Search

Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a symmetric matrix with a product of a nonnegative, low-rank matrix and its transpose. To design faster and more scalable algorithms for SymNMF we develop two randomized algorithms for its computation. The first algorithm uses randomized matrix sketching to compute an initial low-rank input matrix and proceeds to use this input to rapidly compute a SymNMF. The second algorithm uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. We prove theoretically that leverage score sampling can approximately solve nonnegative least squares problems to a chosen accuracy with high probability. Finally we demonstrate that both methods work well in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.

In the rapidly evolving healthcare industry, platforms now have access to not only traditional medical records, but also diverse data sets encompassing various patient interactions, such as those from healthcare web portals. To address this rich diversity of data, we introduce WellFactor: a method that derives patient profiles by integrating information from these sources. Central to our approach is the utilization of constrained low-rank approximation. WellFactor is optimized to handle the sparsity that is often inherent in healthcare data. Moreover, by incorporating task-specific label information, our method refines the embedding results, offering a more informed perspective on patients. One important feature of WellFactor is its ability to compute embeddings for new, previously unobserved patient data instantaneously, eliminating the need to revisit the entire data set or recomputing the embedding. Comprehensive evaluations on real-world healthcare data demonstrate WellFactor's effectiveness. It produces better results compared to other existing methods in classification performance, yields meaningful clustering of patients, and delivers consistent results in patient similarity searches and predictions., Comment: 2023 IEEE International Conference on Big Data (IEEE BigData 2023)

We introduce a novel profile-based patient clustering model designed for clinical data in healthcare. By utilizing a method grounded on constrained low-rank approximation, our model takes advantage of patients' clinical data and digital interaction data, including browsing and search, to construct patient profiles. As a result of the method, nonnegative embedding vectors are generated, serving as a low-dimensional representation of the patients. Our model was assessed using real-world patient data from a healthcare web portal, with a comprehensive evaluation approach which considered clustering and recommendation capabilities. In comparison to other baselines, our approach demonstrated superior performance in terms of clustering coherence and recommendation accuracy., Comment: Accepted for the Short Paper track of CIKM'23, October 21-25, 2023, Birmingham, United Kingdom

Cut-based directed graph (digraph) clustering often focuses on finding dense within-cluster or sparse between-cluster connections, similar to cut-based undirected graph clustering methods. In contrast, for flow-based clusterings the edges between clusters tend to be oriented in one direction and have been found in migration data, food webs, and trade data. In this paper we introduce a spectral algorithm for finding flow-based clusterings. The proposed algorithm is based on recent work which uses complex-valued Hermitian matrices to represent digraphs. By establishing an algebraic relationship between a complex-valued Hermitian representation and an associated real-valued, skew-symmetric matrix the proposed algorithm produces clusterings while remaining completely in the real field. Our algorithm uses less memory and asymptotically less computation while provably preserving solution quality. We also show the algorithm can be easily implemented using standard computational building blocks, possesses better numerical properties, and loans itself to a natural interpretation via an objective function relaxation argument., Comment: 21 pages, 7 figures

Existing tensor factorization methods assume that the input tensor follows some specific distribution (i.e. Poisson, Bernoulli, and Gaussian), and solve the factorization by minimizing some empirical loss functions defined based on the corresponding distribution. However, it suffers from several drawbacks: 1) In reality, the underlying distributions are complicated and unknown, making it infeasible to be approximated by a simple distribution. 2) The correlation across dimensions of the input tensor is not well utilized, leading to sub-optimal performance. Although heuristics were proposed to incorporate such correlation as side information under Gaussian distribution, they can not easily be generalized to other distributions. Thus, a more principled way of utilizing the correlation in tensor factorization models is still an open challenge. Without assuming any explicit distribution, we formulate the tensor factorization as an optimal transport problem with Wasserstein distance, which can handle non-negative inputs. We introduce SWIFT, which minimizes the Wasserstein distance that measures the distance between the input tensor and that of the reconstruction. In particular, we define the N-th order tensor Wasserstein loss for the widely used tensor CP factorization and derive the optimization algorithm that minimizes it. By leveraging sparsity structure and different equivalent formulations for optimizing computational efficiency, SWIFT is as scalable as other well-known CP algorithms. Using the factor matrices as features, SWIFT achieves up to 9.65% and 11.31% relative improvement over baselines for downstream prediction tasks. Under the noisy conditions, SWIFT achieves up to 15% and 17% relative improvements over the best competitors for the prediction tasks., Comment: Accepted by AAAI-21

Nonnegative matrix factorization (NMF) is a prominent technique for data dimensionality reduction that has been widely used for text mining, computer vision, pattern discovery, and bioinformatics. In this paper, a framework called ARkNLS (Alternating Rank-k Nonnegativity constrained Least Squares) is proposed for computing NMF. First, a recursive formula for the solution of the rank-k nonnegativity-constrained least squares (NLS) is established. This recursive formula can be used to derive the closed-form solution for the Rank-k NLS problem for any integer k $\ge$ 1. As a result, each subproblem for an alternating rank-k nonnegative least squares framework can be obtained based on this closed form solution. Assuming that all matrices involved in rank-k NLS in the context of NMF computation are of full rank, two of the currently best NMF algorithms HALS (hierarchical alternating least squares) and ANLS-BPP (Alternating NLS based on Block Principal Pivoting) can be considered as special cases of ARkNLS with k = 1 and k = r for rank r NMF, respectively. This paper is then focused on the framework with k = 3, which leads to a new algorithm for NMF via the closed-form solution of the rank-3 NLS problem. Furthermore, a new strategy that efficiently overcomes the potential singularity problem in rank-3 NLS within the context of NMF computation is also presented. Extensive numerical comparisons using real and synthetic data sets demonstrate that the proposed algorithm provides state-of-the-art performance in terms of computational accuracy and cpu time.

We propose a flexible framework for clustering hypergraph-structured data based on recently proposed random walks utilizing edge-dependent vertex weights. When incorporating edge-dependent vertex weights (EDVW), a weight is associated with each vertex-hyperedge pair, yielding a weighted incidence matrix of the hypergraph. Such weightings have been utilized in term-document representations of text data sets. We explain how random walks with EDVW serve to construct different hypergraph Laplacian matrices, and then develop a suite of clustering methods that use these incidence matrices and Laplacians for hypergraph clustering. Using several data sets from real-life applications, we compare the performance of these clustering algorithms experimentally against a variety of existing hypergraph clustering methods. We show that the proposed methods produce higher-quality clusters and conclude by highlighting avenues for future work.

Phenotyping electronic health records (EHR) focuses on defining meaningful patient groups (e.g., heart failure group and diabetes group) and identifying the temporal evolution of patients in those groups. Tensor factorization has been an effective tool for phenotyping. Most of the existing works assume either a static patient representation with aggregate data or only model temporal data. However, real EHR data contain both temporal (e.g., longitudinal clinical visits) and static information (e.g., patient demographics), which are difficult to model simultaneously. In this paper, we propose Temporal And Static TEnsor factorization (TASTE) that jointly models both static and temporal information to extract phenotypes. TASTE combines the PARAFAC2 model with non-negative matrix factorization to model a temporal and a static tensor. To fit the proposed model, we transform the original problem into simpler ones which are optimally solved in an alternating fashion. For each of the sub-problems, our proposed mathematical reformulations lead to efficient sub-problem solvers. Comprehensive experiments on large EHR data from a heart failure (HF) study confirmed that TASTE is up to 14x faster than several baselines and the resulting phenotypes were confirmed to be clinically meaningful by a cardiologist. Using 80 phenotypes extracted by TASTE, a simple logistic regression can achieve the same level of area under the curve (AUC) for HF prediction compared to a deep learning model using recurrent neural networks (RNN) with 345 features., Comment: 19 pages

We consider the problem of low-rank approximation of massive dense non-negative tensor data, for example to discover latent patterns in video and imaging applications. As the size of data sets grows, single workstations are hitting bottlenecks in both computation time and available memory. We propose a distributed-memory parallel computing solution to handle massive data sets, loading the input data across the memories of multiple nodes and performing efficient and scalable parallel algorithms to compute the low-rank approximation. We present a software package called PLANC (Parallel Low Rank Approximation with Non-negativity Constraints), which implements our solution and allows for extension in terms of data (dense or sparse, matrices or tensors of any order), algorithm (e.g., from multiplicative updating techniques to alternating direction method of multipliers), and architecture (we exploit GPUs to accelerate the computation in this work).We describe our parallel distributions and algorithms, which are careful to avoid unnecessary communication and computation, show how to extend the software to include new algorithms and/or constraints, and report efficiency and scalability results for both synthetic and real-world data sets., Comment: arXiv admin note: text overlap with arXiv:1806.07985

Topic modeling is commonly used to analyze and understand large document collections. However, in practice, users want to focus on specific aspects or "targets" rather than the entire corpus. For example, given a large collection of documents, users may want only a smaller subset which more closely aligns with their interests, tasks, and domains. In particular, our paper focuses on large-scale document retrieval with high recall where any missed relevant documents can be critical. A simple keyword matching search is generally not effective nor efficient as 1) it is difficult to find a list of keyword queries that can cover the documents of interest before exploring the dataset, 2) some documents may not contain the exact keywords of interest but may still be highly relevant, and 3) some words have multiple meanings, which would result in irrelevant documents included in the retrieved subset. In this paper, we present TopicSifter, a visual analytics system for interactive search space reduction. Our system utilizes targeted topic modeling based on nonnegative matrix factorization and allows users to give relevance feedback in order to refine their target and guide the topic modeling to the most relevant results.

Understanding narrative content has become an increasingly popular topic. Nonetheless, research on identifying common types of narrative characters, or personae, is impeded by the lack of automatic and broad-coverage evaluation methods. We argue that computationally modeling actors provides benefits, including novel evaluation mechanisms for personae. Specifically, we propose two actor-modeling tasks, cast prediction and versatility ranking, which can capture complementary aspects of the relation between actors and the characters they portray. For an actor model, we present a technique for embedding actors, movies, character roles, genres, and descriptive keywords as Gaussian distributions and translation vectors, where the Gaussian variance corresponds to actors' versatility. Empirical results indicate that (1) the technique considerably outperforms TransE (Bordes et al. 2013) and ablation baselines and (2) automatically identified persona topics (Bamman, O'Connor, and Smith 2013) yield statistically significant improvements in both tasks, whereas simplistic persona descriptors including age and gender perform inconsistently, validating prior research., Comment: Accepted at AAAI 2019

This paper presents a new method, which we call SUSTain, that extends real-valued matrix and tensor factorizations to data where values are integers. Such data are common when the values correspond to event counts or ordinal measures. The conventional approach is to treat integer data as real, and then apply real-valued factorizations. However, doing so fails to preserve important characteristics of the original data, thereby making it hard to interpret the results. Instead, our approach extracts factor values from integer datasets as scores that are constrained to take values from a small integer set. These scores are easy to interpret: a score of zero indicates no feature contribution and higher scores indicate distinct levels of feature importance. At its core, SUSTain relies on: a) a problem partitioning into integer-constrained subproblems, so that they can be optimally solved in an efficient manner; and b) organizing the order of the subproblems' solution, to promote reuse of shared intermediate results. We propose two variants, SUSTain_M and SUSTain_T, to handle both matrix and tensor inputs, respectively. We evaluate SUSTain against several state-of-the-art baselines on both synthetic and real Electronic Health Record (EHR) datasets. Comparing to those baselines, SUSTain shows either significantly better fit or orders of magnitude speedups that achieve a comparable fit (up to 425X faster). We apply SUSTain to EHR datasets to extract patient phenotypes (i.e., clinically meaningful patient clusters). Furthermore, 87% of them were validated as clinically meaningful phenotypes related to heart failure by a cardiologist.

We present a hybrid method for latent information discovery on the data sets containing both text content and connection structure based on constrained low rank approximation. The new method jointly optimizes the Nonnegative Matrix Factorization (NMF) objective function for text clustering and the Symmetric NMF (SymNMF) objective function for graph clustering. We propose an effective algorithm for the joint NMF objective function, based on a block coordinate descent (BCD) framework. The proposed hybrid method discovers content associations via latent connections found using SymNMF. The method can also be applied with a natural conversion of the problem when a hypergraph formulation is used or the content is associated with hypergraph edges. Experimental results show that by simultaneously utilizing both content and connection structure, our hybrid method produces higher quality clustering results compared to the other NMF clustering methods that uses content alone (standard NMF) or connection structure alone (SymNMF). We also present some interesting applications to several types of real world data such as citation recommendations of papers. The hybrid method proposed in this paper can also be applied to general data expressed with both feature space vectors and pairwise similarities and can be extended to the case with multiple feature spaces or multiple similarity measures., Comment: 9 pages, Submitted to a conference, Feb. 2017

The problem of outlier detection is extremely challenging in many domains such as text, in which the attribute values are typically non-negative, and most values are zero. In such cases, it often becomes difficult to separate the outliers from the natural variations in the patterns in the underlying data. In this paper, we present a matrix factorization method, which is naturally able to distinguish the anomalies with the use of low rank approximations of the underlying data. Our iterative algorithm TONMF is based on block coordinate descent (BCD) framework. We define blocks over the term-document matrix such that the function becomes solvable. Given most recently updated values of other matrix blocks, we always update one block at a time to its optimal. Our approach has significant advantages over traditional methods for text outlier detection. Finally, we present experimental results illustrating the effectiveness of our method over competing methods., Comment: Accepted at 2017 SIAM Data Mining Conference

Embedding and visualizing large-scale high-dimensional data in a two-dimensional space is an important problem since such visualization can reveal deep insights out of complex data. Most of the existing embedding approaches, however, run on an excessively high precision, ignoring the fact that at the end, embedding outputs are converted into coarse-grained discrete pixel coordinates in a screen space. Motivated by such an observation and directly considering pixel coordinates in an embedding optimization process, we accelerate Barnes-Hut tree-based t-distributed stochastic neighbor embedding (BH-SNE), known as a state-of-the-art 2D embedding method, and propose a novel method called PixelSNE, a highly-efficient, screen resolution-driven 2D embedding method with a linear computational complexity in terms of the number of data items. Our experimental results show the significantly fast running time of PixelSNE by a large margin against BH-SNE, while maintaining the minimal degradation in the embedding quality. Finally, the source code of our method is publicly available at https://github.com/awesome-davian/PixelSNE

Non-negative matrix factorization (NMF) is the problem of determining two non-negative low rank factors $W$ and $H$, for the given input matrix $A$, such that $A \approx W H$. NMF is a useful tool for many applications in different domains such as topic modeling in text mining, background separation in video analysis, and community detection in social networks. Despite its popularity in the data mining community, there is a lack of efficient parallel algorithms to solve the problem for big data sets. The main contribution of this work is a new, high-performance parallel computational framework for a broad class of NMF algorithms that iteratively solves alternating non-negative least squares (NLS) subproblems for $W$ and $H$. It maintains the data and factor matrices in memory (distributed across processors), uses MPI for interprocessor communication, and, in the dense case, provably minimizes communication costs (under mild assumptions). The framework is flexible and able to leverage a variety of NMF and NLS algorithms, including Multiplicative Update, Hierarchical Alternating Least Squares, and Block Principal Pivoting. Our implementation allows us to benchmark and compare different algorithms on massive dense and sparse data matrices of size that spans for few hundreds of millions to billions. We demonstrate the scalability of our algorithm and compare it with baseline implementations, showing significant performance improvements. The code and the datasets used for conducting the experiments are available online., Comment: arXiv admin note: text overlap with arXiv:1509.09313

Integer data sets frequently appear in many applications in sciences and technology. To analyze these, integer low rank approximation has received much attention due to its capacity of representing the results in integers preserving the meaning of the original data sets. To our knowledge, none of previously proposed techniques developed for real numbers can be successfully applied, since integers are discrete in nature. In this work, we start with a thorough review of algorithms for solving integer least squares problems, {and} then develop a block coordinate descent method based on the integer least squares estimation to obtain the integer low rank approximation of integer matrices. The numerical application on association analysis and numerical experiments on random integer matrices are presented. Our computed results seem to suggest that our method can find a more accurate solution than other existing methods for continuous data sets., Comment: 18 pages, 2 figures

Non-negative matrix factorization (NMF) is the problem of determining two non-negative low rank factors $W$ and $H$, for the given input matrix $A$, such that $A \approx W H$. NMF is a useful tool for many applications in different domains such as topic modeling in text mining, background separation in video analysis, and community detection in social networks. Despite its popularity in the data mining community, there is a lack of efficient parallel software to solve the problem for big datasets. Existing distributed-memory algorithms are limited in terms of performance and applicability, as they are implemented using Hadoop and are designed only for sparse matrices. We propose a distributed-memory parallel algorithm that computes the factorization by iteratively solving alternating non-negative least squares (NLS) subproblems for $W$ and $H$. To our knowledge, our algorithm is the first high-performance parallel algorithm for NMF. It maintains the data and factor matrices in memory (distributed across processors), uses MPI for interprocessor communication, and, in the dense case, provably minimizes communication costs (under mild assumptions). As opposed to previous implementations, our algorithm is also flexible: (1) it performs well for dense and sparse matrices, and (2) it allows the user to choose from among multiple algorithms for solving local NLS subproblems within the alternating iterations. We demonstrate the scalability of our algorithm and compare it with baseline implementations, showing significant performance improvements.

The importance of unsupervised clustering and topic modeling is well recognized with ever-increasing volumes of text data. In this paper, we propose a fast method for hierarchical clustering and topic modeling called HierNMF2. Our method is based on fast Rank-2 nonnegative matrix factorization (NMF) that performs binary clustering and an efficient node splitting rule. Further utilizing the final leaf nodes generated in HierNMF2 and the idea of nonnegative least squares fitting, we propose a new clustering/topic modeling method called FlatNMF2 that recovers a flat clustering/topic modeling result in a very simple yet significantly more effective way than any other existing methods. We implement highly optimized open source software in C++ for both HierNMF2 and FlatNMF2 for hierarchical and partitional clustering/topic modeling of document data sets. Substantial experimental tests are presented that illustrate significant improvements both in computational time as well as quality of solutions. We compare our methods to other clustering methods including K-means, standard NMF, and CLUTO, and also topic modeling methods including latent Dirichlet allocation (LDA) and recently proposed algorithms for NMF with separability constraints. Overall, we present efficient tools for analyzing large-scale data sets, and techniques that can be generalized to many other data analytics problem domains., Comment: This paper has been withdrawn by the author to clarify the authorship

Comments for a product or a news article are rapidly growing and became a medium of measuring quality products or services. Consequently, spammers have been emerged in this area to bias them toward their favor. In this paper, we propose an efficient spammer detection method using structural rank of author specific term-document matrices. The use of structural rank was found effective and far faster than similar methods., Comment: 8 pages, 1 figure

In this paper, we design a hierarchical clustering algorithm for high-resolution hyperspectral images. At the core of the algorithm, a new rank-two nonnegative matrix factorizations (NMF) algorithm is used to split the clusters, which is motivated by convex geometry concepts. The method starts with a single cluster containing all pixels, and, at each step, (i) selects a cluster in such a way that the error at the next step is minimized, and (ii) splits the selected cluster into two disjoint clusters using rank-two NMF in such a way that the clusters are well balanced and stable. The proposed method can also be used as an endmember extraction algorithm in the presence of pure pixels. The effectiveness of this approach is illustrated on several synthetic and real-world hyperspectral images, and shown to outperform standard clustering techniques such as k-means, spherical k-means and standard NMF., Comment: 29 pages, 19 figures. New experiment on Terrain data set. Accepted in IEEE Trans. Geosci. Remote Sens

Discovering the latent structure from many observed variables is an important yet challenging learning task. Existing approaches for discovering latent structures often require the unknown number of hidden states as an input. In this paper, we propose a quartet based approach which is \emph{agnostic} to this number. The key contribution is a novel rank characterization of the tensor associated with the marginal distribution of a quartet. This characterization allows us to design a \emph{nuclear norm} based test for resolving quartet relations. We then use the quartet test as a subroutine in a divide-and-conquer algorithm for recovering the latent tree structure. Under mild conditions, the algorithm is consistent and its error probability decays exponentially with increasing sample size. We demonstrate that the proposed approach compares favorably to alternatives. In a real world stock dataset, it also discovers meaningful groupings of variables, and produces a model that fits the data better.