958 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
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
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
Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas
In this paper we study the identity testing problem of arithmetic read-once formulas (ROF) and some related models. A read-once formula is formula (a circuit whose underlying graph is a tree) in which the operations are {+,x} and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for read-once formulas, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of [Shpilka-Volkovich, 2015
Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas
In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation
between black-box PIT and lower bounds we obtain lower bounds for these models.
For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial.
For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial.
A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a
hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas.
We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of
a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.
Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a
read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs)
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