A Convex Exemplar-based Approach to MAD-Bayes Dirichlet Process Mixture Models

Appendix A. Notation N = {1, 2, ..., N} =: [N ] is the whole set of data points. i, j ∈ N denote points. dij := D(xi,xj). D is the number of data sets. Td ⊆ N denotes the set of points in the d-th dataset, i.e. ∪d=1Td = N . Nd = |Td| is the number of points in Dataset d. d(i) ∈ [D] denotes the dataset index of Point i. M ⊆ N is the set of medoids. k, l ∈ M denote clusters and themselves are medoids. Sk is the set of points in Cluster k. Nk = |Sk| is the number of points in Cluster k. M(i) ∈ M denotes the cluster/representative of Point i. Let Dk ⊆ [D] denote the data sets contained or partially contained in Cluster k. Denote Sk,d := Sk ∩ Td for d ∈ Dk. Thus ∪d∈DkSk,d = Sk. Denote Nk,d := |Sk,d| for d ∈ Dk. B. Proof of Theorem 1 Theorem 1 is a direct corollary of Theorem 2, by setting θ = 0.