[Paper Review] Pseudorandomness via the discrete Fourier transform

Parikshit Gopalan, Daniel M. Kane+1 more|arXiv (Cornell University)|Jun 14, 2015

Complexity and Algorithms in Graphs18 references4 citations

TL;DR

This paper presents a novel pseudorandom generator (PRG) that fools Fourier shapes—functions defined as products of bounded complex-valued functions—using near-logarithmic seed length, specifically $ O( ext{polyloglog}(n/\varepsilon) imes \log(n/\varepsilon)) $. The construction unifies and improves upon prior work by leveraging discrete Fourier analysis and a new analytic framework, achieving the first unconditional PRG with nearly optimal seed length for classes including halfspaces, modular tests, and combinatorial shapes over general domains.

ABSTRACT

We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the input size and the desired error parameter. Our result gives a single pseudorandom generator that fools several important classes of tests computable in logspace that have been considered in the literature, including halfspaces (over general domains), modular tests and combinatorial shapes. For all these classes, our generator is the first that achieves near logarithmic seed-length in both the input length and the error parameter. Getting such a seed-length is a natural challenge in its own right, which needs to be overcome in order to derandomize RL - a central question in complexity theory. Our construction combines ideas from a large body of prior work, ranging from a classical construction of [NN93] to the recent gradually increasing independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some novel analytic machinery which might find other applications.

Motivation & Objective

• To construct an unconditional pseudorandom generator with near-optimal seed length for classes of functions involving linear combinations of inputs.
• To unify and generalize prior work on fooling halfspaces, modular tests, and combinatorial shapes under a single framework based on the discrete Fourier transform.
• To overcome the longstanding barrier of $ O( ext{log}^2 n) $ seed length in logspace derandomization, particularly toward resolving $ \mathsf{RL} = \mathsf{L} $.
• To develop a new analytic machinery that may extend beyond pseudorandomness to other problems in complexity and probability.

Proposed method

• The authors define Fourier shapes as functions $ f(x_1,\dots,x_n) = \prod_{j=1}^n f_j(x_j) $ with $ f_j: [m] \to \mathbb{C}_1 $, generalizing classes like halfspaces and modular tests.
• They reduce the problem of fooling such functions to controlling the $ L^1 $-distance between the Fourier transforms of the uniform and pseudorandom distributions.
• The core technique combines the gradually increasing independence paradigm with a new analytic framework based on bounded moments and Fourier concentration.
• They use the discrete Fourier transform to express linear functions $ w(x) = \sum w_j x_j $ as $ \phi_\alpha(w(x)) = \exp(2\pi i \alpha w(x)) $, which are themselves Fourier shapes.
• The generator is constructed via a hybrid of classical constructions (e.g., [NN93]) and recent independence-based PRG frameworks (e.g., [KMN11, CRSW13, GMR+12]).
• A key innovation is the use of a bounded-moment condition on the Fourier transform to control statistical distance, enabling near-logarithmic seed length.

Experimental results

Research questions

• RQ1 Can a single pseudorandom generator be constructed that fools multiple natural classes of logspace-computable functions, including halfspaces and modular tests, with near-logarithmic seed length?
• RQ2 Is it possible to unify the analysis of diverse function classes—such as combinatorial shapes and modular tests—under a common framework based on the discrete Fourier transform?
• RQ3 Can the discrete Fourier transform be used as a tool to construct PRGs with improved seed length, approaching the conjectured optimal $ O(\log(n/\varepsilon)) $?
• RQ4 What new analytic tools are required to control the Fourier transform of product distributions with bounded moments, enabling near-optimal pseudorandomness?
• RQ5 Can the framework be extended to derandomize $ \mathsf{RL} $, thereby resolving the central open problem $ \mathsf{RL} = \mathsf{L} $?

Key findings

• The paper constructs an explicit pseudorandom generator that fools all $ (m,n) $-Fourier shapes with seed length $ O(\text{polyloglog}(n/\varepsilon) \cdot \log(n/\varepsilon)) $, achieving near-logarithmic dependence on both $ n $ and $ \varepsilon $.
• This generator is the first to achieve such seed length for multiple classes, including halfspaces, modular tests, and combinatorial shapes, over general domains.
• The result implies a PRG for halfspaces under the Gaussian distribution with seed length $ O(\log(n/\varepsilon) \cdot (\log\log(n/\varepsilon))^2) $, nearly matching the best known result up to a $ \log\log $ factor.
• The framework enables a derandomization of the Chernoff bound with seed length $ O(\log(mn/\varepsilon) \cdot (\log\log(mn/\varepsilon))^2) $, improving on prior bounds.
• The construction leverages a new analytic technique based on bounded moments and Fourier concentration, which may find broader applications in pseudorandomness and probability.
• The work provides a unifying framework that captures and improves upon prior results in fooling halfspaces, modular tests, and combinatorial shapes through a single, general-purpose generator.