Significance and Recovery of Block Structures in Binary Matrices with Noise

Frequent itemset mining (FIM) is one of the core problems in the field of Data Mining and occupies a central place in its literature. One equivalent form of FIM can be stated as follows: given a rectangular data matrix with binary entries, find every submatrix of 1s having a minimum number of columns. This paper presents a theoretical analysis of several statistical questions related to this problem when noise is present. We begin by establishing several results concerning the extremal behavior of submatrices of ones in a binary matrix with random entries. These results provide simple significance bounds for the output of FIM algorithms. We then consider the noise sensitivity of FIM algorithms under a simple binary additive noise model, and show that, even at small noise levels, large blocks of 1s leave behind fragments of only logarithmic size. Thus such blocks cannot be directly recovered by FIM algorithms, which search for submatrices of all 1s. On the positive side, we show how, in the presence of noise, an error-tolerant criterion can recover a square submatrix of 1s against a background of 0s, even when the size of the target submatrix is very small.