Let be a continuous map between a compact real analytic Kähler manifold and a compact complex {hyperbolic manifold} . In this paper we give a lower bound of the diastatic entropy of in terms of the diastatic entropy of and the degree of . When the lower bound i…
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Abstract: Study continuous maps between Kähler manifolds, proving a gap theorem.
problem Continuous maps between Kähler manifolds with negative curvature.
method Adapting Besson-Courtois-Gallot techniques to Kähler setting, proving gap theorem.
result Proves a gap theorem in terms of degree and diastatic entropies.
We give un upper bound Ent(Ω, g)<λ of the diastatic entropy Ent(Ω, g) of a complex bounded domain (Ω, g) in terms of the balanced condition (in Donaldson terminology) of the Kaehler metric λg. When (Ω, g) is a homogeneous bounded domain we show that the converse holds true, namely if Ent(Ω, g)<1 then g is balanced. Mor…
We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic Kähler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex hyperbolic space. As a corollary we obtain explicit lower bounds for the first eigenvalu…
Let be a real analytic Kaehler manifold. We say that a smooth map from a neighborhood of the origin of into is a {\em diastatic exponential} at if it satisfies $$(d \E_p)_0=\id_{T_pM},$$ $$D_p(\E_p (v))=g_p(v, v), \forall v\in W,$$ where is Calabi's diastasis function at $…
Enhances RL by controlling policy stochasticity through trajectory entropy constraints.
problem Non-stationary Q-value estimation and short-sighted entropy tuning in maximum entropy RL.
method Proposes TECRL framework with separate Q-functions for reward and entropy, enforcing a trajectory entropy constraint.
result DSAC-E algorithm achieves higher returns and better stability on OpenAI Gym benchmarks.
The paper calculates bounds for risk metrics and entropies under partial information constraints.
problem Analyzing risk metrics and entropies for unimodal, symmetric distributions with limited information.
method Develops lower and upper bounds for worst-case distortion riskmetrics and weighted entropy for unimodal, symmetric distributions with known mean and variance.
result Sharp upper bounds for distortion riskmetrics and weighted entropy for symmetric distributions.
The paper analyzes worst-case distortion risk metrics and weighted entropy under partial information.
problem Analyzing worst-case distortion risk metrics and weighted entropy with limited information.
method General distributions, partial information (mean and variance), various entropies and risk measures.
result Provides worst-case results for distortion risk metrics and weighted entropy.
Entropy measures geodesic flow complexity.
problem Measuring complexity of geodesic flows on manifolds.
method Introduced barcode entropy to measure exponential growth rate of not-too-short bars in Morse-theoretic barcodes.
result Barcode entropy bounds topological entropy and vice versa.
Entropy for uniform hypergraphs defined via tensor theory.
problem Entropy calculation for uniform hypergraphs.
method Probability distribution of generalized singular values from Laplacian tensors, Shannon entropy formula.
result Tensor entropy is a measure of regularity for uniform hypergraphs.
HCLM framework uses entropy regularization for open learning systems.
problem Real-world AI challenges and limitations of deep learning.
method Dynamical and information-theoretic framework with entropy regularization.
result Geometric entropy surrogates, especially log-determinant covariance entropy, induce stronger and more stable information forces.
Coupled entropy corrects flaws in Tsallis entropy for complex systems.
problem Misinterpretation of generalized temperature and entropy.
method Derived from generalized Pareto and Student's t distributions.
result Provides balanced measure of uncertainty for complex systems.
The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under perturbations.
method Study of topological entropy on Riemannian metrics with topology.
result Metrics with contractible closed geodesics have robust entropy.
Entropy rigidity for Finsler flows but collapse for Reeb flows.
problem Entropy behavior of Reeb and Finsler flows on contact manifolds.
method Analysis of topological entropy for Reeb and Finsler flows.
result Uniform positive lower bound for Finsler flows but arbitrarily small topological entropy for Reeb flows.
JES optimizes expensive functions by considering joint entropy over input and output spaces.
problem Optimizing expensive functions with limited evaluations.
method Joint Entropy Search (JES) considers joint entropy over input and output spaces.
result JES outperforms other information-theoretic methods in Bayesian optimization.
Unified framework for maximum entropy RL using Tsallis entropy.
problem Generalizing maximum entropy reinforcement learning with various entropies.
method Tsallis MDPs with Tsallis entropy maximization, controlling entropic index.
result Different entropic indices lead to different optimal policies and exploration tendencies.
We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic p…
Generic low-entropy hypersurfaces in 4-6D flow with only generic singularities.
problem Analyzing mean curvature flow of low-entropy hypersurfaces.
method Proving flow encounters only generic singularities for specific entropy conditions.
result Proves flow encounters only generic singularities for low-entropy initial data.
Researchers explore gauge freedom in entropies of -Gaussian measures.
problem Exploring the gauge freedom of entropies in -Gaussian measures.
method Introducing a refined -logarithmic function to demonstrate gauge freedom.
result Different escort expectations can lead to the same entropy but different relative entropies.
PFES uses entropy of Pareto-frontier for multi-objective Bayesian optimization.
problem Bayesian optimization for multi-objective problems, especially trade-off among objectives.
method Pareto-frontier entropy search (PFES) incorporating trade-off relation.
result PFES effectively incorporates dependency among objectives conditioned on Pareto-frontier.
Study shows rigidity for entropy minimizers in non-monotone cases.
problem Rigidity of entropy minimizers in non-monotone settings.
method Elementary proofs in non-monotone situations.
result Showed rigidity for minimizers of generalized Colding-Minicozzi entropies.
Unique expanders found with vanishing entropy.
problem Finding unique expanders with specific entropy properties.
method Adapting White's work and using Bernstein-Wang results.
result Generic uniqueness of expanders with vanishing relative entropy.
DAC enhances exploration in reinforcement learning with entropy regularization.
problem Improving exploration efficiency in reinforcement learning.
method Sample-aware entropy regularization using replay buffer action distributions.
result DAC significantly outperforms existing algorithms in reinforcement learning tasks.
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.
Study bounds self-shrinker entropy using Li-Yau volume and Colding-Minicozzi entropy.
problem Bounding entropy of self-shrinkers in arbitrary codimensions.
method Introduced stable conformal volume and virtual entropy to prove bounds.
result Entropy bounds are sharp and independent of codimension.
This paper controls a boundary term in Huisken's formula for entropy.
problem Entropy of translators and its behavior under mean curvature flow.
method Geometrically natural control of the boundary term in Huisken's monotonicity formula.
result Entropy of compact translators is bounded by boundary entropy and maximal cone density.
In 1870s, L. Boltzmann proved the famous -theorem for the Boltzmann equation in the kinetic theory of gas and gave the statistical interpretation of the thermodynamic entropy. In 2002, G. Perelman introduced the notion of -entropy and proved the -entropy formula for the Ricci flow. This plays a crucial role in…
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.
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.
We study the problem of discovering the simplest latent variable that can make two observed discrete variables conditionally independent. The minimum entropy required for such a latent is known as common entropy in information theory. We extend this notion to Renyi common entropy by minimizing the Renyi entropy of the …
Low-entropy surfaces can be flowed into spheres and cylinders.
problem Proving mean curvature flow for low-entropy hypersurfaces.
method Low-entropy density drop argument and recent work on hypersurfaces.
result Closed hypersurfaces with entropy ≤ 2 can be flowed into spherical and cylindrical shapes.
Quantum machine learning uses quantum cross entropy to minimize loss, but measurement loss affects this process.
problem Quantum machine learning's loss minimization through cross entropy is affected by measurement outcomes.
method Defined quantum cross entropy, proved its lower bounds, and investigated its relation to quantum fidelity and likelihood.
result Quantum cross entropy is lower-bounded by negative log-likelihood when derived from quantum data, but measurement outcomes can cause loss.
Surveying entropies for negatively curved manifolds and their relations.
problem Understanding entropy in negatively curved manifolds.
method Exploring various entropy concepts and their interconnections.
result Relations between different entropy notions for negatively curved manifolds.
Entropy from Machine Learningcond-mat.stat-mech
Machine learning for entropy calculation from binary signals.
problem Calculating entropy from binary configurations/signals.
method Transformed entropy calculation into supervised classification tasks using machine learning.
result Reproduced entropy and free energy of the 2D Ising model.
On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spacesmath.DG
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's -entropy and Shannon entropy power to super Ricci flows.
result Equivalence between volume non-local collapsing property and lower boundedness of -entropy on RCD spaces.
Entropy and Economicsq-fin.GN
Entropy helps explain disorder in both macro and micro systems.
problem Connecting macro and micro systems with entropy analysis.
method Analyzing entropy from both macroscopic and microscopic perspectives.
result Entropy measures disorder in both macroscopic and microscopic systems.
Structured entropy improves classification performance on structured targets.
problem Cross-entropy loss fails to account for target variable structure.
method Proposes structured entropy, a generalization of entropy using random partitions.
result Structured cross-entropy loss yields better results on classification problems with known structure.
The article presents a new entropy model for assessing stock market interest.
problem Assessing investor interest and market sentiment in exchange-traded securities.
method Intrinsic entropy model using actual trading data, without exogenous factors.
result Empirical evidence supports the model's ability to predict trading activity.
Paper proposes a method to estimate Transfer Entropy using Copula Entropy.
problem Estimating Transfer Entropy for causal discovery.
method Non-parametric method based on Copula Entropy.
result The proposed method effectively infers causality relationships from data.
Entropy data replaces classical charts for smooth manifolds.
problem Establishing smooth structures on topological manifolds.
method Using entropy data to define admissible coordinate functions and reconstruct smooth atlases.
result Entropy-smooth structures are equivalent to classical smooth structures and stable under perturbations.
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.
Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces.
problem Proving uniqueness of measure of maximal entropy for geodesic flows on surfaces.
method Analyzes geodesic flows on closed orientable C^∞ surfaces, proving uniqueness of measure of maximal entropy and at most one SRB measure.
result Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces, covering previous results and new examples.
State entropy regularization improves robustness in reinforcement learning, especially under structured perturbations.
problem Structured and spatially correlated perturbations in reinforcement learning.
method State entropy regularization, compared to policy entropy.
result State entropy regularization provides better robustness to structured and spatially correlated perturbations.
The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied by Slomczynski and Zastawniak (Ann. Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability space…
The paper analyzes Perelman's entropies on manifolds with conical singularities.
problem Analyzing manifolds with conical singularities using Perelman's entropies.
method Employing singular Ricci de Turck flow and Perelman's entropies to study manifolds with conical singularities.
result Entropy is monotone along the singular Ricci de Turck flow and helps in proving properties of Ricci solitons.
Study on relative entropy for hypersurfaces in hyperbolic space.
problem Understanding relative entropy for hypersurfaces in hyperbolic space.
method Relate relative entropy to renormalized area and apply monotonicity formula to mean curvature flows.
result Obtained a monotonicity formula for relative entropy in hyperbolic space.
Minimal volume entropy vanishes for mapping tori over 3-manifolds.
problem Volume entropy of mapping tori over 3-manifolds.
method A variation of amenable category and minimal volume entropy of a homology class.
result Minimal volume entropy vanishes.
The article proves a new entropy formula for surfaces with boundaries.
problem Entropy formula for surfaces with boundaries.
method Established a monotonicity formula of Hamilton type entropy.
result Entropy functional and -functional relation studied.