449 research outputs found
Inference for determinantal point processes without spectral knowledge
Determinantal point processes (DPPs) are point process models that naturally
encode diversity between the points of a given realization, through a positive
definite kernel . DPPs possess desirable properties, such as exact sampling
or analyticity of the moments, but learning the parameters of kernel
through likelihood-based inference is not straightforward. First, the kernel
that appears in the likelihood is not , but another kernel related to
through an often intractable spectral decomposition. This issue is
typically bypassed in machine learning by directly parametrizing the kernel
, at the price of some interpretability of the model parameters. We follow
this approach here. Second, the likelihood has an intractable normalizing
constant, which takes the form of a large determinant in the case of a DPP over
a finite set of objects, and the form of a Fredholm determinant in the case of
a DPP over a continuous domain. Our main contribution is to derive bounds on
the likelihood of a DPP, both for finite and continuous domains. Unlike
previous work, our bounds are cheap to evaluate since they do not rely on
approximating the spectrum of a large matrix or an operator. Through usual
arguments, these bounds thus yield cheap variational inference and moderately
expensive exact Markov chain Monte Carlo inference methods for DPPs
Learning Determinantal Point Processes
Determinantal point processes (DPPs), which arise in random matrix theory and
quantum physics, are natural models for subset selection problems where
diversity is preferred. Among many remarkable properties, DPPs offer tractable
algorithms for exact inference, including computing marginal probabilities and
sampling; however, an important open question has been how to learn a DPP from
labeled training data. In this paper we propose a natural feature-based
parameterization of conditional DPPs, and show how it leads to a convex and
efficient learning formulation. We analyze the relationship between our model
and binary Markov random fields with repulsive potentials, which are
qualitatively similar but computationally intractable. Finally, we apply our
approach to the task of extractive summarization, where the goal is to choose a
small subset of sentences conveying the most important information from a set
of documents. In this task there is a fundamental tradeoff between sentences
that are highly relevant to the collection as a whole, and sentences that are
diverse and not repetitive. Our parameterization allows us to naturally balance
these two characteristics. We evaluate our system on data from the DUC 2003/04
multi-document summarization task, achieving state-of-the-art results
Large-Margin Determinantal Point Processes
Determinantal point processes (DPPs) offer a powerful approach to modeling
diversity in many applications where the goal is to select a diverse subset. We
study the problem of learning the parameters (the kernel matrix) of a DPP from
labeled training data. We make two contributions. First, we show how to
reparameterize a DPP's kernel matrix with multiple kernel functions, thus
enhancing modeling flexibility. Second, we propose a novel parameter estimation
technique based on the principle of large margin separation. In contrast to the
state-of-the-art method of maximum likelihood estimation, our large-margin loss
function explicitly models errors in selecting the target subsets, and it can
be customized to trade off different types of errors (precision vs. recall).
Extensive empirical studies validate our contributions, including applications
on challenging document and video summarization, where flexibility in modeling
the kernel matrix and balancing different errors is indispensable.Comment: 15 page
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