Scalable tensor decompositions in high performance computing environments

Abstract

This dissertation presents novel algorithmic techniques and data structures to help build scalable tensor decompositions on a variety of high-performance computing (HPC) platforms, including multicore CPUs, graphics co-processors (GPUs), and Intel Xeon Phi processors. A tensor may be regarded as a multiway array, generalizing matrices to more than two dimensions. When used to represent multifactor data, tensor methods can help analysts discover latent structure; this capability has found numerous applications in data modeling and mining in such domains as healthcare analytics, social networks analytics, computer vision, signal processing, and neuroscience, to name a few. When attempting to implement tensor algorithms efficiently on HPC platforms, there are several obstacles: the curse of dimensionality, mode orientation, tensor transformation, irregularity, and arbitrary tensor dimensions (or orders). These challenges result in non-trivial computational and storage overheads. This dissertation considers these challenges in the specific context of the two of the most popular tensor decompositions, the CANDECOMP/PARAFAC (CP) and Tucker decompositions, which are, roughly speaking, the tensor analogues to low-rank approximations in standard linear algebra. Within that context, two of the critical computational bottlenecks are the operations known as Tensor-Times-Matrix (TTM) and Matricized Tensor Times Khatri-Rao Product (MTTKRP). We consider these operations in cases when the tensor is dense or sparse. Our contributions include: 1) applying memoization to overcome the curse of dimensionality challenge that exists in a sequence of tensor operations; 2) addressing the challenge of mode orientation through a novel tensor format HICOO and proposing a parallel scheduler to avoid the locks for write-conflict memory; 3) carrying out TTM and MTTKRP operations in-place, for dense and sparse cases, to avoid tensor-matrix conversions; 4) employing different optimization and parameter tuning techniques for CPU and GPU implementations to conquer the challenges of the irregularity and arbitrary tensor orders. To validate these ideas, we have implemented them in three prototype libraries, named AdaTM, InTensLi, and ParTI!, for arbitrary-order tensors. AdaTM is a model-driven framework to generate an adaptive tensor memoization algorithm with the optimal parameters for sparse CP decomposition. InTensLi produces fast single-node implementations of dense TTM of an arbitrary dimension. ParTI! is short for a Parallel Tensor Infrastructure which is written in C, OpenMP, MPI, and NVIDIA CUDA for sparse tensors and supports MATLAB interfaces for application-level users.