Computing sparse approximations deterministically Original Research Article

It is known that for every n × m matrix A with entries taken from the interval [0, 1] and for every probability vector ulbarp, there is a sparse probability vector ulbarq with only O((lnn)/var epsilon2) nonzero entries such that every component of the vector A · ulbarq differs from every component of A · ulbarp in absolute value by at most var epsilon. The existence of such a vector is proved by a probabilistic argument. It has been an open problem whether there is an efficient, i.e. polynomial-time, deterministic algorithm which actually constructs such a vector ulbarq. We provide an algorithm which does so and which takes time polynomial in n, m, and 1/var epsilon. The algorithm is based on the method of “pessimistic estimators” introduced by Raghavan.