Bounded Independence Fools Degree-2 Threshold Functions

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.

💡 Research Summary

The paper addresses the fundamental question of how much randomness is needed to approximate the behavior of degree‑2 polynomial threshold functions (PTFs) under a distribution that is only k‑wise independent. A degree‑2 PTF is a Boolean function of the form f(x)=sgn(p(x)), where p(x)=xᵀAx+bᵀx+c is a quadratic polynomial. The authors prove that for any ε>0, it suffices to take k=poly(1/ε) to guarantee that the expectation of f under a k‑wise independent distribution over {−1,1}ⁿ (or over standard Gaussian variables) differs from its expectation under the fully independent uniform (or Gaussian) distribution by at most ε. This resolves an open problem posed by Diakonikolas et al. (FOCS 2009), who had established the result for linear threshold functions (halfspaces) but left the quadratic case unresolved.

The technical contribution rests on two novel tools. First, the authors develop a generalized hypercontractive inequality for quadratic forms that incorporates the operator norm ‖A‖₂ of the associated matrix. Unlike classical (2,4)‑hypercontractivity, which bounds higher moments solely in terms of the L₂ norm, this inequality tightly controls the fourth moment of p(x) by a combination of its variance and the spectral magnitude of A. This refined control is crucial for handling the additional correlations introduced by k‑wise independence.

Second, they introduce “multivariate FT‑mollification,” a multidimensional extension of the Fourier‑transform mollification technique previously used in streaming algorithm analysis (Kane et al., SODA 2010). The idea is to replace the discontinuous sign function with a smooth approximation fₜ(p(x)) obtained by convolving sgn with a Gaussian kernel of bandwidth t. The Fourier transform of fₜ decays rapidly, allowing the authors to bound the error incurred by truncating high‑frequency components. By carefully choosing t as a function of ε and k, they split the total error into (i) the mollification error, which can be made arbitrarily small, and (ii) the error due to limited independence, which is bounded using the new hypercontractive inequality. The analysis shows that when k=poly(1/ε), both error terms are ≤ε/2, yielding the main approximation guarantee.

From this theoretical result, the paper derives explicit pseudorandom generators (PRGs) for degree‑2 PTFs. Standard constructions of k‑wise independent distributions—e.g., using small‑seed linear hash families—produce generators with seed length O(log n·poly(1/ε)). Consequently, any algorithm that queries a degree‑2 PTF can be simulated with only logarithmic randomness while preserving its output distribution up to ε. The authors also extend the result to the intersection of a constant number of degree‑2 PTFs, showing that the same PRG fools such conjunctions.

An important application discussed is the derandomization of the Goemans‑Williamson hyperplane rounding scheme for Max‑Cut. The rounding step samples a random Gaussian vector and cuts the graph according to the sign of the inner product with each vertex vector. By replacing the fully independent Gaussian vector with a k‑wise independent one (k=poly(1/ε)), the expected cut value changes by at most ε, providing a near‑optimal deterministic approximation with only O(log n·poly(1/ε)) random bits.

Overall, the paper contributes a robust framework for analyzing low‑degree polynomial threshold functions under limited independence, introduces powerful analytical tools (operator‑norm hypercontractivity and multivariate FT‑mollification), and demonstrates concrete algorithmic implications in pseudorandomness and approximation algorithms. The techniques are likely to be useful beyond degree‑2 PTFs, potentially extending to higher‑degree polynomials and other non‑linear Boolean functions.