1. Given some number k, let k-Richness be the following requirement: For any finite X and every partition C = (C1, . . .Ck ) of X (into nonempty subsets) there exists some dissimilarity function d over X such that F(X,d) = C.
Prove that, for every number k, there exists a clustering function that satisfies the three properties: Scale Invariance, k-Richness, and Consistency.
2. In this exercise we show that in the general case, exact recovery of a linear compression scheme is impossible.
3. let A ∈Rn,d be an arbitrary compression matrix where n ≤d−1. Show that there exists u,v ∈ Rn, u _= v such that Au = Av.
