In this work, we show that the class of multivariate degree-d polynomials mapping {0,1}n to any Abelian group G is locally correctable with Õd((log n )d) queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) and gave analogous results.
In this work, we show that the class of multivariate degree-d polynomials mapping {0,1}n to any Abelian group G is locally correctable with Õd((log n )d) queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) and gave analogous results.
We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain {0,1}n over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance 1/2 and we give local-correction algorithms correcting up to nearly 1/4-fraction errors making O(logn) queries. This query complexity is optimal up to poly(loglogn) factors. We also give local list-correcting algorithms correcting (1/2 − ε)-fraction errors with Oε(logn) queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is ℤ2. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). Previous works in the area mostly focused on the case where the domain is a vector space or a group and this lends to tools that exploit symmetry. Since our domains lack such symmetries, we encounter new challenges whose resolution may be of independent interest. The central challenge in constructing the local corrector is constructing “nearly balanced vectors” over {−1,1}n that span 1n — we show how to construct O(logn) vectors that do so, with entries in each vector summing to ±1. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius (1/2−ε) is Oε(1). Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.
Nisan and Szegedy (CC 1994) showed that any Boolean function f : {0, 1}n → {0, 1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x1,..., xn), has degree at least log n − O(log log n). This was improved to a tight (log n − O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log n)1/2−o(1) and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)c, then the minimum possible probabilistic degree of such an f is at least (log n)c/(c+1)−o(1), and we show this is tight up to (log n)o(1) factors.
Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}n→{0,1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x1,…,xn), has degree at least logn−O(loglogn). This was improved to a tight (logn−O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for \emph{Probabilistic degree}. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (logn)1/2−o(1) and O(logn) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, \emph{modulo} our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (logn)c, then the minimum possible probabilistic degree of such an f is at least (logn)c/(c+1)−o(1), and we show this is tight up to (logn)o(1) factors.
We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry base-code of block-length q to a lifted-code of block-length qm, for arbitrary integer m. The construction generalizes the way degree-d, univariate polynomials evaluated over the q-element field (also known as Reed-Solomon codes) are "lifted" to degree-d, m-variate polynomials (Reed-Muller codes). A number of properties are established: The rate of the degree-lifted code is approximately a 1/m!-fraction of the rate of the base-code. The relative distance of the degree-lifted code is at least as large as that of the base-code. This is proved using a generalization of the Schwartz-Zippel Lemma to degree-lifted Algebraic-Geometry codes. [Local correction] If the base code is invariant under a group that is "close" to being doubly-transitive (in a precise manner defined later then the degree-lifted code is locally correctable with query complexity at most q2. The automorphisms of the base-code are crucially used to generate query-sets, abstracting the use of affine-lines in the local correction procedure of Reed-Muller codes. Taking a concrete illustrating example, we show that degree-lifted Hermitian codes form a family of locally correctable codes over an alphabet that is significantly smaller than that obtained by Reed-Muller codes of similar constant rate, message length, and distance.
Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.
We study the effects of random restrictions on low degree functions that are bounded on every point of the Boolean cube. Our main result shows that, with high probability, the restricted function can be approximated by a junta of arity that is just polynomial in the original degree. More precisely, let f: {± 1}ⁿ → [0,1] be a degree d polynomial (d ≥ 2) and let ρ denote a random restriction with survival probability O(log(d)/d). Then, with probability at least 1-d^{-Ω(1)}, there exists a function g: {± 1}ⁿ → [0,1] depending on at most d^O(1) coordinates such that ||f_{ρ}-g||_2^2 ≤ d^{-1-Ω(1)}. Our result has the following consequence for the well known, outstanding conjecture of Aaronson and Ambainis. The Aaronson-Ambainis conjecture was formulated to show that the acceptance probability of a quantum query algorithm can be well approximated almost everywhere by a classical query algorithm with a polynomial blow-up: it speculates that a polynomal f: {± 1}ⁿ → [0,1] with degree d has a coordinate with influence ≥ poly(1/d, Var[f]). Our result shows that this is true for a non-negligible fraction of random restrictions of f assuming Var[f] is not too low. Our work combines the ideas of Dinur, Friedgut, Kindler and O'Donnell [Dinur et al., 2006] with an approximation theoretic result, first reported in the recent work of Filmus, Hatami, Keller and Lifshitz [Yuval Filmus and Hamed Hatami, 2014].