arXiv subjects

Angelika Rohde

Publications and source records attributed to Angelika Rohde.

26 records · Page 2Linked to original sources

arXiv2016

Spectral analysis of high-dimensional sample covariance matrices with missing observations

We study high-dimensional sample covariance matrices based on independent random vectors with missing coordinates. The presence of missing observations is common in modern applications such as climate studies or gene expression micro-arrays. A weak approximation on the spectral distribution in the "large dimension $d$ and large sample size $n$" asymptotics is derived for possibly different observation probabilities in the coordinates. The spectral distribution turns out to be strongly influenced by the missingness mechanism. In the null case under the missing at random scenario where each component is observed with the same probability $p$, the limiting spectral distribution is a Marčenko-Pastur law shifted by $(1-p)/p$ to the left. As $d/n\rightarrow y< 1$, the almost sure convergence of the extremal eigenvalues to the respective boundary points of the support of the limiting spectral distribution is proved, which are explicitly given in terms of $y$ and $p$. Eventually, the sample covariance matrix is positive definite if $p$ is larger than $$ 1-\left(1-\sqrt{y}\right)^2, $$ whereas this is not true any longer if $p$ is smaller than this quantity.

math.PR↗

arXiv2016

Adaptation to lowest density regions with application to support recovery

A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Hölder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Some new type of risk bound with respect to a density-dependent standardized loss of this estimator is established. This bound is fully nonasymptotic and allows to deduce convergence rates at lowest density regions that can be substantially faster than $n^{-1/2}$. It is complemented by a weighted minimax lower bound which splits into two regimes depending on the value of the density. The new estimator adapts into the second regime, and it is shown that simultaneous adaptation into the fastest regime is not possible in principle as long as the Hölder exponent is unknown. Consequences on plug-in rules for support recovery are worked out in detail. In contrast to those with classical density estimators, the plug-in rules based on the new construction are minimax-optimal, up to some logarithmic factor.

math.ST↗

arXiv2012

Accuracy of empirical projections of high-dimensional Gaussian matrices

Let $X=C+\mathrm{E}$ with a deterministic matrix $C\in\R^{M\times M}$ and $\mathrm{E}$ some centered Gaussian $M\times M$-matrix whose entries are independent with variance $σ^2$. In the present work, the accuracy of reduced-rank projections of $X$ is studied. Non-asymptotic universal upper and lower bounds are derived, and favorable and unfavorable prototypes of matrices $C$ in terms of the accuracy of approximation are characterized. The approach does not involve analytic perturbation theory of linear operators and allows for multiplicities in the singular value spectrum. Our main result is some general non-asymptotic upper bound on the accuracy of approximation which involves explicitly the singular values of $C$, and which is shown to be sharp in various regimes of $C$. The results are accompanied by lower bounds under diverse assumptions. Consequences on statistical estimation problems, in particular in the recent area of low-rank matrix recovery, are discussed.

math.PR↗

arXiv2011

Uniform Central Limit Theorems for Multidimensional Diffusions

It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time. Besides establishing strong parallels to classical theory such as Ossiander's bracketing CLT and the general Giné-Zinn CLT for uniformly bounded families of functions, we find increased regularity also for multivariate ergodic diffusions, assuming that the invariant measure is finite with Lebesgue density $π$. The effect is diminishing for growing dimension but always present. The fine differences to the classical iid setting are worked out using exponential inequalities for martingales and additive functionals of continuous Markov processes as well as the characterization of the sample path behavior of Gaussian processes by means of the generic chaining bound. To uncover the phenomenon, we study a smoothed version of the empirical diffusion process. It turns out that uniform weak convergence of the smoothed empirical diffusion process under necessary and sufficient conditions can take place with even exponentially small bandwidth in dimension $d=2$, and with strongly undersmoothing bandwidth choice for parameters $β> d/2$ in case $d\geq 3$, assuming that the coordinates of drift and diffusion coefficient belong to some Hölder ball with parameter $β$.

math.PR↗

arXiv2011

Estimation of high-dimensional low-rank matrices

Suppose that we observe entries or, more generally, linear combinations of entries of an unknown $m\times T$-matrix $A$ corrupted by noise. We are particularly interested in the high-dimensional setting where the number $mT$ of unknown entries can be much larger than the sample size $N$. Motivated by several applications, we consider estimation of matrix $A$ under the assumption that it has small rank. This can be viewed as dimension reduction or sparsity assumption. In order to shrink toward a low-rank representation, we investigate penalized least squares estimators with a Schatten-$p$ quasi-norm penalty term, $p\leq1$. We study these estimators under two possible assumptions---a modified version of the restricted isometry condition and a uniform bound on the ratio "empirical norm induced by the sampling operator/Frobenius norm." The main results are stated as nonasymptotic upper bounds on the prediction risk and on the Schatten-$q$ risk of the estimators, where $q\in[p,2]$. The rates that we obtain for the prediction risk are of the form $rm/N$ (for $m=T$), up to logarithmic factors, where $r$ is the rank of $A$. The particular examples of multi-task learning and matrix completion are worked out in detail. The proofs are based on tools from the theory of empirical processes. As a by-product, we derive bounds for the $k$th entropy numbers of the quasi-convex Schatten class embeddings $S_p^M\hookrightarrow S_2^M$, $p<1$, which are of independent interest.

math.ST↗

arXiv2009

Adaptive Confidence Sets for the Optimal Approximating Model

In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence sets for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral chi-squared distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.

math.ST↗

arXiv2009

Optimal Calibration for Multiple Testing against Local Inhomogeneity in Higher Dimension

Based on two independent samples X_1,...,X_m and X_{m+1},...,X_n drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearest-neighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The properly adjusted multiple testing procedure is shown to be sharp-optimal for typical arrangements of the observation values which appear with probability close to one. The proof relies on a new coupling Bernstein type exponential inequality, reflecting the non-subgaussian tail behavior of a combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis "p=q" to a simple one with the multivariate mixed density (m/n)p+(1-m/n)q as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic Hölder classes. The exact minimax risk asymptotics are obtained in terms of solutions of the optimal recovery.

math.ST↗

arXiv2008

Adaptive goodness-of-fit tests based on signed ranks

Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l=0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against Hölder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to 1 in case of homoscedastic Gaussian errors within a broad range of Hölder classes simultaneously.

math.ST↗