Metric projection explained

In mathematics, a metric projection is a function that maps each element of a metric space to the set of points nearest to that element in some fixed sub-space.^{[1] [2]}

Formal definition

Formally, let X be a metric space with distance metric d, and let M be a fixed subset of X. Then the metric projection associated with M, denoted p_M, is the following set-valued function from X to M:

p _{M(x)} = \arg min _{y\in} d(x,y)

Equivalently:

p _{M(x)} = \{y \in M : d(x,y) \leq d(x,y') \forall y'\in M \} = \{y \in M : d(x,y) = d(x,M) \}

The elements in the set \arg min _{y\in} d(x,y) are also called elements of best approximation. This term comes from constrained optimization: we want to find an element nearer to x, under the constraint that the solution must be a subset of M. The function p_M is also called an operator of best approximation.

Chebyshev sets

In general, p_M is set-valued, as for every x, there may be many elements in M that have the same nearest distance to x. In the special case in which p_M is single-valued, the set M is called a Chebyshev set. As an example, if (X,d) is a Euclidean space (Rⁿ with the Euclidean distance), then a set M is a Chebyshev set if and only if it is closed and convex.^[3]

Continuity

If M is non-empty compact set, then the metric projection p_M is upper semi-continuous, but might not be lower semi-continuous. But if X is a normed space and M is a finite-dimensional Chebyshev set, then p_M is continuous.

Moreover, if X is a Hilbert space and M is closed and convex, then p_M is Lipschitz continuous with Lipschitz constant 1.

Applications

Metric projections are used both to investigate theoretical questions in functional analysis and for practical approximation methods. They are also used in constrained optimization.^[4]

Notes and References

1. Web site: Metric projection - Encyclopedia of Mathematics . 2024-06-13 . encyclopediaofmath.org.
2. Deutsch . Frank . 1982-12-01 . Linear selections for the metric projection . Journal of Functional Analysis . 49 . 3 . 269–292 . 10.1016/0022-1236(82)90070-2 . 0022-1236.
3. Web site: Chebyshev set - Encyclopedia of Mathematics . 2024-06-13 . encyclopediaofmath.org.
4. Gafni . Eli M. . Bertsekas . Dimitri P. . November 1984 . Two-Metric Projection Methods for Constrained Optimization . SIAM Journal on Control and Optimization . en . 22 . 6 . 936–964 . 10.1137/0322061 . 0363-0129.