Using Butterfly-patterned Partial Sums to Draw from Discrete Distributions

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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Analogue of the F α scheme for discrete distributions

Vestnik St Petersburg University Mathematics ◽

10.3103/s1063454115010082 ◽

2015 ◽

Vol 48 (1) ◽

pp. 18-22 ◽

Cited By ~ 1

Author(s):

V. B. Nevzorov

Keyword(s):

Discrete Distributions

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Fast Bilinear Algorithms for Symmetric Tensor Contractions

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2019-0075 ◽

2020 ◽

Vol 0 (0) ◽

Cited By ~ 1

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.

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