Singular Vectors of Sums of Rectangular Random Matrices and Optimal Estimators of High-Rank Signals: The Extensive Spike Model

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, is of fundamental importance. A theoretical keystone to solving these problems is understanding how the singular value structure of a signal is deformed in the presence of 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 $O(1000)$) 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 real data. To address these deficiencies, we analytically compute how the singular values and vectors of an arbitrary signal matrix are deformed by additive noise. We apply these analytical results to study a different 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 the hidden high-rank signal from the 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.Comment: 25 pages, 10 figure

Weak turbulence theory of the non-linear evolution of the ion ring distribution

The nonlinear evolution of an ion ring instability in a low-beta magnetospheric plasma is considered. The evolution of the two-dimensional ring distribution is essentially quasilinear. Ignoring nonlinear processes the time-scale for the quasilinear evolution is the same as for the linear instability 1/t_ql gamma_l. However, when nonlinear processes become important, a new time scale becomes relevant to the wave saturation mechanism. Induced nonlinear scattering of the lower-hybrid waves by plasma electrons is the dominant nonlinearity relevant for plasmas in the inner magnetosphere and typically occurs on the timescale 1/t_ql w(M/m)W/nT, where W is the wave energy density, nT is the thermal energy density of the background plasma, and M/m is the ion to electron mass ratio, which has the consequence that the wave amplitude saturates at a low level, and the timescale for quasilinear relaxation is extended by orders of magnitude

Anomalous Raman scattering from phonons and electrons of superconducting FeSe0.82_{0.82}

We report interesting anomalies in the temperature dependent Raman spectra of FeSe$_{0.82}$ measured from 3K to 300K in the spectral range from 60 to 1800 cm$^{-1}$ and determine their origin using complementary first-principles density functional calculations. A phonon mode near 100 cm$^{-1}$ exhibits a sharp increase by $\sim$ 5% in frequency below a temperature T$_s$ ($\sim$ 100 K) attributed to strong spin-phonon coupling and onset of short-range antiferromagnetic order. In addition, two high frequency modes are observed at 1350 cm$^{-1}$ and 1600 cm$^{-1}$, attributed to electronic Raman scattering from ($x^2-y^2$)to $xz$ / $yz$ $d$-orbitals of Fe.Comment: 19 pages, 4 figures, 1 tabl

Justice and the racial dimensions of health inequalities:A view from COVID-19

In this paper, we take up the call to further examine structural injustice in health, and racial inequalities in particular. We examine the many facets of racism: structural, interpersonal and institutional as they appeared in the COVID‐19 pandemic in the UK, and emphasize the relevance of their systemic character. We suggest that such inequalities were entirely foreseeable, for their causal mechanisms are deeply ingrained in our social structures. It is by recognizing the conventional, un‐extraordinary nature of racism within social systems that we can begin to address socially mediated health inequalities