We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Table Of Integrals Series And Products

The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In particular, it is often desirable in graph signal processing applications that a graph is chosen such that the data admit certain regularity or smoothness on the graph. In this paper, we address the problem of learning graph Laplacians, which is equivalent to learning graph topologies, such that the input data form graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforces such property and is based on minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from signal observations under the smoothness prior.

Learning Laplacian Matrix in Smooth Graph Signal Representations

Free Random Variables

Theory Of Function Spaces Ii

We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction.

Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames

The Large Time Frequency Analysis Toolbox (LTFAT) is a modern Octave/Matlab toolbox for time-frequency analysis, synthesis, coefficient manipulation and visualization. It’s purpose is to serve as a tool for achieving new scientific developments as well as an educational tool. The present paper introduces main features of the second major release of the toolbox which includes: generalizations of the Gabor transform, the wavelets module, the frames framework and the real-time block processing framework.

The Large Time-Frequency Analysis Toolbox 2.0

In the context of railway safety, it is crucial to know the positions of all trains moving along the infrastructure. In this contribution, we present an algorithm that extracts the positions of moving trains for a given point in time from Distributed Acoustic Sensing (DAS) signals. These signals are obtained by injecting light pulses into an optical fiber close to the railway tracks and measuring the Rayleigh backscatter. We show that the vibrations of moving objects can be identified and tracked in real-time yielding train positions every second. To speed up the algorithm, we describe how the calculations can partly be based on graphical processing units. The tracking quality is assessed by counting the inaccurate and lost train tracks for two different types of cable installations.

Real-Time Train Tracking from Distributed Acoustic Sensing Data

This paper presents a computationally realizable approach for the construction of an approximate dual wavelet frame. We are considering wavelet frames on the real line obtained by appropriate translation and dilation of a single given atom. We show asymptotic results in operator norm. Also we present numerical results to demonstrate the realizability of the approximation.

Construction of approximate dual wavelet frames

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group , where and are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order.

/pdf/representing-and-counting-the-subgroups-of-the-group-erm67noyuh.pdf

Christoph Wiesmeyr

Papers

Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames

The Large Time-Frequency Analysis Toolbox 2.0

Real-Time Train Tracking from Distributed Acoustic Sensing Data

Construction of approximate dual wavelet frames

Representing and Counting the Subgroups of the Group