Polynomial inequalities on the Hamming cube

Eskenazis, Alexandros; Ivanisvili, Paata

doi:10.1007/s00440-020-00973-y

Cited by 15 publications

(23 citation statements)

References 38 publications

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“…where T d (t) is the d-th Chebyshev polynomial of the first kind, that is, the unique real polynomial of degree d such that cos(dθ) = T d (cos θ) for every θ ∈ R. Moreover, Iyer, Rao, Reis, Rothvoss and Yehudayoff observed in [11,Proposition 2] that this estimate is asymptotically sharp. We present a simple proof of their inequality (22) (see also [10] for related arguments).…”

Section: Discussionmentioning

confidence: 96%

Learning low-degree functions from a logarithmic number of random queries

Eskenazis¹,

Ivanisvili²

2021

Preprint

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Section: Discussionmentioning

confidence: 96%

Learning low-degree functions from a logarithmic number of random queries

Eskenazis¹,

Ivanisvili²

2021

Preprint

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“…Throughout this paper and [16], we have been studying learning algorithms for F n,d and B n,d equipped with the Hilbertian L 2 -metric. This choice allows us to use Parseval's identity and thus exploit properties of individual Walsh coefficients to study the distance between f and the hypothesis function h. However as the constructed hypothesis functions h are always of degree at most d themselves, this can be generalized to any L p norm, where 0 < p < ∞, since these are equivalent to the L 2 norm on the space of degree-d polynomials up to constants depending only on d (see [36, §9.5] and [7,6,15] for more on moment comparison of polynomials).…”

Section: Exact Learningmentioning

confidence: 99%

Low-degree learning and the metric entropy of polynomials

Eskenazis¹,

Ivanisvili²,

Streck³

2022

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Let F n,d be the class of all functions f : {−1, 1} n → [−1, 1] on the n-dimensional discrete hypercube of degree at most d. In the first part of this paper, we prove that any (deterministic or randomized) algorithm which learns F n,d with L 2 -accuracy ε requires at least Ω((1 − √ ε)2 d log n) queries for large enough n, thus establishing the sharpness as n → ∞ of a recent upper bound of Eskenazis and Ivanisvili (2021). To do this, we show that the L 2 -packing numbers2 Cd log n ε 4 for large enough n, where c, C > 0 are universal constants. In the second part of the paper, we present a logarithmic upper bound for the randomized query complexity of classes of bounded approximate polynomials whose Fourier spectra are concentrated on few subsets. As an application, we prove new estimates for the number of random queries required to learn approximate juntas of a given degree, functions with rapidly decaying Fourier tails and constant depth circuits of given size. Finally, we obtain bounds for the number of queries required to learn the polynomial class F n,d without error in the query and random example models.

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“…Proof of Theorem 9. The proof of Theorem 9 relies on the following result of Mendel and Naor from [MN14] (see also [EI20] for a different proof and further results in this direction).…”

Section: Proof Of Theoremmentioning

confidence: 99%

Talagrand's influence inequality revisited

Cordero–Erausquin¹,

Eskenazis²

2020

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Let Cn = {−1, 1} n be the discrete hypercube equipped with the uniform probability measure σn. Talagrand's influence inequality (1994), also known as the L1 − L2 inequality, asserts that there exists C ∈ (0, ∞) such that for every n ∈ N, every function f :.In this work, we undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach spacevalued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. Moreover, our proof implies vector-valued versions of a general family of L1 − Lp inequalities, each refining the dimension independent Lp-Poincaré inequality on (Cn, σn). We also obtain a joint strengthening of results of Bakry-Meyer (1982) and Naor-Schechtman ( 2002) on the action of negative powers of the hypercube Laplacian on functions f : Cn → E, whose target space (E, • E ) has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem (1984). Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube Cn equipped with the Hamming metric, thus deriving new nonembeddability results for these finite metrics. Our proofs make use of Banach space-valued Itô calculus, Riesz transform inequalities, Littlewood-Paley-Stein theory and hypercontractivity.

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Polynomial inequalities on the Hamming cube

Cited by 15 publications

References 38 publications

Learning low-degree functions from a logarithmic number of random queries

Learning low-degree functions from a logarithmic number of random queries

Low-degree learning and the metric entropy of polynomials

Talagrand's influence inequality revisited

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