323 research outputs found
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
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 Unabhängigkeit innerhalb der algebraischen Komplexitätstheorie. Arithmetische Schaltkreise, die multivariate Polynome über einem Körper berechnen, bilden die Grundlage unserer Komplexitätsbetrachtungen. Wir befassen uns mit dem polynomial identity testing (PIT) Problem, bei dem entschieden werden soll ob ein gegebener Schaltkreis das Nullpolynom berechnet. Für dieses Problem sind effiziente randomisierte Algorithmen bekannt, aber deterministische Polynomialzeitalgorithmen konnten bisher nur für eingeschränkte Klassen von Schaltkreisen angegeben werden. Besonders von Interesse sind Blackbox-Algorithmen, welche den gegebenen Schaltkreis nicht inspizieren, sondern lediglich an Punkten auswerten. Bekannte Ansätze für das PIT Problem basieren auf den Begriffen der linearen Unabhängigkeit und des Rangs von Untervektorräumen des Polynomrings. Wir übertragen diese Methoden auf algebraische Unabhängigkeit und den Transzendenzgrad von Unteralgebren des Polynomrings. Dadurch erhalten wir effiziente Blackbox-PIT-Algorithmen für neue Klassen von Schaltkreisen. Eine effiziente Charakterisierung der algebraischen Unabhängigkeit von Polynomen ist durch das Jacobi-Kriterium gegeben. Dieses Kriterium ist jedoch nur in Charakteristik Null gültig. Wir leiten ein neues Jacobi-artiges Kriterium für die algebraische Unabhängigkeit von Polynomen über endlichen Körpern her. Dieses liefert einen weiteren Blackbox-PIT-Algorithmus und verbessert die Komplexität des Problems arithmetische Schaltkreise über endlichen Körpern auf algebraische Unabhängigkeit zu testen
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
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
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
Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits
Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS\u2708) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (?^[k] ? ? ?) and sum-product-of-constant-degree-polynomials (?^[k] ? ? ?^[?]), for constants k, ?, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC\u2705; Kayal & Saxena, CCC\u2706; Saxena & Seshadhri, FOCS\u2710, STOC\u2711); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP\u2711; Saha,Saxena & Saptharishi, Comput.Compl.\u2713; Forbes, FOCS\u2715; Kumar & Saraf, CCC\u2716); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS\u2709; Shpilka, STOC\u2719; Peleg & Shpilka, CCC\u2720, STOC\u2721). We solve two of the basic underlying open problems in this work.
We give the first polynomial-time PIT for ?^[k] ? ? ?. Further, we give the first quasipolynomial time blackbox PIT for both ?^[k] ? ? ? and ?^[k] ? ? ?^[?]. No subexponential time algorithm was known prior to this work (even if k = ? = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top ?-gate to ?
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