Regularized l1-Graph for Data Clustering

l1-Graph has been proven to be effective in data clustering, which partitions the data space by using the sparse representation of the data as the similarity measure. However, the sparse representation is performed for each datum independently without taking into account the geometric structure of the data. Motivated by l1-Graph and manifold leaning, we propose Regularized l1-Graph (Rl1-Graph) for data clustering. Compared to l1-Graph, the sparse representations of Rl1-Graph are regularized by the geometric information of the data. In accordance with the manifold assumption, the sparse representations vary smoothly along the geodesics of the data manifold through the graph Laplacian constructed by the sparse codes. Experimental results on various data sets demonstrate the superiority of our algorithm compared to l1-Graph and other competing clustering methods. l1-graph [2, 3], which builds the graph by reconstructing each datum with all the other data, has been shown to be robust to noise and capable of producing superior results for high dimensional data, compared to Kmeans and spectral clustering. Compared to k-nearest-neighbor graph and ε-ball graph, l1-graph adaptively determines the neighborhood of each datum by solving sparse representation problem locally. Given the data X = [x1, . . . ,xn] ∈ IRd×n, l1-graph seeks for the robust sparse representation for the entire data by solving the l1-norm optimization problem for each data point: