Approximately Hadamard matrices and Riesz bases in random frames

TL;DR

This paper constructs approximate Hadamard matrices for all dimensions and analyzes the probability of random frames containing Riesz bases, revealing a phase transition based on the number of vectors and their distribution.

Contribution

It introduces a method to construct near-isometric matrices with ±1 entries for all dimensions and establishes a phase transition in the likelihood of random frames containing Riesz bases.

Findings

Constructed approximate Hadamard matrices with bounded condition numbers for all dimensions.

Identified a phase transition at exponential thresholds for the probability of random frames containing Riesz bases.

Showed that subgaussian entries in random frames lead to a high probability of not containing Riesz bases below a certain size.

Abstract

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.