Polynomial Time Algorithms for Dual Volume Sampling

I will talk briefly about our recent work on polynomial time algorithms for dual volume sampling (DVS), accepted by NIPS this year. DVS is a method for selecting k columns from an n × m short and wide matrix (n ≤ k ≤ m) such that the probability of selection is proportional to the volume spanned by the rows of the induced submatrix. This method was proposed by Avron and Boutsidis (2013), who showed it to be a promising method for column subset selection and its multiple applications. However, its wider adoption has been hampered by the lack of polynomial time sampling algorithms. We remove this hindrance by developing an exact (randomized) polynomial time sampling algorithm as well as its derandomization. Thereafter, we study dual volume sampling via the theory of real-stable polynomials and prove that its distribution satisfies the “Strong Rayleigh” property. This result has remarkable consequences, especially because it implies a provably fast-mixing Markov chain sampler that makes dual volume sampling much more attractive to practitioners.