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Horizontal Flows and Manifold Stochastics in Geometric Deep Learning
TLDR
Two constructions in geometric deep learning are introduced for transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy.Abstract:
We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods -- here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.read more
Citations
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A Manifold-based Airfoil Geometric-feature Extraction and Discrepant Data Fusion Learning Method
TL;DR: A manifold-based airfoil geometric- feature extraction and discrepant data fusion learning method (MDF) to extract geometric- features of airfoils in Riemannian space and further fuse the manifold- features with flight conditions to predict aerodynamic performances.
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Challenges and Opportunities in Machine Learning for Geometry
TL;DR: In this article , a new method for extracting geometric information from the point cloud and reconstructing a 2D or a 3D model, based on the novel concept of generalized asymptotes, was proposed.
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Deep Learning and Geometric Deep Learning: an introduction for mathematicians and physicists
Rita Fioresi,F. Zanchetta +1 more
TL;DR: In this article , the authors give a brief introduction to the inner functioning of the new and successfull algorithms of Deep Learning and Geometric Deep Learning with a focus on Graph Neural Networks.
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Bundle geodesic convolutional neural network for diffusion-weighted imaging segmentation
TL;DR: In this article , a tissue classifier based on a Riemannian deep learning framework for single-shell diffusion-weighted imaging (DWI) data is presented. But it is not possible to learn general patterns from a very limited amount of training data if we take advantage of the geometry of the DWI data.
References
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Proceedings Article
Dropout as a Bayesian approximation: representing model uncertainty in deep learning
Yarin Gal,Zoubin Ghahramani +1 more
TL;DR: A new theoretical framework is developed casting dropout training in deep neural networks (NNs) as approximate Bayesian inference in deep Gaussian processes, which mitigates the problem of representing uncertainty in deep learning without sacrificing either computational complexity or test accuracy.
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Geometric Deep Learning: Going beyond Euclidean data
TL;DR: 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 as mentioned in this paper.
Proceedings ArticleDOI
Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs
Federico Monti,Davide Boscaini,Jonathan Masci,Emanuele Rodolà,Jan Svoboda,Michael M. Bronstein +5 more
TL;DR: In this article, a unified framework allowing to generalize CNN architectures to non-Euclidean domains (graphs and manifolds) and learn local, stationary, and compositional task-specific features is proposed.
Journal ArticleDOI
A Riemannian Framework for Tensor Computing
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Book
Natural operations in differential geometry
TL;DR: In this article, the authors present a general theory of Lie Derivatives and their application in a variety of fields and functions, including bundles and bundles of bundles on manifolds.