Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
This paper proves a curvature entropy inequality for non-symmetric convex bodies.
problem Proving a curvature entropy inequality for non-symmetric convex bodies.
method Demonstrated the log-Minkowski inequality of curvature entropy for general convex bodies in 2D.
result Equivalence of cone-volume measure uniqueness, log-Minkowski volume inequality, and curvature entropy inequality for general convex bodies in 2D.
Entropy of Sinai-Ruelle-Bowen measure is continuous in convex projective structures.
problem Understanding the geometry of convex projective structures.
method Study of Sinai-Ruelle-Bowen measure entropy.
result Entropy is a continuous function.
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.
problem Characterizing non-negative curvature in Alexandrov spaces
method Constructing a parallel trivialization of the entropy tensor
result The entropy tensor is matrix displacement convex on Alexandrov spaces
The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.
problem Characterizing the null energy condition in Lorentzian manifolds.
method Characterization via convexity of the relative entropy along displacement interpolations on null hypersurfaces.
result The null energy condition is characterized in terms of convexity of the relative entropy.
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 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.
We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…
Symplectic homology matches dual capacities for convex domains.
problem Understanding symplectic capacities and Reeb flows on convex domains.
method Isomorphic filtered symplectic homology to dual singular homology.
result Gutt-Hutchings capacities match spectral invariants for convex domains.
Study spherical convex bodies using Lp-floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced Lp-floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
Unique entropy measure found for convex projective manifolds.
problem Entropy measure for convex projective manifolds.
method Developed Patterson--Sullivan densities and mixing theory.
result Unique mixing measure of maximal entropy exists.
The article extends mean curvature flow with surgery for low entropy hypersurfaces.
problem Extending mean curvature flow with surgery for hypersurfaces with low entropy.
method Mean curvature flow with surgery for mean convex hypersurfaces with entropy less than Λn−2, without assuming 2-convexity. result Smooth n-dimensional closed self shrinkers with entropy less than Λn−2 are isotopic to the round n-sphere. 2D simply connected translating solitons in slabs are convex and have entropy < 3.
problem Characterizing 2D translating solitons in slabs with entropy constraints.
method Analyzing the properties of translating solitons in slabs with entropy constraints.
result 2D simply connected translating solitons in slabs are convex and have entropy < 3.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
Compact foliations preserve entropy if leaves are strictly convex projective.
problem Entropy rigidity for foliations by strictly convex projective manifolds.
method Analysis of foliated volume entropies and homeomorphisms.
result Equality in foliated volume entropies implies homothetic leaves.
We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weig…
Study on non-archimedean μ-entropy for toric varieties, proving existence and uniqueness.
problem Exploring non-archimedean μ-entropy for toric varieties and its thermodynamical structure.
method Established a Rellich type compactness result for convex functions on simple polytope, proving existence and uniqueness of optimizer.
result Existence and uniqueness of optimizer for toric non-archimedean μ^λ-entropy for λ ≤ 0.
Investigates properties of volume, entropy, and diameter in higher Teichmüller spaces.
problem Properties of volume, entropy, and diameter for representations in mSO(p,q+1). method Uniform lower bound on entropy times volume, upper bound on entropy, finiteness and compactness results.
result Entropy is bounded by p−1 for representations conjugate to mS(mO(p,1)imesmO(q)). Entropy rigidity shows volume bounds for convex projective manifolds.
problem Entropy rigidity of convex projective manifolds.
method Adapting Besson--Courtois--Gallot's entropy rigidity result to Hilbert geometries.
result Hilbert volume is bounded below by a constant depending only on dimension.
Paper develops MRCs for supervised classification using generalized maximum entropy.
problem Developing robust classifiers for decision problems.
method Generalized maximum entropy principle applied to minimax risk classifiers.
result Learning techniques for determining MRCs with performance guarantees.
Abstract shows entropy and convexity definitions of very strict CD(K,N) spaces are equivalent.
problem Equivalence of definitions of very strict CD(K,N) spaces. method Showed equivalence of definitions using entropy functionals and full displacement convexity class.
result Equivalence of definitions of very strict CD(K,N) spaces. Entropy convexity characterizes strong energy condition in spacetimes.
problem Characterizing strong energy condition in nonsmooth spacetimes.
method Lifting fractional powers of Lorentz distance to probability measures and showing geodesic convexity of Boltzmann-Shannon entropy.
result Strong energy condition is equivalent to geodesic convexity of Boltzmann-Shannon entropy.
Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corol…
Proposes MGCE for improved classification performance.
problem Optimizing between robustness and optimization difficulty in classification.
method Minimax formulation of GCE leading to convex optimization over margins.
result MGCE achieves strong accuracy and better calibration, especially in noisy labels.
In an incomplete Brownian-motion market setting, we propose a convex monotonic pricing functional for nonattainable bounded contingent claims which is compatible with prices for attainable claims. The pricing functional is defined as the convex conjugate of a generalized entropy penalty functional and an interpretation…
Paper introduces robust market making using Wasserstein distance and entropy regularization.
problem Market making robustness under uncertainty.
method Wasserstein distance, entropy regularization, convex optimization, optimal radius selection.
result The robust market making problem can be reformulated as a convex optimization problem.
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.
Geometric inequalities for equipotential curves derived from convex entropy.
problem Geometric relations for equipotential curves defined by harmonic functions.
method Constructing an entropy for each level set and proving convexity.
result Geometric inequalities for curvature and gradient magnitude on equipotential curves.
Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary w…
We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and Blaschke metric are comparable.
It is shown that the volume entropy of a Hilbert geometry associated to an n-dimensional convex body of class C1,1 equals n−1. To achieve this result, a new projective invariant of convex bodies, similar to the centro-affine area, is constructed. In the case n=2, and without any assumption on the boundary, i…
Solves a complex geometric problem for symmetric convex bodies.
problem Conditions for a measure to be the dual curvature measure of a symmetric convex body.
method Variational approach using entropy and quermassintegrals, with estimates on entropy and curvature measures.
result Explicit conditions for the measure concentration, leading to a full solution for 1<q<n. We prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.
This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure …
This paper re-examines Bregman functions and their divergences, introducing new properties and functions.
problem Exploring properties and applications of Bregman functions and divergences.
method Re-examination of existing Bregman functions and introduction of new ones, providing sufficient conditions for construction.
result Several known Bregman functions are reclassified, and new Bregman functions are introduced.
New proof of Wulff-Gage inequality with applications.
problem Proving the Wulff-Gage isoperimetric inequality.
method Provided a new proof of the inequality for origin-symmetric convex bodies.
result Uniqueness of log-Minkowski problem and new proof of log-Minkowski inequality.
Proposes a method to solve deep neural networks' local minimum problem.
problem Local minimum problem in deep neural networks training.
method Transforms cross-entropy loss into risk-averse error criterion, adjusts RSI, and uses convexity region.
result Trained deep learning machine is expected to be inside a global minimum's attraction basin.
Proves mean convex neighborhood conjecture for ancient flows near singularities.
problem Proving mean convex neighborhood conjecture for mean curvature flow near singularities.
method General classification of ancient low entropy flows and mean curvature flow through singularities.
result Proves mean convex neighborhood conjecture for ancient flows near singularities.
Lower bound shows no acceleration for specific convex optimization class.
problem Proving lower bounds for convergence rates of convex optimization methods.
method Proving Ω(L/T) lower bound for minimization of convex and L-smooth functions relative to negative entropy. result Mirror descent is optimal up to a logarithmic factor in the class of functions considered.
The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface…
In this paper we study the deformation of strictly convex real projective structures on a closed surface. Specially we study the deformation in terms of the entropy on bulging deformations. As a byproduct we construct a sequence of divergent structures whose topological entropy converges to a designated number between …
The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.
problem Investigating dilation type inequalities on weighted Riemannian manifolds.
method Introducing dilation profile and comparing it with model space under lower weighted Ricci curvature bounds.
result Showed several functional inequalities related to various entropies.
The study constructs pressure form on Margulis spacetimes and proves their infinitesimal rigidity.
problem Understanding the infinitesimal rigidity of Margulis spacetimes.
method Constructing pressure form and studying its properties on the moduli space of Margulis spacetimes.
result Margulis spacetimes are infinitesimally determined by their marked Margulis invariant spectra.
New results on max-entropy distributions with succinct descriptions and stability.
problem Understanding the complexity and stability of max-entropy distributions.
method Polynomial-time algorithms and bounds on bit complexity.
result Polynomial bit complexity of ε-optimal dual solutions to max-entropy convex programs.
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
New insights into CE dynamics reveal how Hadamard initialization simplifies softmax.
problem Understanding the dynamics of cross-entropy training loss in deep learning.
method Analyzing a two-layer linear neural network with standard-basis vectors as inputs.
result Gradient flow on cross-entropy converges to neural collapse geometry, proving global convergence.