  Detection of the number of signals corrupted by high-dimensional noise is a
fundamental problem in signal processing and statistics. This paper focuses on
a general setting where the high-dimensional noise has an unknown complicated
heterogeneous variance structure. We propose a sequential test which utilizes
the edge singular values (i.e., the largest few singular values) of the data
matrix. It also naturally leads to a consistent sequential testing estimate of
the number of signals. We describe the asymptotic distribution of the test
statistic in terms of the Tracy-Widom distribution. The test is shown to be
accurate and have full power against the alternative, both theoretically and
numerically. The theoretical analysis relies on establishing the Tracy-Widom
law for a large class of Gram type random matrices with non-zero means and
completely arbitrary variance profiles, which can be of independent interest.
