Approximately Hadamard matrices and Riesz bases in random frames

An $n \times n$ matrix with $\pm 1$ entries that 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 that act as approximate scaled isometries in ${\mathbb {R}}^{n}$ for all $n \in {\mathbb {N}}$. 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 coordinate having a nondegenerate 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 sub-Gaussian, 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.