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Providing generalization guarantees for stochastic optimization algorithms remains a key challenge in learning theory. Recently, numerous works demonstrated the impact of the geometric properties of optimization trajectories on…
We advocate the use of a notion of entropy that reflects the relative abundances of the symbols in an alphabet, as well as the similarities between them. This concept was originally introduced in theoretical ecology to study the diversity…
The maximum entropy principle can be used to assign utility values when only partial information is available about the decision maker's preferences. In order to obtain such utility values it is necessary to establish an analogy between…
We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form…
We introduce a new compositional framework for generalized variational inference, clarifying the different parts of a model, how they interact, and how they compose. We explain that both exact Bayesian inference and the loss functions…
The local volatility model is a widely used for pricing and hedging financial derivatives. While its main appeal is its capability of reproducing any given surface of observed option prices---it provides a perfect fit---the essential…
This article provides convergence analysis of online stochastic gradient descent algorithms for functional linear models. Adopting the characterizations of the slope function regularity, the kernel space capacity, and the capacity of the…
We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems…
Much research in machine learning involves finding appropriate inductive biases (e.g. convolutional neural networks, momentum-based optimizers, transformers) to promote generalization on tasks. However, quantification of the amount of…
We show that a lattice formulation of density-functional theory (DFT), guided by renormalization-group concepts, can be used to obtain numerical predictions of energy gaps, spin-density profiles, critical exponents, sound velocities,…
In this essay I describe some new results of a framework for composite gravity coupled to matter. These include the Bekenstein-Hawking entropy formula, modifications to the low-$\ell$ moments of the CMB power spectrum, and new perspectives…
Structural reliability methods aim at computing the probability of failure of systems with respect to some prescribed performance functions. In modern engineering such functions usually resort to running an expensive-to-evaluate…
We study several notions of topological pressure and capacities for multi-potentials $\Phi \in \mathcal C(X;\mathbb R)^m$, with respect to finitely generated continuous semigroups $G$ on a compact metric space $X$. We introduce the…
We present a framework to formally describe probabilistic system behavior and symbolically reason about it. In particular we aim at reasoning about possible failures and fault tolerance. We regard systems which are composed of different…
The present paper contains some investigations about a uniform variant of the notion of metric hemiregularity, the latter being a less explored property obtained by weakening metric regularity. The introduction of such a quantitative…
After a short introduction to the generalized uncertainty principle (GUP), we review some of the physical predictions of the GUP, and we focus in particular on the bounds that present experimental tests can put on the value of the…
We explore how much knowing a parametric restriction on propensity scores improves semiparametric efficiency bounds in the potential outcome framework. For stratified propensity scores, considered as a parametric model, we derive explicit…
We introduce a method of exploring potential energy contours in complex dynamical systems based on potentiostatic kinematics wherein the systems are evolved with minimal changes to their potential energy. We construct a simple iterative…
We use so-called geometrical approach in description of transition from regular motion to chaotic in Hamiltonian systems with potential energy surface that has several local minima. Distinctive feature of such systems is coexistence of…
Density-functional theory is a formally exact description of a many-body quantum system in terms of its density; in practice, however, approximations to the universal density functional are required. In this work, a model based on deep…