The paper studies optimal transport in linear quadratic systems and derives interpolation inequalities.
problem Optimal transport problem in Linear Quadratic optimal control systems.
method Well-posedness of the Monge problem, regularity of optimal transport map, displacement interpolation of measures.
result Derivation of general interpolation inequalities for entropy functionals.
In this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature…
The paper connects Ricci curvature to entropy convexity in one dimension.
problem Understanding curvature bounds in one-dimensional spaces.
method Proving equivalence between 1-weighted Ricci curvature and entropy convexity.
result Established equivalence between curvature bounds and entropy convexity.
The paper proves new Harnack inequalities for various nonlinear heat equations on manifolds.
problem Analyzing and proving new Harnack inequalities for nonlinear heat equations.
method Proving constrained trace, matrix, and interpolated Harnack inequalities for specific nonlinear heat equations.
result Derives new differential Harnack inequalities with time-exponential correction terms.
The study proves inequalities linking gradient norms and Laplace operator solutions on Riemannian manifolds.
problem Establishing global Sobolev regularity for solutions of the Poisson equation and magnetic Schrödinger semigroups.
method Proved second order Lp-interpolation inequalities and used abstract functional analytic arguments. result New global Sobolev regularity results for Lp-solutions of the Poisson equation and magnetic Schrödinger semigroups. We show that stochastic interpolation flow maps are Lipschitz with a sharp constant.
problem High dimensional sampling and transport problems.
method Investigating stochastic interpolation flow for generating data samples.
result Stochastic interpolation flow maps are Lipschitz with a sharp constant matching optimal transport maps.
Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.
problem Extend Cheeger inequalities to simplicial complexes and their higher order Laplacians.
method Combining constructions from simplicial topology, signed graphs, Gromov filling radii, and interpolating between 1-Laplacians and 2-Laplacians.
result Developed a general theory for p-Laplacians on simplicial complexes and proved Cheeger-type inequalities.
Ideal sub-Riemannian manifolds support interpolation inequalities for optimal transport.
problem Optimal transport on sub-Riemannian manifolds.
method Sub-Riemannian Jacobi fields and distortion coefficients.
result Ideal sub-Riemannian manifolds support interpolation inequalities.
The paper proves inequalities for optimal transport on sub-Finslerian manifolds.
problem Optimal transport inequalities on sub-Finslerian manifolds.
method Introduction of sub-Finslerian Jacobi fields and optimal transport theory.
result Characterization of generalized distortion coefficients and fundamental geometric inequalities.
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the ε-Ricci flow on a closed surface. These new Harnac…
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
We refine and generalize several interpolation inequalities bounding the Lp norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich distance to μ on a smooth weighted Riemannian manifold satisfying CD(0,∞) condition.
Study bounds for Brownian motion on manifolds with sticky boundary conditions.
problem Proving geometric bounds for Brownian motion on manifolds with sticky boundary conditions.
method Interpolation involving energy interactions between boundary and interior of the manifold.
result Explicit geometric bounds on Steklov eigenvalues, boundary trace operators, and boundary trace logarithmic Sobolev constants.
The study shows that close hypersurfaces have uniformly bounded inequalities.
problem Bounding inequalities for close hypersurfaces.
method Analyzing families of smooth hypersurfaces close to a fixed one.
result Uniformly bounded constants in Sobolev, Gagliardo-Nirenberg, and geometric Calderón-Zygmund inequalities.
We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied…
We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimen…
This paper approximates Gaussian process emulators with constraints and noisy data.
problem Realistic stochastic emulators with inequality constraints and noisy observations.
method Monte Carlo and Markov Chain Monte Carlo methods with noise term.
result Improved performance of MC and MCMC samplers with noisy observations and constraints.
New method proves inequalities for self-shrinkers using perturbation.
problem Proving Łojasiewicz inequalities for self-shrinkers.
method Perturbative analysis of a new auxiliary quantity.
result New method interpolates between higher order and differential geometric approaches.
Magnetic Brunn-Minkowski inequalities on Riemannian manifolds
problem Establishing equivalence between Brunn-Minkowski inequalities and magnetic Ricci curvature
method Using magnetic geodesics
result Proving a sharp, undistorted Brunn-Minkowski inequality
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
Study resolves conjecture on overparameterized linear models' generalization.
problem Asymptotic generalization of multiclass classification with overparameterized models.
method Gaussian covariates bi-level model, Hanson-Wright inequality variant.
result Min-norm interpolating classifier can be suboptimal compared to noninterpolating classifiers.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
The CR Yamabe flow converges exponentially to a contact form with flat curvature.
problem Analyzing the CR Yamabe flow with zero invariant.
method Used CR Poincaré inequality and Gagliardo-Nirenberg type interpolation inequality.
result The flow converges exponentially to a contact form with flat pseudo-Hermitian scalar curvature.
Characterizes a new curvature bound with convexity of entropies.
problem Lowering Ricci curvature bounds with ε-range.
method Characterization through convexity of entropies over Wasserstein space.
result Derives various interpolation and functional inequalities.
The paper establishes prediction bounds for trend filtering with higher order total variation penalties.
problem Estimating signals with jumps of varying orders using total variation regularization.
method Combining oracle inequalities and interpolating vectors to bound effective sparsity.
result The ℓ1-penalty on (k−1)extth order differences allows adaptive estimation for k∈{1,2,3,4}. The paper improves stability estimates for soap bubble theorem in curved domains.
problem Stability estimates for the Soap Bubble Theorem in curved domains.
method Leveraging Gagliardo-Nirenberg-type interpolation inequalities.
result Optimal stability estimates for Lr deviations of mean curvature from being constant. Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.
problem Investigate functional and geometric inequalities on hyperbolic spaces and Riemannian manifolds.
method Employ symmetrization and semigroup approach based on sharp estimates for heat semigroup.
result Developed robust inequalities and methods relying on geometric and isoperimetric properties.
The paper extends Hoeffding's inequality for Markov chains using a generalized concentrability condition.
problem Applying Hoeffding's inequality to non-ergodic Markov chains.
method Integrates generalized concentrability condition via IPM to extend traditional hypotheses.
result Demonstrates utility in machine learning applications such as empirical risk minimization and bandits.
Quantitative estimates for inequalities on sub-Riemannian manifolds.
problem Quantitative estimates for Lp-Poincaré and log-Sobolev inequalities on sub-Riemannian manifolds. method Introducing the Quasi Curvature-Dimension condition and applying it to various sub-Riemannian manifolds.
result Established quantitative estimates independent of the dimension on various sub-Riemannian manifolds.
We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from t…
LMC algorithm receives first convergence guarantees under weak smoothness conditions.
problem Convergence guarantees for LMC under weak smoothness conditions.
method Using Latała--Oleszkiewicz or modified log-Sobolev inequalities.
result First convergence guarantees for LMC under weak smoothness conditions.
Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
New stability bounds for GD in overparameterised shallow nets without NTK assumptions.
problem Generalisation and excess risk bounds for shallow neural networks.
method Oracle inequalities and stability analysis of GD without kernelisation.
result Oracle type bounds reveal GD's generalisation is controlled by an interpolating network with shortest GD path.
New methods using spacetime harmonic functions solve geometric inequalities.
problem Geometric inequalities involving mass in spacetime.
method Utilizing spacetime harmonic functions and other elliptic equations.
result Novel concept of total mass and proof of positive mass theorem.
The paper proves uncertainty principles on Finsler measure spaces.
problem Uncertainty principles on Finsler measure spaces.
method Analyzes Lp-uncertainty principles on Finsler measure spaces with bounded curvatures. result Sharp Lp-uncertainty principles are proven and characterized. Upper bounds on constants for Brownian motion with sticky boundary.
problem Bounding constants for Brownian motion with sticky boundary.
method Interpolation approach based on energy interactions and Reilly formula.
result Upper bounds on Poincaré and Logarithmic Sobolev constants.
A new projection method for convex optimization reduces computation costs.
problem Efficiently projecting points into convex sets for deep learning.
method Interpolation-based projection for cheaper computation.
result The proposed method converges for linear and convex constraints.
New method for handling multi-dimensional singular controls with jump costs in mean-field problems.
problem Handling jump costs in multi-dimensional singular controls.
method Introducing two-layer parametrisations to interpolate jumps on both distributional and pathwise levels.
result Derivation of a DPP and characterisation of the value function as a minimal super-solution to a quasi-variational inequality.
We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's ν-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family …
The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.
problem Nonlinear isocapacitary mass in 3-manifolds with nonnegative scalar curvature.
method Derives positive mass theorems and shows mass coincides with ADM mass under mild conditions.
result Nonlinear masses coincide with ADM mass and prove the Penrose inequality.
Calculates moments of sectional curvature on Riemannian manifolds.
problem Understanding the distribution and moments of sectional curvature.
method Integrating local Riemannian invariants and analyzing sectional curvature on Grassmann bundles.
result Proves a weak version of the Hitchin-Thorpe Inequality.
Graphons connect graph structures to manifold properties.
problem Interpolating between graphs and manifolds.
method Graph-to-graphon and graphon-to-manifold convergence.
result Established monotonicity inequality linking combinatorial and geometric parameters.
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
New curvature-dimension condition for Lagrangians on manifolds.
problem Establishing a curvature-dimension condition for autonomous Lagrangians.
method Generalizing Klartag's needle decomposition technique to Lagrangian setting.
result Equivalence of curvature-dimension condition to displacement convexity of entropy.
Probability distributions of money, income, and energy consumption per capita are studied for ensembles of economic agents. The principle of entropy maximization for partitioning of a limited resource gives exponential distributions for the investigated variables. A non-equilibrium difference of money temperatures betw…
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.
In this article, we introduce a 2-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometri…