Study hypothesis testing under quantized samples with communication constraints, achieving near-optimal sample complexity.
problem Optimizing hypothesis testing with quantized samples and communication constraints.
method Developed a polynomial-time algorithm achieving near-optimal sample complexity under communication constraints.
result Achieved near-optimal sample complexity under communication constraints, with a logarithmic factor increase over unconstrained setting.
Sharp stability results for reverse isoperimetric inequalities in 2D.
problem Reverse isoperimetric inequalities in the plane.
method Stability analysis of λ-convex bodies and convex bodies with smooth boundaries. result Sharp stability results for reverse isoperimetric inequalities, including inradius and Cheeger inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
Proof of reverse isoperimetric inequality for black holes.
problem Reverse isoperimetric inequality for black holes in Einstein gravity.
method Geometric-analytic approach.
result Reversal of the usual isoperimetric inequality is explained by curved backgrounds governed by Einstein's equations.
We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the Γ-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev…
The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.
problem Analyzing the negative gradient flow of the Willmore energy plus volume.
method Proved a quantitative reverse isoperimetric inequality and applied it to the flow.
result Initial surfaces converge to a round point in finite or infinite time.
Established a Hardy inequality on Finsler manifolds.
problem Hardy inequality on Finsler manifolds.
method Used geometric properties of Finsler structures to prove the inequality.
result Depends on reversibility constant and uniformity constant of Finsler structure.
In the last decade, a large body of literature has been developed to explain the universal features of inequality in terms of income and wealth. By now, it is established that the distributions of income and wealth in various economies show a number of statistical regularities. There are several models to explain such …
New optimal isosystolic inequality found for Finsler reversible 2-tori.
problem Optimal isosystolic inequalities on Finsler tori.
method Survey and new inequality derived from prior work.
result Busemann-Hausdorff area of a Finsler reversible 2-torus with unit systole is at least π/4.
New findings show functional inequalities fail on Finsler manifolds with positive S-curvature.
problem Failure of functional inequalities on Finsler manifolds with positive S-curvature.
method Analysis of Finsler metric measure manifolds with reversibility, flag curvature, and S-curvature.
result Functional inequalities fail on Finsler manifolds with positive S-curvature.
Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.
problem Understanding Hardy and spectral gap inequalities on irreversible Finsler manifolds.
method Finslerian extension of the method of Riccati pairs.
result Sharpness of Hardy and spectral gap inequalities on specific Finsler manifolds.
We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.
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.
Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projec…
New inequality for eigenfunctions on curved spaces.
problem Eigenfunctions on non-smooth spaces with Ricci curvature.
method Sharp reverse-Hölder inequality for Dirichlet Laplacian eigenfunctions.
result Generalizes classical comparison theorem to curved spaces.
We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature ⩾c. We also study the equality case.
Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.
problem Establishing reverse Hölder inequalities on Kähler metrics of Fano varieties.
method Using log-concavity and properties of Ricci potentials, the inequality is proven for Fano manifolds with log terminal singularities.
result The inequality holds for Fano varieties with log terminal singularities and the constant depends only on p and the dimension of X.
We generalize Brendle's geometric inequality considered in \cite{B} to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperb…
Starting from a sequence of independent Wright-Fisher diffusion processes on [0,1], we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $MbeacompleteRiemnnianmanifoldandμthedistributionofthediffusionprocessgeneratedby\ff 1 2\DD+ZwhereZ$…
Defines signed quasiregular curves and proves growth theorem.
problem Understanding growth of signed quasiregular curves.
method Proves weak reverse Hölder inequality and uses it to prove growth theorem.
result Proves growth theorem for signed quasiregular curves.
We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures μ,ν where Pt is the heat semigroup and W1 is the ℓ1-Wasserstein distance. This turns out to be an equivalent formulation of a version of…
The paper proves inequalities for Steklov eigenvalues on finite graphs.
problem Eigenvalues of Laplacians for reversible Markov chains and Steklov eigenvalues.
method Generalized Cheeger inequalities, convergence results, and resolvent convergence.
result Sharp estimate for the first non-trivial Steklov eigenvalue.
New approach to concentration inequalities for unbounded state space dynamical systems.
problem Concentration inequalities for unbounded state space dynamical systems.
method Functional analytic framework, transport-entropy inequality.
result Exponential concentration inequalities for sampling from stationary distribution.
We study the validity of the Lp inequality for the Riesz transform when p>2 and of its reverse inequality when p<2 on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.
Sharp inequalities for curved surfaces and cones.
problem Optimizing areas in nonpositively curved spaces.
method Proving inequalities for disks and triangles in cones.
result Minimal area properties for specific shapes in cones.
Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by opti…
We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usual Holder inequality, which may be of independent interest.
This paper investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by \emph{general}, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a u…
The paper develops a new probabilistic framework for denoising diffusion models using free entropy and stochastic analysis.
problem Developing a mathematical framework for denoising diffusion models in noncommutative settings.
method Formulating diffusion and reverse processes governed by operator-valued stochastic dynamics, using tools from free stochastic analysis.
result Establishing an information-geometric link between entropy production, transport, and deconvolution.
Let M be a smooth connected manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying L1=0, and which is symmetric with respect to μ. We show that if L satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced…
Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide t…
The paper proves Hardy inequalities on Finsler manifolds using superharmonicity.
problem Establishing Hardy inequalities on Finsler manifolds.
method Using superharmonicity of a weight function and properties of the Finsler-Laplace operator.
result Generalization of Riemannian Hardy inequalities to Finsler manifolds.
The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.
problem Improving inequalities for Kähler-Einstein manifolds.
method Using invariant theory and curvature conditions to express and improve inequalities.
result Improved inequalities for Kähler-Einstein manifolds with smaller pinching constants.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
In 3D space forms, a lens minimizes volume for a fixed surface area.
problem Finding the shape with minimal volume for a given surface area in 3D space forms.
method Proving a sharp reverse isoperimetric inequality for λ-convex bodies. result The λ-convex lens minimizes volume for a fixed surface area in 3D space forms. We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formula established by Sturm and the author. We show the L1-gradient estimate on Finsler manifolds (under some additional assumptions in the n…
RML improves generative modeling of complex distributions.
problem Learning complex distributions in applications.
method RML defines a forward process to a known distribution, then learns a reverse Markov process.
result RML efficiently captures complex distributions in simulations and climate data.
A reverse Riesz estimate and spectral gap imply a Poincaré inequality.
problem Establishing a Poincaré inequality using a reverse Riesz estimate and spectral gap.
method Combining a reverse Riesz estimate and spectral gap condition to prove a Poincaré inequality.
result A Poincaré inequality is derived from a reverse Riesz estimate and spectral gap condition.
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in R2. result New inequalities and proofs for star bodies.
We study some equivalent properties of the curvature-dimension conditions CD(n,K) inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. And we also obtain one equivalent property of gradient estimate for a new notion o…
Machine learning infers time-reversible dynamics from data.
problem Learn time-reversible dynamics constrained by initial and final conditions.
method Machine learning algorithms solve boundary value problems for deterministic and stochastic dynamics.
result Inferred time-reversible dynamics for various types of systems.
Sharp 2-Wasserstein bounds for DDPMs derived from Föllmer process.
problem Sampling error bounds for DDPMs in 2-Wasserstein distance.
method Lipschitz-type conditions on score function, Föllmer process, and log-concave target distributions.
result Sharp upper bounds for DDPMs in 2-Wasserstein distance, optimal in dimension and steps.
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
Unified technique for sequential estimation of convex divergences.
problem Estimating convex divergences between distributions.
method Martingale methods and maximal inequalities for reverse submartingales.
result Valid time-uniform confidence sequences for arbitrary stopping times.
New algorithm learns bridged diffusion processes without time-reversals.
problem Learning bridged diffusion processes efficiently and accurately.
method Score matching with Doob's h-transform, avoiding time-reversals.
result Outperforms existing methods in learning bridged diffusion processes.
Minimal stretch factors for certain pseudo-Anosov maps are bounded.
problem Bounding the stretch factor of orientation-reversing pseudo-Anosov maps.
method Using the silver ratio and properties of puncture orbits to derive bounds.
result The bound λ(f)−χ(S)≥σ2 is asymptotically sharp. In this paper, we study the sharp constants of quantitative Hardy and Rellich inequalities on nonreversible Finsler manifolds equipped with arbitrary measures. In particular, these inequalities can be globally refined by adding remainder terms like the Brezis-Vázquez improvement, if Finsler manifolds are of strictly ne…
Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality for Finsler manifolds with specific curvature properties.
method Analyzing measured Finsler manifolds with non-negative Ricci curvature and Euclidean volume growth.
result Sharp isoperimetric inequality and rigidity results for the inequality.