  We present a new piecewise linear regression methodology that utilizes
fitting a difference of convex functions (DC functions) to the data. These are
functions $f$ that may be represented as the difference $\phi_1 - \phi_2$ for a
choice of convex functions $\phi_1, \phi_2$. The method proceeds by estimating
piecewise-liner convex functions, in a manner similar to max-affine regression,
whose difference approximates the data. The choice of the function is
regularised by a new seminorm over the class of DC functions that controls the
$\ell_\infty$ Lipschitz constant of the estimate. The resulting methodology can
be efficiently implemented via Quadratic programming even in high dimensions,
and is shown to have close to minimax statistical risk. We empirically validate
the method, showing it to be practically implementable, and to have comparable
performance to existing regression/classification methods on real-world
datasets.
