We construct a simple and total Boolean function F = f ○ XOR on 2 n variables that has only O (√ n ) spectral norm, O ( n 2 ) 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.

中文翻译：

---

对数近似秩猜想是错误的

我们构造了一个简单而完整的布尔函数F=F○ 异或 2n变量只有○(√n) 谱范数，○(n 2) 近似等级，和○(n 2.5) 近似非负秩。我们证明它具有多项式大的随机有界误差通信复杂度 Ω(√n）。这产生了近似秩的对数和总函数的随机通信复杂度之间的第一个指数差距。因此，F见证了对 Lee 和 Shraibman 提出的对数近似秩猜想的驳斥，该猜想作为随机通信的非常自然的类似物，用于确定性通信的仍未解决的对数秩猜想。对于这两种度量之间的任何总函数，最著名的先前差距是 Göös 等人最近的 4 次幂分离。此外，我们的功能F驳斥了 Grolmusz 的猜想和 Kol 等人最近提出的对数近似非负秩猜想的变体，这两者都被对数近似秩猜想所暗示。的补充F具有指数级大的近似非负秩。这回答了 Lee [32] 的一个问题，表明近似非负秩可以比近似秩大指数倍。内部函数F也证伪了 Tsang 等人关于布尔函数奇偶度量的猜想。后一个猜想暗示了 XOR 函数的对数秩猜想。我们很高兴地注意到，在我们发表结果后不久，两个独立的研究小组 Anshu 等人。Sinha 和 de Wolf 使用了我们的函数F证明量子对数秩猜想也是错误的F有Ω(n 1/6) 量子通信复杂度。