metricadapted

Metricadapted is a term used in mathematics to describe objects, constructions, or mappings that are explicitly designed to be compatible with a given metric on a space. The idea is that the metric structure governs the behavior or properties of the object.

Formally, a map f: (X, d_X) -> (Y, d_Y) is metricadapted if there exists an increasing function φ: [0, ∞) -> [0, ∞) with φ(0) = 0 such that for all x, x' in X, d_Y(f(x), f(x')) ≤ φ(d_X(x, x')). In the special case where φ(t) = C t, f is Lipschitz with constant C and is often described as metric-adapted in a strong sense. More generally, the concept can be extended to families of maps, embeddings, or operators that respect a prescribed distortion bound determined by the metric.

Examples include metric-adapted embeddings into Euclidean space, where the goal is to preserve distances up to φ, and metric-based kernels in machine learning that depend only on the metric via k(x, x') = ψ(d_X(x, x')), for a nondecreasing ψ. In analysis on metric spaces, metricadapted operators are those whose action is stable under changes of metric within a specified class.

Applications span geometry, functional analysis, and data science, especially in contexts where the choice of metric is essential to the problem, such as clustering, manifold learning, and geometric group theory. The term is informal in many texts, but the underlying principle is a deliberate alignment between a construct and the metric structure.

See also: metric space, Lipschitz map, isometry, kernel method, coarse geometry, metric geometry.