Raz's recent result \cite{Raz2010} has rekindled people's interest in the study of \emph{tensor rank}, the generalization of matrix rank to high dimensions, by showing its connections to arithmetic formulas. In this paper, we follow Raz's work and show that \emph{monotone rank}, the monotone variant of tensor rank and matrix rank, has applications in algebraic complexity, quantum computing and communication complexity. This paper differs from Raz's paper in that it leverages existing results to show unconditional bounds while Raz's result relies on some assumptions.
We show a super-exponential separation between monotone and non-monotone computation in the non-commutative model, and thus provide a strong solution to Nisan's question \cite{Nis1991} in algebraic complexity. More specifically, we exhibit that there exists a homogeneous algebraic function $f$ of degree $d$ ($d$ even) on $n$ variables with the monotone algebraic branching program (ABP) complexity $Ω(d^2\log n)$ and the non-monotone ABP complexity $O(d^2)$.
In Bell's theorem\cite{Bel1964, CHSH1969}, a basic assumption is that players have free will, and under such an assumption, local hidden variable theory still cannot predict the correlations produced by quantum mechanics. Using tools from monotone rank, we show that even if we disallow the players to have free will, local hidden variable theory still cannot predict the correlations produced by quantum mechanics.
We generalize the log-rank conjecture \cite{LS1988} in communication complexity to the multiparty case, and prove that for super-polynomial parties, there is a super-polynomial separation between the deterministic communication complexity and the logarithm of the rank of the communication tensor. This means that the log-rank conjecture does not hold in high dimensions.
Approximate resilience, monotonicity, and the complexity of agnostic learning
University of Maryland（马里兰大学）；IBM Research - Almaden（IBM阿尔马登研究中心）；Columbia University（哥伦比亚大学）；Simons Institute, UC Berkeley（加州大学伯克利分校西蒙斯研究所）；Duquesne University（迪尤肯大学）
A function $f$ is $d$ -resilient if all its Fourier coefficients of degree at most $d$ are zero, i.e., $f$ is uncorrelated with all low-degree parities. We study the notion of $\mathit{approximate}$ $\mathit{resilience}$ of Boolean functions, where we say that $f$ is $α$ -approximately $d$ -resilient if $f$ is $α$ -close to a $[-1,1]$ -valued $d$ -resilient function in $\ell_1$ distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class $C$ over the uniform distribution. Roughly speaking, if all functions in a class $C$ are far…
