Section: New Results

Learning Determinantal Point Processes in Sublinear Time

In [21], we propose a new class of determinantal point processes (DPPs) which can be manipulated for inference and parameter learning in potentially sublinear time in the number of items. This class, based on a specific low-rank factorization of the marginal kernel, is particularly suited to a subclass of continuous DPPs and DPPs defined on exponentially many items. We apply this new class to modelling text documents as sampling a DPP of sentences, and propose a conditional maximum likelihood formulation to model topic proportions, which is made possible with no approximation for our class of DPPs. We present an application to document summarization with a DPP on $2^{500}$ items.

We make the following contributions:

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  We propose a new class of determinantal point processes (DPPs) which is based on a particular low-rank factorization of the marginal kernel. Through the availability of a particular second-moment matrix, the complexity for inference and learning tasks is polynomial in the rank of the factorization and thus often sublinear in the total number of items (with exact likelihood computations).

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  As shown in this work, these new DPPs are particularly suited to a subclass of continuous DPPs (infinite number of items), such as on ${[0,1]}^m$, and DPPs defined on the $V$-dimensional hypercube, which has $2^V$ elements.

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  We propose a model of documents as sampling a DPP of sentences, and propose a conditional maximum likelihood formulation to model topic proportions. We present an application to document summarization with a DPP on $2^{500}$ items.