, 05C70, 15A15.

Solving the Matrix Nearness Problem in the Maximum Norm by Applying a Projection and Contraction Method

Let <i >S</i> be a closed convex set of matrices and <i >C</i> be a given matrix. The matrix nearness problem considered in this paper is to find a matrix <i >X</i> in the set <i >S</i> at which max <svg style="vertical-align:-4.77652pt;width:71.987503px;" id="M1" height="16.700001" version="1.1" viewBox="0 0 71.987503 16.700001" width="71.987503" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.7)"> <g transform="translate(72,-58.64)"> <text transform="matrix(1,0,0,-1,-71.95,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0">{</tspan> <tspan style="font-size: 12.50px; " x="6.00144" y="0">|</tspan> <tspan style="font-size: 12.50px; " x="8.5020399" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-56.57,60.32)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-46.64,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="11.327718" y="0">𝑐</tspan> </text> <text transform="matrix(1,0,0,-1,-30.11,60.32)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-22.96,63.45)"> <tspan style="font-size: 12.50px; " x="0" y="0">|</tspan> <tspan style="font-size: 12.50px; " x="2.5006001" y="0">}</tspan> </text> </g> </g> </svg> reaches its minimum value. In order to solve the matrix nearness problem, the problem is reformulated to a min-max problem firstly, then the relationship between the min-max problem and a monotone linear variational inequality (LVI) is built. Since the matrix in the LVI problem has a special structure, a projection and contraction method is suggested to solve this LVI problem. Moreover, some implementing details of the method are presented in this paper. Finally, preliminary numerical results are reported, which show that this simple algorithm is promising for this matrix nearness problem.

Homogenization of nonconvex integrals with convex growth

We study homogenization by Γ-convergence of periodic multiple integrals of the calculus of variations when the integrand can take infinite values outside a convex set of matrices.

Convex solution of a permutation problem

In this paper, we show that a problem of finding a permuted version of k vectors from RN such that they belong to a prescribed rank r subset, can be solved by convex optimization. We prove that under certain generic conditions, the wanted permutation matrix is unique in the convex set of doubly-stochastic matrices. In particular, this implies a solution of the classical correspondence problem of finding a permutation that transforms one collection of points in Rk into the another one. Solutions to these problems have a wide set of applications in Engineering and Computer Science.

Trace-Inequalities and Matrix-Convex Functions

A real-valued continuous function on an interval gives rise to a map via functional calculus from the convex set of Hermitian matrices all of whose eigenvalues belong to the interval. Since the subpace of Hermitian matrices is provided with the order structure induced by the cone of positive semidefinite matrices, one can consider convexity of this map. We will characterize its convexity by the following trace-inequalities: for . A related topic will be also discussed.

Tangles of superpositions and the convex-roof extension

We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, $\mathrm{SL}(2,\mathbb{C})$ invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the zero polytope is a convex set of density matrices with vanishing tangle whereas the convex characteristic curve readily provides a nontrivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.

A note on infinite extreme correlation matrices

We give a characterization for the extreme points of the convex set of correlation matrices with a countable index set. A Hermitian matrix is called a correlation matrix if it is positive semidefinite with unit diagonal entries. Using the characterization we show that there exist extreme points of any rank.

Extreme points of the set of density matrices with positive partial transpose

We present a necessary and sufficient condition for a finite-dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.