Institut de Mathématiques, Université de Toulouse et CNRS (UMR 5219), 118 route de Narbonne, 31062 Toulouse cedex 9, France Abstract Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts. This work was partially supported by ANR Project "Graph-Comp". Preprint submitted to Neurocomputing 19 March 2018 1 Introduction Complex networks are large graphs with a non trivial organization. They arise naturally in numerous context [7], such as, to name a few, the World Wide Web (which gives a perfect example of how large and complex such a network may grow), metabolic pathways, citation networks between scientific articles or more general social networks that model interaction between individuals and/or organizations, etc. Complex networks share common properties that have allowed the emergence of mathematical descriptions such as small world graphs or power law graphs. The structure of these graphs often gives some keys to understand the complex network underlined. To study such a structure, one often begins with a metrology process applied to the graph that describes the degree distribution, the number of components, the density, etc. However, it should be noted that dealing with very large graphs (millions of vertices) is still an open question (see [9] for an example of an efficient algorithm to explore that kind of data sets). Several ways have been explored to cluster the vertices of the graph into communities [43] and some of them have in common the use of the Laplacian matrix. Indeed, there are important relationships between the spectrum of the Laplacian and the graph invariants that characterize its structure (see, e.g. These properties can be used for building, from the eigen-decomposition of the Laplacian, a similarity measure or a metric space such that the induced dissimilarities between vertices of the graph are related to its community structure (see [13], among others).

Gaussian data is pervasive and many learning algorithms (e.g., k-means) model their inputs as a single sample drawn from a multivariate Gaussian. However, in many real-life settings, each input object is best described by multiple samples drawn from a multivariate Gaussian. Such data can arise, for example, in a movie review database where each movie is rated by several users, or in time-series domains such as sensor networks. Here, each input can be naturally described by both a mean vector and covariance matrix which parameterize the Gaussian distribution. In this paper, we consider the problem of clustering such input objects, each represented as a multivariate Gaussian. We formulate the problem using an information theoretic approach and draw several interesting theoretical connections to Bregman divergences and also Bregman matrix divergences. We evaluate our method across several domains, including synthetic data, sensor network data, and a statistical debugging application.

Finite mixture model is a powerful tool in many statistical learning problems. In this paper, we propose a general, structure-preserving approach to reduce its model complexity, which can bring significant computational benefits in many applications. The basic idea is to group the original mixture components into compact clusters, and then minimize an upper bound on the approximation error between the original and simplified models.

We present a local learning approach for clustering. The basic idea is that a good clustering result should have the property that the cluster label of each data point can be well predicted based on its neighboring data and their cluster labels, using current supervised learning methods. An optimization problem is formulated such that its solution has the above property. Relaxation and eigen-decomposition are applied to solve this optimization problem. We also briefly investigate the parameter selection issue and provide a simple parameter selection method for the proposed algorithm. Experimental results are provided to validate the effectiveness of the proposed approach.

We present a general model-independent approach to the analysis of data in cases when these data do not appear in the form of co-occurrence of two variables X,Y, but rather as a sample of values of an unknown (stochastic) function Z(X,Y). For example, in gene expression data, the expression level Z is a function of gene X and condition Y; or in movie ratings data the rating Z is a function of viewer X and movie Y. The approach represents a consistent extension of the Information Bottleneck method that has previously relied on the availability of co-occurrence statistics. By altering the relevance variable we eliminate the need in the sample of joint distribution of all input variables. This new formulation also enables simple MDL-like model complexity control and prediction of missing values of Z. The approach is analyzed and shown to be on a par with the best known clustering algorithms for a wide range of domains. For the prediction of missing values (collaborative filtering) it improves the currently best known results.

While for a finite number of minimizers this result follows from multinomial distribution estimates, the case of infinite minimizers requires more refined tools.

Spectral clustering methods are common graph-based approaches to clustering of data. Spectral clustering algorithms typically start from local information encoded in a weighted graph on the data and cluster according to the global eigenvectors of the corresponding (normalized) similarity matrix. One contribution of this paper is to present fundamental limitations of this general local to global approach. We show that based only on local information, the normalized cut functional is not a suitable measure for the quality of clustering. Further, even with a suitable similarity measure, we show that the first few eigenvectors of such adjacency matrices cannot successfully cluster datasets that contain structures at different scales of size and density. Based on these findings, a second contribution of this paper is a novel diffusion based measure to evaluate the coherence of individual clusters. Our measure can be used in conjunction with any bottom-up graph-based clustering method, it is scale-free and can determine coherent clusters at all scales. We present both synthetic examples and real image segmentation problems where various spectral clustering algorithms fail. In contrast, using this coherence measure finds the expected clusters at all scales.

Dirichlet Process (DP) mixture models are promising candidates for clustering applications where the number of clusters is unknown a priori. Due to computational considerations these models are unfortunately unsuitable for large scale data-mining applications. We propose a class of deterministic accelerated DP mixture models that can routinely handle millions of data-cases. The speedup is achieved by incorporating kd-trees into a variational Bayesian algorithm for DP mixtures in the stick-breaking representation, similar to that of Blei and Jordan (2005). Our algorithm differs in the use of kd-trees and in the way we handle truncation: we only assume that the variational distributions are fixed at their priors after a certain level. Experiments show that speedups relative to the standard variational algorithm can be significant.

We propose a new kernel-based data transformation technique. It is founded on the principle of maximum entropy (MaxEnt) preservation, hence named kernel MaxEnt. The key measure is Renyi's entropy estimated via Parzen windowing. We show that kernel MaxEnt is based on eigenvectors, and is in that sense similar to kernel PCA, but may produce strikingly different transformed data sets. An enhanced spectral clustering algorithm is proposed, by replacing kernel PCA by kernel MaxEnt as an intermediate step. This has a major impact on performance.

Pyramid intersection is an efficient method for computing an approximate partial matching between two sets of feature vectors. We introduce a novel pyramid embedding based on a hierarchy of non-uniformly shaped bins that takes advantage of the underlying structure of the feature space and remains accurate even for sets with high-dimensional feature vectors. The matching similarity is computed in linear time and forms a Mercer kernel. Whereas previous matching approximation algorithms suffer from distortion factors that increase linearly with the feature dimension, we demonstrate that our approach can maintain constant accuracy even as the feature dimension increases. When used as a kernel in a discriminative classifier, our approach achieves improved object recognition results over a state-of-the-art set kernel.