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Series: Stochastics Seminar

Series: Stochastics Seminar

Neural networks have led to new and state of the art approaches for image recovery. They provide a contrast to standard image processing methods based on the ideas of sparsity and wavelets. In this talk, we will study two different random neural networks. One acts as a model for a learned neural network that is trained to sample from the distribution of natural images. Another acts as an unlearned model which can be used to process natural images without any training data. In both cases we will use high dimensional concentration estimates to establish theory for the performance of random neural networks in imaging problems.

Series: Stochastics Seminar

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Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

In this talk I will first recall some general facts about the parabolic Anderson model (PAM), which can be briefly described as a simple heat equation in a random environment. The key phenomenon which has to be observed in this context is called localization. I will review some ways to express this phenomenon, and then single out the so called eigenvectors localization for the Anderson operator. This particular instance of localization motivates our study of large time asymptotics for the stochastic heat equation. In the second part of the talk I will describe the Gaussian environment we consider, which is rougher than white noise, then I will give an account on the asymptotic exponents we obtain as time goes to infinity. If time allows it, I will also give some elements of proof.

Series: Stochastics Seminar

We identify principal component analysis (PCA) as an empirical risk minimization problem with respect to the reconstruction error and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. We also discuss how our results can be transferred to the subspace distance and, for instance, how our approach leads to a sharp $\sin \Theta$ theorem for empirical covariance operators. The proof is based on a novel contraction property, contrasting previous spectral perturbation approaches. This talk is based on joint works with Markus Reiß and Moritz Jirak.

Series: Stochastics Seminar

We present the joint distribution of the Busemann functions, in all directions of growth, of the exactly solvable corner growth model (CGM). This gives a natural coupling of all stationary CGMs and leads to new results about geodesics. Properties of this joint distribution are accessed by identifying it as the unique invariant distribution of a multiclass last passage percolation model. This is joint work with Timo Seppäläinen.

Series: Stochastics Seminar

<p>Wiener-Hopf factorization (WHf) encompasses several important results in probability and stochastic processes, as well as in operator theory. The importance of the WHf stems not only from its theoretical appeal, manifested, in part, through probabilistic interpretation of analytical results, but also from its practical applications in a wide range of fields, such as fluctuation theory, insurance and finance. The various existing forms of the WHf for Markov chains, strong Markov processes, Levy processes, and Markov additive process, have been obtained only in the time-homogeneous case. However, there are abundant real life dynamical systems that are modeled in terms of time-inhomogenous processes, and yet the corresponding Wiener-Hopf factorization theory is not available for this important class of models. In this talk, I will first provide a survey on the development of Wiener-Hopf factorization for time-homogeneous Markov chains, Levy processes, and Markov additive processes. Then, I will discuss our recent work on WHf for time-inhomogensous Markov chains. To the best of our knowledge, this study is the first attempt to investigate the WHf for time-inhomogeneous Markov processes.</p>

Series: Stochastics Seminar

Consider a measurable dense family of semi-infinite nearest-neighbor paths on the integer lattice in d dimensions. If the measure on the paths is translation invariant, we completely classify their collective behavior in d=2 under mild assumptions. We use our theory to classify the behavior of families of semi-infinite geodesics in first- and last-passage percolation that come from Busemann functions. For d>=2, we describe the behavior of bi-infinite trajectories, and show that they carry an invariant measure. We also construct several examples displaying unexpected behavior. One of these examples lets us answer a question of C. Hoffman's from 2016. (joint work with Jon Chaika)