Group-theoretic approach to Fast Matrix Multiplication

Jenya Kirshtein 1 Group-theoretic approach to Fast Matrix Multiplication Jenya Kirshtein Department of Mathematics University of Denver

Jenya Kirshtein 2 References 1. H. Cohn, C. Umans. Group-theoretic approach to Fast Matrix Multiplication. Proceedings of the 44th Annual Symposium on Foundations of Computer Science, 2003. 2. H. Cohn, R. Kleinberg, B. Szegedy, C. Umans. Group-theoretic Algorithms for Matrix Multiplication. Proceedings of the 46th Annual Symposium on Foundations of Computer Science, 2005. 3. James and Liebeck. Representations and Characters of Groups. Cambridge University Press, 1993. 4. I. Martin Isaacs. Character Theory of Finite Groups. AMS Bookstore, 2006.

Jenya Kirshtein 3 Matrix multiplication Standard matrix multiplication takes 2n 3 arithmetic operations Strassen suggested O(n 2.81 ) recursive algorithm in 1969 Question: What is the smallest number w such that ǫ > 0 matrix multiplication can be performed in at most O(n w+ǫ )? Question: Is w = 2? The best known bound is w < 2.38 due to Coppersmith and Winograd (1990)

Jenya Kirshtein 4 Outline Embed matrices A, B into the elements A, B of the group algebra C[G] Multiplication of A and B in the group algebra is carried out in the Fourier domain after performing the Discrete Fourier Transform (DFT) of A and B The product A B is found by performing the inverse DFT Entries of the matrix AB can be read off from the group algebra product A B We ll consider square matrices and finite groups.

Jenya Kirshtein 5 Group Algebra Def: Let G be a finite group and F be a field. A group algebra F[G] of G over F is the set of all linear combinations of elements of G with coefficients in F, with the natural addition and scalar multiplication, and multiplication defined by a g g b h h a g b h f g G,a g F h G,b h F = f G gh=f

Jenya Kirshtein 6 Discrete Fourier Transform (DFT) Embed polynomials as elements of the group algebra C[G]: Let G =< z > be a cyclic group of order m 2n. Define D(A) = A = n 1 i=0 a i z i and B = n 1 i=0 b i z i Discrete Fourier Transform is an invertible linear transformation D : C[G] C G, such that ( n 1 ) i=0 a i x i 0, n 1 i=0 a i x i 1,..., n 1 i=0 where x k = e 2πi n k is the n th root of unity. Then A B = D 1 ( D(A)D(B) ) a i x i n 1 Fast Fourier Transform algorithm computes DFT in O(nlog(n)) arithmetic operations.,

Jenya Kirshtein 7 Matrices as elements of the group algebra Let F be a field and S, T and U be subsets of G. A = (a s,t ) s S,t T and B = (b t,u ) t T,u U are S T and T U matrices, indexed by elements of S, T and T, U, respectively. Then embed A,B as elements A, B F[G]: A = a s,t s 1 t and B = b t,u t 1 u s S,t T t T,u U

Jenya Kirshtein 8 Triple product property Let S G. Let Q(S) denote the right quotient set of S, i.e. Q(S) = { s 1 s 1 2 : s 1 s 2 S } Def: A group G realizes n 1, n 3, n 3 if subsets S 1, S 2, S 3 G s.t. S i = n i and for q i Q(S i ), if q 1 q 2 q 3 = 1 then q 1 = q 2 = q 3 = 1 (equivalently, if q 1 q 2 = q 3 then q 1 = q 2 = q 3 = 1) This condition on S 1, S 2, S 3 is the triple product property.

Jenya Kirshtein 9 Theorem [1] Let F be a field. If G realizes < n, n, n >, then the number of field operations required to multiply two n n matrices over F is at most the number of operations required to multiply two elements of F[G]. Proof. Index A with the sets S, T; B with T, U and AB with S, U. Consider the product in F[G] a st s 1 t b t u 1 t u s S,t T t T,u U By triple product property, ( s 1 t ) ( t 1 u) = s 1 u iff s = s, t = t, u = u, so the coefficient of s 1 u in the product is a st b tu = (AB) su t T

Jenya Kirshtein 10 Characters and character degrees of G Def: Let F be R or C. The set of all invertible n n matrices with entries in F, under matrix multiplication, forms a group called the general linear group GL(n, F). Def: The trace of an n n matrix A = (a ij ) is tra = n i=1 a ii. Def: A representation of G over F is a homomorphism ρ : G GL(n, F) for some integer n (n is the degree of ρ). Def: Suppose ρ is a representation of G. With each n n matrix ρ(g) (g G) we associate the complex number tr(ρ(g)), which we call χ(g). The function χ : G C is called the character of the representation ρ of G. Def: If χ is a character of G, then χ(1) is called a degree of χ.

Jenya Kirshtein 11 Theorem [2] Suppose G realizes n, n, n and the character degrees of G are {d i }. Then n w i d w i Theorem [3] Let G be a group with the character degrees {d i }. Then G = i d2 i Corollary [2] Suppose G realizes n, n, n and has largest character degree d. Then n w d w 2 G Question [1] Does there exist a finite group G that realizes n, n, n and has character degrees {d i } such that n 3 > i d 3 i, i.e. G gives a nontrivial bound on w.

Jenya Kirshtein 12 Example Def: If G is a group and X is a set, then a (left) group action of G on X is a binary function G X X denoted (g, x) g x, which satisfies the following two axioms: 1. (gh) x = g (h x) g, h G, x X; 2. 1 x = x x X (1 is the identity element of G). Def: If G and H are groups with a left action of G on H, then the semidirect product H G is the set H G with the multiplication law (h 1, g 1 ) (h 2, g 2 ) = (h 1 (g 1 h 2 ), g 1 g 2 ).

Jenya Kirshtein 13 Example Denote Cyc n the cyclic group of order n. Let H = Cyc 3 n and G = H 2 Cyc 2 Cyc 2 = {1, z} acts on H 2 by switching the two factors, then G = { (a, b)z i : a, b H, i {0, 1} } Let H 1, H 2, H 3 H be the three factors of Cyc n in H, H 4 = H 1. Define subsets S 1, S 2, S 3 G by S i = { (a, b)z j : a H i \ {0}, b H i+1, j {0, 1} } S 1, S 2, S 3 satisfy the triple product property.

Jenya Kirshtein 14 Example H = Cyc 3 n and G = H 2 Cyc 2 S i = { (a, b)z j : a H i \ {0},b H i+1, j {0, 1} } Theorem [4] Let G be nonabelian group and m be an integer. If there exists abelian H G of index m, then character degrees of G are 1 and m. Theorem If H G and [G : H] = 2, then H G. Since H 2 is an abelian subgroup of G of index [G : H] = G H = 2, character degrees of G are at most 2. Then since i d2 i = G, i d3 i max {d i} i d2 i 2 G = 4n6 Also, S i = 2n(n 1) and S 1 S 2 S 3 = 8n 3 (n 1) 3, so we have S 1 S 2 S 3 > i d3 i for n 5.

Jenya Kirshtein 15 Example By Corollary, S 1 w d w 2 G or 2n(n 1) w 2 w 2 2n 6, which leads to w 6lnn ln2 ln(n(n 1)). The best bound on w (w 2.9088) is achieved by setting n = 17.

Jenya Kirshtein 16 Other results Uniquely Solvable Puzzles prove w < 2.48 Local Uniquely Solvable Puzzles prove w < 2.41 and w < 2.376 Conjecture that a Strong Uniquely Solvable Puzzle capacity equals 3/2 2 3 would imply that w = 2 Conjecture about the existence for any n of an abelian group G with n pairs of subsets A i, B i satisfying the simultaneous double product property such that G = n 2+o(1) and A i B i n 2 o(1) would imply that w = 2

Jenya Kirshtein 17 Thank you!