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Zwiernik P.,Institute for Pure and Applied Mathematics
Journal of Machine Learning Research | Year: 2011

The standard Bayesian Information Criterion (BIC) is derived under regularity conditions which are not always satisfied in the case of graphical models with hidden variables. In this paper we derive the BIC for the binary graphical tree models where all the inner nodes of a tree represent binary hidden variables. This provides an extension of a similar formula given by Rusakov and Geiger for naive Bayes models. The main tool used in this paper is the connection between the growth behavior of marginal likelihood integrals and the real log-canonical threshold. © 2011 Piotr Zwiernik.

Samanta A.,Princeton University | Chen M.,New York University | Yu T.-Q.,Courant Institute of Mathematical Sciences | Tuckerman M.,Courant Institute of Mathematical Sciences | And 4 more authors.
Journal of Chemical Physics | Year: 2014

Many problems in biology, chemistry, and materials science require knowledge of saddle points on free energy surfaces. These saddle points act as transition states and are the bottlenecks for transitions of the system between different metastable states. For simple systems in which the free energy depends on a few variables, the free energy surface can be precomputed, and saddle points can then be found using existing techniques. For complex systems, where the free energy depends on many degrees of freedom, this is not feasible. In this paper, we develop an algorithm for finding the saddle points on a high-dimensional free energy surface "on-the-fly" without requiring a priori knowledge the free energy function itself. This is done by using the general strategy of the heterogeneous multi-scale method by applying a macro-scale solver, here the gentlest ascent dynamics algorithm, with the needed force and Hessian values computed on-the-fly using a micro-scale model such as molecular dynamics. The algorithm is capable of dealing with problems involving many coarse-grained variables. The utility of the algorithm is illustrated by studying the saddle points associated with (a) the isomerization transition of the alanine dipeptide using two coarse-grained variables, specifically the Ramachandran dihedral angles, and (b) the beta-hairpin structure of the alanine decamer using 20 coarse-grained variables, specifically the full set of Ramachandran angle pairs associated with each residue. For the alanine decamer, we obtain a detailed network showing the connectivity of the minima obtained and the saddle-point structures that connect them, which provides a way to visualize the gross features of the high-dimensional surface. © 2014 AIP Publishing LLC.

Marsalek O.,New York University | Chen P.-Y.,New York University | Dupuis R.,University Paul Sabatier | Benoit M.,CNRS Toulouse Center for Materials Elaboration and Structural Studies | And 6 more authors.
Journal of Chemical Theory and Computation | Year: 2014

The problem of computing free energy differences due to isotopic substitution in chemical systems is discussed. The shift in the equilibrium properties of a system upon isotopic substitution is a purely quantum mechanical effect that can be quantified using the Feynman path integral approach. In this paper, we explore two developments that lead to a highly efficient path integral scheme. First, we employ a mass switching function inspired by the work of Ceriotti and Markland [ J. Chem. Phys. 2013, 138, 014112] that is based on the inverse square root of the mass and which leads to a perfectly constant free energy derivative with respect to the switching parameter in the harmonic limit. We show that even for anharmonic systems, this scheme allows a single-point thermodynamic integration approach to be used in the construction of free energy differences. In order to improve the efficiency of the calculations even further, however, we derive a set of free energy derivative estimators based on the fourth-order scheme of Takahashi and Imada [ J. Phys. Soc. Jpn. 1984, 53, 3765]. The Takahashi-Imada procedure generates a primitive fourth-order estimator that allows the number of imaginary time slices in the path-integral approach to be reduced substantially. However, as with all primitive estimators, its convergence is plagued by numerical noise. In order to alleviate this problem, we derive a fourth-order virial estimator based on a transferring of the difference between second- and fourth-order primitive estimators, which remains relatively constant as a function of the number of configuration samples, to the second-order virial estimator. We show that this new estimator converges as smoothly as the second-order virial estimator but requires significantly fewer imaginary time points. © 2014 American Chemical Society.

Raclariu A.-M.,University of Cambridge | Deshpande S.,University of Pennsylvania | Bruggemann J.,Scripps Research Institute | Zhuge W.,University of Tokyo | And 3 more authors.
Computational Materials Science | Year: 2015

Abstract In order to address an important problem in Computational Materials Design, we have demonstrated the feasibility of an algorithm which rapidly scans through various surface configurations of two single crystals selected from a crystal structures database and identifies those pairs which are likely to form stable heterointerfaces. Any two crystals are cut along different planes and all possible heterocrystal interfaces are generated based on geometric criteria and predicted bond directions of atoms on both constituent surfaces. Each configuration is assigned two scores derived using deviations of interfacial bond lengths from ideal and electronegativity differences between atoms on each side of the interface. We present some results to illustrate our method for PtNi3 on Pt3Co, GaP on Si, and Si on SiO2. This technique can be used as a fast filter for further analysis by more detailed ab initio-based methods, and is meant to address the higher throughput methods needed in the Materials Genome Initiative for complex material structures. © 2015 Elsevier B.V.

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