Average-case linear matrix factorization and reconstruction of low width algebraic branching programs
A matrix X is called a linear matrix if its entries are affine forms, i.e., degree one polynomials in n variables. What is a minimal-sized representation of a given matrix F as a product of linear matrices? Finding such a minimal representation is closely related to finding an optimal way to compute...
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| Vydané v: | Computational complexity Ročník 28; číslo 4; s. 749 - 828 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Cham
Springer International Publishing
01.12.2019
Springer Nature B.V |
| Predmet: | |
| ISSN: | 1016-3328, 1420-8954 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | A matrix
X
is called a
linear matrix
if its entries are affine forms, i.e., degree one polynomials in
n
variables. What is a minimal-sized representation of a given matrix
F
as a product of linear matrices? Finding such a minimal representation is closely related to finding an optimal way to compute a given polynomial via an algebraic branching program. Here we devise an efficient algorithm for an average-case version of this problem. Specifically, given
w
,
d
,
n
∈
N
and blackbox access to the
w
2
entries of a matrix product
F
=
X
1
⋯
X
d
, where each
X
i
is a
w
×
w
linear matrix over a given finite field
F
q
, we wish to recover a factorization
F
=
Y
1
⋯
Y
d
′
, where every
Y
i
is also a linear matrix over
F
q
(or a small extension of
F
q
). We show that when the input
F
is sampled from a distribution defined by choosing random linear matrices
X
1
,
…
,
X
d
over
F
q
independently and taking their product and
n
≥
4
w
2
and
char
(
F
q
)
=
(
d
n
)
Ω
(
1
)
, then an equivalent factorization
F
=
Y
1
⋯
Y
d
can be recovered in (randomized) time
(
d
n
log
q
)
O
(
1
)
. In fact, we give a (worst-case) polynomial time randomized algorithm to factor any non-degenerate or
pure
matrix product (a notion we define in the paper) into linear matrices; a matrix product
F
=
X
1
⋯
X
d
is pure with high probability when the
X
i
's are chosen independently at random. We also show that in this situation, if we are instead given a single entry of
F
rather than its
w
2
correlated entries, then the recovery can be done in (randomized) time
(
d
w
3
n
log
q
)
O
(
1
)
. |
|---|---|
| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1016-3328 1420-8954 |
| DOI: | 10.1007/s00037-019-00189-0 |