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.
The sum-of-correlations (SUMCOR) formulation of generalized canonical correlation analysis (GCCA) seeks highly correlated low-dimensional representations of different views via maximizing pairwise latent similarity of the views. SUMCOR is considered arguably the most natural extension of classical two-view CCA to the multiview case, and thus has numerous applications in signal processing and data analytics. Recent work has proposed effective algorithms for handling the SUMCOR problem at very large scale. However, the existing scalable algorithms cannot incorporate structural regularization and prior information -- which are critical for good performance in real-world applications. In this work, we propose a new computational framework for large-scale SUMCOR GCCA that can easily incorporate a suite of structural regularizers which are frequently used in data analytics. The updates of the proposed algorithm are lightweight and the memory complexity is also low. In addition, the proposed algorithm can be readily implemented in a parallel fashion. We show that the proposed algorithm converges to a Karush-Kuhn-Tucker (KKT) point of the regularized SUMCOR problem. Judiciously designed simulations and real-data experiments are employed to demonstrate the effectiveness of the proposed algorithm.
We study the problem of nonnegative rank-one approximation of a nonnegative tensor, and show that the globally optimal solution that minimizes the generalized Kullback-Leibler divergence can be efficiently obtained, i.e., it is not NP-hard. This result works for arbitrary nonnegative tensors with an arbitrary number of modes (including two, i.e., matrices). We derive a closed-form expression for the KL principal component, which is easy to compute and has an intuitive probabilistic interpretation. For generalized KL approximation with higher ranks, the problem is for the first time shown to be equivalent to multinomial latent variable modeling, and an iterative algorithm is derived that resembles the expectation-maximization algorithm. On the Iris dataset, we showcase how the derived results help us learn the model in an \emph{unsupervised} manner, and obtain strikingly close performance to that from supervised methods.
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.
Most learning approaches treat dimensionality reduction (DR) and clustering separately (i.e., sequentially), but recent research has shown that optimizing the two tasks jointly can substantially improve the performance of both. The premise behind the latter genre is that the data samples are obtained via linear transformation of latent representations that are easy to cluster; but in practice, the transformation from the latent space to the data can be more complicated. In this work, we assume that this transformation is an unknown and possibly nonlinear function. To recover the `clustering-friendly' latent representations and to better cluster the data, we propose a joint DR and K-means clustering approach in which DR is accomplished via learning a deep neural network (DNN). The motivation is to keep the advantages of jointly optimizing the two tasks, while exploiting the deep neural network's ability to approximate any nonlinear function. This way, the proposed approach can work well for a broad class of generative models. Towards this end, we carefully design the DNN structure and the associated joint optimization criterion, and propose an effective and scalable algorithm to handle the formulated optimization problem. Experiments using different real datasets are employed to showcase the effectiveness of the proposed approach.
Generalized canonical correlation analysis (GCCA) aims at finding latent low-dimensional common structure from multiple views (feature vectors in different domains) of the same entities. Unlike principal component analysis (PCA) that handles a single view, (G)CCA is able to integrate information from different feature spaces. Here we focus on MAX-VAR GCCA, a popular formulation which has recently gained renewed interest in multilingual processing and speech modeling. The classic MAX-VAR GCCA problem can be solved optimally via eigen-decomposition of a matrix that compounds the (whitened) correlation matrices of the views; but this solution has serious scalability issues, and is not directly amenable to incorporating pertinent structural constraints such as non-negativity and sparsity on the canonical components. We posit regularized MAX-VAR GCCA as a non-convex optimization problem and propose an alternating optimization (AO)-based algorithm to handle it. Our algorithm alternates between {\em inexact} solutions of a regularized least squares subproblem and a manifold-constrained non-convex subproblem, thereby achieving substantial memory and computational savings. An important benefit of our design is that it can easily handle structure-promoting regularization. We show that the algorithm globally converges to a critical point at a sublinear rate, and approaches a global optimal solution at a linear rate when no regularization is considered. Judiciously designed simulations and large-scale word embedding tasks are employed to showcase the effectiveness of the proposed algorithm.
There has recently been considerable interest in completing a low-rank matrix or tensor given only a small fraction (or few linear combinations) of its entries. Related approaches have found considerable success in the area of recommender systems, under machine learning. From a statistical estimation point of view, the gold standard is to have access to the joint probability distribution of all pertinent random variables, from which any desired optimal estimator can be readily derived. In practice high-dimensional joint distributions are very hard to estimate, and only estimates of low-dimensional projections may be available. We show that it is possible to identify higher-order joint PMFs from lower-order marginalized PMFs using coupled low-rank tensor factorization. Our approach features guaranteed identifiability when the full joint PMF is of low-enough rank, and effective approximation otherwise. We provide an algorithmic approach to compute the sought factors, and illustrate the merits of our approach using rating prediction as an example.
Tensors or {\em multi-way arrays} are functions of three or more indices $(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing, statistics, data mining and machine learning. This overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors. As such, it focuses on fundamentals and motivation (using various application examples), aiming to strike an appropriate balance of breadth {\em and depth} that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and/or developing tensor algorithms and software. Some background in applied optimization is useful but not strictly required. The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties (including fairly good coverage of identifiability); broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.
In topic modeling, many algorithms that guarantee identifiability of the topics have been developed under the premise that there exist anchor words -- i.e., words that only appear (with positive probability) in one topic. Follow-up work has resorted to three or higher-order statistics of the data corpus to relax the anchor word assumption. Reliable estimates of higher-order statistics are hard to obtain, however, and the identification of topics under those models hinges on uncorrelatedness of the topics, which can be unrealistic. This paper revisits topic modeling based on second-order moments, and proposes an anchor-free topic mining framework. The proposed approach guarantees the identification of the topics under a much milder condition compared to the anchor-word assumption, thereby exhibiting much better robustness in practice. The associated algorithm only involves one eigen-decomposition and a few small linear programs. This makes it easy to implement and scale up to very large problem instances. Experiments using the TDT2 and Reuters-21578 corpus demonstrate that the proposed anchor-free approach exhibits very favorable performance (measured using coherence, similarity count, and clustering accuracy metrics) compared to the prior art.
This paper considers \emph{volume minimization} (VolMin)-based structured matrix factorization (SMF). VolMin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum-volume simplex that encloses all the columns of the data matrix. Recent work showed that VolMin guarantees the identifiability of the factor matrices under mild conditions that are realistic in a wide variety of applications. This paper focuses on both theoretical and practical aspects of VolMin. On the theory side, exact equivalence of two independently developed sufficient conditions for VolMin identifiability is proven here, thereby providing a more comprehensive understanding of this aspect of VolMin. On the algorithm side, computational complexity and sensitivity to outliers are two key challenges associated with real-world applications of VolMin. These are addressed here via a new VolMin algorithm that handles volume regularization in a computationally simple way, and automatically detects and {iteratively downweights} outliers, simultaneously. Simulations and real-data experiments using a remotely sensed hyperspectral image and the Reuters document corpus are employed to showcase the effectiveness of the proposed algorithm.