Direct Reading Algorithm for Hierarchical Clustering

Reading the clusters from a data set such that the overall computational complexity is linear in both data dimensionality and in the number of data elements has been carried out through filtering the data in wavelet transform space. This objective is also carried out after an initial transforming of the data to a canonical order. Including high dimensional, high cardinality data, such a canonical order is provided by row and column permutations of the data matrix. In our recent work, we induce a hierarchical clustering from seriation through unidimensional representation of our observations. This linear time hierarchical classification is directly derived from the use of the Baire metric, which is simultaneously an ultrametric. In our previous work, the linear time construction of a hierarchical clustering is studied from the following viewpoint: representing the hierarchy initially in an m-adic, m = 10, tree representation, followed by decreasing m to smaller valued representations that include p-adic representations, where p is prime and m is a non-prime positive integer. This has the advantage of facilitating a more direct visualization and hence interpretation of the hierarchy. In this work we present further case studies and examples of how this approach is very advantageous for such an ultrametric topological data mapping.