Invariance transformations of polyadic decompositions matrix multiplication tensors define an equivalence relation on the set such decompositions. In this paper, we present algorithm to efficiently decide whether two a given tensor are equivalent. With algorithm, analyze classes several tensors. This analysis is relevant for study fast as it relates question how many essentially different algor...

This paper describes a method for accelerating large scale Artificial Neural Networks (ANN) training using multi-GPUs by reducing the forward and backward passes to matrix multiplication. We propose an out-of-core multi-GPU matrix multiplication and integrate the algorithm with the ANN training. The experiments demonstrate that our matrix multiplication algorithm achieves linear speedup on mult...

Divide and conquer is an important concept in computer science. It is used ubiquitously to simplify and speed up programs. However, it needs to be optimized, with respect to parameter settings for example, in order to achieve the best performance. The problem boils down to searching for the best implementation choice on a given set of requirements, such as which machine the program is running o...

Motivated by the symmetric version of matrix multiplication we study the plethysm $S^k(\mathfrak{sl}_n)$ of the adjoint representation $\mathfrak{sl}_n$ of the Lie group $SL_n$. In particular, we describe the decomposition of this representation into irreducible components for $k=3$, and find highest weight vectors for all irreducible components. Relations to fast matrix multiplication, in part...

In a typical MapReduce job, each map task processing one piece of the input file. If two input matrices are stored in separate HDFS files, one map task would not be able to access the two input matrices at the same time. To deal with this problem, we propose a efficient matrix multiplication in Hadoop. For dense matrices, we use plain row major order to store the matrices on HDFS; For sparse ma...

A novel user-level scheduling, along with a specific data alignment method is presented for matrix multiplication in cache-coherent Non-Uniform Memory Access (ccNUMA) architectures. Addressing the data locality problem that occurs in such systems alleviates memory bottlenecks in problems with large input data sets. It is shown experimentally that a large number of cache misses occur when using ...

Quasi-Monte Carlo (QMC) rules 1/N ∑N−1 n=0 f(ynA) can be used to approximate integrals of the form ∫ [0,1]s f(yA) dy, where A is a matrix and y is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications fr...

3 We consider the problem of multiplying two U×U matrices A and C of elements from a field F. We present a new randomized algorithm that can use the known fast square matrix multiplication algorithms to perform fewer arithmetic operations than the current state of the art for output matrices that are sparse. In particular, let ω be the best known constant such that two dense U×U matrices can be...

We consider the Online Boolean Matrix-Vector Multiplication (OMV) problem studied by Henzinger et al. [STOC’15]: given an n× n Boolean matrix M, we receive n Boolean vectors v1, . . . ,vn one at a time, and are required to output Mvi (over the Boolean semiring) before seeing the vector vi+1, for all i. Previous known algorithms for this problem are combinatorial, running in O(n3/ log2 n) time. ...

X and Y in multiple different ways as long as the matrix multiplication remains valid. For example, we could assign the multiplication modes in both tensors to columns, in this case the matrix product becomes Z = XY . Alternatively, the tensor Y could be matrisized with the multiplication modes corresponding to rows, resulting in the product Z = XY. In a series of tensor multiplications the ord...