Grassmann’s Outer Product Algebra

2015 ◽  
pp. 55-76
Keyword(s):  
Outer Product ◽  
Product Algebra

Fast Bilinear Algorithms for Symmetric Tensor Contractions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Edgar Solomonik ◽  
James Demmel
Keyword(s):  
Symmetric Matrix ◽  
Matrix Multiplication ◽  
Computational Cost ◽  
Symmetric Tensor ◽  
Partial Sums ◽  
Symmetric Tensors ◽  
Outer Product ◽  
Bilinear Complexity ◽  
Special Cases ◽  
Tensor Contractions

AbstractIn matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order {s+v} and {v+t} by a factor of {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

scholarly journals A basis for the GLn tensor product algebra

2005 ◽  
Vol 196 (2) ◽  
pp. 531-564 ◽  
Author(s):  
Roger E. Howe ◽  
Eng-Chye Tan ◽  
Jeb F. Willenbring
Keyword(s):  
Tensor Product ◽  
Product Algebra

scholarly journals A SCALABLE HYBRID FPGA/GPU FX CORRELATOR

2014 ◽  
Vol 03 (01) ◽  
pp. 1450002 ◽  
Author(s):  
J. KOCZ ◽  
L. J. GREENHILL ◽  
B. R. BARSDELL ◽  
G. BERNARDI ◽  
A. JAMESON ◽  
...  
Keyword(s):  
Graphics Processing Units ◽  
Field Testing ◽  
Outer Product ◽  
Gate Arrays ◽  
Large Numbers ◽  
Field Programmable ◽  
Programmable Gate Arrays ◽  
Hybrid Fpga ◽  
Graphics Processing ◽  
O 10

Radio astronomical imaging arrays comprising large numbers of antennas, O(102–103), have posed a signal processing challenge because of the required O (N2) cross correlation of signals from each antenna and requisite signal routing. This motivated the implementation of a Packetized Correlator architecture that applies Field Programmable Gate Arrays (FPGAs) to the O (N) "F-stage" transforming time domain to frequency domain data, and Graphics Processing Units (GPUs) to the O (N2) "X-stage" performing an outer product among spectra for each antenna. The design is readily scalable to at least O(103) antennas. Fringes, visibility amplitudes and sky image results obtained during field testing are presented.

scholarly journals Identification‐ and singularity‐robust inference for moment condition models

2019 ◽  
Vol 10 (4) ◽  
pp. 1703-1746 ◽  
Author(s):  
Donald W. K. Andrews ◽  
Patrik Guggenberger
Keyword(s):  
Moment Condition ◽  
Outer Product ◽  
Moment Conditions ◽  
Variance Matrix ◽  
Asymptotic Size ◽  
Moment Functions ◽  
Linear And Nonlinear ◽  
The Moment ◽  
Asymptotically Efficient ◽  
Iv Model

This paper introduces a new identification‐ and singularity‐robust conditional quasi‐likelihood ratio (SR‐CQLR) test and a new identification‐ and singularity‐robust Anderson and Rubin (1949) (SR‐AR) test for linear and nonlinear moment condition models. Both tests are very fast to compute. The paper shows that the tests have correct asymptotic size and are asymptotically similar (in a uniform sense) under very weak conditions. For example, in i.i.d. scenarios, all that is required is that the moment functions and their derivatives have 2 +  γ bounded moments for some γ > 0. No conditions are placed on the expected Jacobian of the moment functions, on the eigenvalues of the variance matrix of the moment functions, or on the eigenvalues of the expected outer product of the (vectorized) orthogonalized sample Jacobian of the moment functions. The SR‐CQLR test is shown to be asymptotically efficient in a GMM sense under strong and semi‐strong identification (for all k ≥  p, where k and p are the numbers of moment conditions and parameters, respectively). The SR‐CQLR test reduces asymptotically to Moreira's CLR test when p = 1 in the homoskedastic linear IV model. The same is true for p ≥ 2 in most, but not all, identification scenarios. We also introduce versions of the SR‐CQLR and SR‐AR tests for subvector hypotheses and show that they have correct asymptotic size under the assumption that the parameters not under test are strongly identified. The subvector SR‐CQLR test is shown to be asymptotically efficient in a GMM sense under strong and semi‐strong identification.

Numerical optical computing in the residue number system with outer-product lookup tables

1989 ◽  
Vol 14 (16) ◽  
pp. 847 ◽  
Author(s):  
Mark L. Heinrich ◽  
Ravindra A. Athale ◽  
Michael W. Haney
Keyword(s):  
Optical Computing ◽  
Number System ◽  
Residue Number System ◽  
Outer Product ◽  
Residue Number ◽  
Lookup Tables

scholarly journals D-Optimal Slope Design for Second Degree Kronecker Model Mixture Experiment With Three Ingredients

2020 ◽  
Vol 9 (2) ◽  
pp. 30
Author(s):  
Ngigi Peter Kung’u ◽  
J. K. Arap Koske ◽  
Josphat K. Kinyanjui
Keyword(s):  
Information Matrix ◽  
Coefficient Matrix ◽  
Three Dimensions ◽  
Matrix Means ◽  
The Matrix ◽  
Optimal Values ◽  
The Moment ◽  
Product Algebra ◽  
Slope Design ◽  
Second Degree

This study presents an investigation of an optimal slope design in the second degree Kronecker model for mixture experiments in three dimensions. The study is restricted to weighted centroid designs, with the second degree Kronecker model. A well-defined coefficient matrix is used to select a maximal parameter subsystem for the model since its full parameter space is inestimable. The information matrix of the design is obtained using a linear function of the moment matrices for the centroids and directly linked to the slope matrix. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region. Eventually the matrix means are used in determining optimal values of the efficient developed design.

scholarly journals HARMONIC POLYLOGARITHMS

2000 ◽  
Vol 15 (05) ◽  
pp. 725-754 ◽  
Author(s):  
E. REMIDDI ◽  
J. A. M. VERMASEREN
Keyword(s):  
Mellin Transforms ◽  
Product Algebra

The harmonic polylogarithms (hpl's) are introduced. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t). The coefficients of their expansions and their Mellin transforms are harmonic sums.

Spectral fusion by Outer Product Analysis (OPA) to improve predictions of soil organic C

Geoderma ◽  
2019 ◽  
Vol 335 ◽  
pp. 35-46 ◽  
Author(s):  
Fabrício S. Terra ◽  
Raphael A. Viscarra Rossel ◽  
José A.M. Demattê
Keyword(s):  
Soil Organic C ◽  
Product Analysis ◽  
Outer Product ◽  
Organic C
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