10,227 research outputs found
An Exponential Lower Bound for Homogeneous Depth-5 Circuits over Finite Fields
In this paper, we show exponential lower bounds for the class of homogeneous depth-5 circuits over all small finite fields. More formally, we show that there is an explicit family {P_d} of polynomials in VNP, where P_d is of degree d in n = d^{O(1)} variables, such that over all finite fields GF(q), any homogeneous depth-5 circuit which computes P_d must have size at least exp(Omega_q(sqrt{d})).
To the best of our knowledge, this is the first super-polynomial lower bound for this class for any non-binary field.
Our proof builds up on the ideas developed on the way to proving lower bounds for homogeneous depth-4 circuits [Gupta et al., Fournier et al., Kayal et al., Kumar-Saraf] and for non-homogeneous depth-3 circuits over finite fields [Grigoriev-Karpinski, Grigoriev-Razborov].
Our key insight is to look at the space of shifted partial derivatives of a polynomial as a space of functions from GF(q)^n to GF(q) as opposed to looking at them as a space of formal polynomials and builds over a tighter analysis of the lower bound of Kumar and Saraf [Kumar-Saraf]
Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity
We say that a circuit over a field functionally computes an
-variate polynomial if for every we have that . This is in contrast to syntactically computing , when as
formal polynomials. In this paper, we study the question of proving lower
bounds for homogeneous depth- and depth- arithmetic circuits for
functional computation. We prove the following results :
1. Exponential lower bounds homogeneous depth- arithmetic circuits for a
polynomial in .
2. Exponential lower bounds for homogeneous depth- arithmetic circuits
with bounded individual degree for a polynomial in .
Our main motivation for this line of research comes from our observation that
strong enough functional lower bounds for even very special depth-
arithmetic circuits for the Permanent imply a separation between and
. Thus, improving the second result to get rid of the bounded individual
degree condition could lead to substantial progress in boolean circuit
complexity. Besides, it is known from a recent result of Kumar and Saptharishi
[KS15] that over constant sized finite fields, strong enough average case
functional lower bounds for homogeneous depth- circuits imply
superpolynomial lower bounds for homogeneous depth- circuits.
Our proofs are based on a family of new complexity measures called shifted
evaluation dimension, and might be of independent interest
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]
Arithmetic Circuit Lower Bounds via MaxRank
We introduce the polynomial coefficient matrix and identify maximum rank of
this matrix under variable substitution as a complexity measure for
multivariate polynomials. We use our techniques to prove super-polynomial lower
bounds against several classes of non-multilinear arithmetic circuits. In
particular, we obtain the following results :
As our main result, we prove that any homogeneous depth-3 circuit for
computing the product of matrices of dimension requires
size. This improves the lower bounds by Nisan and
Wigderson(1995) when .
There is an explicit polynomial on variables and degree at most
for which any depth-3 circuit of product dimension at most
(dimension of the space of affine forms feeding into each
product gate) requires size . This generalizes the lower bounds
against diagonal circuits proved by Saxena(2007). Diagonal circuits are of
product dimension 1.
We prove a lower bound on the size of product-sparse
formulas. By definition, any multilinear formula is a product-sparse formula.
Thus, our result extends the known super-polynomial lower bounds on the size of
multilinear formulas by Raz(2006).
We prove a lower bound on the size of partitioned arithmetic
branching programs. This result extends the known exponential lower bound on
the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page
On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields
Recently, Gupta et.al. [GKKS2013] proved that over Q any -variate
and -degree polynomial in VP can also be computed by a depth three
circuit of size . Over fixed-size
finite fields, Grigoriev and Karpinski proved that any
circuit that computes (or ) must be of size
[GK1998]. In this paper, we prove that over fixed-size finite fields, any
circuit for computing the iterated matrix multiplication
polynomial of generic matrices of size , must be of size
. The importance of this result is that over fixed-size
fields there is no depth reduction technique that can be used to compute all
the -variate and -degree polynomials in VP by depth 3 circuits of
size . The result [GK1998] can only rule out such a possibility
for depth 3 circuits of size .
We also give an example of an explicit polynomial () in
VNP (not known to be in VP), for which any circuit computing
it (over fixed-size fields) must be of size . The
polynomial we consider is constructed from the combinatorial design. An
interesting feature of this result is that we get the first examples of two
polynomials (one in VP and one in VNP) such that they have provably stronger
circuit size lower bounds than Permanent in a reasonably strong model of
computation.
Next, we prove that any depth 4
circuit computing
(over any field) must be of size . To the best of our knowledge, the polynomial is the
first example of an explicit polynomial in VNP such that it requires
size depth four circuits, but no known matching
upper bound
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
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