1,161 research outputs found
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
The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent
Polynomial identity testing and arithmetic circuit lower bounds are two
central questions in algebraic complexity theory. It is an intriguing fact that
these questions are actually related. One of the authors of the present paper
has recently proposed a "real {\tau}-conjecture" which is inspired by this
connection. The real {\tau}-conjecture states that the number of real roots of
a sum of products of sparse univariate polynomials should be polynomially
bounded. It implies a superpolynomial lower bound on the size of arithmetic
circuits computing the permanent polynomial. In this paper we show that the
real {\tau}-conjecture holds true for a restricted class of sums of products of
sparse polynomials. This result yields lower bounds for a restricted class of
depth-4 circuits: we show that polynomial size circuits from this class cannot
compute the permanent, and we also give a deterministic polynomial identity
testing algorithm for the same class of circuits.Comment: 16 page
Sums of products of polynomials in few variables : lower bounds and polynomial identity testing
We study the complexity of representing polynomials as a sum of products of
polynomials in few variables. More precisely, we study representations of the
form such that each is
an arbitrary polynomial that depends on at most variables. We prove the
following results.
1. Over fields of characteristic zero, for every constant such that , we give an explicit family of polynomials , where
is of degree in variables, such that any
representation of the above type for with requires . This strengthens a recent result of Kayal and Saha
[KS14a] which showed similar lower bounds for the model of sums of products of
linear forms in few variables. It is known that any asymptotic improvement in
the exponent of the lower bounds (even for ) would separate VP
and VNP[KS14a].
2. We obtain a deterministic subexponential time blackbox polynomial identity
testing (PIT) algorithm for circuits computed by the above model when and
the individual degree of each variable in are at most and
for any constant . We get quasipolynomial running
time when . The PIT algorithm is obtained by combining our
lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04].
To the best of our knowledge, this is the first nontrivial PIT algorithm for
this model (even for the case ), and the first nontrivial PIT algorithm
obtained from lower bounds for small depth circuits
Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits
We present a single, common tool to strictly subsume all known cases of
polynomial time blackbox polynomial identity testing (PIT) that have been
hitherto solved using diverse tools and techniques. In particular, we show that
polynomial time hitting-set generators for identity testing of the two
seemingly different and well studied models - depth-3 circuits with bounded top
fanin, and constant-depth constant-read multilinear formulas - can be
constructed using one common algebraic-geometry theme: Jacobian captures
algebraic independence. By exploiting the Jacobian, we design the first
efficient hitting-set generators for broad generalizations of the
above-mentioned models, namely:
(1) depth-3 (Sigma-Pi-Sigma) circuits with constant transcendence degree of
the polynomials computed by the product gates (no bounded top fanin
restriction), and (2) constant-depth constant-occur formulas (no multilinear
restriction).
Constant-occur of a variable, as we define it, is a much more general concept
than constant-read. Also, earlier work on the latter model assumed that the
formula is multilinear. Thus, our work goes further beyond the results obtained
by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et
al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011),
and brings them under one unifying technique.
In addition, using the same Jacobian based approach, we prove exponential
lower bounds for the immanant (which includes permanent and determinant) on the
same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our
results reinforce the intimate connection between identity testing and lower
bounds by exhibiting a concrete mathematical tool - the Jacobian - that is
equally effective in solving both the problems on certain interesting and
previously well-investigated (but not well understood) models of computation
Arithmetic circuits: the chasm at depth four gets wider
In their paper on the "chasm at depth four", Agrawal and Vinay have shown
that polynomials in m variables of degree O(m) which admit arithmetic circuits
of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m).
This theorem shows that for problems such as arithmetic circuit lower bounds or
black-box derandomization of identity testing, the case of depth four circuits
is in a certain sense the general case. In this paper we show that smaller
depth four circuits can be obtained if we start from polynomial size arithmetic
circuits. For instance, we show that if the permanent of n*n matrices has
circuits of size polynomial in n, then it also has depth 4 circuits of size
n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of
polynomial size. These results have potential applications to lower bounds and
deterministic identity testing, in particular for sums of products of sparse
univariate polynomials. We also give an application to boolean circuit
complexity, and a simple (but suboptimal) reduction to polylogarithmic depth
for arithmetic circuits of polynomial size and polynomially bounded degree
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