Hierarchical parametric models consisting of observable and latent variables are widely used for unsupervised learning tasks. For example, a mixture model is a representative hierarchical model for clustering. From the statistical point of view, the models can be regular or singular due to the distribution of data. In the regular case, the models have the identifiability; there is one-to-one relation between a probability density function for the model expression and the parameter. The Fisher information matrix is positive definite, and the estimation accuracy of both observable and latent variables has been studied. In the singular case, on the other hand, the models are not identifiable and the Fisher matrix is not positive definite. Conventional statistical analysis based on the inverse Fisher matrix is not applicable. Recently, an algebraic geometrical analysis has been developed and is used to elucidate the Bayes estimation of observable variables. The present paper applies this analysis to latent-variable estimation and determines its theoretical performance. Our results clarify behavior of the convergence of the posterior distribution. It is found that the posterior of the observable-variable estimation can be different from the one in the latent-variable estimation. Because of the difference, the Markov chain Monte Carlo method based on the parameter and the latent variable cannot construct the desired posterior distribution.

Frequent Itemsets (FIs) mining is a fundamental primitive in data mining. It requires to identify all itemsets appearing in at least a fraction $\theta$ of a transactional dataset $\mathcal{D}$. Often though, the ultimate goal of mining $\mathcal{D}$ is not an analysis of the dataset \emph{per se}, but the understanding of the underlying process that generated it. Specifically, in many applications $\mathcal{D}$ is a collection of samples obtained from an unknown probability distribution $\pi$ on transactions, and by extracting the FIs in $\mathcal{D}$ one attempts to infer itemsets that are frequently (i.e., with probability at least $\theta$) generated by $\pi$, which we call the True Frequent Itemsets (TFIs). Due to the inherently stochastic nature of the generative process, the set of FIs is only a rough approximation of the set of TFIs, as it often contains a huge number of \emph{false positives}, i.e., spurious itemsets that are not among the TFIs. In this work we design and analyze an algorithm to identify a threshold $\hat{\theta}$ such that the collection of itemsets with frequency at least $\hat{\theta}$ in $\mathcal{D}$ contains only TFIs with probability at least $1-\delta$, for some user-specified $\delta$. Our method uses results from statistical learning theory involving the (empirical) VC-dimension of the problem at hand. This allows us to identify almost all the TFIs without including any false positive. We also experimentally compare our method with the direct mining of $\mathcal{D}$ at frequency $\theta$ and with techniques based on widely-used standard bounds (i.e., the Chernoff bounds) of the binomial distribution, and show that our algorithm outperforms these methods and achieves even better results than what is guaranteed by the theoretical analysis.

We consider the scenario where one observes an outcome variable and sets of features from multiple assays, all measured on the same set of samples. One approach that has been proposed for dealing with this type of data is ``sparse multiple canonical correlation analysis'' (sparse mCCA). All of the current sparse mCCA techniques are biconvex and thus have no guarantees about reaching a global optimum. We propose a method for performing sparse supervised canonical correlation analysis (sparse sCCA), a specific case of sparse mCCA when one of the datasets is a vector. Our proposal for sparse sCCA is convex and thus does not face the same difficulties as the other methods. We derive efficient algorithms for this problem, and illustrate their use on simulated and real data.

Due to the success of the bag-of-word modeling paradigm, clustering histograms has become an important ingredient of modern information processing. Clustering histograms can be performed using the celebrated $k$-means centroid-based algorithm. From the viewpoint of applications, it is usually required to deal with symmetric distances. In this letter, we consider the Jeffreys divergence that symmetrizes the Kullback-Leibler divergence, and investigate the computation of Jeffreys centroids. We first prove that the Jeffreys centroid can be expressed analytically using the Lambert $W$ function for positive histograms. We then show how to obtain a fast guaranteed approximation when dealing with frequency histograms. Finally, we conclude with some remarks on the $k$-means histogram clustering.

The question of how to determine the number of independent latent factors (topics) in mixture models such as Latent Dirichlet Allocation (LDA) is of great practical importance. In most applications, the exact number of topics is unknown, and depends on the application and the size of the data set. Bayesian nonparametric methods can avoid the problem of topic number selection, but they can be impracticably slow for large sample sizes and are subject to local optima. We develop a guaranteed procedure for topic number recovery that does not necessitate learning the model's latent parameters beforehand. Our procedure relies on adapting results from random matrix theory. Performance of our topic number recovery procedure is superior to hLDA, a nonparametric method. We also discuss some implications of our results on the sample complexity and accuracy of popular spectral learning algorithms for LDA. Our results and procedure can be extended to spectral learning algorithms for other exchangeable mixture models as well as Hidden Markov Models.

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.

Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of structural network properties. The present paper advocates two graph embedding approaches with centrality considerations to comply with node hierarchy. The problem is formulated first as one of constrained multi-dimensional scaling (MDS), and it is solved via block coordinate descent iterations with successive approximations and guaranteed convergence to a KKT point. In addition, a regularization term enforcing graph smoothness is incorporated with the goal of reducing edge crossings. A second approach leverages the locally-linear embedding (LLE) algorithm which assumes that the graph encodes data sampled from a low-dimensional manifold. Closed-form solutions to the resulting centrality-constrained optimization problems are determined yielding meaningful embeddings. Experimental results demonstrate the efficacy of both approaches, especially for visualizing large networks on the order of thousands of nodes.

In this work we consider the problem of detecting anomalous spatio-temporal behavior in videos. Our approach is to learn the normative multiframe pixel joint distribution and detect deviations from it using a likelihood based approach. Due to the extreme lack of available training samples relative to the dimension of the distribution, we use a mean and covariance approach and consider methods of learning the spatio-temporal covariance in the low-sample regime. Our approach is to estimate the covariance using parameter reduction and sparse models. The first method considered is the representation of the covariance as a sum of Kronecker products as in (Greenewald et al 2013), which is found to be an accurate approximation in this setting. We propose learning algorithms relevant to our problem. We then consider the sparse multiresolution model of (Choi et al 2010) and apply the Kronecker product methods to it for further parameter reduction, as well as introducing modifications for enhanced efficiency and greater applicability to spatio-temporal covariance matrices. We apply our methods to the detection of crowd behavior anomalies in the University of Minnesota crowd anomaly dataset, and achieve competitive results.

Tree structured graphical models are powerful at expressing long range or hierarchical dependency among many variables, and have been widely applied in different areas of computer science and statistics. However, existing methods for parameter estimation, inference, and structure learning mainly rely on the Gaussian or discrete assumptions, which are restrictive under many applications. In this paper, we propose new nonparametric methods based on reproducing kernel Hilbert space embeddings of distributions that can recover the latent tree structures, estimate the parameters, and perform inference for high dimensional continuous and non-Gaussian variables. The usefulness of the proposed methods are illustrated by thorough numerical results.

Canonical correlation analysis (CCA) has been one of the most popular methods for frequency recognition in steady-state visual evoked potential (SSVEP)-based brain-computer interfaces (BCIs). Despite its efficiency, a potential problem is that using pre-constructed sine-cosine waves as the required reference signals in the CCA method often does not result in the optimal recognition accuracy due to their lack of features from the real EEG data. To address this problem, this study proposes a novel method based on multiset canonical correlation analysis (MsetCCA) to optimize the reference signals used in the CCA method for SSVEP frequency recognition. The MsetCCA method learns multiple linear transforms that implement joint spatial filtering to maximize the overall correlation among canonical variates, and hence extracts SSVEP common features from multiple sets of EEG data recorded at the same stimulus frequency. The optimized reference signals are formed by combination of the common features and completely based on training data. Experimental study with EEG data from ten healthy subjects demonstrates that the MsetCCA method improves the recognition accuracy of SSVEP frequency in comparison with the CCA method and other two competing methods (multiway CCA (MwayCCA) and phase constrained CCA (PCCA)), especially for a small number of channels and a short time window length. The superiority indicates that the proposed MsetCCA method is a new promising candidate for frequency recognition in SSVEP-based BCIs.