Unsupervised and Semi-Supervised Learning via ℓ1-Norm Graph

Abstract

In this paper, we propose a novel ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm graph model to perform unsupervised and semi-supervised learning methods. Instead of minimizing the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm of spectral embedding as traditional graph based learning methods, our new graph learning model minimizes the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm of spectral embedding with well motivation. The sparsity produced by the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization results in the solutions with much clearer cluster structures, which are suitable for both image clustering and classification tasks. We introduce a new efficient iterative algorithm to solve the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm of spectral embedding minimization problem, and prove the convergence of the algorithm. More specifically, our algorithm adaptively re-weight the original weights of graph to discover clearer cluster structure. Experimental results on both toy data and real image data sets show the effectiveness and advantages of our proposed method.