Isometry - Generalizations

Generalizations

• Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map between metric spaces such that
  1. for x,x′ ∈ X one has |d_Y(ƒ(x),ƒ(x′))−d_X(x,x′)| < ε, and
  2. for any point y ∈ Y there exists a point x ∈ X with d_Y(y,ƒ(x)) < ε

That is, an ε-isometry preserves distances to within ε and leaves no element of the codomain further than ε away from the image of an element of the domain. Note that ε-isometries are not assumed to be continuous.

• The Restricted isometry property characterizes nearly isometric matrices for sparse vectors.

• Quasi-isometry is yet another useful generalization.