Multiple tensor-times-matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required (under mild conditions) to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that, with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations when the input and output tensors vary greatly in size.
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Communication Lower Bounds for Matricized Tensor Times Khatri-Rao Product
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.
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- Award ID(s):
- 1642385
- NSF-PAR ID:
- 10078535
- Date Published:
- Journal Name:
- 2018 IEEE International Parallel and Distributed Processing Symposium
- Page Range / eLocation ID:
- 557 to 567
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/IPDPS.2018.00065
