234 research outputs found
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
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
Quantum Query Complexity of Multilinear Identity Testing
Motivated by the quantum algorithm in \cite{MN05} for testing commutativity
of black-box groups, we study the following problem: Given a black-box finite
ring where is an additive
generating set for and a multilinear polynomial over
also accessed as a black-box function (where we allow the
indeterminates to be commuting or noncommuting), we study the
problem of testing if is an \emph{identity} for the ring . More
precisely, the problem is to test if for all .
We give a quantum algorithm with query complexity assuming . Towards a lower bound,
we also discuss a reduction from a version of -collision to this problem.
We also observe a randomized test with query complexity and constant
success probability and a deterministic test with query complexity.Comment: 12 page
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
Independence in Algebraic Complexity Theory
This thesis examines the concepts of linear and algebraic independence in algebraic complexity theory. Arithmetic circuits, computing multivariate polynomials over a field, form the framework of our complexity considerations. We are concerned with polynomial identity testing (PIT), the problem of deciding whether a given arithmetic circuit computes the zero polynomial. There are efficient randomized algorithms known for this problem, but as yet deterministic polynomial-time algorithms could be found only for restricted circuit classes. We are especially interested in blackbox algorithms, which do not inspect the given circuit, but solely evaluate it at some points. Known approaches to the PIT problem are based on the notions of linear independence and rank of vector subspaces of the polynomial ring. We generalize those methods to algebraic independence and transcendence degree of subalgebras of the polynomial ring. Thereby, we obtain efficient blackbox PIT algorithms for new circuit classes. The Jacobian criterion constitutes an efficient characterization for algebraic independence of polynomials. However, this criterion is valid only in characteristic zero. We deduce a novel Jacobian-like criterion for algebraic independence of polynomials over finite fields. We apply it to obtain another blackbox PIT algorithm and to improve the complexity of testing the algebraic independence of arithmetic circuits over finite fields.Die vorliegende Arbeit untersucht die Konzepte der linearen und algebraischen UnabhĂ€ngigkeit innerhalb der algebraischen KomplexitĂ€tstheorie. Arithmetische Schaltkreise, die multivariate Polynome ĂŒber einem Körper berechnen, bilden die Grundlage unserer KomplexitĂ€tsbetrachtungen. Wir befassen uns mit dem polynomial identity testing (PIT) Problem, bei dem entschieden werden soll ob ein gegebener Schaltkreis das Nullpolynom berechnet. FĂŒr dieses Problem sind effiziente randomisierte Algorithmen bekannt, aber deterministische Polynomialzeitalgorithmen konnten bisher nur fĂŒr eingeschrĂ€nkte Klassen von Schaltkreisen angegeben werden. Besonders von Interesse sind Blackbox-Algorithmen, welche den gegebenen Schaltkreis nicht inspizieren, sondern lediglich an Punkten auswerten. Bekannte AnsĂ€tze fĂŒr das PIT Problem basieren auf den Begriffen der linearen UnabhĂ€ngigkeit und des Rangs von UntervektorrĂ€umen des Polynomrings. Wir ĂŒbertragen diese Methoden auf algebraische UnabhĂ€ngigkeit und den Transzendenzgrad von Unteralgebren des Polynomrings. Dadurch erhalten wir effiziente Blackbox-PIT-Algorithmen fĂŒr neue Klassen von Schaltkreisen. Eine effiziente Charakterisierung der algebraischen UnabhĂ€ngigkeit von Polynomen ist durch das Jacobi-Kriterium gegeben. Dieses Kriterium ist jedoch nur in Charakteristik Null gĂŒltig. Wir leiten ein neues Jacobi-artiges Kriterium fĂŒr die algebraische UnabhĂ€ngigkeit von Polynomen ĂŒber endlichen Körpern her. Dieses liefert einen weiteren Blackbox-PIT-Algorithmus und verbessert die KomplexitĂ€t des Problems arithmetische Schaltkreise ĂŒber endlichen Körpern auf algebraische UnabhĂ€ngigkeit zu testen
Interpolation of Shifted-Lacunary Polynomials
Given a "black box" function to evaluate an unknown rational polynomial f in
Q[x] at points modulo a prime p, we exhibit algorithms to compute the
representation of the polynomial in the sparsest shifted power basis. That is,
we determine the sparsity t, the shift s (a rational), the exponents 0 <= e1 <
e2 < ... < et, and the coefficients c1,...,ct in Q\{0} such that f(x) =
c1(x-s)^e1+c2(x-s)^e2+...+ct(x-s)^et. The computed sparsity t is absolutely
minimal over any shifted power basis. The novelty of our algorithm is that the
complexity is polynomial in the (sparse) representation size, and in particular
is logarithmic in deg(f). Our method combines previous celebrated results on
sparse interpolation and computing sparsest shifts, and provides a way to
handle polynomials with extremely high degree which are, in some sense, sparse
in information.Comment: 22 pages, to appear in Computational Complexit
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