5,400 research outputs found
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
Adaptation and learning over networks for nonlinear system modeling
In this chapter, we analyze nonlinear filtering problems in distributed
environments, e.g., sensor networks or peer-to-peer protocols. In these
scenarios, the agents in the environment receive measurements in a streaming
fashion, and they are required to estimate a common (nonlinear) model by
alternating local computations and communications with their neighbors. We
focus on the important distinction between single-task problems, where the
underlying model is common to all agents, and multitask problems, where each
agent might converge to a different model due to, e.g., spatial dependencies or
other factors. Currently, most of the literature on distributed learning in the
nonlinear case has focused on the single-task case, which may be a strong
limitation in real-world scenarios. After introducing the problem and reviewing
the existing approaches, we describe a simple kernel-based algorithm tailored
for the multitask case. We evaluate the proposal on a simulated benchmark task,
and we conclude by detailing currently open problems and lines of research.Comment: To be published as a chapter in `Adaptive Learning Methods for
Nonlinear System Modeling', Elsevier Publishing, Eds. D. Comminiello and J.C.
Principe (2018
Graph Laplacians and their convergence on random neighborhood graphs
Given a sample from a probability measure with support on a submanifold in
Euclidean space one can construct a neighborhood graph which can be seen as an
approximation of the submanifold. The graph Laplacian of such a graph is used
in several machine learning methods like semi-supervised learning,
dimensionality reduction and clustering. In this paper we determine the
pointwise limit of three different graph Laplacians used in the literature as
the sample size increases and the neighborhood size approaches zero. We show
that for a uniform measure on the submanifold all graph Laplacians have the
same limit up to constants. However in the case of a non-uniform measure on the
submanifold only the so called random walk graph Laplacian converges to the
weighted Laplace-Beltrami operator.Comment: Improved presentation, typos corrected, to appear in JML
A deep learning approach to diabetic blood glucose prediction
We consider the question of 30-minute prediction of blood glucose levels
measured by continuous glucose monitoring devices, using clinical data. While
most studies of this nature deal with one patient at a time, we take a certain
percentage of patients in the data set as training data, and test on the
remainder of the patients; i.e., the machine need not re-calibrate on the new
patients in the data set. We demonstrate how deep learning can outperform
shallow networks in this example. One novelty is to demonstrate how a
parsimonious deep representation can be constructed using domain knowledge
Kernel-based Inference of Functions over Graphs
The study of networks has witnessed an explosive growth over the past decades
with several ground-breaking methods introduced. A particularly interesting --
and prevalent in several fields of study -- problem is that of inferring a
function defined over the nodes of a network. This work presents a versatile
kernel-based framework for tackling this inference problem that naturally
subsumes and generalizes the reconstruction approaches put forth recently by
the signal processing on graphs community. Both the static and the dynamic
settings are considered along with effective modeling approaches for addressing
real-world problems. The herein analytical discussion is complemented by a set
of numerical examples, which showcase the effectiveness of the presented
techniques, as well as their merits related to state-of-the-art methods.Comment: To be published as a chapter in `Adaptive Learning Methods for
Nonlinear System Modeling', Elsevier Publishing, Eds. D. Comminiello and J.C.
Principe (2018). This chapter surveys recent work on kernel-based inference
of functions over graphs including arXiv:1612.03615 and arXiv:1605.07174 and
arXiv:1711.0930
Density-sensitive semisupervised inference
Semisupervised methods are techniques for using labeled data
together with unlabeled data
to make predictions. These methods invoke some assumptions that link the
marginal distribution of X to the regression function f(x). For example,
it is common to assume that f is very smooth over high density regions of
. Many of the methods are ad-hoc and have been shown to work in specific
examples but are lacking a theoretical foundation. We provide a minimax
framework for analyzing semisupervised methods. In particular, we study methods
based on metrics that are sensitive to the distribution . Our model
includes a parameter that controls the strength of the semisupervised
assumption. We then use the data to adapt to .Comment: Published in at http://dx.doi.org/10.1214/13-AOS1092 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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