New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
The paper explores how information geometry impacts classical CR inequalities.
problem Deriving and generalizing CR inequalities using information geometry.
method Examining Eguchi's theory and applying Amari-Nagoaka's theory to KL-divergence, and then extending to other divergences.
result Generalized CR inequalities derived from various divergences.
In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual a…
New measures generalize existing ones, linking information and risk.
problem Linking information measures and risk in statistical decision problems.
method Introducing new families of divergence measures and deriving an information processing equality.
result Extension of variational φ-divergence representation to multiple distributions. Innovative inequalities for divergences with applications in PAC-Bayesian bounds and Monte Carlo.
problem Developing new inequalities for divergences.
method Introducing novel change of measure inequalities for f-divergences and α-divergences. result Applications in PAC-Bayesian bounds and Monte Carlo estimates.
Study of generalized Csiszár divergences and their application to Cramér-Rao bounds.
problem Deriving lower bounds for estimator variance using generalized divergences.
method Applied Eguchi's theory to derive Fisher information metric and dual affine connections.
result More widely applicable Cramér-Rao inequality for escort distributions.
In this paper, we introduce new classes of divergences by extending the definitions of the Bregman divergence and the skew Jensen divergence. These new divergence classes (g-Bregman divergence and skew g-Jensen divergence) satisfy some properties similar to the Bregman or skew Jensen divergence. We show these g-diverge…
f-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics and information theory such as Kullback-Leibler divergence, chi-squared divergence, squared Hellinger distance, total variation distance e…
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
In this paper, we derive a useful lower bound for the Kullback-Leibler divergence (KL-divergence) based on the Hammersley-Chapman-Robbins bound (HCRB). The HCRB states that the variance of an estimator is bounded from below by the Chi-square divergence and the expectation value of the estimator. By using the relation b…
Study explores relationship between Hölder and FDPD divergences.
problem Understanding the relationship between Hölder and FDPD divergences.
method Intersection and generalization of divergence families, proving nonnegativity, deriving inequalities.
result Established ξ-Hölder divergence and derived inequalities. Study proves symmetry of bounded domains in Riemannian manifolds.
problem Symmetry of bounded domains in Riemannian manifolds.
method Integral identities and P-function method. result Equality implies the domain is isometric to a Euclidean ball.
The paper finds inequalities for eigenvalues of operators on immersed manifolds.
problem Finding inequalities for eigenvalues of operators on immersed manifolds.
method Computing inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space.
result Universal inequalities for eigenvalues of operators are computed.
Study on Lp affine surface areas and their inequalities for convex bodies.
problem Understanding weighted Lp affine surface areas in convex bodies. method Investigating valuations, isoperimetric inequalities, and connections to f divergences. result Established isoperimetric inequalities for weighted Lp affine surface areas. The paper analyzes convergence rates of Langevin dynamics and Proximal Sampler using Φ-divergence.
problem Analyzing convergence rates of Langevin dynamics and Proximal Sampler.
method Extending mixing time analyses to Φ-divergence, using strong data processing inequalities. result Convergence of Φ-divergence to 0 exponentially fast along Unadjusted Langevin Algorithm and Proximal Sampler. Paper relaxes triangle inequality for KL divergence between Gaussian distributions.
problem KL divergence does not satisfy triangle inequality for Gaussian distributions.
method Investigates relaxed triangle inequality and finds supremum.
result Supremum of KL divergence is found and conditions for attaining it are determined.
Study on geometric Jensen-Shannon divergence for Gaussian measures in Hilbert space.
problem Computing divergence between Gaussian measures in infinite-dimensional Hilbert space.
method Closed form expression and regularization for divergence calculation.
result Closed form expression and regularization for Geometric Jensen-Shannon divergence.
Estimating divergences in a consistent way is of great importance in many machine learning tasks. Although this is a fundamental problem in nonparametric statistics, to the best of our knowledge there has been no finite sample exponential inequality convergence bound derived for any divergence estimators. The main cont…
In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues…
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.
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
Unified framework for deriving generalization bounds in supervised learning.
problem Generalization error bounds in supervised learning.
method Data Processing Inequality PAC-Bayesian framework.
result Unified bounds on binary Kullback-Leibler generalization gap for various divergences.
Faster algorithm for sampling logconcave densities in high dimensions.
problem Cubic barrier in sampling logconcave densities from a cold start.
method Two key ingredients: weaker distance sampling and refined log-Sobolev inequality.
result First sub-cubic sampling algorithms for isotropic position.
New PAC-Bayes bounds derived using Legendre transform and f-divergences.
problem Deriving PAC-Bayes bounds under various assumptions.
method Combining Legendre transform and Fenchel--Young inequality to derive change-of-measure inequalities.
result Extended PAC-Bayesian guarantees under tailored assumptions.
New bounds for Neyman-Pearson region using f-divergences.
problem Bounding the Neyman-Pearson region for hypothesis testing.
method Establishing novel lower and upper bounds using f-divergences. result Best possible lower bound for the Neyman-Pearson boundary using hockey-stick f-divergences. Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth-order elliptic operators on Riemannian manifolds.
method Proves inequalities using Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds.
result Generalizes eigenvalue inequalities for the clamped plate problem to complete Riemannian manifolds.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.
problem Eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.
method Extending the LII operator to Lν, establishing a general formula for eigenvalues, and applying it to estimate eigenvalues on Riemannian manifolds. result Established eigenvalue inequalities for the Lν2 operator on translating solitons and other geometric settings. New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.
M-FISHER detects and adapts to streaming data shifts with statistical validity and stability.
problem Detecting and adapting to distributional shifts in streaming data.
method Constructs an exponential martingale from non-conformity scores and applies Ville's inequality for detection. Fisher-preconditioned updates for adaptation.
result Establishes M-FISHER as a principled approach for robust, anytime-valid detection and geometrically stable adaptation.
The paper proves geometric inequalities in sphere using locally constrained flows.
problem Deriving geometric inequalities in sphere.
method Established the longtime existence and convergence of a locally constrained flow.
result Proved new families of three-term geometric inequalities in sphere.
Paper studies eigenvalues of a specific operator on Riemannian manifolds.
problem Eigenvalues of a specific operator on Riemannian manifolds.
method Established a general formula for eigenvalues and derived estimates.
result Obtained universal inequalities for the eigenvalues on translating solitons.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
Method identifies low-dimensional structure in high-dimensional probability measures.
problem Identifying low-dimensional structure in high-dimensional probability measures.
method Extends prior work on minimizing majorizations of the Kullback-Leibler divergence to identify optimal approximations within a specific class of measures.
result Connection between dimensional logarithmic Sobolev inequality and approximations with the ansatz.
This paper sets a lower bound for sample complexity in inverse reinforcement learning.
problem Finding a reward function that generates a desired optimal policy in MDPs.
method Information-theoretic lower bound using geometric construction and Fano's inequality.
result An O(nlogn) sample complexity lower bound for IRL problems. We study the logarithmic L(α)-divergence which extrapolates the Bregman divergence and corresponds to solutions to novel optimal transport problems. We show that this logarithmic divergence is equivalent to a conformal transformation of the Bregman divergence, and, via an explicit affine immersion, is equivalent t…
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
Study geometric inequalities for quasi-Einstein manifolds using new formulas.
problem Investigate geometric inequalities on quasi-Einstein manifolds.
method Use generalized Reilly's formulas and establish new boundary estimates and isoperimetric inequalities.
result Present a Heintze-Karcher type inequality for compact quasi-Einstein manifolds.
Study on electrostatic systems with boundary, proving new geometric inequalities.
problem Electrostatic systems with boundary in higher dimensions.
method Investigation of electrostatic systems on compact manifolds with boundary, establishing new geometric properties.
result Proved sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds.
The paper finds inequalities in Grassmannian geometry.
problem Understanding geometric properties of Grassmannians.
method Analyzes inequalities for elements in Grassmannians.
result Law of Cosines and geodesic triangle inequalities.
The paper establishes sharp geometric inequalities for hypersurfaces in warped product manifolds.
problem Geometric inequalities involving three distinct quantities in warped product manifolds.
method Two families of inequalities comparing three geometric quantities in space forms or warped product manifolds.
result Generalizes and extends previous results on Weinstock-type inequalities and Steklov/Wentzell eigenvalues.
We unify f-divergences, Bregman divergences, surrogate loss bounds (regret bounds), proper scoring rules, matching losses, cost curves, ROC-curves and information. We do this by systematically studying integral and variational representations of these objects and in so doing identify their primitives which all are rela…
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
Rényi divergence is related to Rényi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its or…
The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.
problem Proving geometric inequalities on smooth oriented Riemannian manifolds.
method Introducing symmetric decreasing rearrangement inequalities and testing their applicability to Riemannian manifolds.
result Smooth co-area formula and re-formulated geometric inequalities on Riemannian manifolds.
This paper explains a technique for proving geometric inequalities.
problem Proving various geometric inequalities in different contexts.
method Unified framework based on Alexandrov-Bakelman-Pucci technique.
result Unified approach to proving geometric inequalities.
Warped product affects divergences in information geometry.
problem Warped product's impact on divergences in information geometry.
method Study of warped product on information geometry.
result Warped product does not preserve canonical divergences.
New proof finds three divergence-free vector fields for any 3D manifold.
problem Proving the existence of divergence-free vector fields on 3D manifolds.
method Using geometric properties of eigenspinors in three dimensions.
result Found three divergence-free vector fields that are orthogonal and have the same length at every point.