Chapter 3

Classical Detection

**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 **A**_{i}, then **A*****A** = ∑_{i}**A**_{i}**A**_{i}^{T} 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 [107]. 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*(*n*log*n*) 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*(*r*log*n*) is sufficient for covariance estimation.

There are deep results in random matrix theory. The main motivation of this subsection is to exploit the existing ...

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