The Euclidean distance degree of an algebraic variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic

The Euclidean distance degree of smooth complex projective varieties

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X

A note on ED degrees of group-stable subvarieties in polar representations

In a recent paper, Drusvyatskiy, Lee, Ottaviani, and Thomas establish a "transfer principle" by means of which the Euclidean distance degree of an orthogonally-stable matrix variety can be computed from the Euclidean distance degree of its intersection with a linear subspace. We generalise this principle

Complexity of Model Testing for Dynamical Systems with Toric Steady States

In this paper we investigate the complexity of model selection and model testing for dynamical systems with toric steady states. Such systems frequently arise in the study of chemical reaction networks. We do this by formulating these tasks as a constrained optimization problem in Euclidean space. This optimization problem is known as a Euclidean distance problem; the complexity of solving this problem is measured by an invariant called the Euclidean distance (ED) degree. We determine closed-form expressions for the ED degree of the steady states of several families of chemical reaction networks with toric steady states and arbitrarily many reactions. To illustrate the utility of this work we show how the ED degree can be used as a tool for estimating the computational cost of solving the model testing and model selection problems