We investigate probability measures with finite pluricomplex energy. We give criteria insuring that a given measure has finite energy and test these on various examples. We show that this notion is a biholomorphic but not a bimeromorphic invariant.
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48 results for “finite energy measures”
Synthetic approach to pluripotential theory measures finite energy.
problem Global pluripotential theory on compact Kähler manifolds and projective Berkovich spaces.
method Definition and study of measures of finite energy, introduction of twisted and free energy functionals.
result Coercivity of energy functionals is an open condition with respect to polarization.
Study on finite entropy and energy in Kähler geometry.
problem Finite entropy and energy measures in Kähler geometry.
method Refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions.
result Quasi-plurisubharmonic potentials with finite entropy belong to the finite energy class .
Extends finite entropy measures in Kähler geometry.
problem Analyzing finite entropy measures on compact Kähler manifolds.
method Defining finite -entropy and demonstrating their inclusion in an energy class.
result Stability result for the complex Monge-Ampère equation.
The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.
problem Analyzing strong topologies for complex Monge-Ampère equations on Kähler manifolds.
method Proving the Monge-Ampère operator is a homeomorphism between finite energy potentials and energy measures with their strong topologies.
result The Monge-Ampère operator produces an homeomorphism between sets of finite energy potentials and measures on Kähler manifolds.
We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian…
New calibration energy measures deviation from calibrated geometry, enabling mean curvature flow in infinite volumes.
problem Mean curvature flow in infinite volumes with finite energy.
method Introducing calibration energy and proving its dissipation identity for mean curvature flows.
result Every proper self-expander with finite calibration energy is a plane in all dimensions and codimensions.
New energy measure for isolated systems in general relativity.
problem Quantifying energy in isolated systems in general relativity.
method Optimal isometric embedding and conformal Killing fields.
result Finite quasi-local energies for asymptotically flat spacetimes.
The study constructs universal invariants for non-Archimedean metrics on projective varieties.
problem Understanding the singularity of non-Archimedean metrics on projective varieties.
method Constructing partial Okounkov bodies and Duistermaat--Heckman measures for non-Archimedean metrics.
result Generalization of Duistermaat--Heckman measures to finite energy metrics on Berkovich analytifications.
The paper analyzes Laplace learning for Gaussian measure data in infinite dimensions, proving convergence.
problem Analyzing Laplace learning for infinite-dimensional Gaussian measure data.
method Minimizes Dirichlet energy on a graph constructed from the full dataset.
result Proves pointwise convergence of the graph Dirichlet energy for Gaussian measure data.
A new method quantizes conditional probability measures using deep learning.
problem Quantizing conditional probability measures efficiently.
method DCMQ method using Huber-energy kernel and deep neural network.
result Promising results on various examples.
Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.
problem Investigating minimizers of Ginzburg-Landau functionals in high dimensions with energy bounds.
method Analyzing minimizers with logarithmic energy bounds and considering the vacuum manifold's homotopy classes.
result Normalized energy measures converge to an -rectifiable measure associated with a stationary varifold.
Infinite dimensional measure-valued processes modeled as polynomial diffusions.
problem Modeling term structure in energy markets using measure-valued polynomial diffusions.
method Introduced measure-valued polynomial diffusions, derived moment formulas, and characterized infinitesimal generators.
result Recovery of measure-valued affine diffusions as a special case.
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
This article considers the quasi-local energy in reference to a general static spacetime. We follow the approach developed by the authors in [19, 20, 7, 9] and define the quasi-local energy as a difference of surface Hamiltonians, which are derived from the Einstein-Hilbert action. The new quasi-local energy provides a…
The paper proves an energy identity for harmonic maps near singularities.
problem Analyzing the behavior of harmonic maps near singular points.
method Analyzes sequences of stationary harmonic maps with bounded energy, proving an energy identity near singularities.
result The energy density of the defect measure is the sum of the energies of the bubbling maps.
Gradient Ricci solitons can be extended to non-gradient Ricci solitons using energy function.
problem Extending the geometry of gradient Ricci solitons to non-gradient Ricci solitons.
method Using energy function to study the geometry.
result A non-steady Ricci soliton with symmetric covariant derivative is gradient.
We study minimal energy problems for strongly singular Riesz kernels on a manifold. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseud…
EBM model for functional data using path measure tilting.
problem Modeling distributions of functions from irregularly sampled data.
method Spectral decomposition of an energy-based model with a Gaussian Process path measure to reweight the distribution.
result Effective approach to up-scaling functional data.
Mean-field neural nets approximate functions using a free energy functional and controlled dynamics.
problem Function approximation by two-layer neural nets in the mean-field regime.
method Phrasing function approximation as global minimization of a free energy functional, examining dynamics in the space of probability measures over weights.
result Characterization of the unique global minimizer and dynamics achieving it, including the Föllmer drift.
We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account of area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied…
Algorithm finds best Dirac mass approximation of target measure.
problem Finding optimal Dirac mass approximation of target measure.
method Minimizes statistical distance between original measure and quantized version using Huber-energy kernel.
result HEMQ algorithm robust and versatile, matches intuitive behavior.
The paper proves topological finiteness for surfaces with finite Willmore energy.
problem Understanding the topology of surfaces with finite Willmore energy.
method Combining Allard regularity theorem and Reifenberg's topological disk theorem.
result Topological finiteness for a class of properly immersed surfaces with finite Willmore energy.
Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.
problem Proving the Wulff theorem for crystalline integrands.
method Direct approach using Minkowski Theory to exploit convex properties.
result Simpler proof of the Wulff theorem for crystalline shapes.
New findings on Mabuchi energy and stability of manifolds.
problem Understanding the Mabuchi energy and its coercive property.
method Analyzing asymptotic stability and polarized manifolds.
result Properness of Mabuchi energy on Kahler metrics in the first Chern class.
We associate certain probability measures on to geodesics in the space $\H_L$ of positively curved metrics on a line bundle , and to geodesics in the finite dimensional symmetric space of hermitian norms on . We prove that the measures associated to the finite dimensional spaces converge weakly to t…
We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate…
Approximate Bayesian computation (ABC) has become an essential part of the Bayesian toolbox for addressing problems in which the likelihood is prohibitively expensive or entirely unknown, making it intractable. ABC defines a pseudo-posterior by comparing observed data with simulated data, traditionally based on some su…
Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.
problem Characterizing the qualitative behavior of 4-harmonic and ES-4-harmonic maps.
method Proving triviality of finite energy solutions for both maps.
result Finite energy solutions of both 4-harmonic and ES-4-harmonic maps are trivial.
Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.
problem Bounding the index of codimension 2 minimal submanifolds.
method Second inner variation of energy, convergence of energy measures, and stress-energy tensors.
result Bound the Morse index of the submanifold by the index of critical points.
Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.
problem Understanding the asymptotic behavior of finite energy SU(2) monopoles on AC 3-manifolds.
method Analysis of critical points of the SU(2) Yang--Mills--Higgs energy on asymptotically conical 3-manifolds.
result Proves integrality of the monopole number and quadratic decay of curvature, among other findings.
Study finite-energy metrics over complex manifold degenerations.
problem Finite-energy metrics on complex manifolds with singularities.
method Investigate spaces of plurisubharmonic metrics with finite-energy conditions.
result Complete and geodesic metric structure on finite-energy metrics space.
Our work is motivated by a desire to study the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of neural networks. The key insight, already observed in the works of Mei, Montanari and Nguyen (2018), Chizat and Bach (2018) as …
Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.
problem Investigate semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces.
method Analyze eigenfunctions in low, critical, and high energy regimes using quantum ergodicity and equidistribution.
result Eigenfunctions in different regimes converge to distinct measures: invariant, Liouville, or equidistributed.
Develops BV function and finite perimeter set theory on Riemannian manifolds.
problem Theory of BV functions and finite perimeter sets on arbitrary Riemannian manifolds.
method Localization framework combining Euclidean and metric measure space techniques.
result Recovery of key Euclidean results in Riemannian setting.
We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's -entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the -noncollapsing property. Finally, we us…
A method smears likelihood to reveal physical energy scales in jet identification.
problem Interpretability of machine learning in particle physics.
method Smearing or averaging over events within a metric energy distance.
result Discrimination power increases as resolution decreases, showing sensitivity to all energy scales.
In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded -dimensional Lipschitz submanifolds in . It turns out that due to a smoothing effect any seq…
Geodesics in non-Archimedean metrics are continuous.
problem Understanding geodesics in spaces of non-Archimedean metrics.
method Maximal psh segments are geodesics, and continuity of these segments is proven.
result Maximal psh segments joining continuous psh metrics are continuous.
Study generalizes Möbius energy to non-smooth sets in arbitrary dimensions.
problem Investigate Möbius-invariant energies on non-smooth subsets of arbitrary dimensions.
method Show local finite energy implies embedded Lipschitz submanifold, and low fractional Sobolev regularity guarantees finite energy.
result Local graph structure of low fractional Sobolev regularity on a set is sufficient to guarantee finite energy.
The study finds critical points of Yang-Mills-Higgs energy on 3-manifolds.
problem Finding critical points of Yang-Mills-Higgs energy on 3-manifolds.
method 2-parameter min-max construction and energy gap analysis.
result Existence of non-trivial critical points on 3-manifolds with bounded geometry.
Energy distance measures feature heterogeneity in federated learning.
problem Heterogeneity across data sources hinders model aggregation in federated learning.
method Introduced Taylor approximations of energy distance for efficient computation.
result Taylor approximations accurately capture feature discrepancies, improving convergence.
Adversarial attacks can fool ML energy theft detection models.
problem Vulnerability of ML-based energy theft detection models to adversarial attacks.
method Design of an adversarial measurement generation algorithm.
result ML models can be significantly fooled by adversarial attacks, reducing their detection accuracy.
The paper classifies energy-minimizing sets in specific domains.
problem Classifying volume-constraint local energy-minimizing sets.
method Proved a Poincaré-type inequality for stable sets.
result Relative boundary of energy-minimizing sets is smooth.
Framework for energy markets using measure-valued processes.
problem Arbitrage-free modeling of energy futures markets.
method Translation of Heath-Jarrow-Morton approach to measure-valued processes, derivation of HJM-drift condition, analysis of measure-valued diffusions.
result Existence of non-negative measure-valued diffusions satisfying the HJM-drift condition.
We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity should be reduced with respect to the group action. About this point we explain …
Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
The paper extends entropy concepts to Monge-Ampère measures with prescribed singularities.
problem Investigating entropy for Monge-Ampère measures with specific singularities.
method Generalizing entropy for potentials, studying stability under blow-ups and perturbations, proving Moser-Trudinger inequalities.
result Functions with finite entropy belong to a specific energy class and maintain singularities of the model potential.