The Sparse Basis Problem and Multilinear Algebra

Let $A$ be a $k \times n$ underdetermined matrix. The sparse basis problem for the row space $W$ of $A$ is to find a basis of $W$ with the fewest number of nonzeros. Suppose that all the entries of $A$ are nonzero, and that they are algebraically independent over the rational number field. Then every nonzero vector in $W$ has at least $n - k + 1$ nonzero entries. Those vectors in $W$ with exactly $n - k + 1$ nonzero entries are the elementary vectors of $W$. A simple combinatorial condition that is both necessary and sufficient for a set of $k$ elementary vectors of $W$ to form a basis of $W$ is presented here. A similar result holds for the null space of $A$ where the elementary vectors now have exactly $k + 1$ nonzero entries. These results follow from a theorem about nonzero minors of order $m$ of the $(m - 1)$st compound of an $m \times n$ matrix with algebraically independent entries, which is proved using multilinear algebra techniques. This combinatorial condition for linear independence is a first step towards the design of algorithms that compute sparse bases for the row and null space without imposing artificial structure constraints to ensure linear independence.