Affiliation:
1. IT University of Copenhagen, København S, Denmark
Abstract
We present a simple algorithm that approximates the product of
n
-by-
n
real matrices
A
and
B
. Let ‖AB‖
F
denote the Frobenius norm of
AB
, and
b
be a parameter determining the time/accuracy trade-off. Given 2-wise independent hash functions
h
1
,
h
2
: [
n
]→ [
b
], and
s
1
,
s
2
: [
n
]→ {−1,+1} the algorithm works by first “compressing” the matrix product into the polynomial
p
(
x
) = ∑
k
=1
n
\left(∑
i
=1
n
A
ik
s
1
(
i
)
x
h
1
(
i
)
\right) \left(∑
j
=1
n
B
kj
s
2
(
j
)
x
h
2
(
j
)
\right). Using the fast Fourier transform to compute polynomial multiplication, we can compute
c
0
,…,
c
b
−1
such that ∑
i
c
i
x
i
= (
p
(
x
) mod
x
b
) + (
p
(
x
) div
x
b
) in time Õ(
n
2
+
nb
). An unbiased estimator of (
AB
)
ij
with variance at most ‖
AB
‖
F
2
/
b
can then be computed as:
C
ij
=
s
1
(
i
)
s
2
(
j
)
c
(
h
1
(
i
)+
h
2
(
j
)) mod
b
. Our approach also leads to an algorithm for computing
AB
exactly, with high probability, in time Õ(
N
+
nb
) in the case where
A
and
B
have at most
N
nonzero entries, and
AB
has at most
b
nonzero entries.
Funder
Danish National Research Foundation
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science
Reference32 articles.
1. Tracking Join and Self-Join Sizes in Limited Storage
2. The Space Complexity of Approximating the Frequency Moments
3. AMS without 4-wise independence on product domains. In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS);Braverman V.;Leibniz International Proceedings in Informatics (LIPIcs) Series,2010
4. Universal classes of hash functions
5. Finding frequent items in data streams
Cited by
57 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献