Semi-supervised Eigenvectors for Large-scale Locally-biased Learning

In many applications, one has side information, e.g., labels that are provided in a semi-supervised manner, about a specific target region of a large data set, and one wants to perform machine learning and data analysis tasks "nearby" that prespecified target region. For example, one might be interested in the clustering structure of a data graph near a prespecified "seed set" of nodes, or one might be interested in finding partitions in an image that are near a prespecified "ground truth" set of pixels. Locally-biased problems of this sort are particularly challenging for popular eigenvector-based machine learning and data analysis tools. At root, the reason is that eigenvectors are inherently global quantities, thus limiting the applicability of eigenvector-based methods in situations where one is interested in very local properties of the data. In this paper, we address this issue by providing a methodology to construct semi-supervised eigenvectors of a graph Laplacian, and we illustrate how these locally-biased eigenvectors can be used to perform locally-biased machine learning. These semi-supervised eigenvectors capture successively-orthogonalized directions of maximum variance, conditioned on being well-correlated with an input seed set of nodes that is assumed to be provided in a semi-supervised manner. We show that these semi-supervised eigenvectors can be computed quickly as the solution to a system of linear equations; and we also describe several variants of our basic method that have improved scaling properties. We provide several empirical examples demonstrating how these semi-supervised eigenvectors can be used to perform locally-biased learning; and we discuss the relationship between our results and recent machine learning algorithms that use global eigenvectors of the graph Laplacian

Identifying networks with common organizational principles

Many complex systems can be represented as networks, and the problem of network comparison is becoming increasingly relevant. There are many techniques for network comparison, from simply comparing network summary statistics to sophisticated but computationally costly alignment-based approaches. Yet it remains challenging to accurately cluster networks that are of a different size and density, but hypothesized to be structurally similar. In this paper, we address this problem by introducing a new network comparison methodology that is aimed at identifying common organizational principles in networks. The methodology is simple, intuitive and applicable in a wide variety of settings ranging from the functional classification of proteins to tracking the evolution of a world trade network.Comment: 26 pages, 7 figure

A Statistical Toolbox For Mining And Modeling Spatial Data

Most data mining projects in spatial economics start with an evaluation of a set of attribute variables on a sample of spatial entities, looking for the existence and strength of spatial autocorrelation, based on the Moran’s and the Geary’s coefficients, the adequacy of which is rarely challenged, despite the fact that when reporting on their properties, many users seem likely to make mistakes and to foster confusion. My paper begins by a critical appraisal of the classical definition and rational of these indices. I argue that while intuitively founded, they are plagued by an inconsistency in their conception. Then, I propose a principled small change leading to corrected spatial autocorrelation coefficients, which strongly simplifies their relationship, and opens the way to an augmented toolbox of statistical methods of dimension reduction and data visualization, also useful for modeling purposes. A second section presents a formal framework, adapted from recent work in statistical learning, which gives theoretical support to our definition of corrected spatial autocorrelation coefficients. More specifically, the multivariate data mining methods presented here, are easily implementable on the existing (free) software, yield methods useful to exploit the proposed corrections in spatial data analysis practice, and, from a mathematical point of view, whose asymptotic behavior, already studied in a series of papers by Belkin & Niyogi, suggests that they own qualities of robustness and a limited sensitivity to the Modifiable Areal Unit Problem (MAUP), valuable in exploratory spatial data analysis

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation