  The fused lasso, also known as total-variation denoising, is a
locally-adaptive function estimator over a regular grid of design points. In
this paper, we extend the fused lasso to settings in which the points do not
occur on a regular grid, leading to an approach for non-parametric regression.
This approach, which we call the $K$-nearest neighbors ($K$-NN) fused lasso,
involves (i) computing the $K$-NN graph of the design points; and (ii)
performing the fused lasso over this $K$-NN graph. We show that this procedure
has a number of theoretical advantages over competing approaches: specifically,
it inherits local adaptivity from its connection to the fused lasso, and it
inherits manifold adaptivity from its connection to the $K$-NN approach. We
show that excellent results are obtained in a simulation study and on an
application to flu data. For completeness, we also study an estimator that
makes use of an $\epsilon$-graph rather than a $K$-NN graph, and contrast this
with the $K$-NN fused lasso.
