arXiv subjects

Raphael Zimmer

Publications and source records attributed to Raphael Zimmer.

7 recordsLinked to original sources

arXiv2024

Generative AI Models: Opportunities and Risks for Industry and Authorities

Generative AI models are capable of performing a wide variety of tasks that have traditionally required creativity and human understanding. During training, they learn patterns from existing data and can subsequently generate new content such as texts, images, audio, and videos that align with these patterns. Due to their versatility and generally high-quality results, they represent, on the one hand, an opportunity for digitalisation. On the other hand, the use of generative AI models introduces novel IT security risks that must be considered as part of a comprehensive analysis of the IT security threat landscape. In response to this risk potential, companies or authorities intending to use generative AI should conduct an individual risk analysis before integrating it into their workflows. The same applies to developers and operators, as many risks associated with generative AI must be addressed during development or can only be influenced by the operating organisation. Based on this, existing security measures can be adapted, and additional measures implemented.

cs.AI↗

arXiv2018

An Elementary Approach To Uniform In Time Propagation Of Chaos

Based on a coupling approach, we prove uniform in time propagation of chaos for weakly interacting mean-field particle systems with possibly non-convex confinement and interaction potentials. The approach is based on a combination of reflection and synchronous couplings applied to the individual particles. It provides explicit quantitative bounds that significantly extend previous results for the convex case.

math.PR↗

arXiv2018

Coupling and Convergence for Hamiltonian Monte Carlo

Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L1 Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal target distributions are included. Explicit quantitative bounds for the number of steps required to approximate the stationary distribution up to a given error are a direct consequence of contractivity. These bounds show that HMC can overcome diffusive behaviour if the duration of the Hamiltonian dynamics is adjusted appropriately.

math.PR↗

arXiv2017

Quantitative Harris type theorems for diffusions and McKean-Vlasov processes

We consider $\mathbb{R}^d$-valued diffusion processes of type \begin{align*} dX_t\ =\ b(X_t)dt\, +\, dB_t. \end{align*} Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich ($L^1$ Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattingly's extension of Harris' Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.

math.PR↗

arXiv2017

Couplings and quantitative contraction rates for Langevin dynamics

We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behavior in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance $a$, we obtain a lower bound for the contraction rate of order $\Omega (a^{-1})$ provided the friction coefficient is of order $\Theta (a^{-1})$.

math.PR↗

arXiv2017

Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions

Given a separable and real Hilbert space $\mathbb{H}$ and a trace-class, symmetric and non-negative operator $\mathcal{G}:\mathbb{H}\rightarrow\mathbb{H}$, we examine the equation \begin{align*} dX_t = -X_t\, dt + b(X_t) \, dt + \sqrt{2} \, dW_t, \qquad X_0=x\in\mathbb{H}, \end{align*} where $(W_t)$ is a $\mathcal{G}$-Wiener process on $\mathbb{H}$ and $b:\mathbb{H}\rightarrow\mathbb{H}$ is Lipschitz. We assume there is a splitting of $\mathbb{H}$ into a finite-dimensional space $\mathbb{H}^l$ and its orthogonal complement $\mathbb{H}^h$ such that $\mathcal{G}$ is strictly positive definite on $\mathbb{H}^l$ and the non-linearity $b$ admits a contraction property on $\mathbb{H}^h$. Assuming a geometric drift condition, we derive a Kantorovich ($L^1$ Wasserstein) contraction with an explicit rate for the corresponding Markov kernels. The estimates for the rate are based on the eigenvalues of $\mathcal{G}$ on the space $\mathbb{H}^l$, a Lipschitz bound on $b$ and a geometric drift condition. The results are derived using coupling methods.

math.PR↗

arXiv2016

Sticky couplings of multidimensional diffusions with different drifts

We present a novel approach of coupling two multidimensional and non-degenerate Itô processes $(X_t)$ and $(Y_t)$ which follow dynamics with different drifts. Our coupling is sticky in the sense that there is a stochastic process $(r_t)$, which solves a one-dimensional stochastic differential equation with a sticky boundary behavior at zero, such that almost surely $|X_t-Y_t|\leq r_t$ for all $t\geq 0$. The coupling is constructed as a weak limit of Markovian couplings. We provide explicit, non-asymptotic and long-time stable bounds for the probability of the event $\{X_t=Y_t\}$.

math.PR↗