Geodesic Distance Function Learning via Heat Flow on Vector Fields

Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if the original manifold is not Euclidean. Note that the distance function on a manifold can always be well-defined. In this paper, we propose to learn the distance function directly on the manifold without embedding. We first provide a theoretical characterization of the distance function by its gradient field. Based on our theoretical analysis, we propose to first learn the gradient field of the distance function and then learn the distance function itself. Specifically, we set the gradient field of a local distance function as an initial vector field. Then we transport it to the whole manifold via heat flow on vector fields. Finally, the geodesic distance function can be obtained by requiring its gradient field to be close to the normalized vector field. Experimental results on both synthetic and real data demonstrate the effectiveness of our proposed algorithm

4-Amino-1-(2-benzoyl-1-phenyl­eth­yl)-3-phenyl-1H-1,2,4-triazol-5(4H)-thione

In the title compound, C23H20N4OS, the two phenyl rings of the diphenyl­propanone fragment form a dihedral angle of 86.8 (1)°, and the third phenyl ring attached to the triazole ring is twisted from the latter at 40.1 (1)°. In the crystal, mol­ecules are paired into centrosymmetric dimers via pairs of inter­molecular N—H⋯O and N—H⋯S hydrogen bonds

Sub-quadratic scaling real-space random-phase approximation correlation energy calculations for periodic systems with numerical atomic orbitals

The random phase approximation (RPA) as formulated as an orbital-dependent, fifth-rung functional within the density functional theory (DFT) framework offers a promising approach for calculating the ground-state energies and the derived properties of real materials. Its widespread use to large-size, complex materials is however impeded by the significantly increased computational cost, compared to lower-rung functionals. The standard implementation exhibits an $\mathcal{O}(N^4)$-scaling behavior with respect to system size $N$. In this work, we develop a low-scaling RPA algorithm for periodic systems, based on the numerical atomic orbital (NAO) basis-set framework and a localized variant of the resolution of identity (RI) approximation. The rate-determining step for RPA calculations -- the evaluation of non-interacting response function matrix, is reduced from $\mathcal{O}(N^4)$ to $\mathcal{O}(N^2)$ by just exploiting the sparsity of the RI expansion coefficients, resultant from localized RI (LRI) scheme and the strict locality of NAOs. The computational cost of this step can be further reduced to linear scaling if the decay behavior of the Green's function in real space can be further taken into account. Benchmark calculations against existing $\textbf k$-space based implementation confirms the validity and high numerical precision of the present algorithm and implementation. The new RPA algorithm allows us to readily handle three-dimensional, closely-packed solid state materials with over 1000 atoms. The algorithm and numerical techniques developed in this work also have implications for developing low-scaling algorithms for other correlated methods to be applicable to large-scale extended materials

Alternative Splicing and Expression Profile Analysis of Expressed Sequence Tags in Domestic Pig

Domestic pig (Sus scrofa domestica) is one of the most important mammals to humans. Alternative splicing is a cellular mechanism in eukaryotes that greatly increases the diversity of gene products. Expression sequence tags (ESTs) have been widely used for gene discovery, expression profile analysis, and alternative splicing detection. In this study, a total of 712,905 ESTs extracted from 101 different non-normalized EST libraries of the domestic pig were analyzed. These EST libraries cover the nervous system, digestive system, immune system, and meat production related tissues from embryo, newborn, and adult pigs, making contributions to the analysis of alternative splicing variants as well as expression profiles in various stages of tissues. A modified approach was designed to cluster and assemble large EST datasets, aiming to detect alternative splicing together with EST abundance of each splicing variant. Much efforts were made to classify alternative splicing into different types and apply different filters to each type to get more reliable results. Finally, a total of 1,223 genes with average 2.8 splicing variants were detected among 16,540 unique genes. The overview of expression profiles would change when we take alternative splicing into account