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This book is concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry; it shows the intriguing interplay between finite and infinite length modules. See All Customer Reviews.

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  6. Shop Books. Add to Wishlist. USD Sign in to Purchase Instantly. Overview This book is concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry; it shows the intriguing interplay between finite and infinite length modules.

    Table of contents

    We can see that the variational Gaussian mixture with a Dirichlet process prior is able to limit itself to only 2 components whereas the Gaussian mixture fits the data with a fixed number of components that has to be set a priori by the user. Note that with very little observations, the variational Gaussian mixture models with a Dirichlet process prior can take a conservative stand, and fit only one component. On the following figure we are fitting a dataset not well-depicted by a Gaussian mixture. We also present on the last two plots a random sampling generated from the two resulting mixtures.

    Here we describe variational inference algorithms on Dirichlet process mixture.

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    The Dirichlet process is a prior probability distribution on clusterings with an infinite, unbounded, number of partitions. Variational techniques let us incorporate this prior structure on Gaussian mixture models at almost no penalty in inference time, comparing with a finite Gaussian mixture model. An important question is how can the Dirichlet process use an infinite, unbounded number of clusters and still be consistent. The stick breaking process is a generative story for the Dirichlet process. We start with a unit-length stick and in each step we break off a portion of the remaining stick.

    Each time, we associate the length of the piece of the stick to the proportion of points that falls into a group of the mixture. The length of each piece is a random variable with probability proportional to the concentration parameter. Smaller value of the concentration will divide the unit-length into larger pieces of the stick defining more concentrated distribution.

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    Larger concentration values will create smaller pieces of the stick increasing the number of components with non zero weights. Previous 2.

    Unsupervised learning. Next 2.

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    Manifold learning. Examples: See GMM covariances for an example of using the Gaussian mixture as clustering on the iris dataset. See Density Estimation for a Gaussian mixture for an example on plotting the density estimation. Examples: See Gaussian Mixture Model Selection for an example of model selection performed with classical Gaussian mixture. This makes it possible to let the model choose a suitable number of effective components automatically. Only an upper bound of this number needs to be provided.

    Regularization: due to the incorporation of prior information, variational solutions have less pathological special cases than expectation-maximization solutions.

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    Bias: there are many implicit biases in the inference algorithms and also in the Dirichlet process if used , and whenever there is a mismatch between these biases and the data it might be possible to fit better models using a finite mixture. Show this page source. As this algorithm maximizes only the likelihood, it will not bias the means towards zero, or bias the cluster sizes to have specific structures that might or might not apply. When one has insufficiently many points per mixture, estimating the covariance matrices becomes difficult, and the algorithm is known to diverge and find solutions with infinite likelihood unless one regularizes the covariances artificially.

    This algorithm will always use all the components it has access to, needing held-out data or information theoretical criteria to decide how many components to use in the absence of external cues.

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