Wald Statistics in high-dimensional PCA

In this note we consider PCA for Gaussian observations $X_1,\dots, X_n$ with covariance $\Sigma=\sum_i \lambda_i P_i$ in the 'effective rank' setting with model complexity governed by $\mathbf{r}(\Sigma):=\text{tr}(\Sigma)/\| \Sigma \|$. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector $\hat P_r$. This can be used to construct non-asymptotic confidence ellipsoids and tests for spectral projectors $P_r$. Using higher order pertubation theory we are able to show that our Theorem remains valid even when $\mathbf{r}(\Sigma) \gg \sqrt{n}$.Comment: 11 page

Constructing confidence sets for the matrix completion problem

In the present note we consider the problem of constructing honest and adaptive confidence sets for the matrix completion problem. For the Bernoulli model with known variance of the noise we provide a realizable method for constructing confidence sets that adapt to the unknown rank of the true matrix

Spectral thresholding for the estimation of Markov chain transition operators

We consider nonparametric estimation of the transition operator $P$ of a Markov chain and its transition density $p$ where the singular values of $P$ are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency. We investigate the performance of a spectral hard thresholded Galerkin-type estimator for $P$ and ${p}$, discarding most of the estimated singular triples. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establishing matching minimax upper and lower bounds in $L^2$-loss. Particularly, the effect of the dimensionality $d$ of the state space on the nonparametric rate improves from $2d$ to $d$ compared to the case without singular value decay.Comment: 28 pages, 2 figure

Statistical inference in high-dimensional matrix models

Matrix models are ubiquitous in modern statistics. For instance, they are used in finance to assess interdependence of assets, in genomics to impute missing data and in movie recommender systems to model the relationship between users and movie ratings. Typically such models are either high-dimensional, meaning that the number of parameters may exceed the number of data points by many orders of magnitudes, or nonparametric in the sense that the quantity of interest is an infinite dimensional operator. This leads to new algorithms and also to new theoretical phenomena that may occur when estimating a parameter of interest or functionals of it or when constructing confidence sets. In this thesis, we will exemplarily consider three such matrix models and develop statistical theory for them: Matrix completion, Principal Component Analysis (PCA) with Gaussian data and transition operators of Markov chains. \\ \\ We start with matrix completion and investigate the existence of adaptive confidence sets in the 'Bernoulli' and 'trace-regression' models. In the 'Bernoulli' model we show that adaptive confidence sets do not exist when the variance of the errors is unknown, whereas we give an explicit construction in the ’trace-regression’ model. Finally, in the known variance case, we show that adaptive confidence sets do also exist in the 'Bernoulli' model based on a testing argument. \\ \\ Next, we consider PCA in a Gaussian observation model with complexity measured by the effective rank, the reciprocal of the percentage of variance explained by the first principal component. We investigate estimation of linear functionals of eigenvectors and prove Berry-Essen type bounds. Due to the high-dimensionality of the problem we discover a new phenomenon: The plug-in estimator based on the sample eigenvector can have non-negligible bias and hence may be not $\sqrt{n}$-consistent anymore. We show how to de-bias this estimator, achieving $\sqrt{n}$-convergence rates, and prove exact matching minimax lower bounds. \\ \\ Finally, we consider nonparametric estimation of the transition operator of a Markov chain and its transition density. We assume that the singular values of the transition operator decay exponentially. For example, this assumption is fulfilled by discrete, low frequency observations of periodised, reversible stochastic differential equations. Using penalization techniques from low rank matrix estimation we develop a new algorithm and show improved convergence rates.Financial support of ERC grant UQMSI/647812 and EPSRC grant EP/L016516/

Optimality of Spectral Clustering in the Gaussian Mixture Model

Spectral clustering is one of the most popular algorithms to group high dimensional data. It is easy to implement and computationally efficient. Despite its popularity and successful applications, its theoretical properties have not been fully understood. In this paper, we show that spectral clustering is minimax optimal in the Gaussian Mixture Model with isotropic covariance matrix, when the number of clusters is fixed and the signal-to-noise ratio is large enough. Spectral gap conditions are widely assumed in the literature to analyze spectral clustering. On the contrary, these conditions are not needed to establish optimality of spectral clustering in this paper

Enhancing of catalytic properties of vanadia via surface doping with phosphorus using atomic layer deposition

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Vac. Sci. Technol. A 34, 01A135 (2016) and may be found at https://doi.org/10.1116/1.4936390.Atomic layer deposition is mainly used to deposit thin films on flat substrates. Here, the authors deposit a submonolayer of phosphorus on V2O5 in the form of catalyst powder. The goal is to prepare a model catalyst related to the vanadyl pyrophosphate catalyst (VO)2P2O7 industrially used for the oxidation of n-butane to maleic anhydride. The oxidation state of vanadium in vanadyl pyrophosphate is 4+. In literature, it was shown that the surface of vanadyl pyrophosphate contains V5+ and is enriched in phosphorus under reaction conditions. On account of this, V2O5 with the oxidation state of 5+ for vanadium partially covered with phosphorus can be regarded as a suitable model catalyst. The catalytic performance of the model catalyst prepared via atomic layer deposition was measured and compared to the performance of catalysts prepared via incipient wetness impregnation and the original V2O5 substrate. It could be clearly shown that the dedicated deposition of phosphorus by atomic layer deposition enhances the catalytic performance of V2O5 by suppression of total oxidation reactions, thereby increasing the selectivity to maleic anhydride.DFG, 53182490, EXC 314: Unifying Concepts in Catalysi

Characterization of Aspergillus terreus Accessory Conidia and Their Interactions With Murine Macrophages

All Aspergillus species form phialidic conidia (PC) when the mycelium is in contact with the air. These small, asexual spores are ideally suited for an airborne dissemination in the environment. Aspergillus terreus and a few closely related species from section Terrei can additionally generate accessory conidia (AC) that directly emerge from the hyphal surface. In this study, we have identified galactomannan as a major surface antigen on AC that is largely absent from the surface of PC. Galactomannan is homogeneously distributed over the entire surface of AC and even detectable on nascent AC present on the hyphal surface. In contrast, β-glucans are only accessible in distinct structures that occur after separation of the conidia from the hyphal surface. During germination, AC show a very limited isotropic growth that has no detectable impact on the distribution of galactomannan. The AC of the strain used in this study germinate much faster than the corresponding PC, and they are more sensitive to desiccation than PC. During infection of murine J774 macrophages, AC are readily engulfed and trigger a strong tumor necrosis factor-alpha (TNFα) response. Both processes are not hampered by the presence of laminarin, which indicates that β-glucans only play a minor role in these interactions. In the phagosome, we observed that galactomannan, but not β-glucan, is released from the conidial surface and translocates to the host cell cytoplasm. AC persist in phagolysosomes, and many of them initiate germination within 24 h. In conclusion, we have identified galactomannan as a novel and major antigen on AC that clearly distinguishes them from PC. The role of this fungal-specific carbohydrate in the interactions with the immune system remains an open issue that needs to be addressed in future research