An Efficient Algorithm to Test Potential Bipartiteness of Graphical Degree Sequences

As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in polynomial time, at the expense of possibly producing an erroneous output on some ``yes\u27\u27 instances but with very low error rate

Sublinear Computation Paradigm

This open access book gives an overview of cutting-edge work on a new paradigm called the “sublinear computation paradigm,” which was proposed in the large multiyear academic research project “Foundations of Innovative Algorithms for Big Data.” That project ran from October 2014 to March 2020, in Japan. To handle the unprecedented explosion of big data sets in research, industry, and other areas of society, there is an urgent need to develop novel methods and approaches for big data analysis. To meet this need, innovative changes in algorithm theory for big data are being pursued. For example, polynomial-time algorithms have thus far been regarded as “fast,” but if a quadratic-time algorithm is applied to a petabyte-scale or larger big data set, problems are encountered in terms of computational resources or running time. To deal with this critical computational and algorithmic bottleneck, linear, sublinear, and constant time algorithms are required. The sublinear computation paradigm is proposed here in order to support innovation in the big data era. A foundation of innovative algorithms has been created by developing computational procedures, data structures, and modelling techniques for big data. The project is organized into three teams that focus on sublinear algorithms, sublinear data structures, and sublinear modelling. The work has provided high-level academic research results of strong computational and algorithmic interest, which are presented in this book. The book consists of five parts: Part I, which consists of a single chapter on the concept of the sublinear computation paradigm; Parts II, III, and IV review results on sublinear algorithms, sublinear data structures, and sublinear modelling, respectively; Part V presents application results. The information presented here will inspire the researchers who work in the field of modern algorithms

GP 2: Efficient Implementation of a Graph Programming Language

The graph programming language GP (Graph Programs) 2 and its implementation is the subject of this thesis. The language allows programmers to write visual graph programs at a high level of abstraction, bringing the task of solving graph-based problems to an environment in which the user feels comfortable and secure. Implementing graph programs presents two main challenges. The first challenge is translating programs from a high-level source code representation to executable code, which involves bridging the gap from a non-deterministic program to deterministic machine code. The second challenge is overcoming the theoretically impractical complexity of applying graph transformation rules, the basic computation step of a graph program. The work presented in this thesis addresses both of these challenges. We tackle the first challenge by implementing a compiler that translates GP 2 graph programs directly to C code. Implementation strategies concerning the storage and access of internal data structures are empirically compared to determine the most efficient approach for executing practical graph programs. The second challenge is met by extending the double-pushout approach to graph transformation with root nodes to support fast execution of graph transformation rules by restricting the search to the local neighbourhood of the root nodes in the host graph. We add this theoretical construct to the GP 2 language in order to support rooted graph transformation rules, and we identify a class of rooted rules that are applicable in constant time on certain classes of graphs. Finally, we combine theory and practice by writing rooted graph programs to solve two common graph algorithms, and demonstrate that their execution times are capable of matching the execution times of tailored C solutions

Symmetries & tensor networks in two-dimensional quantum physics

The most general description of a quantum many-body system is given by a wave- function that lives in a Hilbert space with dimension exponential in the number of particles. This makes it extremely hard to study strongly correlated phenomena like the fractional quantum Hall effect and high-temperature superconductivity. Whenever interactions are sufficiently local and temperature is low, the system does not explore the full Hilbert space, but its ground state resides in the small corner of Hilbert space described by the area law. Containing little entanglement, the states can then be expressed as tensor networks, a family of wavefunctions with a polynomial number of parameters. On the one hand, tensor networks can be used as a variational manifold in nu- merical computations. On the other hand, they allow building model wavefunctions much like locality allows writing down physically realistic Hamiltonians. Besides allowing for an analytical treatment, these models grant access both to the physical and the entanglement degrees of freedom. This is particularly useful in classifying phases of matter. A large number of phases can be explained in terms of Landau’s symmetry-breaking paradigm. This framework, however, is not complete, as exemplified by the existence of phases with intrinsic topological order in two dimensions. It was a major conceptual advance when tensor networks could explain (non-chiral) topological phases as those where the symmetry resides in the entanglement degrees of freedom. The symmetries corresponding to those topological phases act as discrete, finite groups on the virtual degrees of freedom. The purpose of this Thesis is to generalize this program to include other symmetries. We investigate a class of tensor networks with continuous symmetries and find that they cannot describe gapped physics with a unique ground state. The abelian case is found to describe a non-Lorentz invariant phase transition point into a topologically ordered phase. The physics of the non- abelian case is that of a plaquette state that spontaneously breaks the translation symmetry of the lattice. The non-abelian PEPS arises as the ground state of a local parent Hamiltonian whose ground state manifold is completely characterized by the tensor network. In both cases, we find two types of corrections to the entanglement entropy: first there is a correction that is logarithmic in the size of the boundary and independent of the shape. A further correction depends only on the shape of the partition, imposing further restrictions on regions that are suffciently thin. Finally, we investigate symmetries that mix the virtual with the physical degrees of freedom and are furthermore anisotropic. Their physics is described by subsystem symmetry protected topological order. In particular, we focus on the entanglement entropy in the cluster phase and show that there is a universal constant correction to the entropy throughout the phase. This is important in the program of establishing the entanglement entropy as a detection mechanism for topologically ordered phases. We put forward a numerical algorithm to compute the correction and use it to discover a novel phase of matter in which the cluster phase is embedded.Die allgemeinste Beschreibung eines Quanten-Vielteilchensystems ergibt sich aus einer Wellenfunktion, die in einem Hilbert-Raum lebt, dessen Dimension exponentiell in der Anzahl der Teilchen ist. Dies macht es äußerst schwierig, stark korrelierte Phänomene wie den fraktionalen Quanten-Hall-Effekt und die Hochtemperatursupraleitung zu untersuchen. Wenn die Wechselwirkungen ausreichend lokal sind und die Temperatur niedrig ist, steht dem System nicht der gesamte Hilbert-Raum zur Verfügung. Sein Grundzustand befindet sich in der kleinen "Ecke" des Hilbert-Raums, die durch das area law beschrieben wird. Mit wenig Verschränkung können die Zustände dann als Tensornetzwerke ausgedrückt werden, eine Familie von Wellenfunktionen mit einer polynomiellen Anzahl von Parametern. Einerseits können Tensornetzwerke als variationelle Ansätze bei numerischen Berechnungen verwendet werden. Auf der anderen Seite ermöglichen sie das Erstellen von Modellwellenfunktionen. Diese Modelle ermöglichen nicht nur eine analytische Behandlung, sondern gewähren auch Zugang zu den physikalischen und den Verschränkungsfreiheitsgraden. Dies ist besonders nützlich bei der Klassifizierung von Phasen der Materie. Eine große Anzahl von Phasen kann mit Landaus Theorie der Symmetriebrechung erklärt werden. Diese Beschreibung ist jedoch nicht vollständig, was durch die Existenz von Phasen mit intrinsischer topologischer Ordnung in zwei Dimensionen veranschaulicht wird. Es war ein großer konzeptioneller Fortschritt, als Tensornetzwerke (nicht-chirale) topologische Phasen als solche identifizieren konnten, bei denen die Symmetrie in den Verschränkungsfreiheitsgraden liegt. Die diesen topologischen Phasen entsprechenden Symmetrien wirken als diskrete, endliche Gruppen auf den virtuellen Freiheitsgraden. Der Zweck dieser Arbeit ist es, dieses Programm auf andere Symmetrien zu verallgemeinern. Wir untersuchen eine Klasse von Tensornetzwerken mit kontinuierlichen Symmetrien und stellen fest, dass sie keine mit eindeutigen Grundzustand unter einer Energielücke beschreiben können. Der abelsche Fall beschreibt einen nicht-Lorentz-invarianten Phasenübergangspunkt in eine topologisch geordnete Phase. Die Physik des nicht-abelschen Falls ist die eines Plaquette-Zustands, der spontan die Translationssymmetrie des Gitters bricht. Der nicht-abelsche PEPS entsteht als Grundzustand eines lokalen \textit{parent}-Hamiltonians, dessen Grundzustandsunterraum vollständig durch das Tensornetzwerk beschrieben wird. In beiden Fällen finden wir zwei Arten von Korrekturen an der Verschränkungsentropie: Erstens gibt es eine Korrektur, die in der Größe der Grenze logarithmisch und unabhängig von der Form ist. Eine weitere Korrektur hängt nur von der Form des Schnitts ab ab, wodurch ausreichend dünne Bereiche weiter eingeschränkt werden. Schließlich untersuchen wir Symmetrien, die virtuelle und physikalische Freiheitsgraden mischen und darüber hinaus anisotrop sind. Ihre Physik wird durch topologische Ordnung beschrieben, die stabil ist solange bestimmte Subsystem-Symmetrien nicht gebrochen werden. Insbesondere konzentrieren wir uns auf die Verschränkungsentropie in der Clusterphase und zeigen, dass die Entropie in der gesamten Phase universell eine konstante Korrektur erhält. Dies ist wichtig im Programm zur Etablierung der Verschränkungsentropie als Detektionsmechanismus für topologisch geordnete Phasen. Wir schlagen einen numerischen Algorithmus vor, um die Korrektur zu berechnen und entdecken eine neue Phase der Materie, in die die Clusterphase eingebettet ist

Representation Learning for Words and Entities

This thesis presents new methods for unsupervised learning of distributed representations of words and entities from text and knowledge bases. The first algorithm presented in the thesis is a multi-view algorithm for learning representations of words called Multiview Latent Semantic Analysis (MVLSA). By incorporating up to 46 different types of co-occurrence statistics for the same vocabulary of english words, I show that MVLSA outperforms other state-of-the-art word embedding models. Next, I focus on learning entity representations for search and recommendation and present the second method of this thesis, Neural Variational Set Expansion (NVSE). NVSE is also an unsupervised learning method, but it is based on the Variational Autoencoder framework. Evaluations with human annotators show that NVSE can facilitate better search and recommendation of information gathered from noisy, automatic annotation of unstructured natural language corpora. Finally, I move from unstructured data and focus on structured knowledge graphs. I present novel approaches for learning embeddings of vertices and edges in a knowledge graph that obey logical constraints.Comment: phd thesis, Machine Learning, Natural Language Processing, Representation Learning, Knowledge Graphs, Entities, Word Embeddings, Entity Embedding

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum