121,470 research outputs found
Improved bounds for reduction to depth 4 and depth 3
Koiran showed that if a -variate polynomial of degree (with
) is computed by a circuit of size , then it is also computed by
a homogeneous circuit of depth four and of size
. Using this result, Gupta, Kamath, Kayal and
Saptharishi gave a upper bound for the
size of the smallest depth three circuit computing a -variate polynomial of
degree given by a circuit of size .
We improve here Koiran's bound. Indeed, we show that if we reduce an
arithmetic circuit to depth four, then the size becomes
. Mimicking Gupta, Kamath, Kayal and
Saptharishi's proof, it also implies the same upper bound for depth three
circuits.
This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal
and Saptharishi also showed a lower bound for the size
of homogeneous depth four circuits such that gates at the bottom have fan-in at
most . Finally, we show that this last lower bound also holds if the
fan-in is at least
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
On the power of homogeneous depth 4 arithmetic circuits
We prove exponential lower bounds on the size of homogeneous depth 4
arithmetic circuits computing an explicit polynomial in . Our results hold
for the {\it Iterated Matrix Multiplication} polynomial - in particular we show
that any homogeneous depth 4 circuit computing the entry in the product
of generic matrices of dimension must have size
.
Our results strengthen previous works in two significant ways.
Our lower bounds hold for a polynomial in . Prior to our work, Kayal et
al [KLSS14] proved an exponential lower bound for homogeneous depth 4 circuits
(over fields of characteristic zero) computing a poly in . The best known
lower bounds for a depth 4 homogeneous circuit computing a poly in was the
bound of by [LSS, KLSS14].Our exponential lower bounds
also give the first exponential separation between general arithmetic circuits
and homogeneous depth 4 arithmetic circuits. In particular they imply that the
depth reduction results of Koiran [Koi12] and Tavenas [Tav13] are tight even
for reductions to general homogeneous depth 4 circuits (without the restriction
of bounded bottom fanin).
Our lower bound holds over all fields. The lower bound of [KLSS14] worked
only over fields of characteristic zero. Prior to our work, the best lower
bound for homogeneous depth 4 circuits over fields of positive characteristic
was [LSS, KLSS14]
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Let C be a depth-3 circuit with n variables, degree d and top fanin k (called
sps(k,d,n) circuits) over base field F. It is a major open problem to design a
deterministic polynomial time blackbox algorithm that tests if C is identically
zero. Klivans & Spielman (STOC 2001) observed that the problem is open even
when k is a constant. This case has been subjected to a serious study over the
past few years, starting from the work of Dvir & Shpilka (STOC 2005).
We give the first polynomial time blackbox algorithm for this problem. Our
algorithm runs in time poly(nd^k), regardless of the base field. The only field
for which polynomial time algorithms were previously known is F=Q (Kayal &
Saraf, FOCS 2009, and Saxena & Seshadhri, FOCS 2010). This is the first
blackbox algorithm for depth-3 circuits that does not use the rank based
approaches of Karnin & Shpilka (CCC 2008).
We prove an important tool for the study of depth-3 identities. We design a
blackbox polynomial time transformation that reduces the number of variables in
a sps(k,d,n) circuit to k variables, but preserves the identity structure.Comment: 14 pages, 1 figure, preliminary versio
Progress on Polynomial Identity Testing - II
We survey the area of algebraic complexity theory; with the focus being on
the problem of polynomial identity testing (PIT). We discuss the key ideas that
have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve
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