28,951 research outputs found
Spatial Aggregation: Theory and Applications
Visual thinking plays an important role in scientific reasoning. Based on the
research in automating diverse reasoning tasks about dynamical systems,
nonlinear controllers, kinematic mechanisms, and fluid motion, we have
identified a style of visual thinking, imagistic reasoning. Imagistic reasoning
organizes computations around image-like, analogue representations so that
perceptual and symbolic operations can be brought to bear to infer structure
and behavior. Programs incorporating imagistic reasoning have been shown to
perform at an expert level in domains that defy current analytic or numerical
methods. We have developed a computational paradigm, spatial aggregation, to
unify the description of a class of imagistic problem solvers. A program
written in this paradigm has the following properties. It takes a continuous
field and optional objective functions as input, and produces high-level
descriptions of structure, behavior, or control actions. It computes a
multi-layer of intermediate representations, called spatial aggregates, by
forming equivalence classes and adjacency relations. It employs a small set of
generic operators such as aggregation, classification, and localization to
perform bidirectional mapping between the information-rich field and
successively more abstract spatial aggregates. It uses a data structure, the
neighborhood graph, as a common interface to modularize computations. To
illustrate our theory, we describe the computational structure of three
implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the
spatial aggregation generic operators by mixing and matching a library of
commonly used routines.Comment: See http://www.jair.org/ for any accompanying file
Learning SO(3) Equivariant Representations with Spherical CNNs
We address the problem of 3D rotation equivariance in convolutional neural
networks. 3D rotations have been a challenging nuisance in 3D classification
tasks requiring higher capacity and extended data augmentation in order to
tackle it. We model 3D data with multi-valued spherical functions and we
propose a novel spherical convolutional network that implements exact
convolutions on the sphere by realizing them in the spherical harmonic domain.
Resulting filters have local symmetry and are localized by enforcing smooth
spectra. We apply a novel pooling on the spectral domain and our operations are
independent of the underlying spherical resolution throughout the network. We
show that networks with much lower capacity and without requiring data
augmentation can exhibit performance comparable to the state of the art in
standard retrieval and classification benchmarks.Comment: Camera-ready. Accepted to ECCV'18 as oral presentatio
Multi-copy and stochastic transformation of multipartite pure states
Characterizing the transformation and classification of multipartite
entangled states is a basic problem in quantum information. We study the
problem under two most common environments, local operations and classical
communications (LOCC), stochastic LOCC and two more general environments,
multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two
transformable multipartite states under LOCC or SLOCC are also transformable
under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in
the sense that two transformable states under MCLOCC are also transformable
under MCSLOCC, and vice versa. Based on these environments we classify the
multipartite pure states into a few inequivalent sets and orbits, between which
we build the partial order to decide their transformation. In particular, we
investigate the structure of SLOCC-equivalent states in terms of tensor rank,
which is known as the generalized Schmidt rank. Given the tensor rank, we show
that GHZ states can be used to generate all states with a smaller or equivalent
tensor rank under SLOCC, and all reduced separable states with a cardinality
smaller or equivalent than the tensor rank under LOCC. Using these concepts, we
extended the concept of "maximally entangled state" in the multi-partite
system.Comment: 8 pages, 1 figure, revised version according to colleagues' comment
Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics
We give a classification of generic coadjoint orbits for the groups of
symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic
surface. We also classify simple Morse functions on symplectic surfaces with
respect to actions of those groups. This gives an answer to V.Arnold's problem
on describing all invariants of generic isovorticed fields for the 2D ideal
fluids. For this we introduce a notion of anti-derivatives on a measured Reeb
graph and describe their properties.Comment: 38 pages, 11 figures; to appear in Annales de l'Institut Fourie
PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures
Persistence diagrams, the most common descriptors of Topological Data
Analysis, encode topological properties of data and have already proved pivotal
in many different applications of data science. However, since the (metric)
space of persistence diagrams is not Hilbert, they end up being difficult
inputs for most Machine Learning techniques. To address this concern, several
vectorization methods have been put forward that embed persistence diagrams
into either finite-dimensional Euclidean space or (implicit) infinite
dimensional Hilbert space with kernels. In this work, we focus on persistence
diagrams built on top of graphs. Relying on extended persistence theory and the
so-called heat kernel signature, we show how graphs can be encoded by
(extended) persistence diagrams in a provably stable way. We then propose a
general and versatile framework for learning vectorizations of persistence
diagrams, which encompasses most of the vectorization techniques used in the
literature. We finally showcase the experimental strength of our setup by
achieving competitive scores on classification tasks on real-life graph
datasets
Borel reductions of profinite actions of SL(n,Z)
Greg Hjorth and Simon Thomas proved that the classification problem for
torsion-free abelian groups of finite rank \emph{strictly increases} in
complexity with the rank. Subsequently, Thomas proved that the complexity of
the classification problems for -local torsion-free abelian groups of fixed
rank are \emph{pairwise incomparable} as varies. We prove that if
and are distinct primes, then the complexity of the
classification problem for -local torsion-free abelian groups of rank is
again incomparable with that for -local torsion-free abelian groups of rank
- …