The DFS Fused Lasso: Linear-Time Denoising over General Graphs; Oscar Hernan Madrid Padilla, James Sharpnack, James G. Scott, Ryan J. Tibshirani

The fused lasso, also known as (anisotropic) total variation
denoising, is widely used for piecewise constant signal
estimation with respect to a given undirected graph. The fused
lasso estimate is highly nontrivial to compute when the
underlying graph is large and has an arbitrary structure. But
for a special graph structure, namely, the chain graph, the
fused lasso—or simply, 1d fused lasso—can be computed in
linear time. In this paper, we revisit a result recently
established in the online classification literature
(Herbster et al., 2009; Cesa-Bianchi et al., 2013) and show that it has
important implications for signal denoising on graphs. The
result can be translated to our setting as follows. Given a
general graph, if we run the standard depth-first search (DFS)
traversal algorithm, then the total variation of any signal over
the chain graph induced by DFS is no more than twice its total
variation over the original graph. This result leads to several
interesting theoretical and computational conclusions. Letting
$m$ and $n$ denote the number of edges and nodes, respectively,
of the graph in consideration, it implies that for an underlying
signal with total variation $t$ over the graph, the fused lasso
(properly tuned) achieves a mean squared error rate of
$t^{2/3} n^{-2/3}$. Moreover, precisely the same mean
squared error rate is achieved by running the 1d fused lasso on
the DFS-induced chain graph. Importantly, the latter estimator
is simple and computationally cheap, requiring $O(m)$ operations
to construct the DFS-induced chain and $O(n)$ operations to
compute the 1d fused lasso solution over this chain. Further,
for trees that have bounded maximum degree, the error rate of
$t^{2/3} n^{-2/3}$ cannot be improved, in the sense that
it is the minimax rate for signals that have total variation $t$
over the tree. Finally, several related results also hold—for
example, the analogous result holds for a roughness measure
defined by the $\ell_0$ norm of differences across edges in
place of the total variation metric.

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