Tropp’s matrix concentration inequalities are widely used in applied mathematics and other areas. All these inequalities involve a log-dimension factor. As pointed out in Tropp’s monograph in 2015, “It remains a major open question to find a simple quantity, from the coefficients B_k, that decides whether E||Z||^2 contains a dimensional factor or not.” I will present a solution to this problem.

Joint work with Afonso Bandeira and Ramon van Handel. Paper: https://arxiv.org/pdf/2108.06312.pdf