2,172 research outputs found
Geometric deep learning: going beyond Euclidean data
Many scientific fields study data with an underlying structure that is a
non-Euclidean space. Some examples include social networks in computational
social sciences, sensor networks in communications, functional networks in
brain imaging, regulatory networks in genetics, and meshed surfaces in computer
graphics. In many applications, such geometric data are large and complex (in
the case of social networks, on the scale of billions), and are natural targets
for machine learning techniques. In particular, we would like to use deep
neural networks, which have recently proven to be powerful tools for a broad
range of problems from computer vision, natural language processing, and audio
analysis. However, these tools have been most successful on data with an
underlying Euclidean or grid-like structure, and in cases where the invariances
of these structures are built into networks used to model them. Geometric deep
learning is an umbrella term for emerging techniques attempting to generalize
(structured) deep neural models to non-Euclidean domains such as graphs and
manifolds. The purpose of this paper is to overview different examples of
geometric deep learning problems and present available solutions, key
difficulties, applications, and future research directions in this nascent
field
Localized Manifold Harmonics for Spectral Shape Analysis
The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis. We show how to efficiently construct localized orthogonal bases by solving an optimization problem that in turn can be posed as the eigendecomposition of a new operator obtained by a modification of the standard Laplacian. We study the theoretical and computational aspects of the proposed framework and showcase our new construction on the classical problems of shape approximation and correspondence. We obtain significant improvement compared to classical Laplacian eigenbases as well as other alternatives for constructing localized bases
(MS)SM-like models on smooth Calabi-Yau manifolds from all three heterotic string theories
We perform model searches on smooth Calabi-Yau compactifications for both the
supersymmetric E8xE8 and SO(32) as well as for the non-supersymmetric
SO(16)xSO(16) heterotic strings simultaneously. We consider line bundle
backgrounds on both favorable CICYs with relatively small h_11 and the Schoen
manifold. Using Gram matrices we systematically analyze the combined
consequences of the Bianchi identities and the tree-level
Donaldson-Uhlenbeck-Yau equations inside the Kahler cone. In order to evaluate
the model building potential of the three heterotic theories on the various
geometries, we perform computer-aided scans. We have generated a large number
of GUT-like models (up to over a few hundred thousand on the various geometries
for the three heterotic theories) which become (MS)SM-like upon using a freely
acting Wilson line. For all three heterotic theories we present tables and
figures summarizing the potentially phenomenologically interesting models which
were obtained during our model scans.Comment: 1+39 pages latex, 4 figures, 16 tables. v3: Scans and statistics for
the non-supersymmetric models redone in light of the corrected flux
quantization conditio
Representation Learning: A Review and New Perspectives
The success of machine learning algorithms generally depends on data
representation, and we hypothesize that this is because different
representations can entangle and hide more or less the different explanatory
factors of variation behind the data. Although specific domain knowledge can be
used to help design representations, learning with generic priors can also be
used, and the quest for AI is motivating the design of more powerful
representation-learning algorithms implementing such priors. This paper reviews
recent work in the area of unsupervised feature learning and deep learning,
covering advances in probabilistic models, auto-encoders, manifold learning,
and deep networks. This motivates longer-term unanswered questions about the
appropriate objectives for learning good representations, for computing
representations (i.e., inference), and the geometrical connections between
representation learning, density estimation and manifold learning
PyDEC: Software and Algorithms for Discretization of Exterior Calculus
This paper describes the algorithms, features and implementation of PyDEC, a
Python library for computations related to the discretization of exterior
calculus. PyDEC facilitates inquiry into both physical problems on manifolds as
well as purely topological problems on abstract complexes. We describe
efficient algorithms for constructing the operators and objects that arise in
discrete exterior calculus, lowest order finite element exterior calculus and
in related topological problems. Our algorithms are formulated in terms of
high-level matrix operations which extend to arbitrary dimension. As a result,
our implementations map well to the facilities of numerical libraries such as
NumPy and SciPy. The availability of such libraries makes Python suitable for
prototyping numerical methods. We demonstrate how PyDEC is used to solve
physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability,
removed redundant text, emphasized the role of matrix based algorithms,
shortened length of pape
Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Equivariant Projected Kernels
Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge equivariant kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners
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