Given an n × n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0, 1, . . . , n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such att...

Let V be a finite dimensional complex linear space and let G be an irreducible finite subgroup of GL(V). For a G-invariant lattice Λ in V of maximal rank, we give a description of structure of the complex torus V /Λ. In particular, we prove that for a wide class of groups, V /Λ is isogenous to a self-product of an elliptic curve, and that in many cases V /Λ is isomorphic to a product of mutuall...

This paper presents an algorithm that solves optimization problems on a matrix manifold M ⊆ Rm×n with an additional rank inequality constraint. The algorithm resorts to well-known Riemannian optimization schemes on fixed-rank manifolds, combined with new mechanisms to increase or decrease the rank. The convergence of the algorithm is analyzed and a weighted low-rank approximation problem is use...

Let K be the class of atomic models of a countable first order theory. We prove that ifK is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies thatK is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over m...

Higher-order tensors are generalizations of vectors and matrices to thirdor even higher-order arrays of numbers. We consider a generalization of column and row rank of a matrix to tensors, called multilinear rank. Given a higher-order tensor, we are looking for another tensor, as close as possible to the original one and with multilinear rank bounded by prespecified numbers. In this paper, we g...

The BFGS and DFP updates are perhaps the most successful Hessian and inverse Hessian approximations respectively for unconstrained minimization problems. This paper describes these methods in terms of two successive steps: rank reduction and rank restoration. From rank subtractivity and a powerful spectral result, the first step must necessarily result in a positive semidefinite matrix; and the...

We study an estimator with a convex formulation for recovery of low-rank matrices from rank-one projections. Using initial estimates the factors target $d_1\times d_2$ matrix rank-$r$, admits practical subgradient method operating in space dimension $r(d_1+d_2)$. This property makes significantly more scalable than estimators based on lifting and semidefinite programming. Furthermore, we presen...