Limits of spiked random matrices II

The top eigenvalues of rank $r$ spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and P\'{e}ch\'{e} [Duke Math. J. (2006) 133 205-235]. The starting point is a new $(2r+1)$-diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schr\"{o}dinger operator on the half-line with $r\times r$ matrix-valued potential. The perturbation determines the boundary condition and the low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. We treat the real, complex and quaternion ($\beta=1,2,4$) cases simultaneously. We further characterize the limit laws in terms of a diffusion related to Dyson's Brownian motion, or alternatively a linear parabolic PDE; here $\beta$ appears simply as a parameter. At $\beta=2$, the PDE appears to reconcile with known Painlev\'{e} formulas for these $r$-parameter deformations of the GUE Tracy-Widom law.Comment: Published at http://dx.doi.org/10.1214/15-AOP1033 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices

We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v, (X^* X - z)^{-1} w \rangle - \langle v,w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w \in \mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $Im z \geq N^{-1+\epsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices

On the principal components of sample covariance matrices

We introduce a class of M×MM×M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Σ= Σ=EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ−IMΣ−IM may depend on MM in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of Q . We derive precise large deviation estimates on the generalized components ⟨w,ξξi⟩⟨w,ξξi⟩ of the outlier and non-outlier eigenvectors ξξiξξi . Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of ΣΣ . The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q , satisfying =IMEQ=IM , as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.Mathematic

Reply to : On powerful GWAS in admixed populations

Non peer reviewe

Isotropic local laws for sample covariance and generalized Wigner matrices

We consider sample covariance matrices of the form X ∗X, where X is an M × N matrix with independent random entries. We prove the isotropic local MarchenkoPastur law, i.e. we prove that the resolvent (X ∗X − z) −1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity hv,(X ∗X − z) −1wi − hv, wim(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v, w ∈ C N . We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Im z > N −1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.Mathematic

Spiking Tracy-Widom (β)

Non UBCUnreviewedAuthor affiliation: Harvard UniversityPostdoctora

On the principal components of sample covariance matrices

We introduce a class of $M \times M$ sample covariance matrices $\mathcal Q$ which subsumes and generalizes several previous models. The associated population covariance matrix $\Sigma = \mathbb E \cal Q$ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of $\Sigma - I_M$ may depend on $M$ in an arbitrary fashion. We investigate the principal components, i.e.\ the top eigenvalues and eigenvectors, of $\mathcal Q$. We derive precise large deviation estimates on the generalized components $\langle \mathbf w, \boldsymbol \xi_i \rangle$ of the outlier and non-outlier eigenvectors $\boldsymbol \xi_i$. Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the \emph{subcritical} spikes of $\Sigma$. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix $\mathcal Q$, satisfying $\mathbb E \mathcal Q = I_M$, as input: the isotropic local Marchenko-Pastur law established in [9], level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution

PINES: phenotype-informed tissue weighting improves prediction of pathogenic noncoding variants

Abstract Functional characterization of the noncoding genome is essential for biological understanding of gene regulation and disease. Here, we introduce the computational framework PINES (Phenotype-Informed Noncoding Element Scoring), which predicts the functional impact of noncoding variants by integrating epigenetic annotations in a phenotype-dependent manner. PINES enables analyses to be customized towards genomic annotations from cell types of the highest relevance given the phenotype of interest. We illustrate that PINES identifies functional noncoding variation more accurately than methods that do not use phenotype-weighted knowledge, while at the same time being flexible and easy to use via a dedicated web portal

Modulating entropic driving forces to promote high lithium mobility in solid organic electrolytes

As large-scale lithium-ion battery deployment accelerates, continued use of flammable organic electrolytes exacerbate issues associated with battery fires during operation and disposal. While ionic liquid-derived electrolytes promise safe, nonflammable alternatives to carbonate electrolytes, use of ionic liquids in batteries is hindered by poor lithium transport due to formation of long-lived lithium-anion complexes. We report the design and characterization of novel ionic liquid-inspired organic electrolytes that leverage unique self-assembly properties of molecular diamond templates, called “diamondoids.” Combining thermodynamic characterization, vibrational and magnetic spectroscopy, and single-crystal X-ray analysis, we determine that diamondoid-functionalized cations can facilitate formation of molecularly porous phases that resist restructuring upon dissolution of lithium salts. These electrolytes can suppress lithium-anion coordination, manifesting in substantially enhanced lithium-ion mobility in the organic ion matrix. Our results provide a new paradigm for enhancing lithium mobility in solid electrolytes by tuning entropic self-assembly to enhance organic cation-anion interactions, suppress lithium-anion coordination, and increase lithium mobility in solid electrolytes