arXiv research

New method finds minimum in noisy data, useful for model selection.

New method learns binary decision trees efficiently.

Suppose $f(x,y) + \fracκ{2} \|x\|^2 - \fracσ{2}\|y\|^2$ is convex where $σ>0$, and the argmin function $γ(x) = \{ γ: \inf_y f(x,y) = f(x,γ)\}$ exists and is single valued. We will prove $γ$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.

Proposes a new risk measurement method for risk-averse stochastic optimization.

Paper proposes new loss functions for training energy networks.

EGO-MDA identifies optimal spectral-bands for process discrimination.

We describe a novel family of models of multi- layer feedforward neural networks in which the activation functions are encoded via penalties in the training problem. Our approach is based on representing a non-decreasing activation function as the argmin of an appropriate convex optimiza- tion problem. The new framewor…

This paper introduces a novel generative encoder (GE) model for generative imaging and image processing with applications in compressed sensing and imaging, image compression, denoising, inpainting, deblurring, and super-resolution. The GE model consists of a pre-training phase and a solving phase. In the pre-training …

New discrete models for constant mean curvature surfaces and tori.

Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.

The study classifies singularities in discrete improper affine spheres.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells havin…

New method solves discrete mKdV equation from curve motions.

We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.

This paper explores geometric insights into discrete R-congruences and their envelopes.

Permutability of surface transforms yields discrete analogs.

New representations for discrete surfaces derived from dual transforms.

In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric conditio…

Study on discrete surfaces with constant principal curvature for nanocarbon applications.

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $τ$ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …

New method linearizes Darboux transformations of discrete curves.

The paper introduces discrete Dirac structures for mechanics, simplifying dynamics.

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …

Paper proposes a new generative model for discrete distributions using flows on submanifolds.

In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uni…

Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of …

We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal …

Study proposes Local Linear Encoding for better feature discretization.

Exact discrete mechanics for nonholonomic systems defined.

Paper generalizes discrete CMC surfaces and shows how they can be derived.

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invar…

Discrete analogues of ellipsoids with preserved circular cross sections.

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding t…

Plane triangulations remain rigid under discrete conformal changes.

Study integrable discretizations of cyclic systems with circular coordinate lines.

In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computa…

Survey on discrete minimal surfaces and their properties.

Study on discrete Gaussian curvature for polyhedral surfaces.

Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.

Unified discrete diffusion for categorical data simplifies training and sampling.

Discrete geometry model approximates Willmore energy.

Study discretizes Dirac and port-Hamiltonian systems using manifolds.

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter rectangles. For discrete planar quadrilateral nets, circular nets, $Q^*$-nets and conical…

Plane Delaunay triangulations are rigid under Luo's discrete conformal change.

In this paper we propose a process of lagrangian reduction and reconstruction for nonholonomic discrete mechanical systems where the action of a continuous symmetry group makes the configuration space a principal bundle. The result of the reduction process is a discrete dynamical system that we call the discrete reduce…

Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.

We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.

We present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data, defined at vertices, on edges or on faces, are associated with a discrete channel surface that may be used to reconstruct the underlying particular discrete L…