Abstract We propose an efficient distributed out-of-memory implementation of the non-negative matrix factorization (NMF) algorithm for heterogeneous high-performance-computing systems. The proposed implementation is based on prior work on NMFk, which can perform automatic model selection and extract latent variables and patterns from data. In this work, we extend NMFk by adding support for dense and sparse matrix operation on multi-node, multi-GPU systems. The resulting algorithm is optimized for out-of-memory problems where the memory required to factorize a given matrix is greater than the available GPU memory. Memory complexity is reduced by batching/tiling strategies, and sparse and dense matrix operations are significantly accelerated with GPU cores (or tensor cores when available). Input/output latency associated with batch copies between host and device is hidden using CUDA streams to overlap data transfers and compute asynchronously, and latency associated with collective communications (both intra-node and inter-node) is reduced using optimized NVIDIA Collective Communication Library (NCCL) based communicators. Benchmark results show significant improvement, from 32X to 76x speedup, with the new implementation using GPUs over the CPU-based NMFk. Good weak scaling was demonstrated on up to 4096 multi-GPU cluster nodes with approximately 25,000 GPUs when decomposing a dense 340 Terabyte-size matrix and an 11 Exabyte-size sparse matrix of density $$10^{-6}$$ 10 - 6 .

With the boom in the development of computer hardware and software, social media, IoT platforms, and communications, there has been exponential growth in the volume of data produced worldwide. Among these data, relational datasets are growing in popularity as they provide unique insights regarding the evolution of communities and their interactions. Relational datasets are naturally non-negative, sparse, and extra-large. Relational data usually contain triples (subject, relation, object) and are represented as graphs/multigraphs, called knowledge graphs, which need to be embedded into a low-dimensional dense vector space. Among various embedding models, RESCAL allows the learning of relational data to extract the posterior distributions over the latent variables and to make predictions of missing relations. However, RESCAL is computationally demanding and requires a fast and distributed implementation to analyze extra-large real-world datasets. Here we introduce a distributed non-negative RESCAL algorithm for heterogeneous CPU/GPU architectures with automatic selection of the number of latent communities (model selection), called pyDRESCALk. We demonstrate the correctness of pyDRESCALk with real-world and large synthetic tensors and the efficacy showing near-linear scaling that concurs with the theoretical complexities. Finally, pyDRESCALk determines the number of latent communities in an 11-terabyte dense and 9-exabyte sparse synthetic tensor.

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and ℤ 2 ). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization X = W ∧ H has a unique W , a unique H (for a fixed W ), and when both W and H are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMF k , and show on numerical examples that BMF k not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.

Non-negative Matrix Factorization (NMF) models the topics of a text corpus by decomposing the matrix of term frequency-inverse document frequency (TF-IDF) representation, X, into two low-rank non-negative matrices: W , representing the topics and H, mapping the documents onto space of topics. One challenge, common to all topic models, is the determination of the number of latent topics (aka model determination). Determining the correct number of topics is important: underestimating the number of topics results in a poor topic separation, under-fitting, while overestimating leads to noisy topics, over-fitting. Here, we introduce SeNMFk, a semantic-assisted NMF-based topic modeling method, which incorporates semantic correlations in NMF by using a word-context matrix, and employs a method for determination of the number of latent topics. SeNMFk first creates a random ensemble of matrices based on the initial TF-IDF matrix and a word-context matrix, and then applies a coupled factorization to acquire sets of stable coherent topics that are robust to noise. The latent dimension is determined based on the stability of these topics. We show that SeNMFk accurately determines the number of high-quality topics in benchmark text corpora, which leads to an accurate document clustering.

Data heterogeneity can pose a great challenge to process and systematically fuse low-level data from different modalities with no recourse to heuristics and manual adjustments and refinements. In this paper, a new methodology is introduced for the fusion of measured data for detecting and predicting weather-driven natural hazards. The proposed research introduces a robust theoretical and algorithmic framework for the fusion of heterogeneous data in near real time. We establish a flexible information-based fusion framework with a target optimality criterion of choice, which for illustration, is specialized to a maximum entropy principle and a least effort principle for semisupervised learning with noisy labels. We develop a methodology to account for multimodality data and a solution for addressing inherent sensor limitations. In our case study of interest, namely, that of flood density estimation, we further show that by fusing remote sensing and social media data, we can develop well founded and actionable flood maps. This capability is valuable in situations where environmental hazards, such as hurricanes or severe weather, affect very large areas. Relative to the state of the art working with such data, our proposed information-theoretic solution is principled and systematic, while offering a joint exploitation of any set of heterogeneous sensor modalities with minimally assuming priors. This flexibility is coupled with the ability to quantitatively and clearly state the fusion principles with very reasonable computational costs. The proposed method is tested and substantiated with the multimodality data of a 2013 Boulder Colorado flood event.

Hierarchical dictionary learning seeks multiple dictionaries at different image scales to capture complementary coherent characteristics. We propose a method to learn a hierarchy of two overcomplete synthesis dictionaries with an image classification goal. The classification objective in some sense regularizes the joint optimization of the hierarchical dictionaries and injects refinement feedback. The validation of the proposed approach is based on its classification performance using two well-known data sets.

Data fusion is the operation of integrating data from different modalities to construct a single consistent representation. This paper proposes variations of coupled dictionary learning through an additional factorization. One variation of this model is applicable to the pansharpening data fusion problem. Real world pansharpening data was applied to train and test our proposed formulation. The results demonstrate that the data fusion model can successfully be applied to the pan-sharpening problem.

Analysis procedures for higher-dimensional data are generally computationally costly; thereby justifying the high research interest in the area. Entropy-based divergence measures have proven their effectiveness in many areas of computer vision and pattern recognition. However, the complexity of their implementation might be prohibitive in resource-limited applications, as they require estimates of probability densities which are very difficult to compute directly for high-dimensional data. In this paper, we investigate the usage of a non-parametric distribution-free metric, known as the Henze-Penrose test statistic, to estimate the divergence between different classes of vehicles. In this regard, we apply some common feature extraction techniques to further characterize the distributional separation relative to the original data. Moreover, we employ the Henze-Penrose metric to obtain bounds for the Nearest Neighbor (NN) classification accuracy. Simulation results demonstrate the effectiveness and the reliability of this metric in estimating the inter-class separability. In addition, the proposed bounds are exploited for selecting the least number of features that would retain sufficient discriminative information.