5,210 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
The goal of these course notes is to describe the main mathematical ideas behind geometric deep learning and to provide implementation details for several applications in shape analysis and synthesis, computer vision and computer graphics. The text in the course materials is primarily based on previously published work. With these notes we gather and provide a clear picture of the key concepts and techniques that fall under the umbrella of geometric deep learning, and illustrate the applications they enable. We also aim to provide practical implementation details for the methods presented in these works, as well as suggest further readings and extensions of these ideas
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
Deep Shape Representations for 3D Object Recognition
Deep learning is a rapidly growing discipline that models high-level features in data as multilayered
neural networks. The recent trend toward deep neural networks has been driven, in large part, by
a combination of affordable computing hardware, open source software, and the availability of
pre-trained networks on large-scale datasets.
In this thesis, we propose deep learning approaches to 3D shape recognition using a multilevel
feature learning paradigm. We start by comprehensively reviewing recent shape descriptors,
including hand-crafted descriptors that are mostly developed in the spectral geometry setting and
also the ones obtained via learning-based methods. Then, we introduce novel multi-level feature
learning approaches using spectral graph wavelets, bag-of-features and deep learning. Low-level
features are first extracted from a 3D shape using spectral graph wavelets. Mid-level features are
then generated via the bag-of-features model by employing locality-constrained linear coding as a
feature coding method, in conjunction with the biharmonic distance and intrinsic spatial pyramid
matching in a bid to effectively measure the spatial relationship between each pair of the bag-offeature
descriptors.
For the task of 3D shape retrieval, high-level shape features are learned via a deep auto-encoder
on mid-level features. Then, we compare the deep learned descriptor of a query shape to the
descriptors of all shapes in the dataset using a dissimilarity measure for 3D shape retrieval. For the
task of 3D shape classification, mid-level features are represented as 2D images in order to be fed
into a pre-trained convolutional neural network to learn high-level features from the penultimate
fully-connected layer of the network. Finally, a multiclass support vector machine classifier is
trained on these deep learned descriptors, and the classification accuracy is subsequently computed.
The proposed 3D shape retrieval and classification approaches are evaluated on three standard 3D
shape benchmarks through extensive experiments, and the results show compelling superiority of
our approaches over state-of-the-art methods
Wavelet Lifting over Information-Based EEG Graphs for Motor Imagery Data Classification
The imagination of limb movements offers an intuitive paradigm for the control of electronic devices via brain computer interfacing (BCI). The analysis of electroencephalographic (EEG) data related to motor imagery potentials has proved to be a difficult task. EEG readings are noisy, and the elicited patterns occur in different parts of the scalp, at different instants and at different frequencies. Wavelet transform has been widely used in the BCI field as it offers temporal and spectral capabilities, although it lacks spatial information. In this study we propose a tailored second generation wavelet to extract features from these three domains. This transform is applied over a graph representation of motor imaginary trials, which encodes temporal and spatial information. This graph is enhanced using per-subject knowledge in order to optimise the spatial relationships among the electrodes, and to improve the filter design. This method improves the performance of classifying different imaginary limb movements maintaining the low computational resources required by the lifting transform over graphs. By using an online dataset we were able to positively assess the feasibility of using the novel method in an online BCI context
Wavelets and graph -algebras
Here we give an overview on the connection between wavelet theory and
representation theory for graph -algebras, including the higher-rank
graph -algebras of A. Kumjian and D. Pask. Many authors have studied
different aspects of this connection over the last 20 years, and we begin this
paper with a survey of the known results. We then discuss several new ways to
generalize these results and obtain wavelets associated to representations of
higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets"
associated to a higher-rank graph. Here, we generalize this construction to
build wavelets of arbitrary shapes. We also present a different but related
construction of wavelets associated to a higher-rank graph, which we anticipate
will have applications to traffic analysis on networks. Finally, we generalize
the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a
third family of wavelets associated to higher-rank graphs
- …