Geometric and Combinatorial Tiles in 0-1 Data

In this paper we introduce a simple probabilistic model, hierarchical tiles, for 0–1 data. A basic tile (X, Y, p) specifies a subset X of the rows and a subset Y of the columns of the data, i.e., a rectangle, and gives a probability p for the occurrence of 1s in the cells of X × Y . A hierarchical tile has additionally a set of exception tiles that specify the probabilities for subrectangles of the original rectangle. If the rows and columns are ordered and X and Y consist of consecutive elements in those orderings, then the tile is geometric; otherwise it is combinatorial. We give a simple randomized algorithm for finding good geometric tiles. Our main result shows that using spectral ordering techniques one can find good orderings that turn combinatorial tiles into geometric tiles. We give empirical results on the performance of the methods.