Approximately Hadamard matrices and Riesz bases in random frames
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An n × n matrix with ± 1 entries which acts on 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 ± 1 entries which act as approximate scaled isometries in 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 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 ≥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 ≤exp(cn), where c<C are constants depending on the distribution of the entries.
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