# Semi-Supervised Eigenvectors for Large-Scale Locally-Biased Learning

Toke J. Hansen, Michael W. Mahoney.

Year: 2014, Volume: 15, Issue: 114, Pages: 3871−3914

#### Abstract

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.