On Obtaining Upper Bounds on the Complexity of Matrix Multiplication

1972 ◽  
pp. 31-40 ◽  
Author(s):  
Charles M. Fiduccia
Keyword(s):  
Matrix Multiplication ◽  
Upper Bounds

scholarly journals UPPER BOUNDS FOR SUNFLOWER-FREE SETS

2017 ◽  
Vol 5 ◽  
Author(s):  
ERIC NASLUND ◽  
WILL SAWIN
Keyword(s):  
Arithmetic Progression ◽  
Matrix Multiplication ◽  
Upper Bounds ◽  
Polynomial Method ◽  
Combinatorial Properties ◽  
Image Position ◽  
Free Set ◽  
Free Sets ◽  
Exponentially Small

A collection of $k$ sets is said to form a $k$-sunflower, or $\unicode[STIX]{x1D6E5}$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets ${\mathcal{F}}$sunflower-free if it contains no $3$-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt (‘On large subsets of $\mathbb{F}_{q}^{n}$ with no three-term arithmetic progression’, Ann. of Math. (2) 185 (2017), 339–343); (‘Progression-free sets in $\mathbb{Z}_{4}^{n}$ are exponentially small’, Ann. of Math. (2) 185 (2017), 331–337) we apply the polynomial method directly to Erdős–Szemerédi sunflower problem (Erdős and Szemerédi, ‘Combinatorial properties of systems of sets’, J. Combin. Theory Ser. A 24 (1978), 308–313) and prove that any sunflower-free family ${\mathcal{F}}$ of subsets of $\{1,2,\ldots ,n\}$ has size at most $$\begin{eqnarray}|{\mathcal{F}}|\leqslant 3n\mathop{\sum }_{k\leqslant n/3}\binom{n}{k}\leqslant \left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}.\end{eqnarray}$$ We say that a set $A\subset (\mathbb{Z}/D\mathbb{Z})^{n}=\{1,2,\ldots ,D\}^{n}$ for $D>2$ is sunflower-free if for every distinct triple $x,y,z\in A$ there exists a coordinate $i$ where exactly two of $x_{i},y_{i},z_{i}$ are equal. Using a version of the polynomial method with characters $\unicode[STIX]{x1D712}:\mathbb{Z}/D\mathbb{Z}\rightarrow \mathbb{C}$ instead of polynomials, we show that any sunflower-free set $A\subset (\mathbb{Z}/D\mathbb{Z})^{n}$ has size $$\begin{eqnarray}|A|\leqslant c_{D}^{n}\end{eqnarray}$$ where $c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$. This can be seen as making further progress on a possible approach to proving the Erdős and Rado sunflower conjecture (‘Intersection theorems for systems of sets’,J. Lond. Math. Soc. (2) 35 (1960), 85–90), which by the work of Alon et al. (‘On sunflowers and matrix multiplication’, Comput. Complexity22 (2013), 219–243; Theorem 2.6) is equivalent to proving that $c_{D}\leqslant C$ for some constant $C$ independent of $D$.

scholarly journals Rank and border rank of Kronecker powers of tensors and Strassen's laser method

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Austin Conner ◽  
Fulvio Gesmundo ◽  
Joseph M. Landsberg ◽  
Emanuele Ventura
Keyword(s):  
Matrix Multiplication ◽  
Upper Bounds ◽  
Positive Direction ◽  
Laser Method ◽  
Waring Rank ◽  
Border Rank ◽  
Symmetric Version

AbstractWe prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor $$T_{cw,q}$$ T c w , q is the square of its border rank for $$q > 2$$ q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for $$q > 4$$ q > 4 . This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range.In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, $$T_{skewcw,q}$$ T s k e w c w , q . For $$q = 2$$ q = 2 , the Kronecker square of this tensor coincides with the $$3\times 3$$ 3 × 3 determinant polynomial, $$\det_{3} \in \mathbb{C}^{9} \otimes \mathbb{C}^{9} \otimes \mathbb{C}^{9}$$ det 3 ∈ C 9 ⊗ C 9 ⊗ C 9 , regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two.We determine new upper bounds for the (Waring) rank and the (Waring) border rank of $$\det_3$$ det 3 , exhibiting a strict submultiplicative behaviour for $$T_{skewcw,2}$$ T s k e w c w , 2 which is promising for the laser method.We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in $$\mathbb{C}^{3} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3}$$ C 3 ⊗ C 3 ⊗ C 3 .

scholarly journals A note on the multiplication of sparse matrices

2014 ◽  
Vol 4 (1) ◽  
pp. 1-11
Author(s):  
Keivan Borna ◽  
Sohrab Fard
Keyword(s):  
Uniform Distribution ◽  
Sparse Matrices ◽  
Matrix Multiplication ◽  
Upper Bounds ◽  
Practical Case ◽  
Expected Number ◽  
Practical Algorithm

AbstractWe present a practical algorithm for multiplication of two sparse matrices. In fact if A and B are two matrices of size n with m 1 and m 2 non-zero elements respectively, then our algorithm performs O(min{m 1 n, m 2 n, m 1 m 2}) multiplications and O(k) additions where k is the number of non-zero elements in the tiny matrices that are obtained by the columns times rows matrix multiplication method. Note that in the useful case, k ≤ m 2 n. However, in Proposition 3.3 and Proposition 3.4 we obtain tight upper bounds for the complexity of additions. We also study the complexity of multiplication in a practical case where non-zero elements of A (resp. B) are distributed independently with uniform distribution among columns (resp. rows) of them and show that the expected number of multiplications is O(m 1 m 2/n). Finally a comparison of number of required multiplications in the naïve matrix multiplication, Strassen’s method and our algorithm is given.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Stable and Supported Semantics in Continuous Vector Spaces

2020 ◽  
Author(s):  
Yaniv Aspis ◽  
Krysia Broda ◽  
Alessandra Russo ◽  
Jorge Lobo
Keyword(s):  
Logic Program ◽  
Matrix Multiplication ◽  
Vector Spaces ◽  
Continuous Space ◽  
Continuous Vector ◽  
Novel Approach ◽  
Gradient Based ◽  
Normal Logic ◽  
Parameter Values ◽  
Normal Logic Program

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

Modular Matrix Multiplication on a Linear Array.

1983 ◽  
Author(s):  
I. V. Ramakrishnan ◽  
P. J. Varman
Keyword(s):  
Linear Array ◽  
Matrix Multiplication
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