i. approaches to bounding the exponent of matrix ... · • ambainis-filmus 2014: n-th tensor power...
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I. Approaches to
bounding the exponent of
matrix multiplication
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
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Introduction
• Standard method: O(n3) operations
• Strassen (1969): O(n2.81) operations
X = A B C
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Strassen’s Algorithm
• linear combos
of A, B entries
• 7 mults
• linear combos
of results
X = a1, 1 a1, 2
a2, 2 a2, 1
b1, 1 b1, 2
b2, 2 b2, 1
c1, 1 c1, 2
c2, 2 c2, 1
(a1,1 - a1,2) (b2,1 + b2,2)
(a1,1 + a2,2) (b1,1 + b2,2)
(a1,1 - a2,1) (b1,1 + b1,2)
(a1,1 + a1,2) (b2,2)
(a1,1) (b1,2 + b2,2)
(a2,2) (b2,1 - b1,1)
(a2,1 - a2,2) (b1,1)
P1 = (a1,1 - a1,2) × (b2,1 + b2,2)
P2 = (a1,1 + a2,2) × (b1,1 + b2,2)
P3 = (a1,1 - a2,1) × (b1,1 + b1,2)
P4 = (a1,1 + a1,2) × (b2,2)
P5 = (a1,1) × (b1,2 + b2,2)
P6 = (a2,2) × (b2,1 - b1,1)
P7 = (a2,1 - a2,2) × (b1,1)
c1,1 = P1 + P2 - P4 + P6
c1,2 = P4 + P5
c2,1 = P6 + P7
c2,2 = P2 - P3 + P5 - P7
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Strassen’s Algorithm
• linear combos
of A, B entries
• 7 mults
• linear combos
of results
X = a1, 1 a1, 2
a2, 2 a2, 1
b1, 1 b1, 2
b2, 2 b2, 1
c1, 1 c1, 2
c2, 2 c2, 1
T(n) = # operations to
multiply n x n
matrices:
T(n) ≤ 7T(n/2) + O(n2)
T(n) ≤ O(nlog27)
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Introduction
• Standard method: O(n3) operations
• Strassen (1969): O(n2.81) operations
X = A B C
The exponent of matrix multiplication:
smallest number such that for all >0
O(n + ) operations suffice July 31, 2014 5
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History • Standard algorithm ≤ 3
• Strassen (1969) < 2.81
• Pan (1978) < 2.79
• Bini; Bini et al. (1979) < 2.78
• Schönhage (1981) < 2.55
• Pan; Romani; Coppersmith
+ Winograd (1981-1982) < 2.50
• Strassen (1987) < 2.48
• Coppersmith + Winograd (1987) < 2.375
• Stothers (2010) < 2.3737
• Williams (2011) < 2.3729
• Le Gall (2014) < 2.37286
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Introduction
• Other important problems have same
algorithmic complexity as matrix
multiplication (via reductions)
– determinants
– LUP decompositions
– matrix inversion
– solving systems of linear equations
– basic graph problems …
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Outline
1. crash course on main ideas from
Strassen 1969 through Le Gall 2014
2. conjectures implying ! = 2
Two more lectures:
II. group-theoretic approach
III. extending to coherent configurations
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Bilinear algorithms & tensor rank
• Bilinear computation of complexity m:
– m products of form (L.C. of A) x (L.C. of B)
– result matrix entries = L.C.’s of these products
• equivalent: rank of matrix multiplication
tensor “<n,n,n>” is at most m
• Strassen: “bilinear w.l.o.g.”
! = inf {¿ : n x n matmult in O(n¿) size}
= inf {¿ : rank(<n,n,n>) · O(n¿)}
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Tensors
• tensor: 3-D array of
complex numbers
• rank 1 tensor: (i,j,k) entry
= xiyjzk
• rank r tensor: sum of r
rank 1 tensors
X1
x2
x3
x4
y1 y2 y3 y4
equivalent: slices all L.C.’s of r rank 1 matrices
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Tensors
• tensor: 3-D array of complex numbers
• tensor T with entries Ti,j,k
– equivalent: trilinear form
i,j,k
Ti,j,k
Xi Yj Zk
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The matrix multiplication tensor
1
1
1
1
1
1
1
1
b11 b12 b21 b22
a2
2 a
12 a
21 a
11
a11 a12
a21 a22
b11 b12
b21 b22 x =
c11 c12
c21 c22
<n,n,n> is a n2 x n2 x n2 tensor described
by trilinear form i,j,kXi,jYj,kZk,i
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The matrix multiplication tensor
1
1
1
1
1
1
1
1
b11 b12 b21 b22
a2
2 a
12 a
21 a
11
a11 a12
a21 a22
b11 b12
b21 b22 x =
c11 c12
c21 c22
<n,n,n> is a n2 x n2 x n2 tensor described
by trilinear form i,j,kXi,jYj,kZk,i
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The matrix multiplication tensor
1
1
1
1
1
1
1
1
b11 b12 b21 b22
a2
2 a
12 a
21 a
11
a11 a12
a21 a22
b11 b12
b21 b22 x =
c11 c12
c21 c22
<n,n,n> is a n2 x n2 x n2 tensor described
by trilinear form i,j,kXi,jYj,kZk,i
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The matrix multiplication tensor
1
1
1
1
1
1
1
1
b11 b12 b21 b22
a2
2 a
12 a
21 a
11
a11 a12
a21 a22
b11 b12
b21 b22 x =
c11 c12
c21 c22
<n,n,n> is a n2 x n2 x n2 tensor described
by trilinear form i,j,kXi,jYj,kZk,i
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The matrix multiplication tensor
1
1
1
1
1
1
1
1
b11 b12 b21 b22
a2
2 a
12 a
21 a
11
a11 a12
a21 a22
b11 b12
b21 b22 x =
c11 c12
c21 c22
<n,n,n> is a n2 x n2 x n2 tensor described
by trilinear form i,j,kXi,jYj,kZk,i
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The matrix multiplication tensor
<n,m,p> is a nm £ mp £ pn tensor
described by trilinear form i,j,kXi,jYj,kZk,i
X = A B C n
m
p
m n
p
1
1
Each of
np slices of
<n,m,p>:
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Strategies
for upper bounding the rank
of the
matrix multiplication tensor
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Upper bounds on rank
• Observation: <n,n,n>i = <ni, ni, ni>
) R(<ni, ni, ni>) · R(<n,n,n>)i
• Strategy I: bound rank for small n by hand
– R(<2,2,2>) = 7 ! < 2.81
– R(<3,3,3>) 2 [19..23] (worse bound)
– even computer search infeasible…
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Strassen’s example
1
1
1
1
1
1
1
1
1 1
1 1
1 -
1 1
1
1
1 -
1 1
1 1
-
1
-
1 -
1
-
1
1 1 July 31, 2014 20
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Upper bounds on rank
• Border rank = rank of sequence of tensors
approaching target tensor entrywise
• Strategy II: bound border rank for small n
– Lemma: R(<n,n,n>) < r ) ! < logn r
Idea: t-th tensor power is degree O(t) polynomial in ²;
interpolate to recover coefficient on ²0
– R(<2,2,3>) · 10 ! < 2.79
1 1
1
²-1 1
1 ²
1
rank = 3
border rank = 2: ²
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Upper bounds on rank
• Strategy III: bound (border) rank of direct
sums of small matrix multiplication tensors
– R(<4,1,3> © <1,6,1>) · 13 ! < 2.55
• Direct sum of tensors
<n,n,n> © <m,m,m>
<n,n,n>
<m,m,m>
(multiple matrix multiplications in parallel)
R(<n1,n1,n1> © … © <nk,nk,nk>) < r ) ini! < r
“Asymptotic Sum Inequality” and
example (Schönhage 1981)
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• Strategy IV: Strassen “laser method”
– tensor with “course structure” of MM and “fine
structure” components isomorphic to MM
(many independent MMs in high tensor powers)
1
1
Upper bounds on rank
1
1
1
1
…
scalar x row vector
col vector x scalar
q
<1,2,1> coarse structure fine =
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• Strategy IV: Strassen “laser method”
– tensor with “course structure” of MM and “fine
structure” components isomorphic to MM
(many independent MMs in high tensor powers)
border rank = q + 1; q = 5 yields ! < 2.48
1
1
Upper bounds on rank
1
1
1
1
…
q
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Upper bounds on rank
• Strategy V : (C-W) border rank of this tensor
is q+2:
– zero-out variables leaving many independent
MMs in high tensor power (sophisticated proof)
– q = 6 yields ! < 2.41
i=1…q X0YiZi + XiY0Zi + XiYiZ0
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Upper bounds on rank
• Strategy VI: (C-W) border rank of this tensor
is q+2:
– 6 “pieces”: target proportions in high tensor
power affect # and size of independent MMs
– optimize by hand
– q = 6 yields ! < 2.388
i=1…q X0YiZi + XiY0Zi + XiYiZ0 +
X0Y0Zq+1 + X0Yq+1Z0 + Xq+1Y0Z0
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Upper bounds on rank
• Strategy VII: border rank of tensor powers
of T:
T = i=1…q X0YiZi + XiY0Zi + XiYiZ0 +
X0Y0Zq+1 + X0Yq+1Z0 + Xq+1Y0Z0
Tensor power # pieces bound reference
2 36 2.375 C-W
4 1296 2.3737 Stothers
8 1679616 2.3729 Williams
16 2.82 x 10^12 2.3728640 Le Gall
32 7.95 x 10^24 2.3728639 Le Gall
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Upper bounds on rank
• Strategy VII: border rank of tensor powers
• Ambainis-Filmus 2014: N-th tensor power
cannot beat bound of 2.3078
Tensor power # pieces bound reference
2 36 2.375 C-W
4 1296 2.3737 Stothers
8 1679616 2.3729 Williams
16 2.82 x 10^12 2.3728640 Le Gall
32 7.95 x 10^24 2.3728639 Le Gall
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Conjectures implying
! = 2
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“Asymptotic Rank” conjecture [CW90]
– border rank = 4
– asymptotic rank of T = lim n ! 1 R(Tn)1/n
1
1
1
1
1
1 T slices
T = i=1,2 X0YiZi + XiY0Zi + XiYiZ0
conjecture: asymptotic rank of T = 3 July 31, 2014 30
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“no 3 disjoint equivoluminous
subsets” conjecture [CW90]
one way to achieve asymptotic rank 3:
• m1, m2, …, mn 2 H (abelian group)
• for all disjoint S, T, U 2 [n] the sums
i 2 S mi i 2 U mi i 2 T mi
are not all equal
conjecture: can take |H| · 2o(n)
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Strong Uniquely Solvable Puzzle
Conjecture [CKSU05]
1
1
1
1
1
1 T
1
1
1
1
1
1
1
1
1
T’
rank = 4
rank = 3
Strong Uniquely Solvable Puzzle:
gives a way to zero-out variables
leaving many independent MMs in high
tensor power of T’ (instead of T)
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Strong Uniquely Solvable Puzzle
Conjecture [CKSU05]
Uniquely Solvable Puzzle:
every unintended way of
assembling pieces has overlap
of 2 or 3 in some cell
Strong Uniquely Solvable Puzzle:
every unintended way of
assembling pieces has overlap
of exactly 2 in some cell
0 0 1 1 1 2
0 1 0 1 2 1
0 1 1 1 2 2
1 0 0 2 1 1
1 0 1 2 1 2
1 1 0 2 2 1
à w !
N rows
conjecture: Strong USPs exist with
N = (w choose w/3)1 - o(1) rows
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“Two Families” conjecture [CKSU05]
• subsets A1, A2, …, An, B1, B2, …, Bn of
Abelian group H
1. |Ai + Bi| = |Ai|¢|Bi|
2. (Ai + Bi) Å (Aj + Bk) = ;
if j k
A1 A2 An
all
distinct
all
distinct
all
distinct
B1
B2
Bn
conjecture: can achieve
• n = |H|1/2 – o(1)
• |Ai| = |Bi| = |H|1/2 – o(1)
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Status of conj’s implying ! = 2
• asymptotic rank(T) = R(Tn)1/n ! 3?
• no 3 disjoint equivoluminous subsets [CW90]
• strong uniquely solvable puzzle [CKSU05]
• two families [CKSU05]
1
1
1
1
1
1 T slices
sunflower conj. #1 false
)
sunflower conj. #2 false
)
[CW90]
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Sunflower conjecture #1
• s!2s known
• conjecture widely believed
3-sunflower
Erdos-Rado conjecture: every collection of
consts s-subsets contains a 3-sunflower
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Status of conj’s implying ! = 2
• asymptotic rank(T) = R(Tn)1/n ! 3?
• no 3 disjoint equivoluminous subsets [CW90]
• strong uniquely solvable puzzle [CKSU05]
• two families [CKSU05]
1
1
1
1
1
1 T slices
sunflower conj. #1 false
)
sunflower conj. #2 false
)
[CW90]
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Sunflower conjecture #2
• 50% chance of being false?
• 60% chance of being false?
Every collection of 3n(1- const) vectors in Z3n
contains u,v,w such that u + v + w = 0
Every 3-colored collection* of 3n(1- const)
vectors in Z3n contains u,v,w s.t. u+v+w=0
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Status of conj’s implying ! = 2
• asymptotic rank(T) = R(Tn)1/n ! 3?
• no 3 disjoint equivoluminous subsets [CW90]
• strong uniquely solvable puzzle [CKSU05]
• two families [CKSU05]
1
1
1
1
1
1 T slices
sunflower conj. #1 false
)
sunflower conj. #2 false
)
[CW90]
July 31, 2014 39
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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
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