Boolean functions with small spectral norm

有界分解范数布尔矩阵中的块结构

Block structure in boolean matrices of bounded factorization norm

本文基于已有研究，证明了对于γ2范数至多为λ的布尔矩阵，存在行与列互不相交的1-单色子矩阵集合，至少覆盖1/2^{2^{O(λ)}}比例的1项。

Boolean functions with small spectral norm, revisited

We show that if $f$ is an integer-valued function with spectral norm at most $M$ then there are subspaces $V_1,\dots,V_L$ and signs $σ_1,\dots,σ_L \in \{-1,1\}$ such that $f=σ_1 1_{V_1} + \dots + σ_L 1_{V_L}$ where $L < \exp(M^{3+o(1)})$. This note extracts out the argument from arXiv:1610.07092 to the model setting of $\mathbb{F}_2^n$. The hope is that it will clarify the arguments of that paper.

On the Structure of Boolean Functions with Small Spectral Norm

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is $\|\hat{f}\|_1=\sum_α|\hat{f}(α)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n \to \{0,1\}$ with $\|\hat{f}\|_1=A$. 1. There is a subspace $V$ of co-dimension at most $A^2$ such that $f|_V$ is constant. 2. f can be computed by a parity decision tree of size $2^{A^2}n^{2A}$. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. If in addition f has at most s nonzero Fourier coefficients, then f can be compu…