254 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
Compressive PCA for Low-Rank Matrices on Graphs
We introduce a novel framework for an approxi- mate recovery of data matrices
which are low-rank on graphs, from sampled measurements. The rows and columns
of such matrices belong to the span of the first few eigenvectors of the graphs
constructed between their rows and columns. We leverage this property to
recover the non-linear low-rank structures efficiently from sampled data
measurements, with a low cost (linear in n). First, a Resrtricted Isometry
Property (RIP) condition is introduced for efficient uniform sampling of the
rows and columns of such matrices based on the cumulative coherence of graph
eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is
suggested for the sampled data. Finally, several efficient, parallel and
parameter-free decoders are presented along with their theoretical analysis for
decoding the low-rank and cluster indicators for the full data matrix. Thus, we
overcome the computational limitations of the standard linear low-rank recovery
methods for big datasets. Our method can also be seen as a major step towards
efficient recovery of non- linear low-rank structures. For a matrix of size n X
p, on a single core machine, our method gains a speed up of over Robust
Principal Component Analysis (RPCA), where k << p is the subspace dimension.
Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times
faster than Robust PCA
Geometric Algebra Attention Networks for Small Point Clouds
Much of the success of deep learning is drawn from building architectures
that properly respect underlying symmetry and structure in the data on which
they operate - a set of considerations that have been united under the banner
of geometric deep learning. Often problems in the physical sciences deal with
relatively small sets of points in two- or three-dimensional space wherein
translation, rotation, and permutation equivariance are important or even vital
for models to be useful in practice. In this work, we present rotation- and
permutation-equivariant architectures for deep learning on these small point
clouds, composed of a set of products of terms from the geometric algebra and
reductions over those products using an attention mechanism. The geometric
algebra provides valuable mathematical structure by which to combine vector,
scalar, and other types of geometric inputs in a systematic way to account for
rotation invariance or covariance, while attention yields a powerful way to
impose permutation equivariance. We demonstrate the usefulness of these
architectures by training models to solve sample problems relevant to physics,
chemistry, and biology
Deep learning in remote sensing: a review
Standing at the paradigm shift towards data-intensive science, machine
learning techniques are becoming increasingly important. In particular, as a
major breakthrough in the field, deep learning has proven as an extremely
powerful tool in many fields. Shall we embrace deep learning as the key to all?
Or, should we resist a 'black-box' solution? There are controversial opinions
in the remote sensing community. In this article, we analyze the challenges of
using deep learning for remote sensing data analysis, review the recent
advances, and provide resources to make deep learning in remote sensing
ridiculously simple to start with. More importantly, we advocate remote sensing
scientists to bring their expertise into deep learning, and use it as an
implicit general model to tackle unprecedented large-scale influential
challenges, such as climate change and urbanization.Comment: Accepted for publication IEEE Geoscience and Remote Sensing Magazin
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