Piecewise Linear Regression via a Difference of Convex Functions
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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 ℓ_∞ 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.
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