The Log-Approximate-Rank Conjecture Is False

Abstract

We construct a simple and total Boolean function F = f ○ XOR on 2n variables that has only O(√n) spectral norm, O(n²) approximate rank, and O(n^2.5) approximate nonnegative rank. We show it has polynomially large randomized bounded-error communication complexity of Ω(√n). This yields the first exponential gap between the logarithm of the approximate rank and randomized communication complexity for total functions. Thus, F witnesses a refutation of the log-approximate-rank conjecture that was posed by Lee and Shraibman as a very natural analogue for randomized communication of the still unresolved log-rank conjecture for deterministic communication. The best known previous gap for any total function between the two measures is a recent 4th-power separation by Göös et al. Additionally, our function F refutes Grolmusz’s conjecture and a variant of the log-approximate-nonnegative-rank conjecture suggested recently by Kol et al., both of which are implied by the log-approximate-rank conjecture. The complement of F has exponentially large approximate nonnegative rank. This answers a question of Lee [32], showing that approximate nonnegative rank can be exponentially larger than approximate rank. The inner function F also falsifies a conjecture about parity measures of Boolean functions made by Tsang et al. The latter conjecture implied the log-rank conjecture for XOR functions. We are pleased to note that shortly after we published our results, two independent groups of researchers, Anshu et al. and Sinha and de Wolf, used our function F to prove that the quantum-log-rank conjecture is also false by showing that F has Ω(n^1/6) quantum communication complexity.