6,335 research outputs found
Statistical guarantees for the EM algorithm: From population to sample-based analysis
We develop a general framework for proving rigorous guarantees on the
performance of the EM algorithm and a variant known as gradient EM. Our
analysis is divided into two parts: a treatment of these algorithms at the
population level (in the limit of infinite data), followed by results that
apply to updates based on a finite set of samples. First, we characterize the
domain of attraction of any global maximizer of the population likelihood. This
characterization is based on a novel view of the EM updates as a perturbed form
of likelihood ascent, or in parallel, of the gradient EM updates as a perturbed
form of standard gradient ascent. Leveraging this characterization, we then
provide non-asymptotic guarantees on the EM and gradient EM algorithms when
applied to a finite set of samples. We develop consequences of our general
theory for three canonical examples of incomplete-data problems: mixture of
Gaussians, mixture of regressions, and linear regression with covariates
missing completely at random. In each case, our theory guarantees that with a
suitable initialization, a relatively small number of EM (or gradient EM) steps
will yield (with high probability) an estimate that is within statistical error
of the MLE. We provide simulations to confirm this theoretically predicted
behavior
Split Sampling: Expectations, Normalisation and Rare Events
In this paper we develop a methodology that we call split sampling methods to
estimate high dimensional expectations and rare event probabilities. Split
sampling uses an auxiliary variable MCMC simulation and expresses the
expectation of interest as an integrated set of rare event probabilities. We
derive our estimator from a Rao-Blackwellised estimate of a marginal auxiliary
variable distribution. We illustrate our method with two applications. First,
we compute a shortest network path rare event probability and compare our
method to estimation to a cross entropy approach. Then, we compute a
normalisation constant of a high dimensional mixture of Gaussians and compare
our estimate to one based on nested sampling. We discuss the relationship
between our method and other alternatives such as the product of conditional
probability estimator and importance sampling. The methods developed here are
available in the R package: SplitSampling
A spatially distributed model for foreground segmentation
Foreground segmentation is a fundamental first processing stage for vision systems which monitor real-world activity. In this paper we consider the problem of achieving robust segmentation in scenes where the appearance of the background varies unpredictably over time. Variations may be caused by processes such as moving water, or foliage moved by wind, and typically degrade the performance of standard per-pixel background models.
Our proposed approach addresses this problem by modeling homogeneous regions of scene pixels as an adaptive mixture of Gaussians in color and space. Model components are used to represent both the scene background and moving foreground objects. Newly observed pixel values are probabilistically classified, such that the spatial variance of the model components supports correct classification even when the background appearance is significantly distorted. We evaluate our method over several challenging video sequences, and compare our results with both per-pixel and Markov Random Field based models. Our results show the effectiveness of our approach in reducing incorrect classifications
Scene modelling using an adaptive mixture of Gaussians in colour and space
We present an integrated pixel segmentation and region
tracking algorithm, designed for indoor environments. Visual monitoring systems often use frame differencing techniques to independently classify each image pixel as either foreground or background. Typically, this level of processing does not take account of the global image structure, resulting in frequent misclassification.
We use an adaptive Gaussian mixture model in colour and space to represent background and foreground regions of the scene. This model is used to probabilistically classify observed pixel values, incorporating the global scene structure into pixel-level segmentation. We evaluate our system over 4 sequences and show that it successfully segments foreground pixels and tracks major foreground regions as they move through the scene
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