82,470 research outputs found
Inverse problem in cylindrical electrical networks
In this paper we study the inverse Dirichlet-to-Neumann problem for certain
cylindrical electrical networks. We define and study a birational
transformation acting on cylindrical electrical networks called the electrical
-matrix. We use this transformation to formulate a general conjectural
solution to this inverse problem on the cylinder. This conjecture extends work
of Curtis, Ingerman, and Morrow, and of de Verdi\`ere, Gitler, and Vertigan for
circular planar electrical networks. We show that our conjectural solution
holds for certain "purely cylindrical" networks. Here we apply the grove
combinatorics introduced by Kenyon and Wilson.Comment: 22 pages, 15 figure
Deep Learning on Lie Groups for Skeleton-based Action Recognition
In recent years, skeleton-based action recognition has become a popular 3D
classification problem. State-of-the-art methods typically first represent each
motion sequence as a high-dimensional trajectory on a Lie group with an
additional dynamic time warping, and then shallowly learn favorable Lie group
features. In this paper we incorporate the Lie group structure into a deep
network architecture to learn more appropriate Lie group features for 3D action
recognition. Within the network structure, we design rotation mapping layers to
transform the input Lie group features into desirable ones, which are aligned
better in the temporal domain. To reduce the high feature dimensionality, the
architecture is equipped with rotation pooling layers for the elements on the
Lie group. Furthermore, we propose a logarithm mapping layer to map the
resulting manifold data into a tangent space that facilitates the application
of regular output layers for the final classification. Evaluations of the
proposed network for standard 3D human action recognition datasets clearly
demonstrate its superiority over existing shallow Lie group feature learning
methods as well as most conventional deep learning methods.Comment: Accepted to CVPR 201
Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants
We construct a generalization of pure lattice gauge theory (LGT) where the
role of the gauge group is played by a tensor category. The type of tensor
category admissible (spherical, ribbon, symmetric) depends on the dimension of
the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the
category is the (symmetric) category of representations of a compact Lie group.
In the weak coupling limit we recover discretized BF-theory in terms of a
coordinate free version of the spin foam formulation. We work on general
cellular decompositions of the underlying manifold.
In particular, we are able to formulate LGT as well as spin foam models of
BF-type with quantum gauge group (in dimension <=4) and with supersymmetric
gauge group (in any dimension).
Technically, we express the partition function as a sum over diagrams
denoting morphisms in the underlying category. On the LGT side this enables us
to introduce a generalized notion of gauge fixing corresponding to a
topological move between cellular decompositions of the underlying manifold. On
the BF-theory side this allows a rather geometric understanding of the state
sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we
recover.
The construction is extended to include Wilson loop and spin network type
observables as well as manifolds with boundaries. In the topological (weak
coupling) case this leads to TQFTs with or without embedded spin networks.Comment: 58 pages, LaTeX with AMS and XY-Pic macros; typos corrected and
references update
Learning Unitary Operators with Help From u(n)
A major challenge in the training of recurrent neural networks is the
so-called vanishing or exploding gradient problem. The use of a norm-preserving
transition operator can address this issue, but parametrization is challenging.
In this work we focus on unitary operators and describe a parametrization using
the Lie algebra associated with the Lie group of unitary matrices. The exponential map provides a correspondence
between these spaces, and allows us to define a unitary matrix using real
coefficients relative to a basis of the Lie algebra. The parametrization is
closed under additive updates of these coefficients, and thus provides a simple
space in which to do gradient descent. We demonstrate the effectiveness of this
parametrization on the problem of learning arbitrary unitary operators,
comparing to several baselines and outperforming a recently-proposed
lower-dimensional parametrization. We additionally use our parametrization to
generalize a recently-proposed unitary recurrent neural network to arbitrary
unitary matrices, using it to solve standard long-memory tasks.Comment: 9 pages, 3 figures, 5 figures inc. subfigures, to appear at AAAI-1
An Integrated Approach to Seismic Event Location: 1. Evaluating How Method of Location Affects the Volume of Groups of Hypocenters
When seismic events occur in spatially compact clusters, the volume and geometric characteristics of these clusters often provides information about the relative effectiveness of different location methods, or about physical processes occurring within the hypocentral region. This report defines and explains how to determine the convex polyhedron of minimum volume (CPMV) surrounding a set of points. We evaluate both single-event and joint hypocenter determination (JHD) relocations for three rather different clusters of seismic events: 1) nuclear explosions from Mururoa relocated using P and PKP phases reported by the ISC, 2) intermediate depth earthquakes near Bucaramanga, Colombia, relocated using P and PKP phases reported by the ISC, and 3) shallow earthquakes near Vanuatu (formerly, the New Hebrides), relocated using P and S phases from a local station network. This analysis demonstrates that different location methods markedly affect the volume of the CPMV, however, volumes for JHD relations are �not always smaller than volumes for single-event relocations.Phillips Laboratory, Directorate of Geophysics, Air Force Material Command, Hanscom Air Force Base, MassachusettsInstitute for Geophysic
Box Graphs and Singular Fibers
We determine the higher codimension fibers of elliptically fibered Calabi-Yau
fourfolds with section by studying the three-dimensional N=2 supersymmetric
gauge theory with matter which describes the low energy effective theory of
M-theory compactified on the associated Weierstrass model, a singular model of
the fourfold. Each phase of the Coulomb branch of this theory corresponds to a
particular resolution of the Weierstrass model, and we show that these have a
concise description in terms of decorated box graphs based on the
representation graph of the matter multiplets, or alternatively by a class of
convex paths on said graph. Transitions between phases have a simple
interpretation as `flopping' of the path, and in the geometry correspond to
actual flop transitions. This description of the phases enables us to enumerate
and determine the entire network between them, with various matter
representations for all reductive Lie groups. Furthermore, we observe that each
network of phases carries the structure of a (quasi-)minuscule representation
of a specific Lie algebra. Interpreted from a geometric point of view, this
analysis determines the generators of the cone of effective curves as well as
the network of flop transitions between crepant resolutions of singular
elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types
in codimensions two and three, and we find new, non-Kodaira, fiber types for
E_6, E_7 and E_8.Comment: 107 pages, 44 figures, v2: added case of E7 monodromy-reduced fiber
- …