Finite mixture models for clustering can often be improved by adding a regularization that is specific to the topology of the data. For instance, mixtures are common in unsupervised image segmentation, and typically rely on averaging the posterior mixing probabilities of spatially adjacent data points (i. e. smoothing). However, this approach has had limited success with natural images. Herewe make three contributions. First, we show that a Dirichlet prior with an appropriate choice of parameters allows -- using the  Expectation-Maximization approach -- to define any linear update rule for the mixing probabilities, including many smoothing regularizations as special cases. Second, we demonstrate how to use this flexible design of the update rule to propagate segmentation information across layers of a deep network, and to train mixtures jointly across layers. Third, we compare the standard Gaussian mixture and the Student-t mixture, which is known to better capture the statistics of low-level visual features. We show that our models achieve competitive performance in natural image segmentation , with the Student-t mixtures reaching state-of-the art on boundaries scores . We also demonstrate how to exploit the resulting multilayer probabilistic generative model to synthesize naturalistic images beyond uniform textures .
Probabilistic finite mixture models are widely used for unsupervised clustering. These models can often be improved by adapting them to the topology of the data. For instance, in order to classify spatially adjacent data points similarly, it is common to introduce a Laplacian constraint on the posterior probability that each data point belongs to a class. Alternatively, the mixing probabilities can be treated as free parameters, while assuming Gauss-Markov or more complex priors to regularize those mixing probabilities. However, these approaches are constrained by the shape of the prior and often lead to complicated or intractable inference. Here, we propose a new parametrization of the Dirichlet distribution to flexibly regularize the mixing probabilities of over-parametrized mixture distributions. Using the Expectation-Maximization algorithm, we show that our approach allows us to define any linear update rule for the mixing probabilities, including spatial smoothing regularization as a special case. We then show that this flexible design can be extended to share class information between multiple mixture models. We apply our algorithm to artificial and natural image segmentation tasks, and we provide quantitative and qualitative comparison of the performance of Gaussian and Student-t mixtures on the Berkeley Segmentation Dataset . We also demonstrate how to propagate class information across the layers of deep convolutional neural networks in a probabilistically optimal way, suggesting a new interpretation for feedback signals in biological visual systems. Our flexible approach can be easily generalized to adapt probabilistic mixture models to arbitrary data topologies .
R
Finite mixture models for clustering
Probabilistic finite mixture models are widely used for unsupervised clustering. These models
R
adding a regularization that is specific
adapting them
R
mixtures are common in unsupervised image segmentation, and typically rely on averaging the posterior mixing probabilities of spatially adjacent data points (i. e. smoothing). However, this approach has had limited success with natural images. Herewe make three contributions. First, we show that a Dirichlet prior with an appropriate choice of parameters allows -- using
in order to classify spatially adjacent data points similarly, it is common to introduce a Laplacian constraint on
A
posterior probability that each data point belongs to a class. Alternatively, the mixing probabilities can be treated as free parameters, while assuming Gauss-Markov or more complex priors to regularize those mixing probabilities. However, these approaches are constrained by the shape of the prior and often lead to complicated or intractable inference. Here, we propose a new parametrization of the Dirichlet distribution to flexibly regularize the mixing probabilities of over-parametrized mixture distributions. Using the
R
approach --
algorithm, we show that our approach allows us
R
many smoothing regularizations as special cases. Second, we demonstrate how to use
spatial smoothing regularization as a special case. We then show that
R
of the update rule to propagate segmentation information across layers of a deep network, and to train mixtures jointly across layers. Third, we compare the standard Gaussian mixture and the Student-t mixture, which is known to better capture the statistics of low-level visual features. We show that our models achieve competitive performance in
can be extended to share class information between multiple mixture models. We apply our algorithm to artificial and
R
, with the
tasks, and we provide quantitative and qualitative comparison of the performance of Gaussian and
R
reaching state-of-the art on boundaries scores
on the Berkeley Segmentation Dataset
R
exploit the resulting multilayer probabilistic generative model to synthesize naturalistic images beyond uniform textures
propagate class information across the layers of deep convolutional neural networks in a probabilistically optimal way, suggesting a new interpretation for feedback signals in biological visual systems. Our flexible approach can be easily generalized to adapt probabilistic mixture models to arbitrary data topologies