Linear projection equations arise in many optimization problems and have important applications in science and engineering. In this paper, we present a recurrent neural network for solving linear projection equations in real time. The proposed neural network has two layers and is amenable to parallel implementation with simple hardware. In the theoretical aspect, we prove that the proposed neur...

An element A of the n× n copositive cone C is called irreducible with respect to the nonnegative cone N if it cannot be written as a nontrivial sum A = C + N of a copositive matrix C and an elementwise nonnegative matrix N . This property was studied by Baumert [2] who gave a characterisation of irreducible matrices. We demonstrate here that Baumert’s characterisation is incorrect and give a co...

We consider the nonlinear matrix equation X = Q+A (I⊗X−C)A (0 < δ ≤ 1), where Q is an n× n positive definite matrix, C is an mn×mn positive semidefinite matrix, I is the m×m identity matrix, and A is an arbitrary mn×n matrix. We prove the existence and uniqueness of the solution which is contained in some subset of the positive definite matrices under the condition that I ⊗Q > C. Two bounds for...

First, note that by assumption rank{A} > 0. Let Ω1 = ρ1 × ρ1 and Ω2 = ρ2 × ρ2 be the two index sets in the theorem. By assumption we have ρ1 × ρ1 ∪ ρ2 × ρ2 = Ω and Ω 6= [n]× [n]. If A1 is not met, then ρ1 ∪ ρ2 6= [n], and from lemma 6 we can conclude recovery of A is impossible. If ρ1 ∪ ρ2 = [n], but A2 is not met then ι2 = |ρ1 ∩ ρ2| < r so it must be that rank{A(ι2, ι2)} < r. Further, by assum...

We provide proofs that were skipped in the main paper. We also provide some additional experimental results and related work concerning multi-armed bandits that was skipped in the main paper.

We examine the problem of approximating a positive, semidefinite matrix Σ by a dyad xxT , with a penalty on the cardinality of the vector x. This problem arises in sparse principal component analysis, where a decomposition of Σ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, the largest ei...

Approximating the nearest positive semidefinite Hankel matrix in the Frobenius norm to an arbitrary data covariance matrix is useful in many areas of engineering, including signal processing and control theory. In this paper, interior point primal-dual path-following method will be used to solve our problem after reformulating it into different forms, first as a semidefinite programming problem...

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do this we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator which ground state is determined by the lowest ...

Linear projection equations arise in many optimization problems and have important applications in science and engineering. In this paper, we present a recurrent neural network for solving linear projection equations in real time. The proposed neural network has two layers and is amenable to parallel implementation with simple hardware. In the theoretical aspect, we prove that the proposed neur...

Unconstrained zero-one quadratic maximization problems can be solved in polynomial time when the symmetric matrix describing the objective function is positive semidefinite of fixed rank with known spectral decomposition.