Fundamental Fact: The noise lies in a high-dimensional space; the signal, by contrast, lies in a much lower-dimensional space.
If a random matrix A has i.i.d rows Ai, then A*A = ∑iAiAiT where A* is the adjoint matrix of A. We often study A through the n × n symmetric, positive semidefinite matrix, the matrix A*A. The eigenvalues of are therefore nonnegative real numbers.
An immediate application of random matrices is the fundamental problem of estimating covariance matrices of high-dimensional distributions . The analysis of the row-independent models can be interpreted as a study of sample covariance matrices. For a general distribution in , its covariance can be estimated from a sample size of N = O(nlogn) drawn from the distribution. For sub-Gaussian distributions, we have an even better bound N = O(n). For low-dimensional distributions, much fewer samples are needed: if a distribution lies close to a subspace of dimension r in , then a sample of size N = O(rlogn) is sufficient for covariance estimation.
There are deep results in random matrix theory. The main motivation of this subsection is to exploit the existing ...