Piecewise Linear Regression via a Difference of Convex Functions
This provides a new regression method for data analysis, but it appears incremental as it builds on existing concepts like max-affine regression and shows comparable performance to existing methods.
The authors tackled piecewise linear regression by fitting a difference of convex functions, regularized with a new seminorm, and showed it has close to minimax statistical risk and is efficiently implementable via quadratic programming.
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 $φ_1 - φ_2$ for a choice of convex functions $φ_1, φ_2$. 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.