Learning the joint probability of random variables (RVs) lies at the heart of statistical signal processing and machine learning. However, direct nonparametric estimation for high-dimensional joint probability is in general impossible, due to the curse of dimensionality. Recent work has proposed to recover the joint probability mass function (PMF) of an arbitrary number of RVs from three-dimensional marginals, leveraging the algebraic properties of low-rank tensor decomposition and the (unknown) dependence among the RVs. Nonetheless, accurately estimating three-dimensional marginals can still be costly in terms of sample complexity, affecting the performance of this line of work in practice in the sample-starved regime. Using three-dimensional marginals also involves challenging tensor decomposition problems whose tractability is unclear. This work puts forth a new framework for learning the joint PMF using only pairwise marginals, which naturally enjoys a lower sample complexity relative to the third-order ones. A coupled nonnegative matrix factorization (CNMF) framework is developed, and its joint PMF recovery guarantees under various conditions are analyzed. Our method also features a Gram-Schmidt (GS)-like algorithm that exhibits competitive runtime performance. The algorithm is shown to provably recover the joint PMF up to bounded error in finite iterations, under reasonable conditions. It is also shown that a recently proposed economical expectation maximization (EM) algorithm guarantees to improve upon the GS-like algorithm's output, thereby further lifting up the accuracy and efficiency. Real-data experiments are employed to showcase the effectiveness.

Our interest lies in the recoverability properties of compressed tensors under the \textit{canonical polyadic decomposition} (CPD) model. The considered problem is well-motivated in many applications, e.g., hyperspectral image and video compression. Prior work studied this problem under somewhat special assumptions---e.g., the latent factors of the tensor are sparse or drawn from absolutely continuous distributions. We offer an alternative result: We show that if the tensor is compressed by a subgaussian linear mapping, then the tensor is recoverable if the number of measurements is on the same order of magnitude as that of the model parameters---without strong assumptions on the latent factors. Our proof is based on deriving a \textit{restricted isometry property} (R.I.P.) under the CPD model via set covering techniques, and thus exhibits a flavor of classic compressive sensing. The new recoverability result enriches the understanding to the compressed CP tensor recovery problem; it offers theoretical guarantees for recovering tensors whose elements are not necessarily continuous or sparse.

The data deluge comes with high demands for data labeling. Crowdsourcing (or, more generally, ensemble learning) techniques aim to produce accurate labels via integrating noisy, non-expert labeling from annotators. The classic Dawid-Skene estimator and its accompanying expectation maximization (EM) algorithm have been widely used, but the theoretical properties are not fully understood. Tensor methods were proposed to guarantee identification of the Dawid-Skene model, but the sample complexity is a hurdle for applying such approaches---since the tensor methods hinge on the availability of third-order statistics that are hard to reliably estimate given limited data. In this paper, we propose a framework using pairwise co-occurrences of the annotator responses, which naturally admits lower sample complexity. We show that the approach can identify the Dawid-Skene model under realistic conditions. We propose an algebraic algorithm reminiscent of convex geometry-based structured matrix factorization to solve the model identification problem efficiently, and an identifiability-enhanced algorithm for handling more challenging and critical scenarios. Experiments show that the proposed algorithms outperform the state-of-art algorithms under a variety of scenarios.

--Multiview analysis aims at extracting shared latent components from data samples that are acquired in different domains, e.g., image, text, and audio. Classic multiview analysis, e.g., canonical correlation analysis (CCA), tackles this problem via matching the linearly transformed views in a certain latent domain. More recently, powerful nonlinear learning tools such as kernel methods and neural networks are utilized for enhancing the classic CCA. However, unlike linear CCA whose theoretical aspects are clearly understood, nonlinear CCA approaches are largely intuition-driven. In particular, it is unclear under what conditions the shared latent components across the veiws can be identified--while identifiability plays an essential role in many applications. In this work, we revisit nonlinear multiview analysis and address both the theoretical and computational aspects. We take a nonlinear multiview mixture learning viewpoint, which is a natural extension of the classic generative models for linear CCA. From there, we derive a nonlinear multiview analysis criteron. We show that minimizing this criterion leads to identification of the latent shared components up to certain ambiguities, under reasonable conditions. On the computation side, we propose an effective algorithm with simple and scalable update rules. A series of simulations and real-data experiments corroborate our theoretical analysis. Multiview analysis has been an indispensable tool in statistical signal processing, machine learning, and data analytics. In the context of multiview learning, a view can be understood as measurements of data entities (e.g., a cat) in a certain domain (e.g., text, image, and audio). Most data entities naturally appear in different domains. Multiview analysis aims at extracting essential and common information from different views. Compared with single-view analysis tools like principal component analysis (PCA), independent component analysis (ICA) [1], and nonnegative matrix factorization (NMF) [2], multiview analysis tools such as canonical correlation analysis (CCA) [3] have an array of unique features. For example, CCA has been shown to be more robust to noise and view-specific strong interference [4], [5]. The classic CCA has been extensively studied in the literature, ever since its proposal in statistics in the 1930s [3], [6]. Q. Lyu and X. Fu are with the School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR 97331, United States. The transformations are supposed to'project' the views to a domain where the views share similar representations. Interestingly, the formulated optimization problem, although being nonconvex, can be recast into a generalized eigende-composition problem and solved efficiently [3], [7].

Linear mixture models have proven very useful in a plethora of applications, e.g., topic modeling, clustering, and source separation. As a critical aspect of the linear mixture models, identifiability of the model parameters is well-studied, under frameworks such as independent component analysis and constrained matrix factorization. Nevertheless, when the linear mixtures are distorted by an unknown nonlinear functions -- which is well-motivated and more realistic in many cases -- the identifiability issues are much less studied. This work proposes an identification criterion for a nonlinear mixture model that is well grounded in many real-world applications, and offers identifiability guarantees. A practical implementation based on a judiciously designed neural network is proposed to realize the criterion, and an effective learning algorithm is proposed. Numerical results on synthetic and real-data corroborate effectiveness of the proposed method.

Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communities. NMF identifiability has a great impact on many aspects in practice, such as ill-posed formulation avoidance and performance-guaranteed algorithm design. On the other hand, there is no tutorial paper that introduces NMF from an identifiability viewpoint. In this paper, we aim at filling this gap by offering a comprehensive and deep tutorial on model identifiability of NMF as well as the connections to algorithms and applications. This tutorial will help researchers and graduate students grasp the essence and insights of NMF, thereby avoiding typical `pitfalls' that are often times due to unidentifiable NMF formulations. This paper will also help practitioners pick/design suitable factorization tools for their own problems.

We present a new algorithm for identifying the transition and emission probabilities of a hidden Markov model (HMM) from the emitted data. Expectation-maximization becomes computationally prohibitive for long observation records, which are often required for identification. The new algorithm is particularly suitable for cases where the available sample size is large enough to accurately estimate second-order output probabilities, but not higher-order ones. We show that if one is only able to obtain a reliable estimate of the pairwise co-occurrence probabilities of the emissions, it is still possible to uniquely identify the HMM if the emission probability is \emph{sufficiently scattered}. We apply our method to hidden topic Markov modeling, and demonstrate that we can learn topics with higher quality if documents are modeled as observations of HMMs sharing the same emission (topic) probability, compared to the simple but widely used bag-of-words model.

Epanechnikov Mean Shift is a simple yet empirically very effective algorithm for clustering. It localizes the centroids of data clusters via estimating modes of the probability distribution that generates the data points, using the "optimal" Epanechnikov kernel density estimator. However, since the procedure involves non-smooth kernel density functions,the convergence behavior of Epanechnikov mean shift lacks theoretical support as of this writing---most of the existing analyses are based on smooth functions and thus cannot be applied to Epanechnikov Mean Shift. In this work, we first show that the original Epanechnikov Mean Shift may indeed terminate at a non-critical point, due to the non-smoothness nature. Based on our analysis, we propose a simple remedy to fix it. The modified Epanechnikov Mean Shift is guaranteed to terminate at a local maximum of the estimated density, which corresponds to a cluster centroid, within a inite number of iterations. We also propose a way to avoid running the Mean Shift iterates from every data point, while maintaining good clustering accuracies under non-overlapping spherical Gaussian mixture models. This further pushes Epanechnikov Mean Shift to handle very large and high-dimensional data sets. Experiments show surprisingly good performance compared to the Lloyd's K-means algorithm and the EM algorithm.

Estimating the joint probability mass function (PMF) of a set of random variables lies at the heart of statistical learning and signal processing. Without structural assumptions, such as modeling the variables as a Markov chain, tree, or other graphical model, joint PMF estimation is often considered mission impossible - the number of unknowns grows exponentially with the number of variables. But who gives us the structural model? Is there a generic, 'non-parametric' way to control joint PMF complexity without relying on a priori structural assumptions regarding the underlying probability model? Is it possible to discover the operational structure without biasing the analysis up front? What if we only observe random subsets of the variables, can we still reliably estimate the joint PMF of all? This paper shows, perhaps surprisingly, that if the joint PMF of any three variables can be estimated, then the joint PMF of all the variables can be provably recovered under relatively mild conditions. The result is reminiscent of Kolmogorov's extension theorem - consistent specification of lower-order distributions induces a unique probability measure for the entire process. The difference is that for processes of limited complexity (rank of the high-order PMF) it is possible to obtain complete characterization from only third-order distributions. In fact not all third order PMFs are needed; and under more stringent conditions even second-order will do. Exploiting multilinear (tensor) algebra, this paper proves that such higher-order PMF completion can be guaranteed - several pertinent identifiability results are derived. It also provides a practical and efficient algorithm to carry out the recovery task. Judiciously designed simulations and real-data experiments on movie recommendation and data classification are presented to showcase the effectiveness of the approach.

Epanechnikov Mean Shift is a simple yet empirically very effective algorithm for clustering. It localizes the centroids of data clusters via estimating modes of the probability distribution that generates the data points, using the `optimal' Epanechnikov kernel density estimator. However, since the procedure involves non-smooth kernel density functions, the convergence behavior of Epanechnikov mean shift lacks theoretical support as of this writing---most of the existing analyses are based on smooth functions and thus cannot be applied to Epanechnikov Mean Shift. In this work, we first show that the original Epanechnikov Mean Shift may indeed terminate at a non-critical point, due to the non-smoothness nature. Based on our analysis, we propose a simple remedy to fix it. The modified Epanechnikov Mean Shift is guaranteed to terminate at a local maximum of the estimated density, which corresponds to a cluster centroid, within a finite number of iterations. We also propose a way to avoid running the Mean Shift iterates from every data point, while maintaining good clustering accuracies under non-overlapping spherical Gaussian mixture models. This further pushes Epanechnikov Mean Shift to handle very large and high-dimensional data sets. Experiments show surprisingly good performance compared to the Lloyd's K-means algorithm and the EM algorithm.