Luís Daniel Abreu
In this brief note, we obtain a recurrence relation for the average singular value α ( d ) of a complex valued d × d matrix 1 d X with Gaussian entries, of the form: α C d + 1 = α C d + R ( d ) , where R(d) can be written as the linear combination of two integrals involving Laguerre polynomials. This reduces to showing that R ( d ) < 0 (a non-trivial analysis problem) a 2016 conjecture of Bandeira, Kennedy and Singer, where it is predicted that α ( d ) decreases monotonically with d, from α C 1 = π 4 , to the d → ∞ limit 8 3 π provided by the Marchenko-Pastur distribution. While this may point to a possible path to solving the conjecture, it is still far from a solution, since the two integrals R(d) are non-summable hypergeometric functions, and showing that R ( d ) < 0 is a non-trivial analysis problem.