论文标题
大约在随机帧中的Hadamard矩阵和Riesz碱基
Approximately Hadamard matrices and Riesz bases in random frames
论文作者
论文摘要
带有$ \ pm 1 $条目的$ n \ times n $矩阵，该条目在$ \ mathbb {r}^n $上作用为缩放等轴测，称为Hadamard。这样的矩阵存在于某些（但不是全部）。结合数字理论和概率工具，我们将矩阵与$ \ pm 1 $条目构建，这些条目是所有$ n $的$ \ mathbb {r}^n $中近似缩放的等法。更确切地说，我们构建的矩阵具有条件号，由$ n $的常数独立界定。 使用这种结构，我们为随机帧包含Riesz的概率建立了一个相变。也就是说，我们表明，由$ n $ vectors形成的$ \ mathbb {r}^n $中的随机框架具有独立分布的坐标，其非分数对称分布包含许多riesz基础，只要$ n \ ge \ exp（cn）$。另一方面，我们证明，如果条目是subgaussian，则随机帧将不包含riesz基础，只要$ n \ le \ exp（cn）$，概率接近$ 1 $，其中$ c <c $是常数，根据条目的分布。
An $n \times n$ matrix with $\pm 1$ entries which acts on $\mathbb{R}^n$ as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools we construct matrices with $\pm 1$ entries which act as approximate scaled isometries in $\mathbb{R}^n$ for all $n$. More precisely, the matrices we construct have condition numbers bounded by a constant independent of $n$. Using this construction, we establish a phase transition for the probability that a random frame contains a Riesz basis. Namely, we show that a random frame in $\mathbb{R}^n$ formed by $N$ vectors with independent identically distributed coordinates having a non-degenerate symmetric distribution contains many Riesz bases with high probability provided that $N \ge \exp(Cn)$. On the other hand, we prove that if the entries are subgaussian, then a random frame fails to contain a Riesz basis with probability close to $1$ whenever $N \le \exp(cn)$, where $c<C$ are constants depending on the distribution of the entries.
