Large Homogeneous Submatrices

Finding large average submatrices in high dimensional data

The search for sample-variable associations is an important problem in the exploratory analysis of high dimensional data. Biclustering methods search for sample-variable associations in the form of distinguished submatrices of the data matrix. (The rows and columns of a submatrix need not be contiguous.) In this paper we propose and evaluate a statistically motivated biclustering procedure (LAS...

Large Homogeneous Compact Spaces

Homogeneous on Large Scales ?

This morning's debate is focused on the question of the large-scale homo-geneity of the Universe. I shall present the aarmative position, that there is overwhelming evidence for large scale homogeneity on scales in excess of approximately 50h ?1 Mpc, with a fractal distribution of matter on smaller scales. My worthy opponent, Dr. Luciano Pietronero, will present the counter-argument, that the f...

On linear forbidden submatrices

In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) li...

An Observation about Submatrices

Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer ≤ n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of M of order k. The proof uses results about random walks on symmetric groups and concentration of measure. In a similar way, we also show that almost all k × n submatrices of M h...