Julian Yarkony

Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration

The random phase approximation (RPA) is an increasingly popular post-Kohn-Sham correlation method, but its high computational cost has limited molecular applications to systems with few atoms. Here we present an efficient implementation of RPA correlation energies based on a combination of resolution of the identity (RI) and imaginary frequency integration techniques. We show that the RI approximation to four-index electron repulsion integrals leads to a variational upper bound to the exact RPA correlation energy if the Coulomb metric is used. Auxiliary basis sets optimized for second-order Møller-Plesset (MP2) calculations are well suitable for RPA, as is demonstrated for the HEAT [A. Tajti et al., J. Chem. Phys. 121, 11599 (2004)] and MOLEKEL [F. Weigend et al., Chem. Phys. Lett. 294, 143 (1998)] benchmark sets. Using imaginary frequency integration rather than diagonalization to compute the matrix square root necessary for RPA, evaluation of the RPA correlation energy requires O(N(4) log N) operations and O(N(3)) storage only; the price for this dramatic improvement over existing algorithms is a numerical quadrature. We propose a numerical integration scheme that is exact in the two-orbital case and converges exponentially with the number of grid points. For most systems, 30-40 grid points yield muH accuracy in triple zeta basis sets, but much larger grids are necessary for small gap systems. The lowest-order approximation to the present method is a post-Kohn-Sham frequency-domain version of opposite-spin Laplace-transform RI-MP2 [J. Jung et al., Phys. Rev. B 70, 205107 (2004)]. Timings for polyacenes with up to 30 atoms show speed-ups of two orders of magnitude over previous implementations. The present approach makes it possible to routinely compute RPA correlation energies of systems well beyond 100 atoms, as is demonstrated for the octapeptide angiotensin II.

Fast planar correlation clustering for image segmentation

We describe a new optimization scheme for finding high-quality clusterings in planar graphs that uses weighted perfect matching as a subroutine. Our method provides lower-bounds on the energy of the optimal correlation clustering that are typically fast to compute and tight in practice. We demonstrate our algorithm on the problem of image segmentation where this approach outperforms existing global optimization techniques in minimizing the objective and is competitive with the state of the art in producing high-quality segmentations.

Covering trees and lower-bounds on quadratic assignment

Many computer vision problems involving feature correspondence among images can be formulated as an assignment problem with a quadratic cost function. Such problems are computationally infeasible in general but recent advances in discrete optimization such as tree-reweighted belief propagation (TRW) often provide high-quality solutions. In this paper, we improve upon these algorithms in two ways. First, we introduce covering trees, a variant of TRW which provide the same bounds on the MAP energy as TRW with far fewer variational parameters. Optimization of these parameters can be carried out efficiently using either fixed–point iterations (as in TRW) or sub-gradient based techniques. Second, we introduce a new technique that utilizes bipartite matching applied to the min-marginals produced with covering trees in order to compute a tighter lower-bound for the quadratic assignment problem. We apply this machinery to the problem of finding correspondences with pairwise energy functions, and demonstrate the resulting hybrid method outperforms TRW alone and a recent related subproblem decomposition algorithm on benchmark image correspondence problems.

Hierarchical Planar Correlation Clustering for Cell Segmentation

We introduce a novel algorithm for hierarchical clustering on planar graphs we call “Hierarchical Greedy Planar Correlation Clustering” (HGPCC). We formulate hierarchical image segmentation as an ultrametric rounding problem on a superpixel graph where there are edges between superpixels that are adjacent in the image. We apply coordinate descent optimization where updates are based on planar correlation clustering. Planar correlation clustering is NP hard but the efficient PlanarCC solver allows for efficient and accurate approximate inference. We demonstrate HGPCC on problems in segmenting images of cells.

Accelerating Dynamic Programs via Nested Benders Decomposition with Application to Multi-Person Pose Estimation

We present a novel approach to solve dynamic programs (DP), which are frequent in computer vision, on tree-structured graphs with exponential node state space. Typical DP approaches have to enumerate the joint state space of two adjacent nodes on every edge of the tree to compute the optimal messages. Here we propose an algorithm based on Nested Benders Decomposition (NBD) that iteratively lower-bounds the message on every edge and promises to be far more efficient. We apply our NBD algorithm along with a novel Minimum Weight Set Packing (MWSP) formulation to a multi-person pose estimation problem. While our algorithm is provably optimal at termination it operates in linear time for practical DP problems, gaining up to 500\({\times }\) speed up over traditional DP algorithm which have polynomial complexity.