arXiv research

Neural density estimators are flexible families of parametric models which have seen widespread use in unsupervised machine learning in recent years. Maximum-likelihood training typically dictates that these models be constrained to specify an explicit density. However, this limitation can be overcome by instead using …

The paper classifies biharmonic quadratic maps between spheres, proving their energy density properties.

The study classifies biharmonic submanifolds in a sphere using specific eigenmaps.

Derives continuum model from discrete $\varepsilon$-graphs with connectivity functional.

Extends Penrose's method to null shells with pressure and energy flux.

In this paper, we introduce a new energy density function $\mathscr Y$ on the projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$ between Riemannian manifolds $\mathscr Y=g_{ij}f^i_αf^j_β\frac{W^αW^β}{\sum h_{γδ} W^γW^δ}.$ We get new Hessian estimates to this energy density and obtain various new…

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson…

Study on the $L^p$ behavior of helix curves' energy density.

We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hy…

An anisotropic surface energy is the integral of an energy density that depends on the normal at each point over the considered surface, and it is a generalization of surface area. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wul…

We consider a vector bundle $E$ over a compact Riemannian manifold $M$=$M^{n}$,$n\geq 4$,and $A$ is a Yang-Mills connection with $L^{\frac{n}{2}}$ curvature $F_{A}$ on $E$.Then we prove a mean value inequality for the density $|F_{A}|^{\frac{n}{2}}$.This inequality give rise to an energy concentrate principle for seque…

New method trains EBMs using NFs for more accurate likelihood estimation.

The study provides optimal estimates for surfaces close to constant mean curvature.

New optimization method for sampling from unknown density measures.

A new method normalizes EBM training by introducing a learnable parameter.

FNFs model parameter-dependent densities by combining a fixed flow with a polynomial parameter-dependent transformation.

This paper studies the convergence of penalized energy to harmonic maps in Riemannian manifolds.

In this paper, we analyzed the physical meaning of scalar curvatures for a generalized Riemannian space. It is developed the Madsen's formulae for pressures and energy-densities with respect to the corresponding energy-momentum tensors. After that, the energy-momentum tensors, pressures, energy-densities and state-para…

Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and tot…

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with larg…

Outlier detection amounts to finding data points that differ significantly from the norm. Classic outlier detection methods are largely designed for single data type such as continuous or discrete. However, real world data is increasingly heterogeneous, where a data point can have both discrete and continuous attribute…

Efficiently samples and learns densities with symmetries using equivariant methods.

EBMs are flexible but hard to train; this paper explains methods.

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

The integral of the energy density function $\mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $Λ$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this …

Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and highly accurate energies are achieved. Accurate {\em constrained optimal densitie…

The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.

We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_…

We examine Generative Adversarial Networks (GANs) through the lens of deep Energy Based Models (EBMs), with the goal of exploiting the density model that follows from this formulation. In contrast to a traditional view where the discriminator learns a constant function when reaching convergence, here we show that it ca…

Stable solutions to a specific equation are one-dimensional.

A new training method for normalizing flows without samples.

New method trains deep neural networks for non-interacting kinetic-energy functionals in DFT.

We prove that capillary surfaces converge to a specific energy density as the angle approaches zero.

iEFM trains CNF models from unnormalized densities efficiently.

New proof confirms De Giorgi's conjecture about phase-field approximation of Willmore functional.

Proves density and mass theorems for specific initial data sets.

The study shows how energy density of harmonic maps dominates in $n$-Fuchsian fibers, leading to unique minimal surfaces.

The problem of continuous inverse optimal control (over finite time horizon) is to learn the unknown cost function over the sequence of continuous control variables from expert demonstrations. In this article, we study this fundamental problem in the framework of energy-based model, where the observed expert trajectori…

We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivat…

RNE provides a flexible framework for diffusion models, enabling inference-time control and energy-based training.

Wavelet scattering predicts material properties beyond training data.

iDEM generates samples from Boltzmann densities without data.

Study on helix curves and their Möbius energy asymptotics.

Adaptive multi-stage density ratio estimation improves learning of latent space EBM.

For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.

Neural network models improve survival analysis with reduced computation time.

We study, using Mean Curvature Flow methods, 2+1 dimensional cosmologies with a positive cosmological constant and matter satisfying the dominant and the strong energy conditions. If the spatial slices are compact with non-positive Euler characteristic and are initially expanding everywhere, then we prove that the spat…

When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, …