scholarly journals A $2{\mathbf{n}}^2-{\text{log}}_2({\mathbf{n}})-1$ lower bound for the border rank of matrix multiplication

2017 ◽  
Vol 2018 (15) ◽  
pp. 4722-4733 ◽  
Author(s):  
Joseph M Landsberg ◽  
Mateusz Michałek
Keyword(s):  
Lower Bound ◽  
Matrix Multiplication ◽  
Border Rank ◽  
Rank Of Matrix

scholarly journals On the Geometry of Border Rank Algorithms for n × 2 by 2 × 2 Matrix Multiplication

2016 ◽  
Vol 26 (3) ◽  
pp. 275-286 ◽  
Author(s):  
J. M. Landsberg ◽  
Nicholas Ryder
Keyword(s):  
Matrix Multiplication ◽  
Border Rank

scholarly journals Latency Analysis in the 2-Dimensional Systolic Arrays for Matrix Multiplication

2021 ◽  
Vol 15 ◽  
pp. 1-7
Author(s):  
Halil Snopce ◽  
Azir Aliu
Keyword(s):  
Lower Bound ◽  
Systolic Array ◽  
Matrix Multiplication ◽  
Systolic Arrays ◽  
Latency Analysis

This paper deals with the latency analysis in a twodimensional systolic array for matrix multiplication. The latency for all possible connection schemes is discussed. In this way there is obtained the lower bound of the latency that can be achieved using such arrays.

scholarly journals Lower Bounds for Dynamic Algebraic Problems

1998 ◽  
Vol 5 (11) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Johan P. Hansen ◽  
Peter Bro Miltersen
Keyword(s):  
Fourier Transform ◽  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Symmetric Functions ◽  
Matrix Multiplication ◽  
Worst Case ◽  
Dynamic Matrix ◽  
Wide Range ◽  
On Line

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form "change input xi to value v" or "what is the value of output yi?". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent<br />algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal  Omega(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an <br />Omega(sqrt(n)) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and<br />Tate's O(sqrt(n log n)) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Omega(sqrt(n)) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting<br />of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an ((log n)2= log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields.

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 .

A lower bound for the border rank of a bilinear map

CALCOLO ◽  
1986 ◽  
Vol 23 (2) ◽  
pp. 105-114 ◽  
Author(s):  
B. Griesser
Keyword(s):  
Lower Bound ◽  
Bilinear Map ◽  
Border Rank

A Lower Bound for Matrix Multiplication

1989 ◽  
Vol 18 (4) ◽  
pp. 759-765 ◽  
Author(s):  
Nader H. Bshouty
Keyword(s):  
Lower Bound ◽  
Matrix Multiplication
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