The VC-Dimension of Similarity Hypotheses Spaces

Given a set X and a function h : X −→ {0, 1} which labels each element of X with either 0 or 1, we may define a function h to measure the similarity of pairs of points in X according to h. Specifically, for h ∈ {0, 1} we define h ∈ {0, 1} by h(w, x) := 1[h(w) = h(x)]. This idea can be extended to a set of functions, or hypothesis space H ⊆ {0, 1} by defining a similarity hypothesis space H := {h : h ∈ H}. We show that vc-dimension(H) ∈ Θ(vc-dimension(H)).