A unified approach to non-negative matrix factorization and probabilistic latent semantic indexing

Non-negative matrix factorization (NMF) by the multiplicative updates algorithm is a powerful machine learning method for decomposing a high-dimensional nonnegative matrix V into two matrices, W and H, each with nonnegative entries, V ~ WH. NMF has been shown to have a unique parts-based, sparse representation of the data. The nonnegativity constraints in NMF allow only additive combinations of the data which enables it to learn parts that have distinct physical representations in reality. In the last few years, NMF has been successfully applied in a variety of areas such as natural language processing, information retrieval, image processing, speech recognition and computational biology for the analysis and interpretation of large-scale data. We present a generalized approach to NMF based on Renyi\u27s divergence between two non-negative matrices related to the Poisson likelihood. Our approach unifies various competing models and provides a unique framework for NMF. Furthermore, we generalize the equivalence between NMF and probabilistic latent semantic indexing, a well-known method used in text mining and document clustering applications. We evaluate the performance of our method in the unsupervised setting using consensus clustering and demonstrate its applicability using real-life and simulated data

Fast Parallel Randomized Algorithm for Nonnegative Matrix Factorization with KL Divergence for Large Sparse Datasets

Nonnegative Matrix Factorization (NMF) with Kullback-Leibler Divergence (NMF-KL) is one of the most significant NMF problems and equivalent to Probabilistic Latent Semantic Indexing (PLSI), which has been successfully applied in many applications. For sparse count data, a Poisson distribution and KL divergence provide sparse models and sparse representation, which describe the random variation better than a normal distribution and Frobenius norm. Specially, sparse models provide more concise understanding of the appearance of attributes over latent components, while sparse representation provides concise interpretability of the contribution of latent components over instances. However, minimizing NMF with KL divergence is much more difficult than minimizing NMF with Frobenius norm; and sparse models, sparse representation and fast algorithms for large sparse datasets are still challenges for NMF with KL divergence. In this paper, we propose a fast parallel randomized coordinate descent algorithm having fast convergence for large sparse datasets to archive sparse models and sparse representation. The proposed algorithm's experimental results overperform the current studies' ones in this problem

Advanced Probabilistic Models for Clustering and Projection

Probabilistic modeling for data mining and machine learning problems is a fundamental research area. The general approach is to assume a generative model underlying the observed data, and estimate model parameters via likelihood maximization. It has the deep probability theory as the mathematical background, and enjoys a large amount of methods from statistical learning, sampling theory and Bayesian statistics. In this thesis we study several advanced probabilistic models for data clustering and feature projection, which are the two important unsupervised learning problems. The goal of clustering is to group similar data points together to uncover the data clusters. While numerous methods exist for various clustering tasks, one important question still remains, i.e., how to automatically determine the number of clusters. The first part of the thesis answers this question from a mixture modeling perspective. A finite mixture model is first introduced for clustering, in which each mixture component is assumed to be an exponential family distribution for generality. The model is then extended to an infinite mixture model, and its strong connection to Dirichlet process (DP) is uncovered which is a non-parametric Bayesian framework. A variational Bayesian algorithm called VBDMA is derived from this new insight to learn the number of clusters automatically, and empirical studies on some 2D data sets and an image data set verify the effectiveness of this algorithm. In feature projection, we are interested in dimensionality reduction and aim to find a low-dimensional feature representation for the data. We first review the well-known principal component analysis (PCA) and its probabilistic interpretation (PPCA), and then generalize PPCA to a novel probabilistic model which is able to handle non-linear projection known as kernel PCA. An expectation-maximization (EM) algorithm is derived for kernel PCA such that it is fast and applicable to large data sets. Then we propose a novel supervised projection method called MORP, which can take the output information into account in a supervised learning context. Empirical studies on various data sets show much better results compared to unsupervised projection and other supervised projection methods. At the end we generalize MORP probabilistically to propose SPPCA for supervised projection, and we can also naturally extend the model to S2PPCA which is a semi-supervised projection method. This allows us to incorporate both the label information and the unlabeled data into the projection process. In the third part of the thesis, we introduce a unified probabilistic model which can handle data clustering and feature projection jointly. The model can be viewed as a clustering model with projected features, and a projection model with structured documents. A variational Bayesian learning algorithm can be derived, and it turns out to iterate the clustering operations and projection operations until convergence. Superior performance can be obtained for both clustering and projection

No Pattern, No Recognition: a Survey about Reproducibility and Distortion Issues of Text Clustering and Topic Modeling

Extracting knowledge from unlabeled texts using machine learning algorithms can be complex. Document categorization and information retrieval are two applications that may benefit from unsupervised learning (e.g., text clustering and topic modeling), including exploratory data analysis. However, the unsupervised learning paradigm poses reproducibility issues. The initialization can lead to variability depending on the machine learning algorithm. Furthermore, the distortions can be misleading when regarding cluster geometry. Amongst the causes, the presence of outliers and anomalies can be a determining factor. Despite the relevance of initialization and outlier issues for text clustering and topic modeling, the authors did not find an in-depth analysis of them. This survey provides a systematic literature review (2011-2022) of these subareas and proposes a common terminology since similar procedures have different terms. The authors describe research opportunities, trends, and open issues. The appendices summarize the theoretical background of the text vectorization, the factorization, and the clustering algorithms that are directly or indirectly related to the reviewed works

Research on Multilingual News Clustering Based on Cross-Language Word Embeddings

Classifying the same event reported by different countries is of significant importance for public opinion control and intelligence gathering. Due to the diverse types of news, relying solely on transla-tors would be costly and inefficient, while depending solely on translation systems would incur considerable performance overheads in invoking translation interfaces and storing translated texts. To address this issue, we mainly focus on the clustering problem of cross-lingual news. To be specific, we use a combination of sentence vector representations of news headlines in a mixed semantic space and the topic probability distributions of news content to represent a news article. In the training of cross-lingual models, we employ knowledge distillation techniques to fit two semantic spaces into a mixed semantic space. We abandon traditional static clustering methods like K-Means and AGNES in favor of the incremental clustering algorithm Single-Pass, which we further modify to better suit cross-lingual news clustering scenarios. Our main contributions are as follows: (1) We adopt the English standard BERT as the teacher model and XLM-Roberta as the student model, training a cross-lingual model through knowledge distillation that can represent sentence-level bilingual texts in both Chinese and English. (2) We use the LDA topic model to represent news as a combina-tion of cross-lingual vectors for headlines and topic probability distributions for con-tent, introducing concepts such as topic similarity to address the cross-lingual issue in news content representation. (3) We adapt the Single-Pass clustering algorithm for the news context to make it more applicable. Our optimizations of Single-Pass include ad-justing the distance algorithm between samples and clusters, adding cluster merging operations, and incorporating a news time parameter

Advanced Probabilistic Models for Clustering and Projection

Probabilistic modeling for data mining and machine learning problems is a fundamental research area. The general approach is to assume a generative model underlying the observed data, and estimate model parameters via likelihood maximization. It has the deep probability theory as the mathematical background, and enjoys a large amount of methods from statistical learning, sampling theory and Bayesian statistics. In this thesis we study several advanced probabilistic models for data clustering and feature projection, which are the two important unsupervised learning problems. The goal of clustering is to group similar data points together to uncover the data clusters. While numerous methods exist for various clustering tasks, one important question still remains, i.e., how to automatically determine the number of clusters. The first part of the thesis answers this question from a mixture modeling perspective. A finite mixture model is first introduced for clustering, in which each mixture component is assumed to be an exponential family distribution for generality. The model is then extended to an infinite mixture model, and its strong connection to Dirichlet process (DP) is uncovered which is a non-parametric Bayesian framework. A variational Bayesian algorithm called VBDMA is derived from this new insight to learn the number of clusters automatically, and empirical studies on some 2D data sets and an image data set verify the effectiveness of this algorithm. In feature projection, we are interested in dimensionality reduction and aim to find a low-dimensional feature representation for the data. We first review the well-known principal component analysis (PCA) and its probabilistic interpretation (PPCA), and then generalize PPCA to a novel probabilistic model which is able to handle non-linear projection known as kernel PCA. An expectation-maximization (EM) algorithm is derived for kernel PCA such that it is fast and applicable to large data sets. Then we propose a novel supervised projection method called MORP, which can take the output information into account in a supervised learning context. Empirical studies on various data sets show much better results compared to unsupervised projection and other supervised projection methods. At the end we generalize MORP probabilistically to propose SPPCA for supervised projection, and we can also naturally extend the model to S2PPCA which is a semi-supervised projection method. This allows us to incorporate both the label information and the unlabeled data into the projection process. In the third part of the thesis, we introduce a unified probabilistic model which can handle data clustering and feature projection jointly. The model can be viewed as a clustering model with projected features, and a projection model with structured documents. A variational Bayesian learning algorithm can be derived, and it turns out to iterate the clustering operations and projection operations until convergence. Superior performance can be obtained for both clustering and projection

Advances in Nonnegative Matrix Decomposition with Application to Cluster Analysis

Nonnegative Matrix Factorization (NMF) has found a wide variety of applications in machine learning and data mining. NMF seeks to approximate a nonnegative data matrix by a product of several low-rank factorizing matrices, some of which are constrained to be nonnegative. Such additive nature often results in parts-based representation of the data, which is a desired property especially for cluster analysis.  This thesis presents advances in NMF with application in cluster analysis. It reviews a class of higher-order NMF methods called Quadratic Nonnegative Matrix Factorization (QNMF). QNMF differs from most existing NMF methods in that some of its factorizing matrices occur twice in the approximation. The thesis also reviews a structural matrix decomposition method based on Data-Cluster-Data (DCD) random walk. DCD goes beyond matrix factorization and has a solid probabilistic interpretation by forming the approximation with cluster assigning probabilities only. Besides, the Kullback-Leibler divergence adopted by DCD is advantageous in handling sparse similarities for cluster analysis.  Multiplicative update algorithms have been commonly used for optimizing NMF objectives, since they naturally maintain the nonnegativity constraint of the factorizing matrix and require no user-specified parameters. In this work, an adaptive multiplicative update algorithm is proposed to increase the convergence speed of QNMF objectives.  Initialization conditions play a key role in cluster analysis. In this thesis, a comprehensive initialization strategy is proposed to improve the clustering performance by combining a set of base clustering methods. The proposed method can better accommodate clustering methods that need a careful initialization such as the DCD.  The proposed methods have been tested on various real-world datasets, such as text documents, face images, protein, etc. In particular, the proposed approach has been applied to the cluster analysis of emotional data