Nonnegative sectional curvature linked to matrix displacement convexity.
problem Nonnegative sectional curvature in Riemannian manifolds.
method Matrix displacement convexity as a criterion for nonnegative sectional curvature.
result Entropy functional matrix displacement convexity implies nonnegative sectional curvature.
In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study ( κ , N ) (κ,N) ( κ , N ) -convex functions on metric spaces where κ κ κ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that …
Analyzes gradient flows and EVI in metric spaces with structural properties and convergence results.
problem Characterizing and analyzing gradient flows and Evolution Variational Inequalities in metric spaces.
method Study of structural properties, equivalence with maximal slope curves, convergence of Minimizing Movement-JKO scheme.
result Equivalence with maximal slope curves and uniform convergence of Minimizing Movement-JKO scheme.
Study measures inequality in social-economic systems using Fokker-Planck equations and Lotka-Volterra dynamics.
problem Measuring inequality in oscillatory social-economic systems described by Fokker-Planck equations and Lotka-Volterra dynamics.
method Used Fokker-Planck equations and Lotka-Volterra dynamics to model inequality, focusing on coefficient of variation as a measure.
result Inequality initially tends to decrease in oscillatory systems, contrary to steady-state models.
Theory of space-time currents for geometric evolutions.
problem Analysis of geometric evolutions driven by dislocations.
method Development of space-time integral currents with bounded variation, introduction of Lipschitz deformation distance.
result Agreement of Lipschitz deformation distance with integral Whitney flat metric for boundaryless currents.
The paper proves stability of curvature bounds in geometric analysis.
problem Stability of local Riemannian Ricci curvature bounds under convergence.
method Gromov-Hausdorff convergence, Lagrangian approach, heat flow, weak gradients, Evolution Variational Inequality.
result Almost everywhere existence of Euclidean weak tangents.
The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times.…
Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface Σ Σ Σ , we observe that …
Global calculus for manifolds with boundary, solving evolution problems.
problem Global solvability of evolution problems on manifolds with boundary.
method Established global functional calculus and Gårding inequality for pseudo-differential operators without local coordinates.
result Global solvability for a class of evolution problems.
We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a …
Characterizes symplectic and variational operators for scalar evolution equations.
problem Understanding the cohomology spaces and operators for scalar evolution equations.
method Analyzes cohomology spaces and uses isomorphisms to characterize operators.
result Cohomology spaces and operator spaces are isomorphic for certain scalar evolution equations.
The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.
problem Prove global existence of the Willmore flow in higher dimensions.
method Apply Michael-Simon-Sobolev inequality and Gagliardo-Nirenberg inequalities to establish local energy estimates and maximal existence time.
result Global existence of the Willmore flow in higher dimensions is proven.
Study of curve evolution in 2D space forms converging to a circle.
problem Understanding curve evolution in 2D space forms.
method Inverse curvature flow with normal speed defined by weighted inverse curvature and support function.
result Solutions exist for all time and converge exponentially to a standard round geodesic circle.
Survey of methods for solving smooth stochastic variational inequalities.
problem Solving smooth (strongly) monotone stochastic variational inequalities.
method Deterministic foundation, general stochastic formulation, finite sum setup, recent advances.
result Review of various methods for solving smooth stochastic variational inequalities.
Study solves perpetual American option pricing using variational inequality and difference equation.
problem Pricing perpetual American options.
method Proved maximum principle and uniqueness for variational inequality, provided existence and uniqueness for difference equation, and proved convergence of difference equation solution to variational inequality solution.
result Solution to difference equation converges to viscosity solution of variational inequality, showing perpetual American option prices converge as maturity approaches infinity.
Neural networks solve variational inequalities for optimal stopping problems.
problem Solving variational inequalities for optimal stopping problems in finance.
method Proposed neural network approach using loss functions directly incorporating variational inequality on whole domain.
result Existence and convergence of neural networks whose losses converge to zero.
We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savaré, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativi…
This paper combines three techniques to reduce communications in distributed variational inequalities.
problem Efficiently communicating solutions in large-scale distributed variational inequalities.
method Combining similarity, compression, and local steps to reduce communication rounds and cost.
result Best theoretical guarantees of communication complexity and superior performance in adversarial learning experiments.
New method predicts state evolution for non-first-order algorithms on nonconvex problems.
problem Analyzing nonconvex optimization problems with random data.
method Developed a state evolution for a broader class of algorithms including first-order and saddle point updates.
result Established rigorous state evolution predictions and finite-sample guarantees for non-first-order methods.
The geometry of a Lagrangian mechanical system is determined by its associated evolution semispray. We uniquely determine this semispray using the symplectic structure and the energy of the Lagrange space and the external force field. We study the variation of the energy and Lagrangian functions along the evolution and…
Finite time for subsolutions on Riemannian manifolds proved.
problem Finite extinction time for subsolutions of a specific equation on Riemannian manifolds.
method Proved finite extinction time using weighted Sobolev inequality and assumptions on p, q, and ρ.
result Weak subsolutions to the equation have a finite extinction time.
New framework uses PDE for no-regret generative modeling.
problem Developing efficient generative models for complex distributions.
method Iterative refinement of Brenier maps using mirror gradient descent.
result Converges to optimal Brenier map under various step-size schedules.
Using the result by D.Gessler (Differential Geom. Appl. 7 (1997) 303-324, DIPS-9/98, http://diffiety.ac.ru/preprint/98/09_98abs.htm), we show that any invariant variational bivector (resp., variational 2-form) on an evolution equation with nondegenerate right-hand side is Hamiltonian (resp., symplectic).
The paper constructs a Dirichlet form and proves functional inequalities for a specific measure.
problem Investigating functional inequalities for a specific measure in a configuration space.
method Constructing a strongly local symmetric Dirichlet form on the configuration space and proving various inequalities.
result The Dirichlet form satisfies the Bakry-Émery gradient estimate with K = 0 K=0 K = 0 and yields various functional inequalities. The extragradient method fails for hypomonotone variational inequalities.
problem The convergence of the extragradient method for hypomonotone variational inequalities.
method Application of the extragradient method to hypomonotone linear operators.
result The extragradient method diverges for hypomonotone variational inequalities.
Extends tracking guarantees for time-varying variational inequalities.
problem Tracking solutions of time-varying variational inequalities.
method Extends existing results to sublinear solution paths and periodic problems.
result Discrete dynamical systems of periodic time-varying VI can exhibit chaotic behavior or converge to the solution.
Study finds loops with specific curvature exist using Hardy's inequality.
problem Existence of closed planar loops with prescribed curvature.
method Variational approach, Hardy's inequality and associated functional space.
result Existence of loops with specific curvature proven.
New proof of Gaffney's inequality for differential forms on manifolds with boundary.
problem Proving Gaffney's inequality for differential forms on manifolds with boundary.
method Variational approach combined with Bochner's technique.
result New proof of Gaffney's inequality for differential forms.
ES optimization improved by structured control variates.
problem Improving accuracy of Evolution Strategies in RL.
method RL-specific variance reduction through structured control variates.
result Structured control variates outperform general variance reduction methods.
The paper derives oracle inequalities for estimators with fast and slow rates.
problem Developing fast and slow oracle inequalities for estimators.
method Direct study of analysis estimator and adaptation of Dalalyan, Hebiri and Lederer's arguments.
result Constant-friendly rates for (square root) total variation regularized estimators over graphs.
A new method solves variational inequality problems with multiple constraints without needing optimal Lagrange multipliers.
problem Solving variational inequality problems with multiple functional constraints efficiently.
method Constrained Gradient Method (CGM) for Minty variational inequality problems.
result The Constrained Gradient Method achieves complexity similar to projection-based methods but with cheaper oracles.
Stochastic Variational Optimization is a parallelizable method for gradient estimation.
problem Gradient estimation for differentiable objectives in parallel environments.
method Variational Optimization, Natural Evolution Strategies, Gaussian Perturbation, Directional Derivatives.
result Directional Derivatives are preferable to Variational Optimization for parallel Stochastic Gradient Descent.
Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.
problem Efficiently solving constrained convex-concave min-max problems and monotone variational inequalities.
method Higher-order methods achieving iteration complexities of O(1/T^{ rac{p+1}{2}}) for p-th order derivatives.
result Achieved improved convergence rates for min-max and monotone variational inequalities.
New algorithms reduce variance in solving complex mathematical problems.
problem Solving convex-concave saddle point problems, variational inequalities, and inclusions.
method Stochastic variance reduction for extragradient, forward-backward-forward, and forward-reflected-backward methods.
result All proposed methods converge with complexities matching or improving deterministic counterparts.
The paper studies extremal hypersurfaces in ellipsoids using centro-affine geometry.
problem Characterizing extremal hypersurfaces in centro-affine geometry.
method Analyzing invariant submanifold flows and deriving variational formulas.
result Circles on S 2 ( 1 ) \mathbb{S}^2(1) S 2 ( 1 ) with radius 6 / 3 \sqrt{6}/3 6 /3 are equi-centro-affine maximal. Quantum computer method for pricing lookback options with jumps.
problem Pricing lookback options with discrete monitoring and jump conditions.
method Variational Quantum Imaginary Time Evolution (VarQITE) method to solve non-Hermitian Schrodinger equation.
result Quantum algorithm can handle jump conditions in lookback options pricing.
New proof of isoperimetric inequality using Steiner's formula.
problem Proving the isoperimetric inequality in the plane.
method Direct proof using Steiner's formula, bypassing domain existence.
result Establishes the isoperimetric inequality directly.
Unified analysis of efficient local training methods for distributed variational inequalities.
problem Efficient distributed/federated learning for variational inequality problems.
method Unified convergence analysis of communication-efficient local training methods.
result First local gradient descent-accent algorithms with improved communication complexity.
This review explores Ricci soliton inequalities in Riemannian geometry.
problem Understanding geometric and analytic characteristics of Riemannian manifolds.
method Comprehensive study of Ricci soliton inequalities, summarizing historical evolution and current developments.
result Complex interactions between curvature conditions and geometric inequalities.
Unified approach for first-order methods with Markovian noise in stochastic optimization and variational inequalities.
problem Stochastic optimization problems with Markovian noise.
method Unified theoretical analysis of first-order gradient methods using randomized batching and multilevel Monte Carlo.
result Optimal (linear) dependence on the mixing time of the noise sequence, eliminating previous limiting assumptions.
Simplified proof for Frank and Lieb's inequality on Heisenberg group.
problem Proving the sharp Frank-Lieb inequality on the Heisenberg group.
method Simpler proof based on 2nd variation of subcritical functionals.
result A simpler proof of the inequality without the need for minimizer existence.
Unbiased wealth exchanges always lead to inequality.
problem Understanding wealth distribution in unbiased binary exchange systems.
method Analytical demonstration of unbiased binary exchanges leading to perfect inequality.
result Any system driven by unbiased binary exchanges will reach perfect inequality and zero mobility.
The paper proves reverse inequalities in various geometric settings using curvature radius data.
problem Proving reverse Alexandrov-Fenchel inequalities in different geometric settings.
method Using curvature radius data and associated evolute or focal maps.
result Sharp reverse Alexandrov-Fenchel estimates and inequalities in smooth convex curves and hypersurfaces.
We develop variational integrators from discrete Hamiltonian systems with external forces.
problem Creating accurate discrete models of continuous Hamiltonian systems.
method Constructing discrete Hamiltonian systems with external forces, analyzing symplectic structure, and combining methods to build variational integrators.
result We derive variational integrators that approximate continuous Hamiltonian systems with high accuracy.
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …
We give a simple, direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations including the Ricci and cross-curvature flows. The proof, based on a classical argument of Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in the place of Carleman…
New complex structures on jet spaces help explain Fock space dynamics.
problem Understanding dynamics of Fock spaces from variational principles.
method Endowing jet spaces with almost-complex structures, integrating to canonical complex structures, and analyzing Fock spaces.
result Holomorphic approximation explains dynamics of Fock spaces from variational principles.
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.