Neighborhood Regularized l1-Graph

l(1)-Graph, which learns a sparse graph over the data by sparse representation, has been demonstrated to be effective in clustering especially for high dimensional data. Although it achieves compelling performance, the sparse graph generated by l(1)-Graph ignores the geometric information of the data by sparse representation for each datum separately. To obtain a sparse graph that is aligned to the underlying manifold structure of the data, we propose the novel Neighborhood Regularized l(1)-Graph (NRl(1)-Graph). NRl(1)-Graph learns sparse graph with locally consistent neighborhood by encouraging nearby data to have similar neighbors in the constructed sparse graph. We present the optimization algorithm of NRl(1)-Graph with theoretical guarantee on the convergence and the gap between the sub-optimal solution and the globally optimal solution in each step of the coordinate descent, which is essential for the overall optimization of NRl(1)-Graph. Its provable accelerated version, NRl(1)-Graph by Random Projection (NRl(1)-Graph-RP) that employs randomized data matrix decomposition, is also presented to improve the efficiency of the optimization of NRl(1)-Graph. Experimental results on various real data sets demonstrate the effectiveness of both NRl(1)-Graph and NRl(1)-Graph-RP.