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We study the complexity of sampling from a distribution over all index subsets of the set {1, ..., n} with the probability of a subset S proportional to the determinant of the submatrix LS of some n x n positive semidefinite matrix L, where LS corresponds to the entries of L indexed by S. Known as a determinantal point process (DPP), this distribution is used in machine learning to induce diversity in subset selection. When sampling from DDPs, we often wish to sample multiple subsets S with small expected size k = E[S] << n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose we provide DPPVFX, a new algorithm which, given access only to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is n poly(k), i.e., sublinear in the size of L, and (2) its sampling cost is poly(k), i.e., independent of the size of L. Prior to our results, stateoftheart exact samplers required O(n^3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We furthermore give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with n poly(k) time preprocessing. Our implementation of DPPVFX is provided at https://github.com/guilgautier/DPPy/.
Author Information
Michal Derezinski (UC Berkeley)
Daniele Calandriello (LCSL IIT/MIT)
Michal Valko (DeepMind Paris and Inria Lille  Nord Europe)
Michal is a research scientist in DeepMind Paris and SequeL team at Inria Lille  Nord Europe, France, lead by Philippe Preux and Rémi Munos. He also teaches the course Graphs in Machine Learning at l'ENS Cachan. Michal is primarily interested in designing algorithms that would require as little human supervision as possible. This means 1) reducing the “intelligence” that humans need to input into the system and 2) minimising the data that humans need spend inspecting, classifying, or “tuning” the algorithms. Another important feature of machine learning algorithms should be the ability to adapt to changing environments. That is why he is working in domains that are able to deal with minimal feedback, such as semisupervised learning, bandit algorithms, and anomaly detection. The common thread of Michal's work has been adaptive graphbased learning and its application to the real world applications such as recommender systems, medical error detection, and face recognition. His industrial collaborators include Intel, Technicolor, and Microsoft Research. He received his PhD in 2011 from University of Pittsburgh under the supervision of Miloš Hauskrecht and after was a postdoc of Rémi Munos.
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