[Paper Review] Fourier sparsity, spectral norm, and the Log-rank conjecture

Hing Yin Tsang, Chung Hoi Wong+2 more|arXiv (Cornell University)|Apr 4, 2013

Complexity and Algorithms in Graphs42 references6 citations

TL;DR

This paper establishes new upper bounds on the deterministic communication complexity of XOR functions $f(x \oplus y)$ by leveraging the Fourier sparsity and spectral norm of Boolean functions. It proves that for functions with constant $\mathbb{F}_2$-degree, communication complexity is bounded by $O(2^{d^2/2} \log^{d-2} \|\hat{f}\|_1)$, and for general functions, by $O(d \|\hat{f}\|_1)$, improving the trivial linear bound by nearly a quadratic factor. The key contribution is a degree-reduction protocol based on polynomial rank, which supports the conjecture that communication cost is polylogarithmic in the rank.

ABSTRACT

We study Boolean functions with sparse Fourier coefficients or small spectral norm, and show their applications to the Log-rank Conjecture for XOR functions f(x\oplus y) --- a fairly large class of functions including well studied ones such as Equality and Hamming Distance. The rank of the communication matrix M_f for such functions is exactly the Fourier sparsity of f. Let d be the F2-degree of f and D^CC(f) stand for the deterministic communication complexity for f(x\oplus y). We show that 1. D^CC(f) = O(2^{d^2/2} log^{d-2} ||\hat f||_1). In particular, the Log-rank conjecture holds for XOR functions with constant F2-degree. 2. D^CC(f) = O(d ||\hat f||_1) = O(\sqrt{rank(M_f)}\logrank(M_f)). We obtain our results through a degree-reduction protocol based on a variant of polynomial rank, and actually conjecture that its communication cost is already \log^{O(1)}rank(M_f). The above bounds also hold for the parity decision tree complexity of f, a measure that is no less than the communication complexity (up to a factor of 2). Along the way we also show several structural results about Boolean functions with small F2-degree or small spectral norm, which could be of independent interest. For functions f with constant F2-degree: 1) f can be written as the summation of quasi-polynomially many indicator functions of subspaces with \pm-signs, improving the previous doubly exponential upper bound by Green and Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having a small parity decision tree complexity; 3) f depends only on polylog||\hat f||_1 linear functions of input variables. For functions f with small spectral norm: 1) there is an affine subspace with co-dimension O(||\hat f||_1) on which f is a constant; 2) there is a parity decision tree with depth O(||\hat f||_1 log ||\hat f||_0).

Motivation & Objective

• To investigate the communication complexity of XOR functions $f(x \oplus y)$, especially in relation to the Log-rank Conjecture.
• To understand the structural properties of Boolean functions with sparse Fourier spectra or small spectral norm.
• To establish tighter upper bounds on deterministic communication complexity $\sf{D^{CC}}(f \circ \oplus)$ using Fourier-analytic tools.
• To improve upon the trivial linear bound in terms of rank by nearly a quadratic factor.

Proposed method

• Introduce a degree-reduction protocol based on a variant of polynomial rank to analyze the number of parity queries needed to reduce the $\mathbb{F}_2$-degree of $f$.
• Use spectral norm $\|\hat{f}\|_1$ and Fourier sparsity $\|\hat{f}\|_0$ as key parameters to bound communication complexity.
• Apply Lemma 34 from Gopalan et al. to show that functions close to sparse functions in $\ell_2$-norm are also close to Boolean functions of low Fourier sparsity.
• Leverage Lemma 33 to show that functions with small Fourier sparsity can be computed via $\log^{O(1)}\|\hat{f}\|_0$ parity queries.
• Use the fact that the rank of the communication matrix $M_f$ equals the Fourier sparsity of $f$ for XOR functions.
• Conjecture that the communication cost of the protocol is $\log^{O(1)}\mathrm{rank}(M_f)$, supporting the Log-rank Conjecture.

Experimental results

Research questions

• RQ1 Can the communication complexity of XOR functions be bounded in terms of Fourier sparsity and spectral norm?
• RQ2 Does the degree-reduction protocol based on polynomial rank yield a near-optimal communication cost?
• RQ3 Is the communication complexity of XOR functions with constant $\mathbb{F}_2$-degree bounded by $O(2^{d^2/2} \log^{d-2} \|\hat{f}\|_1)$?
• RQ4 Can the trivial linear bound in terms of rank be improved by nearly a quadratic factor using spectral norm?
• RQ5 Is the communication cost of the proposed protocol polylogarithmic in the rank of the communication matrix?

Key findings

• For XOR functions with constant $\mathbb{F}_2$-degree $d$, $\sf{D^{CC}}(f \circ \oplus) = O(2^{d^2/2} \log^{d-2} \|\hat{f}\|_1)$, confirming the Log-rank Conjecture for this class.
• For general functions, $\sf{D^{CC}}(f \circ \oplus) = O(d \|\hat{f}\|_1) = O(\sqrt{\mathrm{rank}(M_f)} \log \mathrm{rank}(M_f))$, improving the trivial linear bound by nearly a quadratic factor.
• Functions with constant $\mathbb{F}_2$-degree can be expressed as the sum of quasi-polynomially many signed indicator functions of subspaces, improving Green and Sanders' doubly exponential bound.
• Fourier sparsity and parity decision tree complexity are polynomially equivalent for functions with constant $\mathbb{F}_2$-degree.
• There exists an affine subspace of co-dimension $O(\|\hat{f}\|_1)$ on which $f$ is constant, for functions with small spectral norm.
• A parity decision tree of depth $O(\|\hat{f}\|_1 \log \|\hat{f}\|_0)$ exists for functions with small spectral norm.