Piecewiselinear

Piecewise linear, often written piecewise-linear, describes a function that is linear on each of a finite collection of subdomains of its domain. In one dimension, a real-valued function f: R -> R is piecewise linear if there exists a finite set of breakpoints x1 < x2 < ... < xk such that on each interval between consecutive breakpoints, as well as on the ends, f agrees with an affine function: f(x) = m_i x + b_i for x in the corresponding interval. In higher dimensions, a function is piecewise linear if its domain can be partitioned into finitely many polyhedral regions on each of which the function is the restriction of an affine map.

Common examples include f(x) = |x|, which equals x for x >= 0 and -x for x < 0, and f(x) = max(0, x), which is 0 for x <= 0 and x for x >= 0. More generally, any combination of affine pieces with breakpoints yields a piecewise linear function.

Properties of piecewise linear functions include a graph composed of straight-line segments joined at breakpoints; the function is continuous at breakpoints when the adjacent linear expressions meet at that point. Derivatives exist and are constant on each linear region, while the derivative may be discontinuous at breakpoints. In multiple dimensions, the gradient is defined within each region and may change across boundaries.

Applications span modeling non-smooth phenomena and optimization. They appear in hinge loss and ReLU activations in machine learning, piecewise linear interpolation, and various methods that favor computationally tractable, linear representations of nonlinear behavior. They are widely used in computational geometry and graphics as well.

See also: piecewise function, affine function, ReLU, hinge loss.