5 research outputs found
Model-Centric Data Manifold: The Data Through the Eyes of the Model
We show that deep ReLU neural network classifiers can see a low-dimensional Riemannian manifold structure on data. Such structure comes via the \sl local data matrix, a variation of the Fisher information matrix, where the role of the model parameters is taken by the data variables. We obtain a foliation of the data domain, and we show that the dataset on which the model is trained lies on a leaf, the \sl data leaf, whose dimension is bounded by the number of classification labels. We validate our results with some experiments with the MNIST dataset: paths on the data leaf connect valid images, while other leaves cover noisy images
Challenges and opportunities in machine learning for geometry
Over the past few decades, the mathematical community has accumulated a significant amount of pure mathematical data, which has been analyzed through supervised, semi-supervised, and unsupervised machine learning techniques with remarkable results, e.g., artificial neural networks, support vector machines, and principal component analysis. Therefore, we consider as disruptive the use of machine learning algorithms to study mathematical structures, enabling the formulation of conjectures via numerical algorithms. In this paper, we review the latest applications of machine learning in the field of geometry. Artificial intelligence can help in mathematical problem solving, and we predict a blossoming of machine learning applications during the next years in the field of geometry. As a contribution, we propose a new method for extracting geometric information from the point cloud and reconstruct a 2D or a 3D model, based on the novel concept of generalized asymptotes.Agencia Estatal de Investigació
Horizontal Flows and Manifold Stochastics in Geometric Deep Learning
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
Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds
Motivated by the vast success of deep convolutional networks, there is a
great interest in generalizing convolutions to non-Euclidean manifolds. A major
complication in comparison to flat spaces is that it is unclear in which
alignment a convolution kernel should be applied on a manifold. The underlying
reason for this ambiguity is that general manifolds do not come with a
canonical choice of reference frames (gauge). Kernels and features therefore
have to be expressed relative to arbitrary coordinates. We argue that the
particular choice of coordinatization should not affect a network's inference
-- it should be coordinate independent. A simultaneous demand for coordinate
independence and weight sharing is shown to result in a requirement on the
network to be equivariant under local gauge transformations (changes of local
reference frames). The ambiguity of reference frames depends thereby on the
G-structure of the manifold, such that the necessary level of gauge
equivariance is prescribed by the corresponding structure group G. Coordinate
independent convolutions are proven to be equivariant w.r.t. those isometries
that are symmetries of the G-structure. The resulting theory is formulated in a
coordinate free fashion in terms of fiber bundles. To exemplify the design of
coordinate independent convolutions, we implement a convolutional network on
the M\"obius strip. The generality of our differential geometric formulation of
convolutional networks is demonstrated by an extensive literature review which
explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general
surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions
is publicly available at https://github.com/mauriceweiler/MobiusCNN