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Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Data Mining - Concepts and Techniques.

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

In this work, we are interested in generalizing convolutional neural networks (CNNs) from low-dimensional regular grids, where image, video and speech are represented, to high-dimensional irregular domains, such as social networks, brain connectomes or words' embedding, represented by graphs. We present a formulation of CNNs in the context of spectral graph theory, which provides the necessary mathematical background and efficient numerical schemes to design fast localized convolutional filters on graphs. Importantly, the proposed technique offers the same linear computational complexity and constant learning complexity as classical CNNs, while being universal to any graph structure. Experiments on MNIST and 20NEWS demonstrate the ability of this novel deep learning system to learn local, stationary, and compositional features on graphs.

Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering

We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.

/pdf/recent-advances-in-graph-partitioning-4eois71ejs.pdf

Recent Advances in Graph Partitioning

Graph partitioning is a well-known optimization problem of great interest in theoretical and applied studies. Since the 1990s, many multilevel schemes have been introduced as a practical tool to solve this problem. A multilevel algorithm may be viewed as a process of graph topology learning at different scales in order to generate a better approximation for any approximation method incorporated at the uncoarsening stage in the framework. In this work we compare two multilevel frameworks based on the geometric and the algebraic multigrid schemes for the partitioning problem.

/pdf/comparison-of-coarsening-schemes-for-multilevel-graph-48low01w6k.pdf

Comparison of Coarsening Schemes for Multilevel Graph Partitioning

In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the "closeness" between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization.

Relaxation-based coarsening and multiscale graph organization

http://www.wisdom.weizmann.ac.il/~achi/LOP162.pdf

Relaxation-Based Coarsening and Multiscale Graph Organization

Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.

/pdf/algebraic-distance-on-graphs-459b0psyn1.pdf

Ilya Safro

Papers

Recent Advances in Graph Partitioning

Comparison of Coarsening Schemes for Multilevel Graph Partitioning

Relaxation-based coarsening and multiscale graph organization

Relaxation-Based Coarsening and Multiscale Graph Organization

Algebraic Distance on Graphs