The DFS fused lasso: nearly optimal linear-time denoising over graphs and trees

The fused lasso is a non-parametric regression estimator commonly used for graph denoising. This has been widely used in applications where the graph structure indicates that neighbor nodes have similar signal values. In this paper, we study the statistical and computational properties of the fused lasso. On the theoretical side, we show, for the fused lasso on arbitrary graphs, an upper bound on the mean squared error that depends on the total variation of the underlying signal on the graph. The key behind our theoretical results is a surprising connection of the depth--first search algorithm with total variation on an arbitrary graph. This has the implication that, with a linear time solver, the fused lasso on trees of bounded degree achieves nearly optimal minimax rates. Moreover, for general graphs with DFS ordering, such linear time solver provides a surrogate estimator of the fused lasso that inherits the attractive statistical properties of the one--dimensional fused lasso.