In this paper we characterize the optimization geometry of a matrix factorization problem where we aim to find n×r and m×r matrices U and V such that UV T approximates a given matrixX. We show that the objective function of the matrix factorization problem has no spurious local minima and obeys the strict saddle property not only for the exact-parameterization case where rank(X) = r, but also f...

We study the duality properties of the well-known DFJP factorization of operators 3] by means of a reenement of it. Given an operator T : X ! Y we consider a decomposition T = jU k , where U : E ! F is an isomorphism, and j , U k are the factors in the DFJP factorization. If T is the conjugate operator of T , and T : X =X ! Y =Y is the operator given by T (x + X) := T x + Y (x 2 X), then we sho...

We study the functorial characterization of bisimulation-based equivalences over a categorical model of labeled trees. We show that, in a setting where all labels are visible, strong bisimilarity can be characterized in terms of enriched functors by relying on reflection of paths with their factorizations. For an enriched functor F , this notion requires that a path (an internal morphism in our...

This paper presents a robust structured multifrontal factorization method for large symmetric positive definite sparse matrices arising from the discretization of partial differential equations (PDEs). For PDEs such as 2D and 3D elliptic equations, the method costs roughly O(n) and O(n4/3) flops, respectively. The algorithm takes advantage of a low-rank property in the direct factorization of s...

This thesis investigates relations over a category C relative to an (E;M)-factorization system of C. In order to establish the 2-categoryRel(C) of relations over C in the rst part we discuss su cient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as de ...

Weak factorization systems Weak factorization systems are of paramount importance to homotopical algebra. This connection is best illustrated by the following definition, due to Joyal and Tierney [JT07]. Definition 1. A Quillen model structure on a category M, with a class of maps W called weak equivalences satisfying the 2-of-3 property, consists of two classes of maps C and F so that (C ∩W,F)...

We present a simple proof of the factorization of (complex) symmetric matrices into a product of a square matrix and its transpose, and discuss its application in establishing a uniqueness property of certain antilinear operators.

A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism). For monoids in this class, the important factorization property of finite elasticity is characterized.

The factorization property of graviton scattering amplitudes is reviewed and shown to be valid only if the “natural” value of the gyromagnetic ratio gS = 2 is employed—independent of spin.

A large class of multiplicative submonoids of the natural numbers is presented, which includes congruence monoids as well as numerical monoids (by isomorphism). For monoids in this class, the important factorization property of finite elasticity is characterized.