94 research outputs found
A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits
In this paper we study the complexity of constructing a hitting set for the
closure of VP, the class of polynomials that can be infinitesimally
approximated by polynomials that are computed by polynomial sized algebraic
circuits, over the real or complex numbers. Specifically, we show that there is
a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the
rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all
n-variate polynomials of degree-r that are the limit of size-s algebraic
circuits. Previously it was known that a random set of this size is a hitting
set, but a construction that is certified to work was only known in EXPSPACE
(or EXPH assuming the generalized Riemann hypothesis). As a corollary we get
that a host of other algebraic problems such as Noether Normalization Lemma,
can also be solved in PSPACE deterministically, where earlier only randomized
algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann
hypothesis) were known.
The proof relies on the new notion of a robust hitting set which is a set of
inputs such that any nonzero polynomial that can be computed by a polynomial
size algebraic circuit, evaluates to a not too small value on at least one
element of the set. Proving the existence of such a robust hitting set is the
main technical difficulty in the proof.
Our proof uses anti-concentration results for polynomials, basic tools from
algebraic geometry and the existential theory of the reals
Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity
Testing whether a set of polynomials has an algebraic dependence
is a basic problem with several applications. The polynomials are given as
algebraic circuits. Algebraic independence testing question is wide open over
finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is
NP (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). In this work we
put the problem in AM coAM. In particular, dependence testing is
unlikely to be NP-hard and joins the league of problems of "intermediate"
complexity, eg. graph isomorphism & integer factoring. Our proof method is
algebro-geometric-- estimating the size of the image/preimage of the polynomial
map over the finite field. A gap in this size is utilized in the
AM protocols.
Next, we study the open question of testing whether every annihilator of
has zero constant term (Kayal, CCC'09). We give a geometric
characterization using Zariski closure of the image of ;
introducing a new problem called approximate polynomials satisfiability (APS).
We show that APS is NP-hard and, using projective algebraic-geometry ideas, we
put APS in PSPACE (prior best was EXPSPACE via Grobner basis computation). As
an unexpected application of this to approximative complexity theory we get--
Over any field, hitting-set for can be designed in PSPACE.
This solves an open problem posed in (Mulmuley, FOCS'12, J.AMS 2017); greatly
mitigating the GCT Chasm (exponentially in terms of space complexity)
Algebraic Methods in Computational Complexity
Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test
are some of the most prominent examples. In some of the most exciting recent progress in Computational Complexity the algebraic theme still plays a central role. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Also the areas of derandomization and coding theory have experimented important advances. The seminar aimed to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and the goal of the seminar was to play an important role in educating a diverse community about the latest new techniques
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