First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper introduces the idea of deep Gaussian mixture models. A GMM can be seen as consisting of a single isotropic unit norm Gaussian, where each of the components of the mixture consists of applying a different linear transformation to that Gaussian. This idea is extended to the case of a multilayer network, where each node in the network corresponds to a linear transformation, and each route through the network corresponds to a sequence of linear transformations. The number of mixture components is then the number of routes through the network.

With modern high-density probes (Jun et al., 2017), neuroscientists can observe the spiking activity

Model-based clustering is widely used for identifying and distinguishing types of diseases. However, modern biomedical data coming with high dimensions make it challenging to perform the model estimation in traditional cluster analysis. The incorporation of factor analyzer into the mixture model provides a way to characterize the large set of data features, but the current estimation method is computationally impractical for massive data due to the intrinsic slow convergence of the embedded algorithms, and the incapability to vary the size of the factor analyzers, preventing the implementation of a generalized mixture of factor analyzers and further characterization of the data clusters. We propose a hybrid matrix-free computational scheme to efficiently estimate the clusters and model parameters based on a Gaussian mixture along with generalized factor analyzers to summarize the large number of variables using a small set of underlying factors. Our approach outperforms the existing method with faster convergence while maintaining high clustering accuracy. Our algorithms are applied to accurately identify and distinguish types of breast cancer based on large tumor samples, and to provide a generalized characterization for subtypes of lymphoma using massive gene records.

This paper develops a novel hybrid approach for estimating the mixture model of $t$-factor analyzers (MtFA) that employs multivariate $t$-distribution and factor model to cluster and characterize grouped data. The traditional estimation method for MtFA faces computational challenges, particularly in high-dimensional settings, where the eigendecomposition of large covariance matrices and the iterative nature of Expectation-Maximization (EM) algorithms lead to scalability issues. We propose a computational scheme that integrates a profile likelihood method into the EM framework to efficiently obtain the model parameter estimates. The effectiveness of our approach is demonstrated through simulations showcasing its superior computational efficiency compared to the existing method, while preserving clustering accuracy and resilience against outliers. Our method is applied to cluster the Gamma-ray bursts, reinforcing several claims in the literature that Gamma-ray bursts have heterogeneous subpopulations and providing characterizations of the estimated groups.

We present an automatic alignment procedure which maps the disparate internal representations learned by several local dimensionality reduction experts into a single, coherent global coordinate system for the original data space. Our algorithm can be applied to any set of experts, each of which produces a low-dimensional local representation of a high- dimensional input. Unlike recent efforts to coordinate such models by modifying their objective functions [1, 2], our algorithm is invoked after training and applies an efficient eigensolver to post-process the trained models. The post-processing has no local optima and the size of the sys- tem it must solve scales with the number of local models rather than the number of original data points, making it more efficient than model-free algorithms such as Isomap [3] or LLE [4]. 1 Introduction: Local vs. Global Dimensionality Reduction Beyond density modelling, an important goal of unsupervised learning is to discover com- pact, informative representations of high-dimensional data. If the data lie on a smooth low dimensional manifold, then an excellent encoding is the coordinates internal to that man- ifold.

Gaussian Mixture Models (GMMs) are a standard tool in data analysis. However, they face problems when applied to high-dimensional data (e.g., images) due to the size of the required full covariance matrices (CMs), whereas the use of diagonal or spherical CMs often imposes restrictions that are too severe. The Mixture of Factor analyzers (MFA) model is an important extension of GMMs, which allows to smoothly interpolate between diagonal and full CMs based on the number of \textit{factor loadings} $l$. MFA has successfully been applied for modeling high-dimensional image data. This article contributes both a theoretical analysis as well as a new method for efficient high-dimensional MFA training by stochastic gradient descent, starting from random centroid initializations. This greatly simplifies the training and initialization process, and avoids problems of batch-type algorithms such Expectation-Maximization (EM) when training with huge amounts of data. In addition, by exploiting the properties of the matrix determinant lemma, we prove that MFA training and inference/sampling can be performed based on precision matrices, which does not require matrix inversions after training is completed. At training time, the methods requires the inversion of $l\times l$ matrices only. Besides the theoretical analysis and proofs, we apply MFA to typical image datasets such as SVHN and MNIST, and demonstrate the ability to perform sample generation and outlier detection.

Ever since Pearl's probability propagation algorithm in graphs with cycles was shown to produce excellent results for error-correcting decoding a few years ago, we have been curious about whether local probability propagation could be used successfully for ma(cid:173) chine learning. One of the simplest adaptive models is the factor analyzer, which is a two-layer network that models bottom layer sensory inputs as a linear combination of top layer factors plus in(cid:173) dependent Gaussian sensor noise. We show that local probability propagation in the factor analyzer network usually takes just a few iterations to perform accurate inference, even in networks with 320 sensors and 80 factors. We derive an expression for the algorithm's fixed point and show that this fixed point matches the exact solu(cid:173) tion in a variety of networks, even when the fixed point is unstable. We also show that this method can be used successfully to perform inference for approximate EM and we give results on an online face recognition task. 1 Factor analysis A simple way to encode input patterns is to suppose that each input can be well(cid:173) approximated by a linear combination of component vectors, where the amplitudes of the vectors are modulated to match the input.

High dimensional data that lies on or near a low dimensional manifold can be de- scribed by a collection of local linear models. Such a description, however, does not provide a global parameterization of the manifold--arguably an important goal of unsupervised learning. In this paper, we show how to learn a collection of local linear models that solves this more difficult problem. Our local linear models are represented by a mixture of factor analyzers, and the "global coordi- nation" of these models is achieved by adding a regularizing term to the standard maximum likelihood objective function. The regularizer breaks a degeneracy in the mixture model's parameter space, favoring models whose internal coor- dinate systems are aligned in a consistent way.

Dependent Dirichlet processes (DDP) have been widely applied to model data from distributions over collections of measures which are correlated in some way. On the other hand, in recent years, increasing research efforts in machine learning and data mining have been dedicated to dealing with data involving interactions from two or more factors. However, few researchers have addressed the heterogeneous relationship in data brought by modulation of multiple factors using techniques of DDP. In this paper, we propose a novel technique, MultiLinear Dirichlet Processes (MLDP), to constructing DDPs by combining DP with a state-of-the-art factor analysis technique, multilinear factor analyzers (MLFA). We have evaluated MLDP on real-word data sets for different applications and have achieved state-of-the-art performance. Dependent Dirichlet processes (DDP) have been widely applied to model data from distributions over collections of measures which are correlated in some way. To introduce dependency into DDP, various techniques have been developed via correlating through components of atomic measures, such as atom sizes [8], [11], [23] and atom locations [6], [10], [28], sampling from a DP with random distributions as atoms [24], operating on underlying compound Poisson processes [18], regulating by Lévy Copulas [16], or constructing those measures through a mixture of several independent measures drawn from DPs [12], [15], [19], [20].

Mining and learning the shape variability of underlying population has benefited the applications including parametric shape modeling, 3D animation, and image segmentation. The current statistical shape modeling method works well on learning unstructured shape variations without obvious pose changes (relative rotations of the body parts). Studying the pose variations within a shape population involves segmenting the shapes into different articulated parts and learning the transformations of the segmented parts. This paper formulates the pose learning problem as mixtures of factor analyzers. The segmentation is obtained by components posterior probabilities and the rotations in pose variations are learned by the factor loading matrices. To guarantee that the factor loading matrices are composed by rotation matrices, constraints are imposed and the corresponding closed form optimal solution is derived. Based on the proposed method, the pose variations are automatically learned from the given shape populations. The method is applied in motion animation where new poses are generated by interpolating the existing poses in the training set. The obtained results are smooth and realistic.