In certain applications there may only be positive samples available to to learn concepts of a class of interest, and this has to be done properly, i.e. the hypothesis space has to coincide with the concept class, and without false positives, i.e. the hypothesis always has be a subset of the real concept (one-sided error). For the well studied class of k-term DNF formulas it has been known that learning is difficult. Unless RP = NP, it is not feasible to learn k-term DNF formulas properly in a distribution-free sense even if both positive and negative samples are available and even if false positives are allowed.

This paper constructs an efficient algorithm that for arbitrary fixed k, if samples are drawn from distributions like uniform or q-bounded ones, properly learns the class of k-term DNFs without false positives from positive samples alone with arbitrarily small relative error.