II. Group-Theoretic Approach Chris Umans Caltech Based on joint - PowerPoint PPT Presentation

II. Group-Theoretic Approach Chris Umans Caltech Based on joint work with Noga Alon, Henry Cohn, Bobby Kleinberg, Amir Shpilka, Balazs Szegedy Modern Applications of Representation Theory, IMA, Chicago July 2014

Introduction A X B = C • Standard method: O(n 3 ) operations • Strassen (1969): O(n 2.81 ) operations July 31, 2014 2

Introduction A X B = C • Standard method: O(n 3 ) operations • Strassen (1969): O(n 2.81 ) operations The exponent of matrix multiplication: smallest number ω such that for all ε >0 O(n ω + ε ) operations suffice July 31, 2014 3

History ω ≤ 3 • Standard algorithm ω < 2.81 • Strassen (1969) ω < 2.79 • Pan (1978) ω < 2.78 • Bini; Bini et al. (1979) ω < 2.55 • Schönhage (1981) • Pan; Romani; Coppersmith ω < 2.50 + Winograd (1981-1982) ω < 2.48 • Strassen (1987) ω < 2.375 • Coppersmith + Winograd (1987) ω < 2.3737 • Stothers (2010) ω < 2.3729 • Williams (2011) ω < 2.37286 • Le Gall (2014) July 31, 2014 4

Outline Lecture I: – crash course on main ideas from Strassen 1969 through Le Gall 2014 – conjectures implying ! = 2 This lecture and next one: II. group-theoretic approach III. extending to coherent configurations July 31, 2014 5

A different approach • So far... – bound border rank of small tensor (by hand) – asymptotic bound from high tensor powers • Disadvantages – limited universe of “starting” tensors – high tensor powers hard to analyze • Next: matrix multiplication via groups July 31, 2014 6

The Group Algebra Also think of elements • Given a group G as vectors a with |G| entries • The group algebra C[G] has elements Σ g a g g with multiplication (Σ g a g g)(Σ h b h h) = Σ f (Σ gh = f a g b h )f July 31, 2014 7

The Group Algebra C[G] ' (C d 1 ×d 1 ) × (C d 2 ×d 2 ) × … × (C d k ×d k ) • d 1 , d 2 , …, d k are character degrees of G • two facts: – Σd i 2 = |G| – all character degrees are 1 for abelian groups July 31, 2014 8

Main idea: multiply in Fourier domain C[G] ' (C d 1 ×d 1 ) × (C d 2 ×d 2 ) × … × (C d k ×d k ) DFT = DFT = a b convolution (with respect to G) becomes block-diagonal matrix multiplication × = DFT -1 = a*b

Matrix Multiplication • Two input matrices: A=(a ij ), B=(b kl ) • “embed” A → A ∈ C[G], B → B ∈ C[G] A DFT = DFT = B × = read off C = AB DFT -1 = A*B from A*B

Can this work? • All depends on choice of group G • need G to permit an embedding A → A ∈ C[G], B → B ∈ C[G] so that we can read off entries of AB from A*B . July 31, 2014 11

The embedding: Subgroups X, Y, Z of G satisfy the triple product property if for all x ∈ X , y ∈ Y , z ∈ Z : xyz = 1 iff x = y = z = 1. July 31, 2014 12

The embedding: Q(S) = {s -1 t: s, t ∈ S} Subsets X, Y, Z of G satisfy the triple product property if for all x ∈ Q(X), y ∈ Q(Y), z ∈ Q(Z): xyz = 1 iff x = y = z = 1. A = Σ a x1,y1 (x 1 y 1 -1 ) B = Σ b y2,z2 (y 2 z 2 -1 ) Claim: (AB) x3,z3 = coeff. on (x 3 z 3 -1 ) in A * B . July 31, 2014 13

The embedding: Q(S) = {s -1 t: s, t ∈ S} Subsets X, Y, Z of G satisfy the triple product property if for all x ∈ Q(X), y ∈ Q(Y), z ∈ Q(Z): xyz = 1 iff x = y = z = 1. A = Σ a x1,y1 (x 1 y 1 -1 ) B = Σ b y2,z2 (y 2 z 2 -1 ) Claim: (AB) x3,z3 = coeff. on (x 3 z 3 -1 ) in A * B . (x 1 y 1 -1 )(y 2 z 2 -1 ) = x 3 z 3 -1 ) x 3 -1 x 1 y 1 -1 y 2 z 2 -1 z 3 =1 July 31, 2014 14

How many multiplications? Fact: method to multiply k × k matrices using m multiplications proves ! ≤ log k m • we use m ≤ Σ d i 3 mults • really m = Σ d i ! mults • at least m ≥ Σ d i 2 = |G| mults First Challenge : embed k × k matrix multiplication in group of size ¼ k 2 July 31, 2014 15

The embedding First Challenge : embed k × k matrix multiplication in group of size ¼ k 2 • simple pigeonhole argument: – embedding in an abelian group requires group to have size k 3 July 31, 2014 16

The triangle construction Theorem : can embed k × k matrix multiplication in symmetric group of size k 2 + o(1) • subgroup X n objects • subgroup Y • subgroup Z need X, Y, Z in S n all with size ≈ |S n | 1/2 July 31, 2014 17

The triangle construction – X moves points within rows – Y moves points within columns – Z moves points within diagonals – want: xyz = 1 ⇒ x = y = z = 1 July 31, 2014 18

Character degrees • We have described a reduction from k × k mat. mult. to block-diagonal mat. mult. Theorem : in group G with character degrees d 1 , d 2 , d 3 ,…, we obtain: k ω · ∑ i d i ω July 31, 2014 19

Potential barrier Can use this framework to prove ω < 3 if and only if can find X, Y, Z subsets of G satisfying the triple product property, and |X||Y||Z| > ∑ d i 3 . “beating the sum of the cubes” July 31, 2014 20

Recall: the triangle construction Theorem : can embed k × k matrix multiplication in symmetric group of size k 2 + o(1) • subgroup X n objects • subgroup Y • subgroup Z unfortunately, d max > |X| (= |Y| = |Z|) July 31, 2014 21

What should we be aiming for? Theorem : in group G supporting k x k matrix multiplication with character degrees d 1 , d 2 , d 3 ,…, we obtain: k ω · ∑ i d i ω • If X, Y, Z µ G satisfy T.P.P. and – (|X|¢|Y|¢|Z|) 1/3 = k ¸ |G| 1/2 – o(1) – d max · |G| 1/2 – ² ∑ i d i ! · d max ! – 2 |G| then ! = 2 July 31, 2014 22

Constructions in linear groups • Good candidate family: SL(n, q) for fixed dimension n because d max · |G| 1/2 - ²n • a non-trivial construction (i.e., k 3 > |G|): X = { } Y = { } Z = { } 1+z z 1 x 1 0 -z 1-z 0 1 y 1 1 x x 1 0 = 1+xy x 0 1 y 1 y 1 July 31, 2014 23

Constructions in linear groups • Good candidate family: SL(n, q) for fixed dimension n because d max · |G| 1/2 - ²n • best we know, in SL(2, q) for q = p 2 : X = { } Y = { } Z = { } 1 x 1 0 z w 0 1 y 1 w z 1 x x 1 0 = 1+xy x 0 1 y 1 y 1 July 31, 2014 24

Constructions in linear groups • Good candidate family: SL(n, q) for fixed dimension n because d max · |G| 1/2 - ²n • best we know, in SL(2, q) for q = p 2 : X = { } Y = { } Z = { } 1 x 1 0 z w 0 1 y 1 w z – (|X|¢|Y|¢|Z|) 1/3 = |G| 18/7 – o(1) July 31, 2014 25

Constructions in linear groups • Good candidate family: SL(n, q) for fixed dimension n • In SL(n, R) these three subgroups satisfy the triple product property: – upper-triangular with ones on the diagonal – lower-triangular with ones on the diagonal – the special orthogonal group SO(n, R) and dim. of each is ½ dim. of G as n ! 1 July 31, 2014 26

an example yielding ω < 3 July 31, 2014 27

Wreath product groups • A abelian group • G semidirect product of (A w ) N and S N (symmetric group) Ã w ! 6 9 8 3 7 3 9 6 1 + = 0 7 4 8 4 5 8 1 9 N rows 8 3 0 1 3 2 9 6 2 0 6 2 0 9 5 0 5 7 July 31, 2014 28

Wreath product groups • A abelian group • G semidirect product of (A w ) N and S N (symmetric group) 6 9 8 0 6 2 π π = 0 7 4 6 9 8 8 3 0 0 7 4 0 6 2 8 3 0 July 31, 2014 29

Beating the sum of the cubes X 0 0 Three subsets of π X = 0 X’ 0 (A 3 ) 2 semidirect S 2 : 0 Y 0 π Y = 0 0 Y’ 0 0 Z π Z = Z’ 0 0 July 31, 2014 30

Beating the sum of the cubes Q(X) Q(Y) Q(Z) X 0 0 X 0 0 0 Y 0 0 Y 0 0 0 Z 0 0 Z 0 0 0 0 X 0 - π + 0 0 Y - ρ + Z 0 0 - τ = 0 X 0 0 0 Y Z 0 0 0 0 0 • Group is (A 3 ) 2 semidirect S 2 • π , ρ , τ 2 S 2 either “flip” or “no flip” • must be even number of flips July 31, 2014 31

Beating the sum of the cubes Q(X) Q(Y) Q(Z) X 0 0 X 0 0 0 Y 0 0 Y 0 0 0 Z 0 0 Z 0 0 0 0 X 0 - + 0 0 Y - + Z 0 0 - = 0 X 0 0 0 Y Z 0 0 0 0 0 • Group is (A 3 ) 2 semidirect S 2 • π , ρ , τ 2 S 2 either “flip” or “no flip” • must be even number of flips – CASE 1: π = ρ = τ = “no flip” July 31, 2014 32

Beating the sum of the cubes Q(X) Q(Y) Q(Z) X 0 0 0 X 0 X 0 0 0 Y 0 0 Y 0 0 0 Z 0 0 Z 0 0 0 0 X 0 - π + 0 0 Y - ρ + Z 0 0 - = X 0 0 0 X 0 0 0 Y Z 0 0 0 0 0 • Group is (A 3 ) 2 semidirect S 2 • π , ρ , τ 2 S 2 either “flip” or “no flip” • must be even number of flips – CASE 2: π = ρ = “flip”; τ = “no flip” July 31, 2014 33

Beating the sum of the cubes Q(X) Q(Y) Q(Z) 0 0 Y X 0 0 0 X 0 0 Y 0 0 Y 0 0 0 Z 0 0 Z 0 0 0 0 X 0 - π + 0 0 Y - ρ 0 Y 0 - + Z 0 0 - = X 0 0 0 0 Y Z 0 0 0 0 0 • Group is (A 3 ) 2 semidirect S 2 • π , ρ , τ 2 S 2 either “flip” or “no flip” • must be even number of flips – CASE 2: π = ρ = “flip”; τ = “no flip” – contradiction. July 31, 2014 34

Beating the sum of the cubes G semidirect product of (A 3 ) 2 and S 2 X = { π : π 2 S 2 } X 0 0 0 X’ 0 Y = { π : π 2 S 2 } 0 Y 0 0 0 Y’ |A| = 17 yields ω < 2.908… Z = { π : π 2 S 2 } 0 0 Z Z’ 0 0 • |X||Y||Z| = 8(|A|-1) 6 > • ∑ i d i 3 · d max ∑ i d i 2 = 2|G| = 4|A| 6 July 31, 2014 35

generalizing the construction via Uniquely Solvable Puzzles July 31, 2014 36