Molecular Embedding via a Second Order Dissimilarity Parameterized Approach

We describe a computational approach to the embedding problem in structural molecular biology. The approach is based on a dissimilarity parameterization of the problem that leads to a large-scale nonconvex bound constrained matrix optimization problem. The underlying idea is that an increased number of independent variables decouples the complicated effects of varying the location of individual atoms in coordinate-based formulations. Numerical tests support this hypothesis and indicate that the optimization problem that results is relatively benign and easy to solve, despite being large and nonconvex. We can solve problems with millions of independent variables in a few dozen to a few score optimization iterations. The nonconvexity arises due to matrix rank constraints in the problem, and we focus on their efficient computational treatment. We present numerical results for a number of synthetic and real protein data sets and comment on features of real experimental data that can cause computational difficulties.