15 research outputs found
Helicity amplitudes for high-energy scattering
We present a prescription to calculate manifestly gauge invariant tree-level
helicity amplitudes for arbitrary scattering processes with off-shell
initial-state gluons within the kinematics of high-energy scattering. We show
that it is equivalent to Lipatov's effective action approach, and show its
computational potential through numerical calculations for scattering processes
with several particles in the final state.Comment: 27 pages, reference adde
Realisations of elliptic operators on compact manifolds with boundary
This paper investigates realisations of elliptic differential operators of
general order on manifolds with boundary following the approach of
B\"ar-Ballmann to first order elliptic operators. The space of possible
boundary values of elements in the maximal domain is described as a Hilbert
space densely sandwiched between two mixed order Sobolev spaces. The
description uses Calder\'on projectors which, in the first order case, is
equivalent to results of B\"ar-Bandara using spectral projectors of an adapted
boundary operator. Boundary conditions that induce Fredholm as well as regular
realisations, and those that admit higher order regularity, are characterised.
In addition, results concerning spectral theory, homotopy invariance of the
Fredholm index, and well-posedness for higher order elliptic boundary value
problems are proven
MODEL SELECTION TECHNIQUES FOR REPEATED MEASURES COVARIANCE STRUCTURES
A parsimonious covariance structure of repeated measures is often sought for purposes of increased power for testing hypotheses about the means, and for insight into the stochastic processes governing the repeated measures. For normal data, model selection is often based upon likelihood ratio tests or information criteria derived from the likelihood, sometimes supplemented with graphical plots of correlations and partial correlations. We exploit the ordered nature of repeated measures to decompose the likelihood ratio goodness-of-fit test statistic, and display graphical fingerprints associated with the covariance structures to help detect covariance structure misspecification, in order to provide guidance in choosing an appropriate structure for the data. The proposed methodology is illustrated with simulated repeated measures data and then applied to an experiment to compare tillage methods of pasture establishment
Planar Doodles: Their Properties, Codes and Classification
We present those properties of planar doodles, especially when regarded as
4-valent graphs, that enable us to classify them into {\it prime} and {\it
super prime} doodles by analogy to a knot sum. We describe a method for
partially characterising a doodle diagram by a {\it doodle code} that describes
the complementary regions of the diagram and use that code to enumerate all
possible prime and super prime doodle diagrams via their dual graph. In
addition we explore the relationship between planar doodles and twin groups,
and note that a theorem of Tutte means that super prime doodles have a
Hamiltonian circuit. We hope to expand upon this last point in a follow-up
paper.Comment: 28 pages, many figures and table
Learning Sparse Graphon Mean Field Games
Although the field of multi-agent reinforcement learning (MARL) has made
considerable progress in the last years, solving systems with a large number of
agents remains a hard challenge. Graphon mean field games (GMFGs) enable the
scalable analysis of MARL problems that are otherwise intractable. By the
mathematical structure of graphons, this approach is limited to dense graphs
which are insufficient to describe many real-world networks such as power law
graphs. Our paper introduces a novel formulation of GMFGs, called LPGMFGs,
which leverages the graph theoretical concept of graphons and provides a
machine learning tool to efficiently and accurately approximate solutions for
sparse network problems. This especially includes power law networks which are
empirically observed in various application areas and cannot be captured by
standard graphons. We derive theoretical existence and convergence guarantees
and give empirical examples that demonstrate the accuracy of our learning
approach for systems with many agents. Furthermore, we extend the Online Mirror
Descent (OMD) learning algorithm to our setup to accelerate learning speed,
empirically show its capabilities, and conduct a theoretical analysis using the
novel concept of smoothed step graphons. In general, we provide a scalable,
mathematically well-founded machine learning approach to a large class of
otherwise intractable problems of great relevance in numerous research fields.Comment: accepted for publication at the International Conference on
Artificial Intelligence and Statistics (AISTATS) 2023; code available at:
https://github.com/ChrFabian/Learning_sparse_GMFG
Learning Decentralized Partially Observable Mean Field Control for Artificial Collective Behavior
Recent reinforcement learning (RL) methods have achieved success in various
domains. However, multi-agent RL (MARL) remains a challenge in terms of
decentralization, partial observability and scalability to many agents.
Meanwhile, collective behavior requires resolution of the aforementioned
challenges, and remains of importance to many state-of-the-art applications
such as active matter physics, self-organizing systems, opinion dynamics, and
biological or robotic swarms. Here, MARL via mean field control (MFC) offers a
potential solution to scalability, but fails to consider decentralized and
partially observable systems. In this paper, we enable decentralized behavior
of agents under partial information by proposing novel models for decentralized
partially observable MFC (Dec-POMFC), a broad class of problems with
permutation-invariant agents allowing for reduction to tractable single-agent
Markov decision processes (MDP) with single-agent RL solution. We provide
rigorous theoretical results, including a dynamic programming principle,
together with optimality guarantees for Dec-POMFC solutions applied to finite
swarms of interest. Algorithmically, we propose Dec-POMFC-based policy gradient
methods for MARL via centralized training and decentralized execution, together
with policy gradient approximation guarantees. In addition, we improve upon
state-of-the-art histogram-based MFC by kernel methods, which is of separate
interest also for fully observable MFC. We evaluate numerically on
representative collective behavior tasks such as adapted Kuramoto and Vicsek
swarming models, being on par with state-of-the-art MARL. Overall, our
framework takes a step towards RL-based engineering of artificial collective
behavior via MFC.Comment: Accepted to ICLR 202