  The spectral $k$-support norm enjoys good estimation properties in low rank
matrix learning problems, empirically outperforming the trace norm. Its unit
ball is the convex hull of rank $k$ matrices with unit Frobenius norm. In this
paper we generalize the norm to the spectral $(k,p)$-support norm, whose
additional parameter $p$ can be used to tailor the norm to the decay of the
spectrum of the underlying model. We characterize the unit ball and we
explicitly compute the norm. We further provide a conditional gradient method
to solve regularization problems with the norm, and we derive an efficient
algorithm to compute the Euclidean projection on the unit ball in the case
$p=\infty$. In numerical experiments, we show that allowing $p$ to vary
significantly improves performance over the spectral $k$-support norm on
various matrix completion benchmarks, and better captures the spectral decay of
the underlying model.
