By (2) the probability of being output by a discourse is determined by the inner product, so one expects that the vector for "clothing" discourse has a high inner product with all of shoe, jacket, tie1, etc., and thus can stand as surrogate for v tie1 in (1)!
it may be sufficient to consider the following global optimization: Given word vectors {v w } in ℜ d and two integers k, m with k < m, find a set of unit vectors where at most k of the coefficients α w,1 , .