778 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
Multiscale Geometric Methods for Data Sets II: Geometric Multi-Resolution Analysis
Data sets are often modeled as point clouds in , for large. It is
often assumed that the data has some interesting low-dimensional structure, for
example that of a -dimensional manifold , with much smaller than .
When is simply a linear subspace, one may exploit this assumption for
encoding efficiently the data by projecting onto a dictionary of vectors in
(for example found by SVD), at a cost for data points. When
is nonlinear, there are no "explicit" constructions of dictionaries that
achieve a similar efficiency: typically one uses either random dictionaries, or
dictionaries obtained by black-box optimization. In this paper we construct
data-dependent multi-scale dictionaries that aim at efficient encoding and
manipulating of the data. Their construction is fast, and so are the algorithms
that map data points to dictionary coefficients and vice versa. In addition,
data points are guaranteed to have a sparse representation in terms of the
dictionary. We think of dictionaries as the analogue of wavelets, but for
approximating point clouds rather than functions.Comment: Re-formatted using AMS styl
Coronal Mass Ejection Detection using Wavelets, Curvelets and Ridgelets: Applications for Space Weather Monitoring
Coronal mass ejections (CMEs) are large-scale eruptions of plasma and
magnetic feld that can produce adverse space weather at Earth and other
locations in the Heliosphere. Due to the intrinsic multiscale nature of
features in coronagraph images, wavelet and multiscale image processing
techniques are well suited to enhancing the visibility of CMEs and supressing
noise. However, wavelets are better suited to identifying point-like features,
such as noise or background stars, than to enhancing the visibility of the
curved form of a typical CME front. Higher order multiscale techniques, such as
ridgelets and curvelets, were therefore explored to characterise the morphology
(width, curvature) and kinematics (position, velocity, acceleration) of CMEs.
Curvelets in particular were found to be well suited to characterising CME
properties in a self-consistent manner. Curvelets are thus likely to be of
benefit to autonomous monitoring of CME properties for space weather
applications.Comment: Accepted for publication in Advances in Space Research (3 April 2010
Adaptive multiresolution analysis based on synchronization
We propose an adaptive multiscale approach to data analysis based on synchronization. The approach is nonlinear, data driven in the sense that it does not rely on a priori chosen basis, and automatically determines the data scale. Numerical results for one- and two-dimensional cases illustrate that the method works effectively for the usual modulated signals such as chirps, etc., as well as for more complicated data with multiple scales. The method extends straightforwardly to functions defined on weighted graphs and grids in high dimensions. Connections with some other recent approaches to multiscale analysis are briefly discussed
The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch
Recent and forthcoming advances in instrumentation, and giant new surveys,
are creating astronomical data sets that are not amenable to the methods of
analysis familiar to astronomers. Traditional methods are often inadequate not
merely because of the size in bytes of the data sets, but also because of the
complexity of modern data sets. Mathematical limitations of familiar algorithms
and techniques in dealing with such data sets create a critical need for new
paradigms for the representation, analysis and scientific visualization (as
opposed to illustrative visualization) of heterogeneous, multiresolution data
across application domains. Some of the problems presented by the new data sets
have been addressed by other disciplines such as applied mathematics,
statistics and machine learning and have been utilized by other sciences such
as space-based geosciences. Unfortunately, valuable results pertaining to these
problems are mostly to be found only in publications outside of astronomy. Here
we offer brief overviews of a number of concepts, techniques and developments,
some "old" and some new. These are generally unknown to most of the
astronomical community, but are vital to the analysis and visualization of
complex datasets and images. In order for astronomers to take advantage of the
richness and complexity of the new era of data, and to be able to identify,
adopt, and apply new solutions, the astronomical community needs a certain
degree of awareness and understanding of the new concepts. One of the goals of
this paper is to help bridge the gap between applied mathematics, artificial
intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in
Astronomy, special issue "Robotic Astronomy
Mapping and monitoring forest remnants : a multiscale analysis of spatio-temporal data
KEYWORDS : Landsat, time series, machine learning, semideciduous Atlantic forest, Brazil, wavelet transforms, classification, change detectionForests play a major role in important global matters such as carbon cycle, climate change, and biodiversity. Besides, forests also influence soil and water dynamics with major consequences for ecological relations and decision-making. One basic requirement to quantify and model these processes is the availability of accurate maps of forest cover. Data acquisition and analysis at appropriate scales is the keystone to achieve the mapping accuracy needed for development and reliable use of ecological models.The current and upcoming production of high-resolution data sets plus the ever-increasing time series that have been collected since the seventieth must be effectively explored. Missing values and distortions further complicate the analysis of this data set. Thus, integration and proper analysis is of utmost importance for environmental research. New conceptual models in environmental sciences, like the perception of multiple scales, require the development of effective implementation techniques.This thesis presents new methodologies to map and monitor forests on large, highly fragmented areas with complex land use patterns. The use of temporal information is extensively explored to distinguish natural forests from other land cover types that are spectrally similar. In chapter 4, novel schemes based on multiscale wavelet analysis are introduced, which enabled an effective preprocessing of long time series of Landsat data and improved its applicability on environmental assessment.In chapter 5, the produced time series as well as other information on spectral and spatial characteristics were used to classify forested areas in an experiment relating a number of combinations of attribute features. Feature sets were defined based on expert knowledge and on data mining techniques to be input to traditional and machine learning algorithms for pattern recognition, viz . maximum likelihood, univariate and multivariate decision trees, and neural networks. The results showed that maximum likelihood classification using temporal texture descriptors as extracted with wavelet transforms was most accurate to classify the semideciduous Atlantic forest in the study area.In chapter 6, a multiscale approach to digital change detection was developed to deal with multisensor and noisy remotely sensed images. Changes were extracted according to size classes minimising the effects of geometric and radiometric misregistration.Finally, in chapter 7, an automated procedure for GIS updating based on feature extraction, segmentation and classification was developed to monitor the remnants of semideciduos Atlantic forest. The procedure showed significant improvements over post classification comparison and direct multidate classification based on artificial neural networks.</p
Graph Signal Processing: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
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
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