Lukas Trottner
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View article: Mini-Workshop: Statistical Challenges for Deep Generative Models
Mini-Workshop: Statistical Challenges for Deep Generative Models Open
Over the last decade, deep generative modelling has emerged as a powerful probabilistic tool in machine learning. The idea behind generative modelling is simple: transform noise to create new data that matches a given training data set. Su…
View article: Beyond Fixed Horizons: A Theoretical Framework for Adaptive Denoising Diffusions
Beyond Fixed Horizons: A Theoretical Framework for Adaptive Denoising Diffusions Open
We introduce a new class of generative diffusion models that, unlike conventional denoising diffusion models, achieve a time-homogeneous structure for both the noising and denoising processes, allowing the number of steps to adaptively adj…
View article: Statistical guarantees for denoising reflected diffusion models
Statistical guarantees for denoising reflected diffusion models Open
In recent years, denoising diffusion models have become a crucial area of research due to their abundance in the rapidly expanding field of generative AI. While recent statistical advances have delivered explanations for the generation abi…
View article: Multivariate change estimation for a stochastic heat equation from local measurements
Multivariate change estimation for a stochastic heat equation from local measurements Open
We study a stochastic heat equation with piecewise constant diffusivity $θ$ having a jump at a hypersurface $Γ$ that splits the underlying space $[0,1]^d$, $d\geq2,$ into two disjoint sets $Λ_-\cupΛ_+.$ Based on multiple spatially localize…
View article: The uniqueness of the Wiener–Hopf factorisation of Lévy processes and random walks
The uniqueness of the Wiener–Hopf factorisation of Lévy processes and random walks Open
We prove that the spatial Wiener–Hopf factorisation of a Lévy process or random walk without killing is unique.
View article: Markov additive friendships
Markov additive friendships Open
The Wiener–Hopf factorisation of a Lévy or Markov additive process describes the way that it attains new extrema in terms of a pair of so-called ladder height processes. Vigon’s theory of friendship for Lévy processes addresses the inverse…
View article: Learning to reflect: A unifying approach for data-driven stochastic control strategies
Learning to reflect: A unifying approach for data-driven stochastic control strategies Open
Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their…
View article: Covariate shift in nonparametric regression with Markovian design
Covariate shift in nonparametric regression with Markovian design Open
Covariate shift in regression problems and the associated distribution mismatch between training and test data is a commonly encountered phenomenon in machine learning. In this paper, we extend recent results on nonparametric convergence r…
View article: Concentration analysis of multivariate elliptic diffusions
Concentration analysis of multivariate elliptic diffusions Open
We prove concentration inequalities and associated PAC bounds for both continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approa…
View article: The uniqueness of the Wiener-Hopf factorisation of Lévy processes and random walks
The uniqueness of the Wiener-Hopf factorisation of Lévy processes and random walks Open
We prove that the spatial Wiener-Hopf factorisation of a Lévy process or random walk without killing is unique.
View article: Stability of overshoots of Markov additive processes
Stability of overshoots of Markov additive processes Open
We prove precise stability results for overshoots of Markov additive\nprocesses (MAPs) with finite modulating space. Our approach is based on the\nMarkovian nature of overshoots of MAPs whose mixing and ergodic properties are\ninvestigated…
View article: Data-driven rules for multidimensional reflection problems
Data-driven rules for multidimensional reflection problems Open
Over the recent past data-driven algorithms for solving stochastic optimal control problems in face of model uncertainty have become an increasingly active area of research. However, for singular controls and underlying diffusion dynamics …
View article: Markov additive friendships
Markov additive friendships Open
The Wiener--Hopf factorisation of a Lévy or Markov additive process describes the way that it attains new maxima and minima in terms of a pair of so-called ladder height processes. Vigon's theory of friendship for Lévy processes addresses …
View article: Change point estimation for a stochastic heat equation
Change point estimation for a stochastic heat equation Open
We study a change point model based on a stochastic partial differential equation (SPDE) corresponding to the heat equation governed by the weighted Laplacian $Δ_\vartheta = \nabla\vartheta\nabla$, where $\vartheta=\vartheta(x)$ is a space…
View article: Covariate shift in nonparametric regression with Markovian design
Covariate shift in nonparametric regression with Markovian design Open
Covariate shift in regression problems and the associated distribution mismatch between training and test data is a commonly encountered phenomenon in machine learning. In this paper, we extend recent results on nonparametric convergence r…
View article: Concentration analysis of multivariate elliptic diffusion processes
Concentration analysis of multivariate elliptic diffusion processes Open
We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach vi…
View article: Learning to reflect: A unifying approach for data-driven stochastic control strategies
Learning to reflect: A unifying approach for data-driven stochastic control strategies Open
Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their…
View article: Stability of overshoots of Markov additive processes
Stability of overshoots of Markov additive processes Open
We prove precise stability results for overshoots of Markov additive processes (MAPs) with finite modulating space. Our approach is based on the Markovian nature of overshoots of MAPs whose mixing and ergodic properties are investigated in…
View article: Mixing it up: A general framework for Markovian statistics
Mixing it up: A general framework for Markovian statistics Open
Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of…
View article: Mixing it up: A general framework for Markovian statistics beyond reversibility and the minimax paradigm
Mixing it up: A general framework for Markovian statistics beyond reversibility and the minimax paradigm Open
Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of…