TY - GEN

T1 - Random low degree polynomials are hard to approximate

AU - Ben-Eliezer, Ido

AU - Hod, Rani

AU - Lovett, Shachar

PY - 2009

Y1 - 2009

N2 - We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over double-struck F 2. We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d, for all degrees d up to Θ (n). That is, a random degree d + ∈1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed-Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed-Muller codes.

AB - We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over double-struck F 2. We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d, for all degrees d up to Θ (n). That is, a random degree d + ∈1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed-Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed-Muller codes.

UR - http://www.scopus.com/inward/record.url?scp=70350589611&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-03685-9_28

DO - 10.1007/978-3-642-03685-9_28

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AN - SCOPUS:70350589611

SN - 3642036848

SN - 9783642036842

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 366

EP - 377

BT - Approximation, Randomization, and Combinatorial Optimization

T2 - 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009

Y2 - 21 August 2009 through 23 August 2009

ER -