3,448 research outputs found
Network Density of States
Spectral analysis connects graph structure to the eigenvalues and
eigenvectors of associated matrices. Much of spectral graph theory descends
directly from spectral geometry, the study of differentiable manifolds through
the spectra of associated differential operators. But the translation from
spectral geometry to spectral graph theory has largely focused on results
involving only a few extreme eigenvalues and their associated eigenvalues.
Unlike in geometry, the study of graphs through the overall distribution of
eigenvalues - the spectral density - is largely limited to simple random graph
models. The interior of the spectrum of real-world graphs remains largely
unexplored, difficult to compute and to interpret.
In this paper, we delve into the heart of spectral densities of real-world
graphs. We borrow tools developed in condensed matter physics, and add novel
adaptations to handle the spectral signatures of common graph motifs. The
resulting methods are highly efficient, as we illustrate by computing spectral
densities for graphs with over a billion edges on a single compute node. Beyond
providing visually compelling fingerprints of graphs, we show how the
estimation of spectral densities facilitates the computation of many common
centrality measures, and use spectral densities to estimate meaningful
information about graph structure that cannot be inferred from the extremal
eigenpairs alone.Comment: 10 pages, 7 figure
Statistical Physics and Representations in Real and Artificial Neural Networks
This document presents the material of two lectures on statistical physics
and neural representations, delivered by one of us (R.M.) at the Fundamental
Problems in Statistical Physics XIV summer school in July 2017. In a first
part, we consider the neural representations of space (maps) in the
hippocampus. We introduce an extension of the Hopfield model, able to store
multiple spatial maps as continuous, finite-dimensional attractors. The phase
diagram and dynamical properties of the model are analyzed. We then show how
spatial representations can be dynamically decoded using an effective Ising
model capturing the correlation structure in the neural data, and compare
applications to data obtained from hippocampal multi-electrode recordings and
by (sub)sampling our attractor model. In a second part, we focus on the problem
of learning data representations in machine learning, in particular with
artificial neural networks. We start by introducing data representations
through some illustrations. We then analyze two important algorithms, Principal
Component Analysis and Restricted Boltzmann Machines, with tools from
statistical physics
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
CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters
The rise of graph-structured data such as social networks, regulatory
networks, citation graphs, and functional brain networks, in combination with
resounding success of deep learning in various applications, has brought the
interest in generalizing deep learning models to non-Euclidean domains. In this
paper, we introduce a new spectral domain convolutional architecture for deep
learning on graphs. The core ingredient of our model is a new class of
parametric rational complex functions (Cayley polynomials) allowing to
efficiently compute spectral filters on graphs that specialize on frequency
bands of interest. Our model generates rich spectral filters that are localized
in space, scales linearly with the size of the input data for
sparsely-connected graphs, and can handle different constructions of Laplacian
operators. Extensive experimental results show the superior performance of our
approach, in comparison to other spectral domain convolutional architectures,
on spectral image classification, community detection, vertex classification
and matrix completion tasks
Automated design of robust discriminant analysis classifier for foot pressure lesions using kinematic data
In the recent years, the use of motion tracking systems for acquisition of functional biomechanical gait data, has received increasing interest due to the richness and accuracy of the measured kinematic information. However, costs frequently restrict the number of subjects employed, and this makes the dimensionality of the collected data far higher than the available samples. This paper applies discriminant analysis algorithms to the classification of patients with different types of foot lesions, in order to establish an association between foot motion and lesion formation. With primary attention to small sample size situations, we compare different types of Bayesian classifiers and evaluate their performance with various dimensionality reduction techniques for feature extraction, as well as search methods for selection of raw kinematic variables. Finally, we propose a novel integrated method which fine-tunes the classifier parameters and selects the most relevant kinematic variables simultaneously. Performance comparisons are using robust resampling techniques such as Bootstrapand k-fold cross-validation. Results from experimentations with lesion subjects suffering from pathological plantar hyperkeratosis, show that the proposed method can lead tocorrect classification rates with less than 10% of the original features
Analysis of Dynamic Mode Decomposition
In this master thesis, a study was conducted on a method known as Dynamic mode decomposition(DMD), an equation-free technique which does not require to know the underlying governing equations of the complex data. As a result of massive datasets from various resources, like experiments, simulation, historical records, etc. has led to an increasing demand for an efficient method for data mining and analysis techniques. The main goals of data mining are the description and prediction. Description involves finding patterns in the data and prediction involves predicting the system dynamics. An important aspect when analyzing an algorithm is testing. In this work, DMD-a data based technique is used to test different cases to find the underlying patterns, predict the system dynamics and for reconstruction of original data. Using real data for analyzing a new algorithm may not be appropriate due to lack of knowledge of the algorithm performance in various cases. So, testing is done on synthetic data for all the cases discussed in this work, as it is useful for visualization and to find the robustness of the new algorithm. Finally, this work makes an attempts to understand the DMD\u27s performance and limitations better for the future applications with real data
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