2,738 research outputs found
Techniques for high-multiplicity scattering amplitudes and applications to precision collider physics
In this thesis, we present state-of-the-art techniques for the computation of scattering amplitudes in Quantum Field Theories. Following an introduction to the topic, we describe a robust framework that enables the calculation of multi-scale two-loop amplitudes directly relevant to modern particle physics phenomenology at the Large Hadron Collider and beyond. We discuss in detail the use of finite fields to bypass the algebraic complexity of such computations, as well as the method of integration-by-parts relations and differential equations. We apply our framework to calculate the two-loop amplitudes contributing to three process: Higgs boson production in association with a bottom-quark pair, W boson production with a photon and a jet, as well as lepton-pair scattering with an off-shell and an on-shell photon. Finally, we draw our conclusions and discuss directions for future progress of amplitude computations
Categorical Invariants of Graphs and Matroids
Graphs and matroids are two of the most important objects in combinatorics.We study invariants of graphs and matroids that behave well with respect to
certain morphisms by realizing these invariants as functors from a category of
graphs (resp. matroids).
For graphs, we study invariants that respect deletions and contractions ofedges. For an integer , we define a category of of graphs of genus at most
g where morphisms correspond to deletions and contractions. We prove that this
category is locally Noetherian and show that many graph invariants form finitely
generated modules over the category . This fact allows us to exihibit many
stabilization properties of these invariants. In particular we show that the torsion
that can occur in the homologies of the unordered configuration space of n points
in a graph and the matching complex of a graph are uniform over the entire family
of graphs with genus .
For matroids, we study valuative invariants of matroids. Given a matroid,one can define a corresponding polytope called the base polytope. Often, the base
polytope of a matroid can be decomposed into a cell complex made up of base
polytopes of other matroids. A valuative invariant of matroids is an invariant that
respects these polytope decompositions. We define a category of matroids
whose morphisms correspond to containment of base polytopes. We then define the
notion of a categorical matroid invariant which categorifies the notion of a valuative
invariant. Finally, we prove that the functor sending a matroid to its Orlik-Solomon
algebra is a categorical valuative invariant. This allows us to derive relations among
the Orlik-Solomon algebras of a matroid and matroids that decompose its base
polytope viewed as representations of any group whose action is compatible with
the polytope decomposition.
This dissertation includes previously unpublished co-authored material
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Quantum ergodicity on the Bruhat-Tits building for in the Benjamini-Schramm limit
We study eigenfunctions of the spherical Hecke algebra acting on
where with a
non-archimedean local field of characteristic zero, with the ring of integers of , and
is a sequence of cocompact torsionfree lattices. We prove a form of
equidistribution on average for eigenfunctions whose spectral parameters lie in
the tempered spectrum when the associated sequence of quotients of the
Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table
BOtied: Multi-objective Bayesian optimization with tied multivariate ranks
Many scientific and industrial applications require joint optimization of
multiple, potentially competing objectives. Multi-objective Bayesian
optimization (MOBO) is a sample-efficient framework for identifying
Pareto-optimal solutions. We show a natural connection between non-dominated
solutions and the highest multivariate rank, which coincides with the outermost
level line of the joint cumulative distribution function (CDF). We propose the
CDF indicator, a Pareto-compliant metric for evaluating the quality of
approximate Pareto sets that complements the popular hypervolume indicator. At
the heart of MOBO is the acquisition function, which determines the next
candidate to evaluate by navigating the best compromises among the objectives.
Multi-objective acquisition functions that rely on box decomposition of the
objective space, such as the expected hypervolume improvement (EHVI) and
entropy search, scale poorly to a large number of objectives. We propose an
acquisition function, called BOtied, based on the CDF indicator. BOtied can be
implemented efficiently with copulas, a statistical tool for modeling complex,
high-dimensional distributions. We benchmark BOtied against common acquisition
functions, including EHVI and random scalarization (ParEGO), in a series of
synthetic and real-data experiments. BOtied performs on par with the baselines
across datasets and metrics while being computationally efficient.Comment: 10 pages (+5 appendix), 9 figures. Submitted to NeurIP
Factor Graph Neural Networks
In recent years, we have witnessed a surge of Graph Neural Networks (GNNs),
most of which can learn powerful representations in an end-to-end fashion with
great success in many real-world applications. They have resemblance to
Probabilistic Graphical Models (PGMs), but break free from some limitations of
PGMs. By aiming to provide expressive methods for representation learning
instead of computing marginals or most likely configurations, GNNs provide
flexibility in the choice of information flowing rules while maintaining good
performance. Despite their success and inspirations, they lack efficient ways
to represent and learn higher-order relations among variables/nodes. More
expressive higher-order GNNs which operate on k-tuples of nodes need increased
computational resources in order to process higher-order tensors. We propose
Factor Graph Neural Networks (FGNNs) to effectively capture higher-order
relations for inference and learning. To do so, we first derive an efficient
approximate Sum-Product loopy belief propagation inference algorithm for
discrete higher-order PGMs. We then neuralize the novel message passing scheme
into a Factor Graph Neural Network (FGNN) module by allowing richer
representations of the message update rules; this facilitates both efficient
inference and powerful end-to-end learning. We further show that with a
suitable choice of message aggregation operators, our FGNN is also able to
represent Max-Product belief propagation, providing a single family of
architecture that can represent both Max and Sum-Product loopy belief
propagation. Our extensive experimental evaluation on synthetic as well as real
datasets demonstrates the potential of the proposed model.Comment: Accepted by JML
Constructing cost-effective infrastructure networks
The need for reliable and low-cost infrastructure is crucial in today's
world. However, achieving both at the same time is often challenging.
Traditionally, infrastructure networks are designed with a radial topology
lacking redundancy, which makes them vulnerable to disruptions. As a result,
network topologies have evolved towards a ring topology with only one redundant
edge and, from there, to more complex mesh networks. However, we prove that
large rings are unreliable. Our research shows that a sparse mesh network with
a small number of redundant edges that follow some design rules can
significantly improve reliability while remaining cost-effective. Moreover, we
have identified key areas where adding redundant edges can impact network
reliability the most by using the SAIDI index, which measures the expected
number of consumers disconnected from the source node. These findings offer
network planners a valuable tool for quickly identifying and addressing
reliability issues without the need for complex simulations. Properly planned
sparse mesh networks can thus provide a reliable and a cost-effective solution
to modern infrastructure challenges
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
Advances and Applications of DSmT for Information Fusion. Collected Works, Volume 5
This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered.
First Part of this book presents some theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes.
Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification.
Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well
A Structural Approach to Tree Decompositions of Knots and Spatial Graphs
Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants and properties of the knot. It was recently proved that there exist knots which do not admit any diagram of low treewidth, and the proof relied on intricate low-dimensional topology techniques. In this work, we initiate a thorough investigation of tree decompositions of knot diagrams (or more generally, diagrams of spatial graphs) using ideas from structural graph theory. We define an obstruction on spatial embeddings that forbids low tree width diagrams, and we prove that it is optimal with respect to a related width invariant. We then show the existence of this obstruction for knots of high representativity, which include for example torus knots, providing a new and self-contained proof that those do not admit diagrams of low treewidth. This last step is inspired by a result of Pardon on knot distortion
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