Fast Algorithm for the Lasso based L1-Graph Construction

The lasso-based L1-graph is used in many applications since it can effectively model a set of data points as a graph. The lasso is a popular regression approach and the L1-graph represents data points as nodes by using the regression result. More specifically, by solving the L1-optimization problem of the lasso, the sparse regression coefficients are used to obtain the weights of the edges in the graph. Conventional graph structures such as k-NN graph use two steps, adjacency searching and weight selection, for constructing the graph whereas the lasso-based L1-graph derives the adjacency structure as well as the edge weights simultaneously by using a coordinate descent. However, the construction cost of the lasso-based L1-graph is impractical for large data sets since the coordinate descent iteratively updates the weights of all edges until convergence. Our proposal, Castnet, can efficiently construct the lasso-based L1-graph. In order to avoid updating the weights of all edges, we prune edges that cannot have nonzero weights before entering the iterations. In addition, we update edge weights only if they are nonzero in the iterations. Experiments show that Castnet is significantly faster than existing approaches.