Nonnegative matrix factorization (NMF) has an established reputation as a useful data analysis technique in numerous applications. However, its usage in practical situations is undergoing challenges in recent years. The fundamental factor to this is the increasingly growing size of the datasets available and needed in the information sciences. To address this, in this work we propose to use structured random compression, that is, random projections that exploit the data structure, for two NMF variants: classical and separable. In separable NMF (SNMF) the left factors are a subset of the columns of the input matrix. We present suitable formulations for each problem, dealing with different representative algorithms within each one. We show that the resulting compressed techniques are faster than their uncompressed variants, vastly reduce memory demands, and do not encompass any significant deterioration in performance. The proposed structured random projections for SNMF allow to deal with arbitrarily shaped large matrices, beyond the standard limit of tall-and-skinny matrices, granting access to very efficient computations in this general setting. We accompany the algorithmic presentation with theoretical foundations and numerous and diverse examples, showing the suitability of the proposed approaches.
Tensor decompositions are powerful tools for large data analytics as they jointly model multiple aspects of data into one framework and enable the discovery of the latent structures and higher-order correlations within the data. One of the most widely studied and used decompositions, especially in data mining and machine learning, is the Canonical Polyadic or CP decomposition. However, today's datasets are not static and these datasets often dynamically growing and changing with time. To operate on such large data, we present OCTen the first ever compression-based online parallel implementation for the CP decomposition. We conduct an extensive empirical analysis of the algorithms in terms of fitness, memory used and CPU time, and in order to demonstrate the compression and scalability of the method, we apply OCTen to big tensor data. Indicatively, OCTen performs on-par or better than state-of-the-art online and online methods in terms of decomposition accuracy and efficiency, while saving up to 40-200 % memory space.
The robust principal component analysis (RPCA) problem seeks to separate low-rank trends from sparse outlierswithin a data matrix, that is, to approximate a $n\times d$ matrix $D$ as the sum of a low-rank matrix $L$ and a sparse matrix $S$.We examine the robust principal component analysis (RPCA) problem under data compression, wherethe data $Y$ is approximately given by $(L + S)\cdot C$, that is, a low-rank $+$ sparse data matrix that has been compressed to size $n\times m$ (with $m$ substantially smaller than the original dimension $d$) via multiplication witha compression matrix $C$. We give a convex program for recovering the sparse component $S$ along with the compressed low-rank component $L\cdot C$, along with upper bounds on the error of this reconstructionthat scales naturally with the compression dimension $m$ and coincides with existing results for the uncompressedsetting $m=d$. Our results can also handle error introduced through additive noise or through missing data.The scaling of dimension, compression, and signal complexity in our theoretical results is verified empirically through simulations, and we also apply our method to a data set measuring chlorine concentration acrossa network of sensors, to test its performance in practice.
Tensor decomposition on big data has attracted significant attention recently. Among the most popular methods is a class of algorithms that leverages compression in order to reduce the size of the tensor and potentially parallelize computations. A fundamental requirement for such methods to work properly is that the low-rank tensor structure is retained upon compression. In lieu of efficient and realistic means of computing and studying the effects of compression on the low rank of a tensor, we study the effects of compression on the core consistency; a widely used heuristic that has been used as a proxy for estimating that low rank. We provide theoretical analysis, where we identify sufficient conditions for the compression such that the core consistency is preserved, and we conduct extensive experiments that validate our analysis. Further, we explore popular compression schemes and how they affect the core consistency.
The low-rank tensor approximation is very promising for the compression of deep neural networks. We propose a new simple and efficient iterative approach, which alternates low-rank factorization with a smart rank selection and fine-tuning. We demonstrate the efficiency of our method comparing to non-iterative ones. Our approach improves the compression rate while maintaining the accuracy for a variety of tasks.