### System Identification through Online Sparse Gaussian Process Regression with Input Noise

There has been a growing interest in using non-parametric regression methods like Gaussian Process (GP) regression for system identification. GP regression does traditionally have three important downsides: (1) it is computationally intensive, (2) it cannot efficiently implement newly obtained measurements online, and (3) it cannot deal with stochastic (noisy) input points. In this paper we present an algorithm tackling all these three issues simultaneously. The resulting Sparse Online Noisy Input GP (SONIG) regression algorithm can incorporate new noisy measurements in constant runtime. A comparison has shown that it is more accurate than similar existing regression algorithms. When applied to non-linear black-box system modeling, its performance is competitive with existing non-linear ARX models.

### Deep Gaussian Mixture Models

Deep learning is a hierarchical inference method formed by subsequent multiple layers of learning able to more efficiently describe complex relationships. In this work, Deep Gaussian Mixture Models are introduced and discussed. A Deep Gaussian Mixture model (DGMM) is a network of multiple layers of latent variables, where, at each layer, the variables follow a mixture of Gaussian distributions. Thus, the deep mixture model consists of a set of nested mixtures of linear models, which globally provide a nonlinear model able to describe the data in a very flexible way. In order to avoid overparameterized solutions, dimension reduction by factor models can be applied at each layer of the architecture thus resulting in deep mixtures of factor analysers.

### Differentially Private Algorithms for Learning Mixtures of Separated Gaussians

Learning the parameters of Gaussian mixture models is a fundamental and widely studied problem with numerous applications. In this work, we give new algorithms for learning the parameters of a high-dimensional, well separated, Gaussian mixture model subject to the strong constraint of differential privacy. In particular, we give a differentially private analogue of the algorithm of Achlioptas and McSherry. Our algorithm has two key properties not achieved by prior work: (1) The algorithm's sample complexity matches that of the corresponding non-private algorithm up to lower order terms in a wide range of parameters. Papers published at the Neural Information Processing Systems Conference.

### Mixtures of Shifted Asymmetric Laplace Distributions

A mixture of shifted asymmetric Laplace distributions is introduced and used for clustering and classification. A variant of the EM algorithm is developed for parameter estimation by exploiting the relationship with the general inverse Gaussian distribution. This approach is mathematically elegant and relatively computationally straightforward. Our novel mixture modelling approach is demonstrated on both simulated and real data to illustrate clustering and classification applications. In these analyses, our mixture of shifted asymmetric Laplace distributions performs favourably when compared to the popular Gaussian approach. This work, which marks an important step in the non-Gaussian model-based clustering and classification direction, concludes with discussion as well as suggestions for future work.

### Estimating LASSO Risk and Noise Level

We study the fundamental problems of variance and risk estimation in high dimensional statistical modeling. In particular, we consider the problem of learning a coefficient vector $\theta_0\in R^p$ from noisy linear observation $y=X\theta_0+w\in R^n$ and the popular estimation procedure of solving an $\ell_1$-penalized least squares objective known as the LASSO or Basis Pursuit DeNoising (BPDN). In this context, we develop new estimators for the $\ell_2$ estimation risk $\|\hat{\theta}-\theta_0\|_2$ and the variance of the noise. These can be used to select the regularization parameter optimally. Our approach combines Stein unbiased risk estimate (Stein'81) and recent results of (Bayati and Montanari'11-12) on the analysis of approximate message passing and risk of LASSO. We establish high-dimensional consistency of our estimators for sequences of matrices $X$ of increasing dimensions, with independent Gaussian entries. We establish validity for a broader class of Gaussian designs, conditional on the validity of a certain conjecture from statistical physics. Our approach is the first that provides an asymptotically consistent risk estimator. In addition, we demonstrate through simulation that our variance estimation outperforms several existing methods in the literature.