501 research outputs found
Approximate Inference in Continuous Determinantal Point Processes
Determinantal point processes (DPPs) are random point processes well-suited
for modeling repulsion. In machine learning, the focus of DPP-based models has
been on diverse subset selection from a discrete and finite base set. This
discrete setting admits an efficient sampling algorithm based on the
eigendecomposition of the defining kernel matrix. Recently, there has been
growing interest in using DPPs defined on continuous spaces. While the
discrete-DPP sampler extends formally to the continuous case, computationally,
the steps required are not tractable in general. In this paper, we present two
efficient DPP sampling schemes that apply to a wide range of kernel functions:
one based on low rank approximations via Nystrom and random Fourier feature
techniques and another based on Gibbs sampling. We demonstrate the utility of
continuous DPPs in repulsive mixture modeling and synthesizing human poses
spanning activity spaces
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, Large Scale Inference, and Temporal Modeling of Determinantal Point Processes
Determinantal Point Processes (DPPs) are random point processes well-suited for modelling repulsion. In discrete settings, DPPs are a natural model for subset selection problems where diversity is desired. For example, they can be used to select relevant but diverse sets of text or image search results. Among many remarkable properties, they offer tractable algorithms for exact inference, including computing marginals, computing certain conditional probabilities, and sampling.
In this thesis, we provide four main contributions that enable DPPs to be used in more general settings. First, we develop algorithms to sample from approximate discrete DPPs in settings where we need to select a diverse subset from a large amount of items.
Second, we extend this idea to continuous spaces where we develop approximate algorithms to sample from continuous DPPs, yielding a method to select point configurations that tend to be overly-dispersed.
Our third contribution is in developing robust algorithms to learn the parameters of the DPP kernels, which is previously thought to be a difficult, open problem.
Finally, we develop a temporal extension for discrete DPPs, where we model sequences of subsets that are not only marginally diverse but also diverse across time
Bayesian Inference for Latent Biologic Structure with Determinantal Point Processes (DPP)
We discuss the use of the determinantal point process (DPP) as a prior for
latent structure in biomedical applications, where inference often centers on
the interpretation of latent features as biologically or clinically meaningful
structure. Typical examples include mixture models, when the terms of the
mixture are meant to represent clinically meaningful subpopulations (of
patients, genes, etc.). Another class of examples are feature allocation
models. We propose the DPP prior as a repulsive prior on latent mixture
components in the first example, and as prior on feature-specific parameters in
the second case. We argue that the DPP is in general an attractive prior model
for latent structure when biologically relevant interpretation of such
structure is desired. We illustrate the advantages of DPP prior in three case
studies, including inference in mixture models for magnetic resonance images
(MRI) and for protein expression, and a feature allocation model for gene
expression using data from The Cancer Genome Atlas. An important part of our
argument are efficient and straightforward posterior simulation methods. We
implement a variation of reversible jump Markov chain Monte Carlo simulation
for inference under the DPP prior, using a density with respect to the unit
rate Poisson process
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