arXiv research

Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.

In this paper we extend DDVV-type inequalities involving the Frobenius norm of commutators from real symmetric and skew-symmetric matrices to Hermitian and skew-Hermitian matrices.

Study of J-Hermitian matrices and geometric mean definition.

Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.

A Hermitian TQFT from non-semisimple quantum sl(2) modules.

New matrix reveals cluster info in sparse directed graphs.

Researchers found sub-Lorentzian geodesics on a specific Lie subgroup.

Study of strictly accretive matrices using Finsler geometry.

In this paper we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system …

Study inequalities for singular values of rectangular matrices.

The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus \mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space which has a Morse theoretic origin. We prove that these strata define closed subana…

Researchers approximate partition functions on Riemannian spaces in the large N limit.

Study of Pascal algebra matrices and their jet bundle map for vector bundles.

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermiti…

Polarimetric Synthetic Aperture Radar (PolSAR) images are establishing as an important source of information in remote sensing applications. The most complete format this type of imaging produces consists of complex-valued Hermitian matrices in every image coordinate and, as such, their visualization is challenging. Th…

Let ${\mathcal S}_m$ be the set of all $m\times m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $ρ\in {\mathcal S}_m$ based on outcomes of $n$ measurements of observables $X_1,\dots, X_n\in {\mathbb H}_m$ (${\mathbb H}_m$ bei…

If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices or C has dimension 8, 14 or 26.

We investigate the poset of strata of a Schubert-like stratification of certain natural compactification of the space of hermitian $n\times n$ matrices. We prove that this poset is a modular ortholattice, we compute its Möbius function and we describe the topology of its order intervals.

Develops interpolation methods for matrix functions in statistics and machine learning.

The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…

Continuity of roots of hyperbolic polynomials with smooth coefficients.

We prove that there is no faithful finite-dimensional representation by skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^\infty(M) containing B.

In the previous work, the first author established an algorithm to compute the Morse index and the nullity of an $n$-periodic minimal surface in $\mathbb{R}^n$. In fact, the Morse index can be translated into the number of negative eigenvalues of a real symmetric matrix and the nullity can be translated into the number…

Overview of high-dimensional dynamical systems and their applications to machine learning.

The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characterist…

Paper connects algebraic and analytic methods for braid group representations.

The paper explores criteria for positivity of forms and proves their strong positivity in specific cases.

The paper explores Cholesky decompositions for symmetric matrices and their geometric properties.

Generically, topological insulators have conical points leading to Dirac-like currents.

In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the tail behavior of various matrix functions, e.g., arbitrary matrix norms, the absol…

Convexity proven for sums of angles of unitary paths.

Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval $[a,b]\subset\mathbb R$, we study the action defined in the Lie group of $n\times n$ unitary matrices $\mathcal{U}(n)$ by $S(α)=\int_a^b L(\dotα(t))\,dt\,,$ where $α:[a,b]\to\mathcal{U}(n)$ is a …

In statistical relational learning, knowledge graph completion deals with automatically understanding the structure of large knowledge graphs---labeled directed graphs---and predicting missing relationships---labeled edges. State-of-the-art embedding models propose different trade-offs between modeling expressiveness, …

Formula establishes determinant majorization for symmetric matrices.

A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negative-definite Hermitian.…

Quantizes geodesics in Kähler and Sasaki geometry.

Defines cross product for m vectors in n-dimensional spaces.

Novel unsupervised MIG detectors improve signal detection in cluttered environments.

We propose a conjugate gradient type optimization technique for the computation of the Karcher mean on the set of complex linear subspaces of fixed dimension, modeled by the so-called Grassmannian. The identification of the Grassmannian with Hermitian projection matrices allows an accessible introduction of the geometr…

We revisit the initialization of deep residual networks (ResNets) by introducing a novel analytical tool in free probability to the community of deep learning. This tool deals with non-Hermitian random matrices, rather than their conventional Hermitian counterparts in the literature. As a consequence, this new tool ena…

Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in {\it quantum state tomography} (QST) is to estimate the unknown l…

Statistic dynamics of financial systems is investigated, basing on a model of randomly coupled equation system driven by stochastic Langevin force. It is found that in stable regime the noise power spectrum of the system is of 1/f^alpha form, with the exponent alpha=3/2 in case of Hermitian coupling matrices, or slight…

Geometrically decomposes Kähler functions on toric manifolds.

Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.

Matrix Chernoff bound for Markov chains applied to co-occurrence matrices.

The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.

Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.

Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.