Jeremiah Birrell

University of Massachusetts Amherst | UMass Amherst · Department of Mathematics and Statistics

We introduce the $ARMOR_D$ methods as novel approaches to enhancing the adversarial robustness of deep learning models. These methods are based on a new class of optimal-transport-regularized divergences, constructed via an infimal convolution between an information divergence and an optimal-transport (OT) cost. We use these as tools to enhance adv...

We offer a survey of the matter-antimatter evolution within the primordial Universe. While the origin of the tiny matter-antimatter asymmetry has remained one of the big questions in modern cosmology, antimatter itself has played a large role for much of the Universe’s early history. In our study of the evolution of the Universe we adopt the positi...

We offer a survey of the matter-antimatter evolution within the primordial Universe. While the origin of the tiny matter-antimatter asymmetry has remained one of the big questions in modern cosmology, antimatter itself has played a large role for much of the Universe's early history. In our study of the evolution of the Universe we adopt the positi...

We propose a new family of regularized R\'enyi divergences parametrized not only by the order $\alpha$ but also by a variational function space. These new objects are defined by taking the infimal convolution of the standard R\'enyi divergence with the integral probability metric (IPM) associated with the chosen function space. We derive a novel du...

Generative adversarial networks (GANs), a class of distribution-learning methods based on a two-player game between a generator and a discriminator, can generally be formulated as a minmax problem based on the variational representation of a divergence between the unknown and the generated distributions. We introduce structure-preserving GANs as a...

Variational representations of divergences and distances between high-dimensional probability distributions offer significant theoretical insights and practical advantages in numerous research areas. Recently, they have gained popularity in machine learning as a tractable and scalable approach for training probabilistic models and for statistically...

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 probabilit...

We develop a general framework for constructing new information-theoretic divergences that rigorously interpolate between $f$-divergences and integral probability metrics (IPMs), such as the Wasserstein distance. These new divergences inherit features from IPMs, such as the ability to compare distributions which are not absolute continuous, as well...

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 informat...

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 dimensio...

We derive a new variational formula for the R\'enyi family of divergences, $R_\alpha(Q\|P)$, generalizing the classical Donsker-Varadhan variational formula for the Kullback-Leibler divergence. The objective functional in this new variational representation is expressed in terms of expectations under $Q$ and $P$, and hence can be estimated using sa...

Variational representations of distances and divergences between high-dimensional probability distributions offer significant theoretical insights and practical advantages in numerous research areas. Recently, they have gained popularity in machine learning as a tractable and scalable approach for training probabilistic models and statistically dif...

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 /...

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 cons...

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 ph...

In this paper we provide performance guarantees for hypocoercive non-reversible MCMC samplers our results apply in particular to the Langevin equation, Hamiltonian Monte-Carlo, and the bouncy particle and zigzag samplers. Specifically, we establish a concentration inequality of Bernstein type for ergodic averages of bounded observables. As a conseq...

Variational-principle-based methods that relate expectations of a quantity of interest to information-theoretic divergences have proven to be effective tools for obtaining robustness bounds under both parametric and non-parametric model-form uncertainty. Here, we extend these ideas to utilize information divergences that are specifically targeted a...

A non-perturbative framework for robust uncertainty bounds via goal-oriented information theory.

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 non-parametric model-form uncertainty. In this work, we combine such variational formulas with functional inequalities (Poincar\'e, $\log$-Sobolev, Liapunov fun...

We show that the non-integer effective number of neutrinos $N^{\mathrm{eff}}_\nu$ can be understood as an effect of lepton $L$ asymmetry in the early Universe carried by the Dirac neutrino cosmic background. We show that $N_\nu^{\mathrm{eff}}=3.36\pm0.34$ (CMB only) and $N_\nu^{\mathrm{eff}}= 3.62\pm0.25$ (CMB and $H_0$) require a ratio between bar...

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-...

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. The present work generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $m\to 0$ limit. Specifically, we develop a hierarchy of...

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 a...

We investigate entropy production in the small mass (or overdamped) limit of Langevin-Kramers dynamics. Our results apply to systems with magnetic field as well as matrix valued drag and diffusion coefficients that satisfy a version of the fluctuation dissipation relation with state dependent temperature. In particular, we generalize the anomalous...

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 ph...

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 con...

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 noise-induced drift term. We prove convergence to the solution of the homogenized equation in probability and,...

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 domin...

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 noise-induced drift term....

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 ener...

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...

We present a novel method for detecting the relic neutrino background that takes advantage of structured quantum degeneracy to amplify the drag force from neutrinos scattering off a detector. Developing this idea, we present a characterization of the present day relic neutrino distribution in an arbitrary frame, including the influence of neutrino...

The effective number of neutrinos, , obtained from CMB fluctuations accounts for all effectively massless degrees of freedom present in the Universe, including but not limited to the three known neutrinos. Using a lattice-QCD derived QGP equation of state, we constrain the observed range of in terms of the freeze-out of unknown degrees of freedom n...

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 formu...

In this dissertation, we study the evolution and properties of the relic (or cosmic) neutrino distribution from neutrino freeze-out at $T=O(1)$ MeV through the free-streaming era up to today, focusing on the deviation of the neutrino spectrum from equilibrium. In particular, we demonstrate the presence of chemical non-equilibrium that continues to...

Analysis of cosmic microwave background radiation fluctuations favors an effective number of neutrinos, $N_\nu>3$. This motivates a reinvestigation of the neutrino freeze-out process. Here we characterize the dependence of $N_\nu$ on the Standard Model (SM) parameters that govern neutrino freeze-out. We show that $N_\nu$ depends on a combination $\...

It is known that the freeze-out of light particles in the early Universe can lead to a non-integer contribution to the effective number of neutrinos, $N_{\text{eff}}$. Using a lattice-QCD derived quark-gluon plasma equations of state we show that freeze-out at the time when the quark-gluon deconfined phase froze into hadrons near $T=150$ MeV leads...

We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature $T(t)$ and phase space...

We study the present day relic neutrino distribution in the Earth frame and characterize the dependence on neutrino mass and freeze-out conditions in early Universe. We present explicitly the neutrino velocity and de Broglie wavelength distributions. We characterize the expected neutrino drag force ${\cal O}(G_F^2)$ of a spherical detector in the h...

We perform a model independent study of the neutrino momentum distribution at freeze-out, treating the freeze-out temperature as a free parameter. Our results imply that measurement of neutrino reheating, as characterized by the measurement of the effective number of neutrinos Nν, amounts to the determination of the neutrino kinetic freeze-out temp...

We clarify in a quantitative way the impact that distinct chemical $T_c$ and kinetic $T_k$ freeze-out temperatures have on the reduction of the neutrino fugacity $\Upsilon_\nu$ below equilibrium, i.e. $\Upsilon_\nu<1$, and the increase of the neutrino temperature $T_\nu$ via partial reheating. We establish the connection between $\Upsilon_\nu$ and...

We present the Einstein-Vlasov solution for the momentum distribution of the relic free-streaming neutrinos. We show that it is possible to explain the rise in the effective number of neutrinos $N_\nu$ from those present at the end of the big bang nucleosynthesis (BBN) $N_\nu(T_{BBN})=3.71^{+0.47}_{-0.45}$ towards $N_\nu(T_{r})=4.34^{+.086}_{-0.88}...

We study models of strong first order `low' temperature electroweak phase transition. To achieve this we propose a class of Higgs effective potential models which preserve the known features of the present day massive phase. However, the properties of the symmetry restored massless phase are modified in a way that for a large parameter domain we fi...

We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we find a critical parameter value $\alpha_{1}=\alpha_{2}$ such that when $\alpha_{2}>\alpha_{1}$ the system is ergodic and when $\alpha_{2}<\alpha_{1}$ solutions are not defin...