The Gale-Berlekamp game for Hadamard matrices

The Gale-Berlekamp game for complex Hadamard matrices

Associated to a complex Hadamard matrix $H\in M_N(\mathbb C)$ is the complex probability measure $μ\in\mathcal P(\mathbb C)$ describing the distribution of $φ(a,b)=<a,Hb>$, where $a,b\in\mathbb T^N$ are random. This measure is called "glow" of the matrix, due to the analogy with the Gale-Berlekamp switching game, where $H,a,b$ are real. We prove here that: (1) $μ$ becomes complex Gaussian in the $N\to\infty$ limit, (2) the universality holds as well at order 2, (3) the order 3 term seems to be quite interesting, particularly for the master Hadamard matrices, (4) in the Fourier matrix case, som…

On a continuous Gale--Berlekamp switching game

We propose a continuous version of the classical Gale--Berlekamp switching game. We also study a weighted version of this new continuous game. The main results of this paper concern growth estimates for the corresponding optimization problems. The methods developed in this article are deterministic in nature and in some special cases the estimates obtained are optimal.

A Gale-Berlekamp permutation-switching problem in higher dimensions

Let an $n\times n$ array $\left( a_{ij}\right) $ of lights be given, each either on (when $a_{ij}=1$) or off (when $a_{ij}=-1$). For each row and each column there is a switch so that if the switch is pulled ($x_{i}=-1$ for row $i$ and $y_{j}=-1$ for column $j$) all of the lights in that line are switched: on to off or off to on. The unbalancing lights problem (Gale-Berlekamp switching game) consists in maximizing the difference between the lights on and off. We obtain the exact parameters for a generalization of the unbalancing lights problem in higher dimensions.

Gale-Berlekamp 切换博弈的纯几何变体

A Purely Geometric Variant of the Gale-Berlekamp Switching Game

本文研究 Gale-Berlekamp 切换博弈的平面点集几何变体，证明可通过选择经过至少两点的直线翻转权重符号，使总权重至少达到 n-o(n) 或 n/3，并给出多项式时间算法。

Variants of the Gale-Berlekamp Switching Game and their Solutions: Balancing the Rectangle and the Cube

The Gale-Berlekamp Light Switching Game is played on a square board of lights. Each light has two states, either on or off. There is a switch to every row and column. Turning this switch would change the state of all the lights on that row or column. The aim of the game is to minimise the imbalance in the board, defined to be the absolute difference between the number of lights switched on and that of lights switched off. We investigate variants of the game for an $m \times n$ matrix with n even and $m \le n$. We provide a constructive proof that for any $m \times n$ rectangle matrix $A$, ther…

Upper bounds for the constants of Bennett's inequality and the Gale--Berlekamp switching game

In $1977$, G. Bennett proved, by means of non-deterministic methods, an inequality which plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for $p_{1},p_{2} \in\lbrack1,\infty]$ and all positive integers $n_{1},n_{2}$, there exists a bilinear form $A_{n_{1},n_{2}}\colon\left( \mathbb{R}^{n_{1}},\left\Vert \cdot\right\Vert _{p_{1}}\right) \times\left( \mathbb{R}^{n_{2}},\left\Vert \cdot\right\Vert _{p_{2}}\right) \longrightarrow\mathbb{R}$ with coefficients $\pm1$ satisfying \[ \left\Vert A_{n_{1},n_{2}}\right\Vert \leq C_{p_{1},p_{2}…