A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles

We use convex polyhedral cones to study a large class of multivariate meromorphic germs, namely those with linear poles, which naturally arise in various contexts in mathematics and physics. We express such a germ as a sum of a holomorphic germ and a linear combination of special non-holomorphic germs called polar germs. In analyzing the supporting cones -- cones that reflect the pole structure of the polar germs -- we obtain a geometric criterion for the non-holomorphicity of linear combinations of polar germs. This yields the uniqueness of the above sum when required to be supported on a suitable family of cones and assigns a Laurent expansion to the germ. Laurent expansions provide various decompositions of such germs and thereby a uniformized proof of known results on decompositions of rational fractions. These Laurent expansions also yield new concepts on the space of such germs, all of which are independent of the choice of the specific Laurent expansion. These include a generalization of Jeffrey-Kirwan's residue, a filtered residue and a coproduct in the space of such germs. When applied to exponential sums on rational convex polyhedral cones, the filtered residue yields back exponential integrals.Comment: 30 page

Knowledge Graph Embedding with Iterative Guidance from Soft Rules

Embedding knowledge graphs (KGs) into continuous vector spaces is a focus of current research. Combining such an embedding model with logic rules has recently attracted increasing attention. Most previous attempts made a one-time injection of logic rules, ignoring the interactive nature between embedding learning and logical inference. And they focused only on hard rules, which always hold with no exception and usually require extensive manual effort to create or validate. In this paper, we propose Rule-Guided Embedding (RUGE), a novel paradigm of KG embedding with iterative guidance from soft rules. RUGE enables an embedding model to learn simultaneously from 1) labeled triples that have been directly observed in a given KG, 2) unlabeled triples whose labels are going to be predicted iteratively, and 3) soft rules with various confidence levels extracted automatically from the KG. In the learning process, RUGE iteratively queries rules to obtain soft labels for unlabeled triples, and integrates such newly labeled triples to update the embedding model. Through this iterative procedure, knowledge embodied in logic rules may be better transferred into the learned embeddings. We evaluate RUGE in link prediction on Freebase and YAGO. Experimental results show that: 1) with rule knowledge injected iteratively, RUGE achieves significant and consistent improvements over state-of-the-art baselines; and 2) despite their uncertainties, automatically extracted soft rules are highly beneficial to KG embedding, even those with moderate confidence levels. The code and data used for this paper can be obtained from https://github.com/iieir-km/RUGE.Comment: To appear in AAAI 201

Quantum resource studied from the perspective of quantum state superposition

Quantum resources,such as discord and entanglement, are crucial in quantum information processing. In this paper, quantum resources are studied from the aspect of quantum state superposition. We define the local superposition (LS) as the superposition between basis of single part, and nonlocal superposition (NLS) as the superposition between product basis of multiple parts. For quantum resource with nonzero LS, quantum operation must be introduced to prepare it, and for quantum resource with nonzero NLS, nonlocal quantum operation must be introduced to prepare it. We prove that LS vanishes if and only if the state is classical and NLS vanishes if and only if the state is separable. From this superposition aspect, quantum resources are categorized as superpositions existing in different parts. These results are helpful to study quantum resources from a unified frame.Comment: 9 pages, 4 figure