We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.
The hierarchical RRBPM (H-RRBPM) proposed in @cite_27 by Thomas and Carrington is a significant improvement of the RRBPM. This method also utilizes sum-of-products representation, but treats strongly coupled coordinates together. Coupled coordinates are further decomposed hierarchically.
@cite_27
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We propose a method for solving the vibrational Schrodinger equation with which one can compute hundreds of energy levels of seven-atom molecules using at most a few gigabytes of memory. It uses nested contractions in conjunction with the reduced-rank block power method (RRBPM) described in J. Chem. Phys. 2014, 140, 174111. Successive basis contractions are organized into a tree, the nodes of which are associated with eigenfunctions of reduced-dimension Hamiltonians. The RRBPM is used recursively to compute eigenfunctions of nodes in bases of products of reduced-dimension eigenfunctions of nodes with fewer coordinates. The corresponding vectors are tensors in what is called CP-format. The final wave functions are therefore represented in a hierarchical CP-format. Computational efficiency and accuracy are significantly improved by representing the Hamiltonian in the same hierarchical format as the wave function. We demonstrate that with this hierarchical RRBPM it is possible to compute energy levels of a 6... Discriminative methods often generate hand poses kinematically implausible, then generative methods are used to correct (or verify) these results in a hybrid method. Estimating 3D hand pose in a hierarchy, where the high-dimensional output space is decomposed into smaller ones, has been shown effective. Existing hierarchical methods mainly focus on the decomposition of the output space while the input space remains almost the same along the hierarchy. In this paper, a hybrid hand pose estimation method is proposed by applying the kinematic hierarchy strategy to the input space (as well as the output space) of the discriminative method by a spatial attention mechanism and to the optimization of the generative method by hierarchical Particle Swarm Optimization (PSO). The spatial attention mechanism integrates cascaded and hierarchical regression into a CNN framework by transforming both the input(and feature space) and the output space, which greatly reduces the viewpoint and articulation variations. Between the levels in the hierarchy, the hierarchical PSO forces the kinematic constraints to the results of the CNNs. The experimental results show that our method significantly outperforms four state-of-the-art methods and three baselines on three public benchmarks. Considers the problem of robot motion planning using an approximate cell decomposition method for Cspace representation. By means of a hierarchical multilevel discretization the authors are able to reduce the complexity of the problem and to provide a practically feasible procedure for its solution. Hierarchical discretization is based on the use of a multisource distance field that is introduced and described in the paper. The developed procedure, used as an heuristic, systematically reduces the amount of memory needed for implementing the motion planning algorithm. Using such a procedure, several instances of the motion planning problem, involving for example two three-links manipulators acting in the same tridimensional environment, have been solved in a reasonable amount of time. > The main stated contribution of the Deformable Parts Model (DPM) detector of (over the Histogram-of-Oriented-Gradients approach of Dalal and Triggs) is the use of deformable parts. A secondary contribution is the latent discriminative learning. Tertiary is the use of multiple components. A common belief in the vision community (including ours, before this study) is that their ordering of contributions reflects the performance of detector in practice. However, what we have experimentally found is that the ordering of importance might actually be the reverse. First, we show that by increasing the number of components, and switching the initialization step from their aspect-ratio, left-right flipping heuristics to appearance-based clustering, considerable improvement in performance is obtained. But more intriguingly, we show that with these new components, the part deformations can now be completely switched off, yet obtaining results that are almost on par with the original DPM detector. Finally, we also show initial results for using multiple components on a different problem -- scene classification, suggesting that this idea might have wider applications in addition to object detection.