Related papers: Perspective Functions with Nonlinear Scaling
The main result states that every convex set-valued function defined on a real interval with compact values in a locally convex space, admits an affine selection. In the case if the target space is a real line and the values are closed real…
Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no general way to transport spectral invariants in a functorial…
We analyze the classical model of compound interest with a constant per-period payment and interest rate. We examine the outstanding balance function as well as the periodic payment function and show that the outstanding balance function is…
In static analysis by abstract interpretation, one often uses widening operators in order to enforce convergence within finite time to an inductive invariant. Certain widening operators, including the classical one over finite polyhedra,…
Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic…
We consider a class of nonconvex energy functionals that lies in the framework of the peridynamics model of continuum mechanics. The energy densities are functions of a nonlocal strain that describes deformation based on pairwise…
We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of…
We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly non-reflexive and non-separable spaces. The pivoting point is to establish a novel variational structure, based…
A functional differential equation related to the logistic equation is studied by a combination of numerical and perturbation methods. Parameter regions are identified where the solution to the nonlinear problem is approximated well by…
This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from `noncommutative functional analysis', more specifically the new field of {\em operator spaces.} In our…
Motivated by the direct method in the calculus of variations in $L^{\infty}$, our main result identifies the notion of convexity characterizing the weakly$^*$ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity.…
The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is…
We introduce and study a new class of generalized convex functions termed star quasiconvex functions. This class includes convex, star-convex, quasiconvex, quasar-convex, and positively homogeneous functions of any degree $p>0$ as special…
The great advances of learning-based approaches in image processing and computer vision are largely based on deeply nested networks that compose linear transfer functions with suitable non-linearities. Interestingly, the most frequently…
Computing explicitly the {\epsilon}-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class…
Classical functional calculus is primarily spectral, capturing eigenvalue information through resolvent methods while largely ignoring nilpotent structure. Building on the projector-nilpotent characterization developed in our companion…
Quantifying uncertainty in predictions or, more generally, estimating the posterior conditional distribution, is a core challenge in machine learning and statistics. We introduce Convex Nonparanormal Regression (CNR), a conditional…
As science and engineering have become increasingly data-driven, the role of optimization has expanded to touch almost every stage of the data analysis pipeline, from signal and data acquisition to modeling and prediction. The optimization…
Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…
The idea of functional programming has played a big role in shaping today's landscape of mainstream programming languages. Another concept that dominates the current programming style is Dijkstra's structured programming. Both concepts have…