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The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. We propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The tensor-train (TT) format is a highly compact low-rank representation for highdimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20\times speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives. [ABSTRACT FROM AUTHOR]

We present efficient and scalable parallel algorithms for performing mathematical operations for low-rank tensors represented in the tensor train (TT) format. We consider algorithms for addition, elementwise multiplication, computing norms and inner products, orthonormalization, and rounding (rank truncation). These are the kernel operations for applications such as iterative Krylov solvers that exploit the TT structure. The parallel algorithms are designed for distributedmemory computation, and we propose a data distribution and strategy that parallelizes computations for individual cores within the TT format. We analyze the computation and communication costs of the proposed algorithms to show their scalability, and we present numerical experiments that demonstrate their efficiency on both shared-memory and distributed-memory parallel systems. For example, we observe better single-core performance than the existing MATLAB TT-Toolbox in rounding a 2GB TT tensor, and our implementation achieves a 34× speedup using all 40 cores of a single node. We also show nearly linear parallel scaling on larger TT tensors up to over 10,000 cores for all mathematical operations. [ABSTRACT FROM AUTHOR]

The CANDECOMP/PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problems. We extend randomized least squares methods to tensors and show that the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri--Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization. [ABSTRACT FROM AUTHOR]

Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently im- plemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical al- gorithm, they are not used even in cases where they provide a performance benefit. We argue in this paper that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoret- ically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulat- ing the algorithms, and reducing input anomalies by various forms of diagonal scaling. Finally, we benchmark performance and demonstrate our improved numerical accuracy. [ABSTRACT FROM AUTHOR]

Sparse matrix-matrix multiplication (or SpGEMM) is a key primitive for many high-performance graph algorithms as well as for some linear solvers, such as algebraic multigrid. The scaling of existing parallel implementations of SpGEMM is heavily bound by communication. Even though 3D (or 2.5D) algorithms have been proposed and theoretically analyzed in the flat MPI model on Erdös--Rényi matrices, those algorithms had not been implemented in practice and their complexities had not been analyzed for the general case. In this work, we present the first implementation of the 3D SpGEMM formulation that exploits multiple (intranode and internode) levels of parallelism, achieving significant speedups over the state-of-the-art publicly available codes at all levels of concurrencies. We extensively evaluate our implementation and identify bottlenecks that should be subject to further research. [ABSTRACT FROM AUTHOR]

We consider the sequence of sparse matrix-matrix multiplications performed during the setup phase of algebraic multigrid. In particular, we show that the most commonly used parallel algorithm is often not the most communication-efficient one for all of the matrix-matrix multiplica- tions involved. By using an alternative algorithm, we show that the communication costs are reduced (in theory and practice), and we demonstrate the performance benefit for both model (structured) and more realistic unstructured problems on large-scale distributed-memory parallel systems. Our theoretical analysis shows that we can reduce communication by a factor of up to 5.4 for a model problem, and we observe in our empirical evaluation communication reductions of factors up to 4.7 for structured problems and 3.7 for unstructured problems. These reductions in communication translate to run-time speedups of factors up to 2.8 and 2.5, respectively. [ABSTRACT FROM AUTHOR]

We describe and analyze a novel symmetric triangular factorization algorithm. The algorithm is essentially a block version of Aasen's triangular tridiagonalization. It factors a dense symmetric matrix A as the product A = PLTLT PT, where P is a permutation matrix, L is lower triangular, and T is block tridiagonal and banded. The algorithm is the first symmetric-indefinite communication-avoiding factorization: it performs an asymptotically optimal amount of communication in a two-level memory hierarchy for almost any cache-line size. Adaptations of the algorithm to parallel computers are likely to be communication efficient as well; one such adaptation has been recently published. The current paper describes the algorithm, proves that it is numerically stable, and proves that it is communication optimal. [ABSTRACT FROM AUTHOR]

In 1981 Hong and Kung proved a lower bound on the amount of communication (amount of data moved between a small, fast memory and large, slow memory) needed to perform dense, n-by-n matrix multiplication using the conventional O(n³) algorithm, where the input matrices were too huge to Bt in the small, East memory. In 2004 Irony, Toledo, and Tiskin gave a new proof of this result and extended it to the parallel case (where communication means the amount of data moved between processors). In both cases the lower bound may be expressed Ω(#arithinetic_operations/√M), where M is the size of the fast memory (or local memory in the parallel case). Here we generalize these results to a much wider variety of algorithms, including LU factorization. Cholesky factorization, LDLT factorization. QR factorization, the Gram--Schmidt algorithm, and algorithms for eigenvalues and singular values, i.e., essentially all direct methods of linear algebra. The proof works for dense or sparse matrices and for sequential or parallel algorithms. In addition to lower bounds on the amount of data moved (bandwidth cost), we get lower bounds on the number of messages required to move it (latency cost). We extend our lower bound technique to compositions of linear algebra operations (like Computing powers of a matrix) to decide whether it is enough to call a sequence of simpler optimal algorithms (like matrix multiplication) to minimize communication, or whether we can do better. We give examples of both. We also show how to extend our lower bounds to certain graph-theoretic problems. We point out recently designed algorithms that attain many of these lower bounds. [ABSTRACT FROM AUTHOR]

Numerical algorithms have two kinds of costs: arithmetic and communication, by which we mean either moving data between levels of a memory hierarchy (in the sequential case) or over a network connecting processors (in the parallel case). Communication costs often dominate arithmetic costs, so it is of interest to design algorithms minimizing communication. In this paper we first extend known lower bounds on the communication cost (both for bandwidth and for latency) of conventional (O(n3)) matrix multiplication to Cholesky factorization, which is used for solving dense symmetric positive definite linear systems. Second, we compare the costs of various Cholesky decomposition implementations to these lower bounds and identify the algorithms and data structures that attain them. In the sequential case, we consider both the two-level and hierarchical memory models. Combined with prior results in [J. Demmel et al., Communication-optimal Parallel and Sequential QR and LU Factorizations, Technical report EECS-2008-89, University of California, Berkeley, CA, 2008], [J. Demmel et al., Implementing Communication-optimal Parallel and Sequential QR and LU Factorizations, SIAM. J. Sci. Comp., submitted], and [J. Demmel, L. Grigori, and H. Xiang, Communication-avoiding Gaussian Elimination, Proceedings of the 2008 ACM/IEEE Conference on Supercomputing, 2008] this gives a set of communication-optimal algorithms for O(n³) implementations of the three basic factorizations of dense linear algebra: LU with pivoting, QR, and Cholesky. But it goes beyond this prior work on sequential LU by optimizing communication for any number of levels of memory hierarchy. [ABSTRACT FROM AUTHOR]