Learning With ℓ1-Graph for Image Analysis

Bin Cheng, Jianchao Yang, Shuicheng YAN, Yun Fu, Thomas S. Huang

Abstract: The graph construction procedure essentially determines the potentials of those graph-oriented learning algorithms for image analysis. In this paper, we propose a process to build the so-called directed ¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -graph, in which the vertices involve all the samples and the ingoing edge weights to each vertex describe its ¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -norm driven reconstruction from the remaining samples and the noise. Then, a series of new algorithms for various machine learning tasks, e.g., data clustering, subspace learning, and semi-supervised learning, are derived upon the ¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -graphs. Compared with the conventional <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -nearest-neighbor graph and ¿-ball graph, the ¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -graph possesses the advantages: (1) greater robustness to data noise, (2) automatic sparsity, and (3) adaptive neighborhood for individual datum. Extensive experiments on three real-world datasets show the consistent superiority of ¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -graph over those classic graphs in data clustering, subspace learning, and semi-supervised learning tasks.