1,708 research outputs found
A hitting set construction, with application to arithmetic circuit lower bounds
14 pagesA polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form . From our algorithm we derive an exponential lower bound for representations of polynomials such as under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the ``hardness from derandomization'' approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers
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
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
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
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
Near-optimal Bootstrapping of Hitting Sets for Algebraic Models
The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel
[Ore22,DL78,Zip79,Sch80] states that any nonzero polynomial of degree at most will evaluate to a nonzero value at some point on a
grid with . Thus, there is an explicit
hitting set for all -variate degree , size algebraic circuits of size
.
In this paper, we prove the following results:
- Let be a constant. For a sufficiently large constant and
all , if we have an explicit hitting set of size
for the class of -variate degree polynomials that are computable by
algebraic circuits of size , then for all , we have an explicit hitting
set of size for -variate circuits of
degree and size . That is, if we can obtain a barely non-trivial
exponent compared to the trivial sized hitting set even for
constant variate circuits, we can get an almost complete derandomization of
PIT.
- The above result holds when "circuits" are replaced by "formulas" or
"algebraic branching programs".
This extends a recent surprising result of Agrawal, Ghosh and Saxena [AGS18]
who proved the same conclusion for the class of algebraic circuits, if the
hypothesis provided a hitting set of size at most
(where is any constant). Hence, our work significantly weakens the
hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial
saving over the trivial hitting set, and also presents the first such result
for algebraic branching programs and formulas.Comment: The main result has been strengthened significantly, compared to the
older version of the paper. Additionally, the stronger theorem now holds even
for subclasses of algebraic circuits, such as algebraic formulas and
algebraic branching program
Algebraic Independence and Blackbox Identity Testing
Algebraic independence is an advanced notion in commutative algebra that
generalizes independence of linear polynomials to higher degree. Polynomials
{f_1, ..., f_m} \subset \F[x_1, ..., x_n] are called algebraically independent
if there is no non-zero polynomial F such that F(f_1, ..., f_m) = 0. The
transcendence degree, trdeg{f_1, ..., f_m}, is the maximal number r of
algebraically independent polynomials in the set. In this paper we design
blackbox and efficient linear maps \phi that reduce the number of variables
from n to r but maintain trdeg{\phi(f_i)}_i = r, assuming f_i's sparse and
small r. We apply these fundamental maps to solve several cases of blackbox
identity testing:
(1) Given a polynomial-degree circuit C and sparse polynomials f_1, ..., f_m
with trdeg r, we can test blackbox D := C(f_1, ..., f_m) for zeroness in
poly(size(D))^r time.
(2) Define a spsp_\delta(k,s,n) circuit C to be of the form \sum_{i=1}^k
\prod_{j=1}^s f_{i,j}, where f_{i,j} are sparse n-variate polynomials of degree
at most \delta. For k = 2 we give a poly(sn\delta)^{\delta^2} time blackbox
identity test.
(3) For a general depth-4 circuit we define a notion of rank. Assuming there
is a rank bound R for minimal simple spsp_\delta(k,s,n) identities, we give a
poly(snR\delta)^{Rk\delta^2} time blackbox identity test for spsp_\delta(k,s,n)
circuits. This partially generalizes the state of the art of depth-3 to depth-4
circuits.
The notion of trdeg works best with large or zero characteristic, but we also
give versions of our results for arbitrary fields.Comment: 32 pages, preliminary versio
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