Fixed points of the EM algorithm and nonnegative rank boundaries

Nonnegative Rank vs. Binary Rank

Motivated by (and using tools from) communication complexity, we investigate the relationship between the following two ranks of a 0-1 matrix: its nonnegative rank and its binary rank (the log of the latter being the unambiguous nondeterministic communication complexity). We prove that for partial 0-1 matrices, there can be an exponential separation. For total 0-1 matrices, we show that if the ...

Fixed Points in Boundaries of Negatively Curved Groups

An important feature of the natural action of a negatively curved (= word hyperbolic) group on its boundary is that the xed points of hyperbolic (= innnite order) elements form a dense subset of the boundary. Gromov states this in his innuential treatise Hyperbolic Groups 4, 8.2 D] and his explanation involves Stalling's theorem on groups with innnitely many ends. This short note provides an el...

Diagonal arguments and fixed points

‎A universal schema for diagonalization was popularized by N.S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fi...

A Singly-Exponential Time Algorithm for Computing Nonnegative Rank

Here, we give an algorithm for deciding if the nonnegative rank of a matrix M of dimension m × n is at most r which runs in time (nm)O(r2). This is the first exact algorithm that runs in time singly-exponential in r. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this ta...

A Nonnegative Projection Based Algorithm for Low-Rank Nonnegative Matrix Approximation