Feature Selection for Unsupervised and Supervised Inference: The Emergence of Sparsity in a Weight-Based Approach

Lior Wolf, Amnon Shashua; 6(Nov):1855--1887, 2005.


The problem of selecting a subset of relevant features in a potentially overwhelming quantity of data is classic and found in many branches of science. Examples in computer vision, text processing and more recently bio-informatics are abundant. In text classification tasks, for example, it is not uncommon to have 104 to 107 features of the size of the vocabulary containing word frequency counts, with the expectation that only a small fraction of them are relevant. Typical examples include the automatic sorting of URLs into a web directory and the detection of spam email.

In this work we present a definition of "relevancy" based on spectral properties of the Laplacian of the features' measurement matrix. The feature selection process is then based on a continuous ranking of the features defined by a least-squares optimization process. A remarkable property of the feature relevance function is that sparse solutions for the ranking values naturally emerge as a result of a "biased non-negativity" of a key matrix in the process. As a result, a simple least-squares optimization process converges onto a sparse solution, i.e., a selection of a subset of features which form a local maximum over the relevance function. The feature selection algorithm can be embedded in both unsupervised and supervised inference problems and empirical evidence show that the feature selections typically achieve high accuracy even when only a small fraction of the features are relevant.


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