2,859 research outputs found
Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
The Euclidean scattering transform was introduced nearly a decade ago to
improve the mathematical understanding of convolutional neural networks.
Inspired by recent interest in geometric deep learning, which aims to
generalize convolutional neural networks to manifold and graph-structured
domains, we define a geometric scattering transform on manifolds. Similar to
the Euclidean scattering transform, the geometric scattering transform is based
on a cascade of wavelet filters and pointwise nonlinearities. It is invariant
to local isometries and stable to certain types of diffeomorphisms. Empirical
results demonstrate its utility on several geometric learning tasks. Our
results generalize the deformation stability and local translation invariance
of Euclidean scattering, and demonstrate the importance of linking the used
filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer
comment
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
Learning to Reconstruct Texture-less Deformable Surfaces from a Single View
Recent years have seen the development of mature solutions for reconstructing
deformable surfaces from a single image, provided that they are relatively
well-textured. By contrast, recovering the 3D shape of texture-less surfaces
remains an open problem, and essentially relates to Shape-from-Shading. In this
paper, we introduce a data-driven approach to this problem. We introduce a
general framework that can predict diverse 3D representations, such as meshes,
normals, and depth maps. Our experiments show that meshes are ill-suited to
handle texture-less 3D reconstruction in our context. Furthermore, we
demonstrate that our approach generalizes well to unseen objects, and that it
yields higher-quality reconstructions than a state-of-the-art SfS technique,
particularly in terms of normal estimates. Our reconstructions accurately model
the fine details of the surfaces, such as the creases of a T-Shirt worn by a
person.Comment: Accepted to 3DV 201
NeuroMorph: Unsupervised Shape Interpolation and Correspondence in One Go
We present NeuroMorph, a new neural network architecture that takes as input
two 3D shapes and produces in one go, i.e. in a single feed forward pass, a
smooth interpolation and point-to-point correspondences between them. The
interpolation, expressed as a deformation field, changes the pose of the source
shape to resemble the target, but leaves the object identity unchanged.
NeuroMorph uses an elegant architecture combining graph convolutions with
global feature pooling to extract local features. During training, the model is
incentivized to create realistic deformations by approximating geodesics on the
underlying shape space manifold. This strong geometric prior allows to train
our model end-to-end and in a fully unsupervised manner without requiring any
manual correspondence annotations. NeuroMorph works well for a large variety of
input shapes, including non-isometric pairs from different object categories.
It obtains state-of-the-art results for both shape correspondence and
interpolation tasks, matching or surpassing the performance of recent
unsupervised and supervised methods on multiple benchmarks.Comment: Published at the IEEE/CVF Conference on Computer Vision and Pattern
Recognition 202
Discovering Representations for Black-box Optimization
The encoding of solutions in black-box optimization is a delicate,
handcrafted balance between expressiveness and domain knowledge -- between
exploring a wide variety of solutions, and ensuring that those solutions are
useful. Our main insight is that this process can be automated by generating a
dataset of high-performing solutions with a quality diversity algorithm (here,
MAP-Elites), then learning a representation with a generative model (here, a
Variational Autoencoder) from that dataset. Our second insight is that this
representation can be used to scale quality diversity optimization to higher
dimensions -- but only if we carefully mix solutions generated with the learned
representation and those generated with traditional variation operators. We
demonstrate these capabilities by learning an low-dimensional encoding for the
inverse kinematics of a thousand joint planar arm. The results show that
learned representations make it possible to solve high-dimensional problems
with orders of magnitude fewer evaluations than the standard MAP-Elites, and
that, once solved, the produced encoding can be used for rapid optimization of
novel, but similar, tasks. The presented techniques not only scale up quality
diversity algorithms to high dimensions, but show that black-box optimization
encodings can be automatically learned, rather than hand designed.Comment: Presented at GECCO 2020 -- v2 (Previous title 'Automating
Representation Discovery with MAP-Elites'
Diskrete Spin-Geometrie für Flächen
This thesis proposes a discrete framework for spin geometry of surfaces. Specifically, we discretize the basic notions in spin geometry, such as the spin structure, spin connection and Dirac operator. In this framework, two types of Dirac operators are closely related as in smooth case. Moreover, they both induce the discrete conformal immersion with prescribed mean curvature half-density.In dieser Arbeit wird ein diskreter Zugang zur Spin-Geometrie vorgestellt. Insbesondere diskretisieren wir die grundlegende Begriffe, wie zum Beispiel die Spin-Struktur, den Spin-Zusammenhang und den Dirac Operator. In diesem Rahmen sind zwei Varianten fĂĽr den Dirac Operator eng verwandt wie in der glatten Theorie. DarĂĽber hinaus induzieren beide die diskret-konforme Immersion mit vorgeschriebener Halbdichte der mittleren KrĂĽmmung
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