Mathematics

Contains reports on three research projects

Iteratively Learning Embeddings and Rules for Knowledge Graph Reasoning

Reasoning is essential for the development of large knowledge graphs, especially for completion, which aims to infer new triples based on existing ones. Both rules and embeddings can be used for knowledge graph reasoning and they have their own advantages and difficulties. Rule-based reasoning is accurate and explainable but rule learning with searching over the graph always suffers from efficiency due to huge search space. Embedding-based reasoning is more scalable and efficient as the reasoning is conducted via computation between embeddings, but it has difficulty learning good representations for sparse entities because a good embedding relies heavily on data richness. Based on this observation, in this paper we explore how embedding and rule learning can be combined together and complement each other's difficulties with their advantages. We propose a novel framework IterE iteratively learning embeddings and rules, in which rules are learned from embeddings with proper pruning strategy and embeddings are learned from existing triples and new triples inferred by rules. Evaluations on embedding qualities of IterE show that rules help improve the quality of sparse entity embeddings and their link prediction results. We also evaluate the efficiency of rule learning and quality of rules from IterE compared with AMIE+, showing that IterE is capable of generating high quality rules more efficiently. Experiments show that iteratively learning embeddings and rules benefit each other during learning and prediction.Comment: This paper is accepted by WWW'1

AGI and the Knight-Darwin Law: why idealized AGI reproduction requires collaboration

Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: “Not without outside help”. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce another, which asexually produces another, and so on forever: that any sequence of organisms (each one a child of the previous) must contain occasional multi-parent organisms, or must terminate. By proving that a certain measure (arguably an intelligence measure) decreases when an idealized parent AGI single-handedly creates a child AGI, we argue that a similar Law holds for AGIs

Graph ambiguity

In this paper, we propose a rigorous way to define the concept of ambiguity in the domain of graphs. In past studies, the classical definition of ambiguity has been derived starting from fuzzy set and fuzzy information theories. Our aim is to show that also in the domain of the graphs it is possible to derive a formulation able to capture the same semantic and mathematical concept. To strengthen the theoretical results, we discuss the application of the graph ambiguity concept to the graph classification setting, conceiving a new kind of inexact graph matching procedure. The results prove that the graph ambiguity concept is a characterizing and discriminative property of graphs. (C) 2013 Elsevier B.V. All rights reserved

Towards an Algebra for Cascade Effects

We introduce a new class of (dynamical) systems that inherently capture cascading effects (viewed as consequential effects) and are naturally amenable to combinations. We develop an axiomatic general theory around those systems, and guide the endeavor towards an understanding of cascading failure. The theory evolves as an interplay of lattices and fixed points, and its results may be instantiated to commonly studied models of cascade effects. We characterize the systems through their fixed points, and equip them with two operators. We uncover properties of the operators, and express global systems through combinations of local systems. We enhance the theory with a notion of failure, and understand the class of shocks inducing a system to failure. We develop a notion of mu-rank to capture the energy of a system, and understand the minimal amount of effort required to fail a system, termed resilience. We deduce a dual notion of fragility and show that the combination of systems sets a limit on the amount of fragility inherited.Comment: 31 page