  We study the statistical properties of the least squares estimator in
unimodal sequence estimation. Although closely related to isotonic regression,
unimodal regression has not been as extensively studied. We show that the
unimodal least squares estimator is adaptive in the sense that the risk scales
as a function of the number of values in the true underlying sequence. Such
adaptivity properties have been shown for isotonic regression by Chatterjee et
al(2015) and Bellec(2015). A technical complication in unimodal regression is
the non-convexity of the underlying parameter space. We develop a general
variational representation of the risk that holds whenever the parameter space
can be expressed as a finite union of convex sets, using techniques that may be
of interest in other settings.
