Quasi-Fuchsian AdS representations are Anosov

In a recent paper, Q. M\'erigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in the boundary of anti-de Sitter space. In the present paper, we prove the reverse implication. It also includes: -- A construction of Dirichlet domains in the context of anti-de Sitter geometry, -- A proof that spatially compact globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit locally CAT(-1) Cauchy hypersurfaces

Fast, Sample-Efficient, Affine-Invariant Private Mean and Covariance Estimation for Subgaussian Distributions

We present a fast, differentially private algorithm for high-dimensional covariance-aware mean estimation with nearly optimal sample complexity. Only exponential-time estimators were previously known to achieve this guarantee. Given $n$ samples from a (sub-)Gaussian distribution with unknown mean $\mu$ and covariance $\Sigma$, our $(\varepsilon,\delta)$-differentially private estimator produces $\tilde{\mu}$ such that $\|\mu - \tilde{\mu}\|_{\Sigma} \leq \alpha$ as long as $n \gtrsim \tfrac d {\alpha^2} + \tfrac{d \sqrt{\log 1/\delta}}{\alpha \varepsilon}+\frac{d\log 1/\delta}{\varepsilon}$. The Mahalanobis error metric $\|\mu - \hat{\mu}\|_{\Sigma}$ measures the distance between $\hat \mu$ and $\mu$ relative to $\Sigma$; it characterizes the error of the sample mean. Our algorithm runs in time $\tilde{O}(nd^{\omega - 1} + nd/\varepsilon)$, where $\omega < 2.38$ is the matrix multiplication exponent. We adapt an exponential-time approach of Brown, Gaboardi, Smith, Ullman, and Zakynthinou (2021), giving efficient variants of stable mean and covariance estimation subroutines that also improve the sample complexity to the nearly optimal bound above. Our stable covariance estimator can be turned to private covariance estimation for unrestricted subgaussian distributions. With $n\gtrsim d^{3/2}$ samples, our estimate is accurate in spectral norm. This is the first such algorithm using $n= o(d^2)$ samples, answering an open question posed by Alabi et al. (2022). With $n\gtrsim d^2$ samples, our estimate is accurate in Frobenius norm. This leads to a fast, nearly optimal algorithm for private learning of unrestricted Gaussian distributions in TV distance. Duchi, Haque, and Kuditipudi (2023) obtained similar results independently and concurrently.Comment: 44 pages. New version fixes typos and includes additional exposition and discussion of related wor

HTself2: Combining p-values to Improve Classification of Differential Gene Expression in HTself

HTself is a web-based bioinformatics tool designed to deal with the classification of differential gene expression for low replication microarray studies. It is based on a statistical test that uses self-self experiments to derive intensity-dependent cutoffs. The method was previously described in V&#xea;ncio et al, (DNA Res. 12: 211- e 214, 2005). In this work we consider an extension of HTself by calculating p-values instead of using a fixed credibility level &#x3b1;. As before, the statistic used to compute single spots p-values is obtained from the gaussian Kernel Density Estimator method applied to self-self data. Different spots corresponding to the same biological gene (replicas) give rise to a set of independent p-values which can be combined by well known statistical methods. The combined p-value can be used to decide whether a gene can be considered differentially expressed or not. HTself2 is a new version of HTself that uses the idea of p-values combination. It was implemented as a user-friendly desktop application to help laboratories without a bioinformatics infrastructure

Some model theory of fibrations and algebraic reductions

Let p=tp(a/A) be a stationary type in an arbitrary finite rank stable theory, and P an A-invariant family of partial types. The following property is introduced and characterised: whenever c is definable over (A,a) and a is not algebraic over (A,c) then \tp(c/A) is almost internal to P. The characterisation involves among other things an apparently new notion of ``descent" for stationary types. Motivation comes partly from results in Section~2 of [Campana, Oguiso, and Peternell. Non-algebraic hyperk\"ahler manifolds. Journal of Differential Geometry, 85(3):397--424, 2010] where structural properties of generalised hyperk\"ahler manifolds are given. The model-theoretic results obtained here are applied back to the complex analytic setting to prove that the algebraic reduction of a nonalgebraic (generalised) hyperk\"ahler manifold does not descend. The results are also applied to the theory of differentially closed fields, where examples coming from differential algebraic groups are given.Comment: Substantially revised and augmented. A new section applying the results to differentially closed fields has been added; title, abstract, and introduction are new, and several new examples are added. 14 page