Matrix Tile Analysis

Many tasks require finding groups of elements in a matrix of numbers, symbols or class likelihoods. One approach is to use efficient bior tri-linear factorization techniques including PCA, ICA, sparse matrix factorization and plaid analysis. These techniques are not appropriate when addition and multiplication of matrix elements are not sensibly defined. More directly, methods like biclustering can be used to classify matrix elements, but these methods make the overlyrestrictive assumption that the class of each element is a function of a row class and a column class. We introduce a general computational problem, ‘matrix tile analysis’ (MTA), which consists of decomposing a matrix into a set of non-overlapping tiles, each of which is defined by a subset of usually nonadjacent rows and columns. MTA does not require an algebra for combining tiles, but must search over an exponential number of discrete combinations of tile assignments. We describe a loopy BP (sum-product) algorithm and an ICM algorithm for performing MTA. We compare the effectiveness of these methods to PCA and the plaid method on hundreds of randomly generated tasks. Using doublegene-knockout data, we show that MTA finds groups of interacting yeast genes that have biologically-related functions.