  Across many disciplines from neuroscience and genomics to machine learning,
atmospheric science and finance, the problems of denoising large data matrices
to recover signals obscured by noise, and of estimating the structure of these
signals, are of fundamental importance. A key to solving these problems lies in
understanding how the singular value structure of a signal is deformed by
noise. This question has been thoroughly studied in the well-known spiked
matrix model, in which data matrices originate from low-rank signals perturbed
by additive noise, in an asymptotic limit where the size of these matrices
tends to infinity but the signal rank remains finite. We first show,
strikingly, that the singular value structure of large finite matrices (of size
$\sim 1000$) with even moderate-rank signals, as low as $10$, is not accurately
predicted by the finite-rank theory, thereby limiting the application of this
theory to real data. To address these deficiencies, we analytically compute how
the singular values and vectors of an arbitrary high-rank signal matrix are
deformed by additive noise. We next study an asymptotic limit corresponding to
an $\textit{extensive}$ spike model, in which the rank of the hidden signal is
proportional to the size of the data matrix, while both tend to infinity. We
map out the phase diagram of the singular value structure of the extensive
spike model as a joint function of signal strength and rank. We further exploit
these analytics to derive optimal rotationally invariant denoisers to recover
hidden $\textit{high}$-rank signals from data, as well as optimal invariant
estimators of the signal covariance structure. Overall, our results provide
fundamental theory governing how high-dimensional signals are deformed by
additive noise, together with practical formulas for optimal denoising and
covariance estimation.
