distanceprojection

Distance projection, also called metric projection or nearest-point projection, is the problem of finding, for a given point x in a metric space (X, d) and a nonempty subset A ⊆ X, the point(s) a ∈ A that are closest to x. The distance from x to A is dist(x, A) = inf_{a∈A} d(x, a).

In general, the projection set P_A(x) = { a ∈ A : d(x, a) = dist(x, A) } may be empty if A is empty, or may be nonunique if the infimum is not attained. When A is nonempty and the infimum is attained, projection exists. In a Hilbert space, every nonempty closed convex A has a unique projection for every x, and the projection operator P_A is nonexpansive.

Examples include projection onto a linear subspace, which yields the orthogonal projection; projection onto a closed convex set, which returns the closest point in the set; and projection onto a sphere, which maps x to the point on the sphere along the ray from the center (when x is not at the center).

Computation in finite-dimensional Euclidean space is often explicit: the projection onto a hyperplane with normal n and offset b is p = x - ((n·x − b)/||n||^2) n. More generally, projections onto convex sets solve minimize ||x − a||^2 subject to a ∈ C, using methods such as gradient projection, proximal algorithms, or alternating projections.

Applications of distance projections appear in approximation, optimization, signal processing, computer graphics, and machine learning, where they enforce feasibility constraints or implement proximal operators.

See also: metric projection; nearest point; best approximation.