65 research outputs found
Weak and strong fillability of higher dimensional contact manifolds
For contact manifolds in dimension three, the notions of weak and strong
symplectic fillability and tightness are all known to be inequivalent. We
extend these facts to higher dimensions: in particular, we define a natural
generalization of weak fillings and prove that it is indeed weaker (at least in
dimension five),while also being obstructed by all known manifestations of
"overtwistedness". We also find the first examples of contact manifolds in all
dimensions that are not symplectically fillable but also cannot be called
overtwisted in any reasonable sense. These depend on a higher-dimensional
analogue of Giroux torsion, which we define via the existence in all dimensions
of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor
edits. v3: exposition improved using referee's comments. Published by Invent.
Mat
Families of contact 3-manifolds with arbitrarily large Stein fillings
We show that there are vast families of contact 3-manifolds each member of
which admits infinitely many Stein fillings with arbitrarily big euler
characteristics and arbitrarily small signatures ---which disproves a
conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework
which generalizes the construction of Stein structures on allowable Lefschetz
fibrations over the 2-disk to those over any orientable base surface, along
with the construction of contact structures via open books on 3-manifolds to
spinal open books introduced in [24].Comment: 36 pages, 9 figures, with an appendix by Samuel Lisi and Chris Wend
The Singular Value Decomposition, Applications and Beyond
The singular value decomposition (SVD) is not only a classical theory in
matrix computation and analysis, but also is a powerful tool in machine
learning and modern data analysis. In this tutorial we first study the basic
notion of SVD and then show the central role of SVD in matrices. Using
majorization theory, we consider variational principles of singular values and
eigenvalues. Built on SVD and a theory of symmetric gauge functions, we discuss
unitarily invariant norms, which are then used to formulate general results for
matrix low rank approximation. We study the subdifferentials of unitarily
invariant norms. These results would be potentially useful in many machine
learning problems such as matrix completion and matrix data classification.
Finally, we discuss matrix low rank approximation and its recent developments
such as randomized SVD, approximate matrix multiplication, CUR decomposition,
and Nystrom approximation. Randomized algorithms are important approaches to
large scale SVD as well as fast matrix computations
Estimating quantum chromatic numbers
We develop further the new versions of quantum chromatic numbers of graphs
introduced by the first and fourth authors. We prove that the problem of
computation of the commuting quantum chromatic number of a graph is solvable by
an SDP algorithm and describe an hierarchy of variants of the commuting quantum
chromatic number which converge to it. We introduce the tracial rank of a
graph, a parameter that gives a lower bound for the commuting quantum chromatic
number and parallels the projective rank, and prove that it is multiplicative.
We describe the tracial rank, the projective rank and the fractional chromatic
numbers in a unified manner that clarifies their connection with the commuting
quantum chromatic number, the quantum chromatic number and the classical
chromatic number, respectively. Finally, we present a new SDP algorithm that
yields a parameter larger than the Lov\'asz number and is yet a lower bound for
the tracial rank of the graph. We determine the precise value of the tracial
rank of an odd cycle.Comment: 34 pages; v2 has improved presentation based after referees'
comments, published versio
Subgroup discovery for structured target concepts
The main object of study in this thesis is subgroup discovery, a theoretical framework for finding subgroups in data—i.e., named sub-populations— whose behaviour with respect to a specified target concept is exceptional when compared to the rest of the dataset. This is a powerful tool that conveys crucial information to a human audience, but despite past advances has been limited to simple target concepts. In this work we propose algorithms that bring this framework to novel application domains. We introduce the concept of representative subgroups, which we use not only to ensure the fairness of a sub-population with regard to a sensitive trait, such as race or gender, but also to go beyond known trends in the data. For entities with additional relational information that can be encoded as a graph, we introduce a novel measure of robust connectedness which improves on established alternative measures of density; we then provide a method that uses this measure to discover which named sub-populations are more well-connected. Our contributions within subgroup discovery crescent with the introduction of kernelised subgroup discovery: a novel framework that enables the discovery of subgroups on i.i.d. target concepts with virtually any kind of structure. Importantly, our framework additionally provides a concrete and efficient tool that works out-of-the-box without any modification, apart from specifying the Gramian of a positive definite kernel. To use within kernelised subgroup discovery, but also on any other kind of kernel method, we additionally introduce a novel random walk graph kernel. Our kernel allows the fine tuning of the alignment between the vertices of the two compared graphs, during the count of the random walks, while we also propose meaningful structure-aware vertex labels to utilise this new capability. With these contributions we thoroughly extend the applicability of subgroup discovery and ultimately re-define it as a kernel method.Der Hauptgegenstand dieser Arbeit ist die Subgruppenentdeckung (Subgroup Discovery), ein theoretischer Rahmen für das Auffinden von Subgruppen in Daten—d. h. benannte Teilpopulationen—deren Verhalten in Bezug auf ein bestimmtes Targetkonzept im Vergleich zum Rest des Datensatzes außergewöhnlich ist. Es handelt sich hierbei um ein leistungsfähiges Instrument, das einem menschlichen Publikum wichtige Informationen vermittelt. Allerdings ist es trotz bisherigen Fortschritte auf einfache Targetkonzepte beschränkt. In dieser Arbeit schlagen wir Algorithmen vor, die diesen Rahmen auf neuartige Anwendungsbereiche übertragen. Wir führen das Konzept der repräsentativen Untergruppen ein, mit dem wir nicht nur die Fairness einer Teilpopulation in Bezug auf ein sensibles Merkmal wie Rasse oder Geschlecht sicherstellen, sondern auch über bekannte Trends in den Daten hinausgehen können. Für Entitäten mit zusätzlicher relationalen Information, die als Graph kodiert werden kann, führen wir ein neuartiges Maß für robuste Verbundenheit ein, das die etablierten alternativen Dichtemaße verbessert; anschließend stellen wir eine Methode bereit, die dieses Maß verwendet, um herauszufinden, welche benannte Teilpopulationen besser verbunden sind. Unsere Beiträge in diesem Rahmen gipfeln in der Einführung der kernelisierten Subgruppenentdeckung: ein neuartiger Rahmen, der die Entdeckung von Subgruppen für u.i.v. Targetkonzepten mit praktisch jeder Art von Struktur ermöglicht. Wichtigerweise, unser Rahmen bereitstellt zusätzlich ein konkretes und effizientes Werkzeug, das ohne jegliche Modifikation funktioniert, abgesehen von der Angabe des Gramian eines positiv definitiven Kernels. Für den Einsatz innerhalb der kernelisierten Subgruppentdeckung, aber auch für jede andere Art von Kernel-Methode, führen wir zusätzlich einen neuartigen Random-Walk-Graph-Kernel ein. Unser Kernel ermöglicht die Feinabstimmung der Ausrichtung zwischen den Eckpunkten der beiden unter-Vergleich-gestelltenen Graphen während der Zählung der Random Walks, während wir auch sinnvolle strukturbewusste Vertex-Labels vorschlagen, um diese neue Fähigkeit zu nutzen. Mit diesen Beiträgen erweitern wir die Anwendbarkeit der Subgruppentdeckung gründlich und definieren wir sie im Endeffekt als Kernel-Methode neu
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