On the local and global minimizers of the smooth stress function in Euclidean Distance Matrix problems

TL;DR

This paper investigates the conditions under which local and global minimizers of a nonconvex stress function in Euclidean Distance Matrix problems exist, providing theoretical results and numerical methods to identify them, with implications for exact recovery.

Contribution

It proves that all second-order stationary points are global minimizers for n ≤ d+1 and demonstrates the existence of local nonglobal minimizers in other cases, introducing a novel approach to find them.

Findings

All second-order stationary points are global minimizers when n ≤ d+1.

Existence of local nonglobal minimizers for certain n and d cases.

A new method exploiting invariances to find and analyze local minimizers.

Abstract

We consider the nonconvex minimization problem, with quartic objective function, that arises in the exact recovery of a configuration matrix $P\in \R^{nd}$ of $n$ points when a Euclidean distance matrix, \EDMp, is given with embedding dimension $d$. It is an open question in the literature whether there are conditions such that the minimization problem admits a local nonglobal minimizer, \lngmp. We prove that all second-order stationary points are global minimizers whenever $n \leq d + 1$. {And, for $d=1$ and $n\geq 7>d+1$, we present an example where we can analytically exhibit a local nonglobal minimizer. For more general cases,} we numerically find a second-order stationary point and then prove that there indeed exists a nearby \lngm for the quartic nonconvex minimization problem. Thus, we answer the previously open question about their existence in the affirmative. Our approach to finding the \lngm is novel in that we first exploit the translation and rotation invariance to remove the singularities of the Hessian, and reduce the size of the problem from $nd$ variables in $P$ to $(n-1)d - d(d-1)/2$ variables. This allows for stabilizing Newton's method, and for finding examples that satisfy the strict second order sufficient optimality conditions. The motivation for being able to find global minima is to obtain \emph{exact recovery} of the configuration matrix, even in the cases where the data is noisy and/or incomplete, without resorting to approximating solutions from convex (semidefinite programming) relaxations. In the process of our work we present new insights into when \lngmp s of the smooth stress function do and do not exist.