An alternating projection algorithm for computing the nearest euclidean distance matrix
SIAM Journal on Matrix Analysis and Applications
Convergence behavior of interior-point algorithms
Mathematical Programming: Series A and B
Distance Matrix Completion by Numerical Optimization
Computational Optimization and Applications
Global Continuation for Distance Geometry Problems
SIAM Journal on Optimization
Distance Geometry Optimization for Protein Structures
Journal of Global Optimization
Journal of Global Optimization
Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment
SIAM Journal on Scientific Computing
Theory of semidefinite programming for sensor network localization
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Journal of Global Optimization
SIAM Journal on Scientific Computing
Molecular Embedding via a Second Order Dissimilarity Parameterized Approach
SIAM Journal on Scientific Computing
Protein structure by semidefinite facial reduction
RECOMB'12 Proceedings of the 16th Annual international conference on Research in Computational Molecular Biology
ACM Transactions on Mathematical Software (TOMS)
RECOMB'13 Proceedings of the 17th international conference on Research in Computational Molecular Biology
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We propose the DISCO algorithm for graph realization in $\mathbb{R}^d$, given sparse and noisy short-range intervertex distances as inputs. Our divide-and-conquer algorithm works as follows. When a group has a sufficiently small number of vertices, the basis step is to form a graph realization by solving a semidefinite program. The recursive step is to break a large group of vertices into two smaller groups with overlapping vertices. These two groups are solved recursively, and the subconfigurations are stitched together, using the overlapping atoms, to form a configuration for the larger group. At intermediate stages, the configurations are improved by gradient descent refinement. The algorithm is applied to the problem of determining protein moleculer structure. Tests are performed on molecules taken from the Protein Data Bank database. For each molecule, given 20-30% of the inter-atom distances less than 6Å that are corrupted by a high level of noise, DISCO is able to reliably and efficiently reconstruct the conformation of large molecules. In particular, given 30% of distances with 20% multiplicative noise, a 13000-atom conformation problem is solved within an hour with a root mean square deviation of 1.6Å.