Grooms, Ian G.; Lewis, Robert Michael; Trosset, Michael W. Molecular embedding via a second order dissimilarity parameterized approach. (English) Zbl 1193.92039 SIAM J. Sci. Comput. 31, No. 4, 2733-2756 (2009). Summary: 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 non-convex 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 non-convex. We can solve problems with millions of independent variables in a few dozen to a few score optimization iterations. The non-convexity 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. Cited in 4 Documents MSC: 92C40 Biochemistry, molecular biology 65K05 Numerical mathematical programming methods 92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.) 90C90 Applications of mathematical programming Keywords:graph embedding; protein folding; rank constraints; molecular conformation; molecular embedding; distance geometry; Euclidean distance matrices; spectral functions Software:ARPACK PDFBibTeX XMLCite \textit{I. G. Grooms} et al., SIAM J. Sci. Comput. 31, No. 4, 2733--2756 (2009; Zbl 1193.92039) Full Text: DOI Link