 Award ID(s):
 1650733
 Publication Date:
 NSFPAR ID:
 10078501
 Journal Name:
 Computational Complexity Conference (CCC)
 Page Range or eLocationID:
 2837
 ISSN:
 18688969
 Sponsoring Org:
 National Science Foundation
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We introduce a new technique for reducing the dimension of the ambient space of lowdegree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilonoptimal noisestable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilonapproximate any joint distribution that can be noninteractively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermannlike to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap noninteractive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for lowdegree polynomials is simple and can be seen as a generalization of the JohnsonLindenstrauss lemma and could be of independent interest.

We introduce a new technique for reducing the dimension of the ambient space of lowdegree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilonoptimal noisestable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilonapproximate any joint distribution that can be noninteractively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermannlike to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap noninteractive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for lowdegree polynomials is simple and can be seen as a generalization of the JohnsonLindenstrauss lemma and could be of independent interest.

Raz, Ran (Ed.)We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs whose lines are circuits from restricted boolean circuit classes. Essentially all of the subsystems considered in this paper can simulate the wellstudied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). Our main contributions are two general methods of converting certain algebraic lower bounds into proof complexity ones. Both require stronger arithmetic lower bounds than common, which should hold not for a specific polynomial but for a whole family defined by it. These may be likened to some of the methods by which Boolean circuit lower bounds are turned into related proofcomplexity ones, especially the "feasible interpolation" technique. We establish algebraic lower bounds of these forms for several explicit polynomials, against a variety of classes, and infer the relevant proofmore »

We prove two new results about the inability of lowdegree polynomials to uniformly approximate constantdepth circuits, even to slightlybetterthantrivial error. First, we prove a tight Omega~(n^{1/2}) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC^0 function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS 2015). Our result also extends to a 2^{Omega~(n^{1/2})} lower bound on the signrank of an AC^0 function, improving on the previous best bound of 2^{Omega(n^{2/5})} (Bun and Thaler, ICALP 2016). Second, for any delta>0, we exhibit a function f : {1,1}^n > {1,1} that is computed by a circuit of depth O(1/delta) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error epsilon=12^{Omega(n^{1delta})}, even by polynomials of degree n^{1delta}. Our recent prior work (Bun and Thaler, FOCS 2017) proved a similar lower bound, but which held only for error epsilon=1/3. Our result implies 2^{Omega(n^{1delta})} lower bounds on the complexity of AC^0 under a variety of basic measures such as discrepancy, margin complexity, and threshold weight. This nearly matches the trivial upper bound of 2^{O(n)} that holds for everymore »

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