On Sufficient Statistics of Least-Squares Superposition of Vector Sets

Abstract

The problem of superposition of two corresponding vector sets by minimizing their sum-of-squares error under orthogonal transformation is a fundamental task in many areas of science, notably structural molecular biology. This problem can be solved exactly using an algorithm whose time complexity grows linearly with the number of correspondences. This efficient solution has facilitated the widespread use of the superposition task, particularly in studies involving macromolecular structures. This article formally derives a set of sufficient statistics for the least-squares superposition problem. These statistics are additive. This permits a highly efficient (constant time) computation of superpositions (and sufficient statistics) of vector sets that are composed from its constituent vector sets under addition or deletion operation, where the sufficient statistics of the constituent sets are already known (that is, the constituent vector sets have been previously superposed). This results in a drastic improvement in the run time of the methods that commonly superpose vector sets under addition or deletion operations, where previously these operations were carried out ab initio (ignoring the sufficient statistics). We experimentally demonstrate the improvement our work offers in the context of protein structural alignment programs that assemble a reliable structural alignment from well-fitting (substructural) fragment pairs. A C++ library for this task is available online under an open-source license.

A new statistical framework to assess structural alignment quality using information compression

“…While the linearity of the first two is clear, that of $I(T|S\mathrm{\&}\mathcal{A})$ is not, as it requires repeated adaptive superpositions. However, we have recently proved sufficient statistics for the orthogonal superposition problem that allows each updated superposition to be computed as a constant-time update over the previous ones (Konagurthu et al , 2014), making the computation of $I(T|S\mathrm{\&}\mathcal{A})$, and I -value under rigid superposition, linear. On the other hand, using the flexible model which allows for hinge rotations and shifts, the computation of $I(\mathcal{A}\mathrm{\&}S\mathrm{\&}T)$ is quadratic, as it is dictated by the complexity of the dynamic program given by Equation 4.…”

An efficient RANSAC hypothesis evaluation using sufficient statistics for RGB-D pose estimation

“…The method we propose is to reduce the time taken for the re-computations of the transformation by using a set of sufficient statistics for the 3D point alignment problem derived in (Konagurthu et al 2014). In this previous work in bio-informatics, this set of sufficient statistics has been proposed for efficiently aligning protein structures.…”

“…Sufficient statistics (Hogg and Craig 1994) are essentially a set of statistics that summarises all of the information in a sample about a certain parameter. A set of sufficient statistics with respect to the least squares alignment was derived in (Konagurthu et al 2014), and furthermore, these statistics were demonstrated to be additive. Therefore, by using these statistics in the RANSAC process, computing transformations of the…”

“…This process is much faster than recomputing transformations from scratch. Below we re-derive the set of sufficient statistics of (Konagurthu et al 2014) with respect to our problem.…”

“…The key idea of the approach lies in the observation that the computation of relative pose depends on a number of 'sufficient statistics,' that is, a number of functions on the input data. These sufficient statistics have been previously proposed in bioinformatics (Konagurthu et al 2014) for efficiently aligning protein structures. Even though pose estimation involves a similar alignment process for computing the transformation between camera poses, the 3-D feature point sets contain noisy data that are normally pruned out by the RANSAC procedure.…”