Space Of Symmetric Matrices Research Articles

LDPC Codes Based on the Space of Symmetric Matrices Over Finite Fields

In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on $\mathbb{S}_{n}(\mathbb{F}_{q})$, the space of $n\times n$ symmetric matrices over $\mathbb{F}_{q}$. Using this method, we obtain two classes of binary LDPC codes, $\cal{C}(n,q)$ and $\cal{C}^{T}(n,q)$, both of which have grith $8$. Then both the minimum distance and the stopping distance of each class are investigated. It is shown that the minimum distance and the stopping distance of $\cal{C}^{T}(n,q)$ are both $2q$. As for $\cal{C}(n,q)$, we determine the minimum distance and the stopping distance for some special cases and obtain the lower bounds for other cases.

Convergence Properties of an Isospectral Saddle-Point Dynamics

We introduce a saddle-point dynamics, evolving isospectrally on the product of two homogeneous space of symmetric matrices, each orthogonally equivalent to a positive diagonal matrix. This dynamics is a saddle-point version of the so-called double-bracket flow and corresponds to a minimax version of the Frobenius norm minimization problem considered in (Brockett, 1989, 1991). We study the set of equilibria of this dynamics and prove its asymptotic convergence properties under suitable conditions. We also demonstrate how, under appropriate conditions, this dynamics provides a learning strategy for a two-player zero-sum game, where the players strategy set is the set of permutations.

Linear spaces of symmetric nilpotent matrices

In 1958 Gerstenhaber showed that if L is a subspace of the vector space of the square matrices of order n over some field F, consisting of nilpotent matrices, and the field F is sufficiently large, then the maximal dimension of L is n(n−1)2, and if this dimension is attained, then the space L is triangularizable. Linear spaces of symmetric matrices seem to be first studied by Meshulam in 1989 in view of the bound of their rank. Although it seems unnatural to ask when a linear space of symmetric matrices is made of nilpotents and when it is triangular, we find a way to do so by going to an equivalent notion for symmetric matrices, i.e. persymmetric matrices. We develop a theory that enables us to prove extensions of some beautiful classical triangularizability results to the case of symmetric matrices. Not only the Gerstenhaber's result, but also Engel, Jacobson and Radjavi theorems can be extended. We also study maximal linear spaces of symmetric nilpotents of smaller dimension.

Joint Estimation of Inverse Covariance Matrices Lying in an Unknown Subspace

We consider the problem of joint estimation of inverse covariance matrices lying in an unknown subspace of the linear space of symmetric matrices. We perform the estimation using groups of measurements with different covariances. Assuming the inverse covariances span a low-dimensional subspace, our aim is to determine this subspace and to exploit this knowledge in order to improve the estimation. We develop a novel optimization algorithm discovering and exploiting the underlying low-dimensional subspace. We provide a computationally efficient algorithm and derive a tight upper performance bound. Numerical simulations on synthetic and real world data are presented to illustrate the performance benefits of the algorithm.

Crofton formulas and indefinite signature

We study the $O(p,q)$-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an $O(p,q)$-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the $(p-1)$ homogeneous case of $O(p,p)$, yielding a Crofton formula for the centro-affine surface area when $p\not\equiv 3\mod 4$. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all $O(p,2)$-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the $|\det X|^s$ family of distributions on the space of symmetric matrices, which we use to construct a family of $O(p,q)$-invariant Crofton distributions. We conjecture there are no others, which we then prove for $O(p,2)$ with $p$ even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.

Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices.

Data encoded as symmetric positive definite (SPD) matrices frequently arise in many areas of computer vision and machine learning. While these matrices form an open subset of the Euclidean space of symmetric matrices, viewing them through the lens of non-Euclidean Riemannian (Riem) geometry often turns out to be better suited in capturing several desirable data properties. Inspired by the great success of dictionary learning and sparse coding (DLSC) for vector-valued data, our goal in this paper is to represent data in the form of SPD matrices as sparse conic combinations of SPD atoms from a learned dictionary via a Riem geometric approach. To that end, we formulate a novel Riem optimization objective for DLSC, in which the representation loss is characterized via the affine-invariant Riem metric. We also present a computationally simple algorithm for optimizing our model. Experiments on several computer vision data sets demonstrate superior classification and retrieval performance using our approach when compared with SC via alternative non-Riem formulations.

Free resolutions of some Schubert singularities in the Lagrangian Grassmannian

In this paper we construct free resolutions of certain class of closed subvarieties of affine space of symmetric matrices (of a given size). Our class covers the symmetric determinantal varieties (i.e., determinantal varieties in the space of symmetric matrices), whose resolutions were first constructed by Jozefiak-Pragacz-Weyman. Our approach follows the techniques developed in Free Resolutions of Some Schubert Singularities by Kummini-Lakshmibai-Sastry-Seshadri, and uses the geometry of Schubert varieties.

Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure

The paper deals with the best unbiased estimators of the blocked compound symmetric covariance structure for m-variate observations over u sites under the assumption of multivariate normality. The free-coordinate approach is used to prove that the quadratic estimation of covariance parameters is equivalent to linear estimation with a properly defined inner product in the space of symmetric matrices. Complete statistics are then derived to prove that the estimators are best unbiased. Finally, strong consistency is proven. The proposed method is implemented with a real data set.

Joint Covariance Estimation With Mutual Linear Structure

We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on the application of principal component analysis in the matrix space. We also derive an upper performance bound of the proposed algorithm in the Gaussian scenario and compare it with the Cramer-Rao lower bound. Numerical simulations are presented to illustrate the performance benefits of the proposed method.

Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds

Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in Euclidean spaces. With the aim of building a bridge between the two realms, we address the problem of sparse coding and dictionary learning in Grassmann manifolds, i.e., the space of linear subspaces. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping. This in turn enables us to extend two sparse coding schemes to Grassmann manifolds. Furthermore, we propose an algorithm for learning a Grassmann dictionary, atom by atom. Lastly, to handle non-linearity in data, we extend the proposed Grassmann sparse coding and dictionary learning algorithms through embedding into higher dimensional Hilbert spaces. Experiments on several classification tasks (gender recognition, gesture classification, scene analysis, face recognition, action recognition and dynamic texture classification) show that the proposed approaches achieve considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelized Affine Hull Method and graph-embedding Grassmann discriminant analysis.

Picard Iterations for Diffusions on Symmetric Matrices

Matrix-valued stochastic processes have been of significant importance in areas such as physics, engineering and mathematical finance. One of the first models studied has been the so-called Wishart process, which is described as the solution of a stochastic differential equation in the space of matrices. In this paper, we analyze natural extensions of this model and prove the existence and uniqueness of the solution. We do this by carrying out a Picard iteration technique in the space of symmetric matrices. This approach takes into account the operator character of the matrices, which helps to corroborate how the Lipchitz conditions also arise naturally in this context.

A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(div)-P k polynomial tensors and the displacement is approximated by C −1-P k−1 polynomial vectors, for all k ⩽ 4. The main ingredients for the analysis are a new basis of the space of symmetric matrices, an intrinsic H(div) bubble function space on each element, and a new technique for establishing the discrete inf-sup condition. In particular, they enable us to prove that the divergence space of the H(div) bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued P k−1 polynomial space on each tetrahedron. The optimal error estimate is proved, verified by numerical examples.

Complementarity problems with respect to Loewnerian cones

This work deals with the analysis and numerical resolution of a broad class of complementarity problems on spaces of symmetric matrices. The complementarity conditions are expressed in terms of the Loewner ordering or, more generally, with respect to a dual pair of Loewnerian cones.

On cell matrices: A class of Euclidean distance matrices

In this paper, we study the set of cell matrices and its relationship with the cone of positive semidefinite diagonal matrices. The set forms a convex polyhedral cone in the linear space of symmetric matrices. We describe the faces of the cone and its polar. We also provide a new linear inequality associated with cell matrices.

Conditional gradient algorithms for norm-regularized smooth convex optimization

Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $$K$$ , a norm $$\Vert \cdot \Vert $$ and a smooth convex function $$f$$ , we want either (1) to minimize the norm over the intersection of the cone and a level set of $$f$$ , or (2) to minimize over the cone the sum of $$f$$ and a multiple of the norm. We focus on the case where (a) the dimension of the problem is too large to allow for interior point algorithms, (b) $$\Vert \cdot \Vert $$ is “too complicated” to allow for computationally cheap Bregman projections required in the first-order proximal gradient algorithms. On the other hand, we assume that it is relatively easy to minimize linear forms over the intersection of $$K$$ and the unit $$\Vert \cdot \Vert $$ -ball. Motivating examples are given by the nuclear norm with $$K$$ being the entire space of matrices, or the positive semidefinite cone in the space of symmetric matrices, and the Total Variation norm on the space of 2D images. We discuss versions of the Conditional Gradient algorithm capable to handle our problems of interest, provide the related theoretical efficiency estimates and outline some applications.

New results on the cp-rank and related properties of co(mpletely )positive matrices

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order and a further improvement for such matrices of order six.

Multivariate Transient Price Impact and Matrix-Valued Positive Definite Functions

We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such strategies is guaranteed by assuming that the decay kernel corresponds to a matrix-valued positive definite function. An example illustrates, however, that positive definiteness alone does not guarantee that optimal strategies are well-behaved. Building on previous results from the one-dimensional case, we investigate a class of nonincreasing, non-negative, and convex decay kernels with values in a space of symmetric matrices. We show that these decay kernels are always positive definite and characterize when they are even strictly positive definite, a result that may be of independent interest. Optimal strategies for kernels from this class are particularly well-behaved if one requires that the decay kernel is also commuting. We show how such decay kernels can be constructed by means of matrix functions and provide a number of examples. In particular, we completely solve the case of matrix exponential decay.

Products of higher degree forms

We consider -linear () spaces of dimension over an algebraically closed field of characteristic 0, whose centre (the analogue of the space of symmetric matrices of a bilinear form) is maximal, as a subalgebra of In a previous work (with S. Ryom-Hansen), we found several properties of the orthogonal group of d-linear spaces whose centre consists of powers of a single lineal map acting cyclically on the underlying vector space, called cyclic spaces. We extend those results to tensor products of cyclic spaces. In particular, we give a description of the orthogonal group of the product of two cyclic spaces.

Action Recognition From Video Using Feature Covariance Matrices

We propose a general framework for fast and accurate recognition of actions in video using empirical covariance matrices of features. A dense set of spatio-temporal feature vectors are computed from video to provide a localized description of the action, and subsequently aggregated in an empirical covariance matrix to compactly represent the action. Two supervised learning methods for action recognition are developed using feature covariance matrices. Common to both methods is the transformation of the classification problem in the closed convex cone of covariance matrices into an equivalent problem in the vector space of symmetric matrices via the matrix logarithm. The first method applies nearest-neighbor classification using a suitable Riemannian metric for covariance matrices. The second method approximates the logarithm of a query covariance matrix by a sparse linear combination of the logarithms of training covariance matrices. The action label is then determined from the sparse coefficients. Both methods achieve state-of-the-art classification performance on several datasets, and are robust to action variability, viewpoint changes, and low object resolution. The proposed framework is conceptually simple and has low storage and computational requirements making it attractive for real-time implementation.

Vlasov moment flows and geodesics on the Jacobi group

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices [Bloch, A. M., Brînzănescu, V., Iserles, A., Marsden, J. E., and Ratiu, T. S., “A class of integrable flows on the space of symmetric matrices,” Commun. Math. Phys. 290, 399–435 (2009)]10.1007/s00220-009-0849-6 as a geodesic flow on the Jacobi group \documentclass[12pt]{minimal}\begin{document}${\rm Jac}(\mathbb {R}^{2n})={\rm Sp}(\mathbb {R}^{2n})\,\circledS \,{\rm H}(\mathbb {R}^{2n})$\end{document} Jac (R2n)= Sp (R2n)ⓈH(R2n). We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered by Ismagilov, Losik, and Michor [“A 2-cocycle on a group of symplectomorphisms,” Mosc. Math. J. 6, 307–315 (2006)]), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.