A treeless absolutely random forest with closed-form estimators of expected proximities

We introduce a simple variant of a Purely Random Forest, an Absolute Random Forest (ARF) for clustering. At every node splits of units are determined by a randomly chosen feature and a random threshold drawn from a uniform distribution whose support, the range of the selected feature in the root node, does not change. This enables closed-form estimators of parameters, such as pairwise proximities, to be obtained without having to grow a forest. The probabilistic structure corresponding to an ARF is called a Treeless Absolute Random Forest (TARF). With high probability, the algorithm will split units whose feature vectors are far apart and keep together units whose feature vectors are similar. Thus, the underlying structure of the data drives the growth of the tree. The expected value of pairwise proximities is obtained for three pathway functions. One, a completely common pathway function, is an indicator of whether a pair of units follow the same path from the root to the leaf node. The properties of TARF-based proximity estimators for clustering and classification are compared to other methods in eight real-world data sets and in simulations. Results show substantial performance and computing efficiencies of particular value for large data sets.