Piecewiselineare

Piecewiselineare refers to a class of functions or mappings that are linear on each piece of a finite partition of their domain. In common mathematical usage, a piecewiselineare map is described as piecewise affine: there exists a finite collection of polyhedral regions that cover the domain, and on each region the function is given by an affine formula.

Formally, let f: D → R^m with D ⊆ R^n. There exists a finite family of polyhedra {P1, P2, ..., Pk} such that D = ⋃i Pi and each Pi has a nonempty interior. On every Pi, f(x) = Ai x + bi for some linear map Ai and vector bi. The regions Pi may intersect at their boundaries, where the defining affine expressions may change. If adjacent regions agree on their common boundary, f is continuous; if they do not, f may have jumps at the boundaries.

Piecewiselinear maps have several notable properties. They are differentiable on the interiors of the regions Pi, with the derivative equal to Ai there, and they are globally Lipschitz on D. At boundary points between pieces, the function may be non-differentiable, though subgradients can often be described. When the function is convex, a convex piecewiselinear function can be represented as the pointwise maximum of a finite set of affine functions.

Common examples include the one-dimensional absolute value function and the two-dimensional hinge-like forms used in optimization. Applications span linear programming with piecewise-linear costs, computational geometry, computer graphics for piecewise-linear surfaces, and neural networks with piecewise-linear activation functions such as ReLU.

See also: piecewise affine, polyhedral geometry, convex analysis, ReLU networks.