Search for: All records

We consider the problem of low-rank approximation of massive dense nonnegative 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 Parallel Low-rank Approximation with Nonnegativity 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. 

Our goal is to establish lower bounds on the communication required to perform the Matricized-Tensor Times Khatri-Rao Product (MTTKRP) computation on a distributed-memory parallel machine. MTTKRP is the bottleneck computation within algorithms for computing the CP tensor decomposition, which is an approximation by a sum of rank-one tensors and frequently used in multidimensional data analysis. The main result of this paper is a communication lower bound that generalizes previous results, tightening the bound so that it is attainable even when the tensor dimensions vary (the tensor is not cubical) and when the number of processors is small relative to the tensor dimensions. The attainability of the bound proves that the algorithm that attains it, which is based on a block distribution of the tensor and communicating only factor matrices, is communication optimal. The proof technique utilizes an established inequality that relates computations to data access as well as a novel approach based on convex optimization. 

The CP tensor decomposition is a low-rank approximation of a tensor. We present a distributed-memory parallel algorithm and implementation of an alternating optimization method for computing a CP decomposition of dense tensors that can enforce nonnegativity of the computed low-rank factors. The principal task is to parallelize the Matricized-Tensor Times Khatri-Rao Product (MTTKRP) bottleneck subcomputation. The algorithm is computation efficient, using dimension trees to avoid redundant computation across MTTKRPs within the alternating method. Our approach is also communication efficient, using a data distribution and parallel algorithm across a multidimensional processor grid that can be tuned to minimize communication. We benchmark our software on synthetic as well as hyperspectral image and neuroscience dynamic functional connectivity data, demonstrating that our algorithm scales well to 100s of nodes (up to 4096 cores) and is faster and more general than the currently available parallel software. 

The matricized-tensor times Khatri-Rao product (MTTKRP) computation is the typical bottleneck in algorithms for computing a CP decomposition of a tensor. In order to develop high performance sequential and parallel algorithms, we establish communication lower bounds that identify how much data movement is required for this computation in the case of dense tensors. We also present sequential and parallel algorithms that attain the lower bounds and are therefore communication optimal. In particular, we show that the structure of the computation allows for less communication than the straightforward approach of casting the computation as a matrix multiplication operation. 

The matricized-tensor times Khatri-Rao product (MTTKRP) is the computational bottleneck for algorithms computing CP decompositions of tensors. In this work, we develop shared-memory parallel algorithms for MTTKRP involving dense tensors. The algorithms cast nearly all of the computation as matrix operations in order to use optimized BLAS subroutines, and they avoid reordering tensor entries in memory. We use our parallel implementation to compute a CP decomposition of a neuroimaging data set and achieve a speedup of up to 7.4X over existing parallel software. 

Non-negative matrix factorization (NMF), the problem of finding two non-negative low-rank factors whose product approximates an input matrix, is a useful tool for many data mining and scientific applications such as topic modeling in text mining and unmixing in microscopy. In this paper, we focus on scaling algorithms for NMF to very large sparse datasets and massively parallel machines by employing effective algorithms, communication patterns, and partitioning schemes that leverage the sparsity of the input matrix. We consider two previous works developed for related problems,one that uses a fine-grained partitioning strategy using a point-to-point communication pattern and one that uses a Cartesian, or checkerboard, partitioning strategy using a collective-based communication pattern. We show that a combination of the previous approaches balances the demands of the various computations within NMF algorithms and achieves high efficiency and scalability. From the experiments, we see that our proposed strategy runs up to 10x faster than the state of the art on real-world datasets.