Supplementary Material: The Emergence of Organizing Structure in Conceptual Representation

A data set is a matrix D = {f (1), ..., f (m)} (D ∈ Rnx×m), where the rows correspond to objects and the columns are features f (i). A data set is generated by a structure, parameterized by a symmetric matrix S ∈ Rnt×nt . Each row/column in S corresponds to a node in the graph. The set of object nodes is called X and cluster nodes is called Z, where nx and nz are their cardinalities, combining for a total of nt = nx + nz nodes in the graph. A key property of the structural sparsity model is that S is sparse, meaning that most of its entries are equal to zero (sij = 0). For the remainder, their values are positive (sij = sji > 0). The sparsity pattern defines the adjacency matrix of the undirected graph, where a non-zero value for sij means that nodes i and j share an edge. Given that each object node x ∈ X connects to only one cluster node, then sxz > 0 for exactly one z ∈ Z. There are no self-edges, so the diagonal of S is also zero. Following Zhu, Lafferty, and Ghahramani (2003) and the setup for the structural forms model (Kemp & Tenenbaum, 2008), we introduce the graph Laplacian ∆. Its off-diagonal elements are