arXiv subjects

Jeremiah Birrell

Publications and source records attributed to Jeremiah Birrell.

At least 37 records · Page 2Linked to original sources

arXiv2021

$(f,Γ)$-Divergences: Interpolating between $f$-Divergences and Integral Probability Metrics

We develop a rigorous and general framework for constructing information-theoretic divergences that subsume both $f$-divergences and integral probability metrics (IPMs), such as the $1$-Wasserstein distance. We prove under which assumptions these divergences, hereafter referred to as $(f,Γ)$-divergences, provide a notion of `distance' between probability measures and show that they can be expressed as a two-stage mass-redistribution/mass-transport process. The $(f,Γ)$-divergences inherit features from IPMs, such as the ability to compare distributions which are not absolutely continuous, as well as from $f$-divergences, namely the strict concavity of their variational representations and the ability to control heavy-tailed distributions for particular choices of $f$. When combined, these features establish a divergence with improved properties for estimation, statistical learning, and uncertainty quantification applications. Using statistical learning as an example, we demonstrate their advantage in training generative adversarial networks (GANs) for heavy-tailed, not-absolutely continuous sample distributions. We also show improved performance and stability over gradient-penalized Wasserstein GAN in image generation.

stat.ML↗

arXiv2021

Variational Representations and Neural Network Estimation of Rényi Divergences

We derive a new variational formula for the Rényi family of divergences, $R_α(Q\|P)$, between probability measures $Q$ and $P$. Our result generalizes the classical Donsker-Varadhan variational formula for the Kullback-Leibler divergence. We further show that this Rényi variational formula holds over a range of function spaces; this leads to a formula for the optimizer under very weak assumptions and is also key in our development of a consistency theory for Rényi divergence estimators. By applying this theory to neural-network estimators, we show that if a neural network family satisfies one of several strengthened versions of the universal approximation property then the corresponding Rényi divergence estimator is consistent. In contrast to density-estimator based methods, our estimators involve only expectations under $Q$ and $P$ and hence are more effective in high dimensional systems. We illustrate this via several numerical examples of neural network estimation in systems of up to 5000 dimensions.

stat.ML↗

arXiv2021

Distributionally Robust Variance Minimization: Tight Variance Bounds over $f$-Divergence Neighborhoods

Distributionally robust optimization (DRO) is a widely used framework for optimizing objective functionals in the presence of both randomness and model-form uncertainty. A key step in the practical solution of many DRO problems is a tractable reformulation of the optimization over the chosen model ambiguity set, which is generally infinite dimensional. Previous works have solved this problem in the case where the objective functional is an expected value. In this paper we study objective functionals that are the sum of an expected value and a variance penalty term. We prove that the corresponding variance-penalized DRO problem over an $f$-divergence neighborhood can be reformulated as a finite-dimensional convex optimization problem. This result also provides tight uncertainty quantification bounds on the variance.

math.OC↗

arXiv2020

Quantification of Model Uncertainty on Path-Space via Goal-Oriented Relative Entropy

Quantifying the impact of parametric and model-form uncertainty on the predictions of stochastic models is a key challenge in many applications. Previous work has shown that the relative entropy rate is an effective tool for deriving path-space uncertainty quantification (UQ) bounds on ergodic averages. In this work we identify appropriate information-theoretic objects for a wider range of quantities of interest on path-space, such as hitting times and exponentially discounted observables, and develop the corresponding UQ bounds. In addition, our method yields tighter UQ bounds, even in cases where previous relative-entropy-based methods also apply, e.g., for ergodic averages. We illustrate these results with examples from option pricing, non-reversible diffusion processes, stochastic control, semi-Markov queueing models, and expectations and distributions of hitting times.

math.PR↗

arXiv2020

Uncertainty Quantification for Markov Processes via Variational Principles and Functional Inequalities

Information-theory based variational principles have proven effective at providing scalable uncertainty quantification (i.e. robustness) bounds for quantities of interest in the presence of nonparametric model-form uncertainty. In this work, we combine such variational formulas with functional inequalities (Poincar{é}, $\log$-Sobolev, Liapunov functions) to derive explicit uncertainty quantification bounds for time-averaged observables, comparing a Markov process to a second (not necessarily Markov) process. These bounds are well-behaved in the infinite-time limit and apply to steady-states of both discrete and continuous-time Markov processes.

math.PR↗

arXiv2020

Langevin equations in the small-mass limit: Higher-order approximations

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order $m^{\ell/2}$ over compact time intervals for any $\ell\in\mathbb{Z}^+$. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $m\to 0$ limit, which result in order $m^{1/2}$ approximations. Our results cover bounded forces, for which we prove convergence in $L^p$ norms, and unbounded forces, in which case we prove convergence in probability.

math-ph↗

arXiv2019

Distributional Robustness and Uncertainty Quantification for Rare Events

Rare events, and more general risk-sensitive quantities-of-interest (QoIs), are significantly impacted by uncertainty in the tail behavior of a distribution. Uncertainty in the tail can take many different forms, each of which leads to a particular ambiguity set of alternative models. Distributional robustness bounds over such an ambiguity set constitute a stress-test of the model. In this paper we develop a method, utilizing Rényi-divergences, of constructing the ambiguity set that captures a user-specified form of tail-perturbation. We then obtain distributional robustness bounds (performance guarantees) for risk-sensitive QoIs over these ambiguity sets, using the known connection between Rényi-divergences and robustness for risk-sensitive QoIs. We also expand on this connection in several ways, including a generalization of the Donsker-Varadhan variational formula to Rényi divergences, and various tightness results. These ideas are illustrated through applications to uncertainty quantification in a model of lithium-ion battery failure, robustness of large deviations rate functions, and risk-sensitive distributionally robust optimization for option pricing.

math.PR↗

arXiv2019

Lepton Number and Expansion of the Universe

We show that the non-integer effective number of neutrinos $N^{\mathrm{eff}}_ν$ can be understood as an effect of lepton $L$ asymmetry in the early Universe carried by the Dirac neutrino cosmic background. We show that $N_ν^{\mathrm{eff}}=3.36\pm0.34$ (CMB only) and $N_ν^{\mathrm{eff}}= 3.62\pm0.25$ (CMB and $H_0$) require a ratio between baryon number $B$ and lepton number to be $1.16 \times 10^{-9}\leqslant B/|L|\leqslant 1.51 \times 10^{-9}$. These values are close to the baryon-to-photon ratio $0.57\times 10^{-9}\leqslant B/N_γ\leqslant 0.67\times10^{-9}$. Thus instead of the usual $|L|\ll N_γ$ and $B\simeq |L|$, we propose to use $0.4 \leqslant |L|/N_γ\leqslant 0.52$ and $B\ll|L|$ as another natural choice, which resolves the tension between Planck-CMB and $H_0$ and leads to a non-integer value of $N_ν^{\mathrm{eff}}>3$.

hep-ph↗

arXiv2018

Entropy Anomaly in Langevin-Kramers Dynamics with a Temperature Gradient, Matrix Drag, and Magnetic Field

We investigate entropy production in the small-mass (or overdamped) limit of Langevin-Kramers dynamics. The results generalize previous works to provide a rigorous derivation that covers systems with magnetic field as well as anisotropic (i.e. matrix-valued) drag and diffusion coefficients that satisfy a fluctuation-dissipation relation with state-dependent temperature. In particular, we derive an explicit formula for the anomalous entropy production which can be estimated from simulated paths of the overdamped system. As a part of this work, we develop a theory for homogenizing a class of integral processes involving the position and scaled-velocity variables. This allows us to rigorously identify the limit of the entropy produced in the environment, including a bound on the convergence rate.

math-ph↗

arXiv2018

A Posteriori Error Bounds for Two Point Boundary Value Problems: A Green's Function Approach

We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.

math.NA↗

arXiv2018

A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics

This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation ("the cell problem"), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature.

math-ph↗

arXiv2018

Small Mass Limit of a Langevin Equation on a Manifold

We study damped geodesic motion of a particle of mass $m$ on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as $m \to 0$, its solutions converge to solutions of a limiting equation which includes a {\it noise-induced drift} term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems.

math-ph↗

arXiv2018

Temperature Dependence of the Neutron Lifespan

Current precision big bang nucleosynthesis (BBN) studies motivate us to revisit the neutron lifespan in the plasma medium of the early universe. The mechanism we explore is the Fermi-blocking of decay electrons and neutrinos by plasma. As result, neutrons live longer and we find a significant 6.4\% modification of neutron abundance in the BBN era arising from in plasma modification of the neutron lifetime. This effect can influence the final abundances of the light elements in BBN.

nucl-th↗

arXiv2017

Phase Space Homogenization of Noisy Hamiltonian Systems

We study the dynamics of an inertial particle coupled to forcing, dissipation, and noise in the small mass limit. We derive an expression for the limiting (homogenized) joint distribution of the position and (scaled) velocity degrees of freedom. In particular, weak convergence of the joint distributions is established, along with a bound on the convergence rate for a wide class of expected values.

math-ph↗

arXiv2017

Homogenization of Dissipative, Noisy, Hamiltonian Dynamics

We study the dynamics of a class of Hamiltonian systems with dissipation, coupled to noise, in a singular (small mass) limit. We derive the homogenized equation for the position degrees of freedom in the limit, including the presence of a {\em noise-induced drift} term. We prove convergence to the solution of the homogenized equation in probability and, under stronger assumptions, in an $L^p$-norm. Applications cover the overdamped limit of particle motion in a time-dependent electromagnetic field, on a manifold with time-dependent metric, and the dynamics of nuclear matter.

math-ph↗

arXiv2016

The hot Hagedorn Universe

In the context of the half-centenary of Hagedorn temperature and the statistical bootstrap model (SBM) we present a short account of how these insights coincided with the establishment of the hot big-bang model (BBM) and helped resolve some of the early philosophical difficulties. We then turn attention to the present day context and show the dominance of strong interaction quark and gluon degrees of freedom in the early stage, helping to characterize the properties of the hot Universe. We focus attention on the current experimental insights about cosmic microwave background (CMB) temperature fluctuation, and develop a much improved understanding of the neutrino freeze-out, in this way paving the path to the opening of a direct connection of quark-gluon plasma (QGP) physics in the early Universe with the QCD-lattice, and the study of the properties of QGP formed in the laboratory.

hep-ph↗

arXiv2015

A Recursive Method for Computing Certain Bessel Function Integrals

We investigate a family of integrals involving modified Bessel functions that arise in the context of neutrino scattering. Recursive formulas are derived for evaluating these integrals and their asymptotic expansions are computed. We prove in certain cases that the asymptotic expansion yields the exact result after a finite number of terms. In each of these cases we derive a formula that bounds the order at which the expansion terminates. The method of calculation developed in this paper is applicable to similar families of integrals that involve Bessel or modified Bessel functions.

math.CA↗

arXiv2015

Dynamical Emergence of the Universe into the False Vacuum

We study how the hot Universe evolves and acquires the prevailing vacuum state, demonstrating that in specific conditions which are believed to apply, the Universe becomes frozen into the state with the smallest value of Higgs vacuum field $v=\langle h\rangle$, even if this is not the state of lowest energy. This supports the false vacuum dark energy $Λ$-model. Under several likely hypotheses we determine the temperature in the evolution of the Universe at which two vacuua $v_1, v_2$ can swap between being true and false. We evaluate the dynamical surface pressure on domain walls between low and high mass vaccua due to the presence of matter and show that the low mass state remains the preferred vacuum of the Universe.

astro-ph.CO↗