Large Structures Seminar: Robert Krone

The non-negative rank of a matrix is useful for algebraic statistics, but difficult to compute. The set of matrices with rank and non-negative rank both equal to r forms a semi-algebraic, Zariski-dense subset of the variety of rank r matrices. Describing all the boundary components of this set can provide a membership test, and this was accomplished for rank 3 by Mond, Smith and van Straten. Using a slightly different approach, we describe some of the boundary components in the higher rank cases, particularly the set of matrices that have a “rigid” nonnegative decomposition.