96 research outputs found
Towards Blackbox Identity Testing of Log-Variate Circuits
Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC\u2718) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few variables, eg. logarithmic in the size s.
We give the first poly(s)-time blackbox identity test for n=O(log s) variate size-s circuits that have poly(s)-dimensional partial derivative space; eg. depth-3 diagonal circuits (or Sigma wedge Sigma^n). The former model is well-studied (Nisan,Wigderson, FOCS\u2795) but no poly(s2^n)-time identity test was known before us. We introduce the concept of cone-closed basis isolation and prove its usefulness in studying log-variate circuits. It subsumes the previous notions of rank-concentration studied extensively in the context of ROABP models
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
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
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
Quasi-polynomial Hitting-set for Set-depth-Delta Formulas
We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition
(X_1,...,X_d) of the variable indices [n] that the top product layer respects,
i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a
sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we
call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi
gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in
C respects a (unknown) partition on the variables; if Delta is even then the
product gates of the bottom-most Pi-layer are allowed to compute arbitrary
monomials.
In this work, we give a hitting-set generator for set-depth-Delta formulas
(over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta -
1}), where s is the size bound on the input set-depth-Delta formula. In other
words, we give a quasi-polynomial time blackbox polynomial identity test for
such constant-depth formulas. Previously, the very special case of Delta=3
(also known as set-multilinear depth-3 circuits) had no known sub-exponential
time hitting-set generator. This was declared as an open problem by Shpilka &
Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson
(FOCS 1995). Our work settles this question, not only for depth-3 but, up to
depth epsilon.log s / loglog s, for a fixed constant epsilon < 1.
The technique is to investigate depth-Delta formulas via depth-(Delta-1)
formulas over a Hadamard algebra, after applying a `shift' on the variables. We
propose a new algebraic conjecture about the low-support rank-concentration in
the latter formulas, and manage to prove it in the case of set-depth-Delta
formulas.Comment: 22 page
Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness
We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field ?, in the blackbox setting.
Our algorithm works for every field ? with | ? | ? d+1, and uses only d log n + log (1/ ?) + O(d log log n) random bits to achieve a success probability 1 - ? for some ? > 0. In the low degree regime that is d ? n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC\u2714 and Bshouty, ITCS\u2714) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound.
We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions
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