Spectral triples from stationary Bratteli diagrams

We construct spectral triples for path spaces of stationary Bratteli diagrams and study their associated mathematical objects, in particular their zeta function, their heat kernel expansion and their Dirichlet forms. One of the main difficulties to properly define a Dirichlet form concerns its domain. We address this question in particular in the context of Pisot substitution tiling spaces for which we find two types of Dirichlet forms: one of transversal type, and one of longitudinal type. Here the eigenfunctions under the translation action can serve as a good core for a non-trivial Dirichlet form. We find that the infinitesimal generators can be interpreted as elliptic differential operators on the maximal equicontinuous factor of the tiling dynamical system.Comment: Version 2, 49 page

Many-Body Expanded Full Configuration Interaction. I. Weakly Correlated Regime

Over the course of the past few decades, the field of computational chemistry has managed to manifest itself as a key complement to more traditional lab-oriented chemistry. This is particularly true in the wake of the recent renaissance of full configuration interaction (FCI)-level methodologies, albeit only if these can prove themselves sufficiently robust and versatile to be routinely applied to a variety of chemical problems of interest. In the present series of works, performance and feature enhancements of one such avenue towards FCI-level results for medium to large one-electron basis sets, the recently introduced many-body expanded full configuration interaction (MBE-FCI) formalism [J. Phys. Chem. Lett., 8, 4633 (2017)], will be presented. Specifically, in this opening part of the series, the capabilities of the MBE-FCI method in producing near-exact ground state energies for weakly correlated molecules of any spin multiplicity will be demonstrated.Comment: 38 pages, 7 tables, 3 figures, 1 SI attached as an ancillary fil

Context-aware Path Ranking for Knowledge Base Completion

Knowledge base (KB) completion aims to infer missing facts from existing ones in a KB. Among various approaches, path ranking (PR) algorithms have received increasing attention in recent years. PR algorithms enumerate paths between entity pairs in a KB and use those paths as features to train a model for missing fact prediction. Due to their good performances and high model interpretability, several methods have been proposed. However, most existing methods suffer from scalability (high RAM consumption) and feature explosion (trains on an exponentially large number of features) problems. This paper proposes a Context-aware Path Ranking (C-PR) algorithm to solve these problems by introducing a selective path exploration strategy. C-PR learns global semantics of entities in the KB using word embedding and leverages the knowledge of entity semantics to enumerate contextually relevant paths using bidirectional random walk. Experimental results on three large KBs show that the path features (fewer in number) discovered by C-PR not only improve predictive performance but also are more interpretable than existing baselines