New flow method solves Christoffel-Minkowski problem.
problem Solving Christoffel-Minkowski problem.
method Entropy preserving curvature flow with global term.
result Entropy preserving flow solves Christoffel-Minkowski problem.
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.
Constructs entropy-minimizing pseudo-Anosov diffeomorphisms on K3 surfaces.
problem Finding minimal entropy diffeomorphisms on K3 surfaces.
method Constructs pseudo-Anosov diffeomorphisms minimizing entropy.
result Obtains infinitely many entropy-minimizing diffeomorphisms.
In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prov…
Survey on Ricci flow on spaces with conical singularities.
problem Analyzing Ricci flow on spaces with isolated conical singularities.
method Ricci de Turck flow preserving conical singularities, stability of Ricci flat metrics, preservation of positive scalar curvature under certain conditions.
result Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow.
Paper introduces entropy for Riemannian foliations and connects it to Ricci flow.
problem Entropy functional for Riemannian foliations and its relation to Ricci flow.
method Introduced entropy functional and related its gradient flow to transverse Ricci flow.
result Entropy functional is monotonic along transverse Ricci flow and related to it.
Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.
problem Uniqueness of dynamical invariants for 3D volume-preserving diffeomorphisms.
method Examined failure of uniqueness on integral homology spheres and arbitrary three-manifolds using local and global invariants.
result Failure of uniqueness is severe, with continuous and non-constant invariants appearing in C1-open sets of nonvanishing exact fields of fixed helicity. Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.
problem Analyzing the generalized Kähler-Ricci flow on toric Fano varieties.
method Establishing global existence, deriving entropy and energy functionals.
result Convergence of nonsingular solutions at infinity and weak convergence of the flow.
Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.
problem Understanding the asymptotic behavior of Goeritz groups for link decompositions.
method Defined Goeritz groups for link decompositions, analyzed their properties, and discussed their asymptotic behavior.
result Discussed the asymptotic behavior of minimal pseudo-Anosov entropies and related it to Goeritz groups of Heegaard splittings.
A new copula, the checkerboard copula, maximizes entropy and preserves dependence.
problem Choosing copula for non-continuous marginal distributions.
method Introducing the checkerboard copula, maximizing Shannon entropy.
result Checkerboard copula maximizes entropy and preserves dependence.
Study proves stability and uniqueness for a specific type of flow.
problem Volume-preserving mean curvature flow stability and uniqueness.
method New gradient flow calibrations for volume preservation, stability estimate in distributional solutions.
result Strong solutions are calibrated and stable under certain conditions.
AECF improves multimodal inference robustness and calibration.
problem Robustness and calibration issues in multimodal systems with missing inputs.
method Adaptive Entropy-Gated Contrastive Fusion (AECF) layer.
result Improves masked-input mAP by +18 pp at a 50% drop rate.
Entrocraft addresses RL performance saturation in LLMs by customizing entropy curves.
problem Performance saturation in RL algorithms for LLMs.
method Entrocraft uses rejection sampling to bias advantage distributions for customized entropy schedules.
result Entrocraft significantly improves generalization, output diversity, and long-term training in 4B models.
When a spacetime has boundaries, the entangling surface does not have to be necessarily compact and it may have boundaries as well. Then there appear a new, boundary, contribution to the entanglement entropy due to the intersection of the entangling surface with the boundary of the spacetime. We study the boundary cont…
A new framework describes dissipation using a metriplectic 4-bracket.
problem Describing dissipation in a way that preserves energy and entropy.
method Using a metriplectic 4-bracket, a quantity like the Poisson bracket with symmetries motivated by Riemannian curvature.
result The metriplectic 4-bracket dynamics includes all known previous binary bracket theories for dissipation.
The paper discovers a hidden component in data using an autoencoder with a discriminator.
problem Discovering a single independent latent variable in data.
method An autoencoder with a discriminator is used to recover the hidden component.
result The approach can recover the hidden component up to entropy-preserving transformations.
Entropy tracking reveals class commitment transitions in diffusion models.
problem Diffusion models lack reliable methods to detect semantic structure transitions.
method Tracking class-conditional entropy of latent variables.
result Entropy isolates noise regimes critical for semantic structure formation.
We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric…
Geodesic flows on compact manifolds without conjugate points are shown to have a unique measure of maximal entropy.
problem Analyzing geodesic flows on compact manifolds without conjugate points and with visibility universal covering.
method Using topological mixing, local product structure, and properties of geodesic flows, the authors prove the existence of an expansive factor and uniqueness of measure of maximal entropy.
result The geodesic flow on compact manifolds without conjugate points has a unique measure of maximal entropy.
Debiased Wasserstein barycenters improve on entropy regularization in OT.
problem Entropy regularization in OT introduces bias, leading to blurred barycenters.
method Propose debiased Wasserstein barycenters using Sinkhorn iterations.
result Debiased barycenters preserve fast Sinkhorn-like iterations without entropy smoothing bias.
Proposes an accuracy-preserving calibration method for DNNs.
problem Calibration of deep neural networks (DNNs) to measure prediction reliability.
method Uses Concrete distribution on the probability simplex to calibrate DNNs without accuracy loss.
result The proposed method outperforms previous methods in accuracy-preserving calibration tasks.
ALIEN improves uncertainty estimation of language models by refining entropy-based methods.
problem Overconfidence in uncertainty estimation for language models, especially for difficult inputs.
method ALIEN refines entropy-based uncertainty by aligning it with prediction reliability, using a lightweight uncertainty head.
result ALIEN consistently outperforms strong baselines in detecting incorrect predictions and achieving the lowest calibration error.
Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.
problem Understanding geodesic flows on compact surfaces without conjugate points.
method Time-preserving semi-conjugation to a continuous expansive flow.
result Geodesic flows on compact surfaces without conjugate points of genus > 1 have a unique measure of maximal entropy.
A quantum state generation method that respects physical constraints.
problem Generating quantum states with complex-valued Hermitian, positive semi-definite, and trace one properties.
method Mirror diffusion model with von Neumann entropy to enforce structural constraints.
result Demonstrated effective generation of quantum states with conditional guidance.
EntroPath learns manifold geometry from diffusion paths.
problem Learning geodesic geometry from data graphs with spurious shortcuts.
method Maximum Entropy Path Ensemble Embedding (MERW) with k-step diffusion paths.
result EntroPath converges to squared geodesic distance in the short-time limit.
Hyperbolic manifolds are stable under volume-preserving metrics.
problem Stability of hyperbolic metrics under volume-preserving deformations.
method Proof of stability using volume entropy and Plateau solutions.
result Hyperbolic metrics are stable under volume-preserving deformations.
The study establishes a curvature-dimension condition for discrete Markov chains.
problem Proving modified logarithmic Sobolev inequalities for discrete Markov chains.
method Identifying and proving a curvature-dimension inequality CDΥ(κ,∞), and showing its compatibility with diffusive settings. result The CDΥ condition preserves curvature bounds under tensorization and leads to Beckner inequalities. In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that 4λ1>h2, where λ1 is the bottom of the L2-spectrum and h the…
This work enhances collaborative inference privacy by minimizing conditional entropy and boosting robustness against model inversion attacks.
problem Privacy leakage in collaborative inference systems via model inversion attacks.
method Theoretical proof and derivation of a differentiable measure for bounding conditional entropy, followed by a CEM algorithm to maximize it.
result Theoretical proof and experimental validation show that CEM consistently boosts inversion robustness without compromising feature utility or efficiency.
This work addresses two main issues of the standard Kernel Entropy Component Analysis (KECA) algorithm: the optimization of the kernel decomposition and the optimization of the Gaussian kernel parameter. KECA roughly reduces to a sorting of the importance of kernel eigenvectors by entropy instead of by variance as in K…
The paper studies entropy calibration in language models and finds that miscalibration improves slowly with scale.
problem The problem is whether language model entropy calibration improves with scale and if it's possible to calibrate without reducing log loss.
method The authors study a simplified theoretical setting to characterize miscalibration scaling behavior and measure it empirically in language models ranging from 0.5B to 70B parameters.
result The observed scaling behavior of miscalibration is similar to theoretical predictions, indicating slow improvement with scale. The authors also prove theoretically that it is possible to reduce entropy while preserving log loss if access to a black box predicting future entropy is available.
We define On-Average KL-Privacy and present its properties and connections to differential privacy, generalization and information-theoretic quantities including max-information and mutual information. The new definition significantly weakens differential privacy, while preserving its minimalistic design features such …
In this paper, we extend the construction of pressure metrics to Teichmüller spaces of surfaces with punctures. This construction recovers Thurston's Riemannian metric on Teichmüller spaces. Moreover, we prove the real analyticity and the convexity of Manhattan curves of the finite area type-preserving Fuchsian represe…
Despite its well-known shortcomings, k-means remains one of the most widely used approaches to data clustering. Current research continues to tackle its flaws while attempting to preserve its simplicity. Recently, the \textit{power k-means} algorithm was proposed to avoid trapping in local minima by annealing throu…
Proposes a link between randomness and compression in deep learning.
problem Improving efficiency in deep learning training.
method Introduces a novel tomographic compression framework called Dual Tomographic Compression (DTC).
result Demonstrates high correlation between learning performance and Gibbs entropy over compression ratios.
A pseudo-Anosov surface automorphism φ has associated to it an algebraic unit λφ called the dilatation of φ. It is known that in many cases λφ appears as the spectral radius of a Perron-Frobenius matrix preserving a symplectic form L. We investigate what algebraic units could potentially appear as dilatatio…
We introduce two new functionals on Sasaki manifolds, inspired by the work of Perelman, which are monotonic along the Sasaki-Ricci flow. We relate their gradient flow, via diffeomorphisms preserving the foliated structure of the manifold, to the transverse Ricci flow. Finally, when the basic first Chern class is positi…
The paper introduces submodular information measures for machine learning applications.
problem Generalizing information-theoretic measures to non-random variables.
method Developing combinatorial information measures based on submodular functions.
result Submodular mutual information is submodular in one argument for certain submodular functions.
Study improves summarization reliability in risky scenarios.
problem Reliability of automatic summarization in high-risk contexts.
method Conditional generation with Bayesian inference and entropy regularization.
result Significant improvement in robustness and reliability of summarization.
A new measure of causal influence quantifies intrinsic contributions in DAGs.
problem Quantifying intrinsic causal contributions in Directed Acyclic Graphs (DAGs).
method Recursive decomposition of node contributions, structure-preserving interventions, Shapley symmetrization.
result A measure of intrinsic causal contribution that is invariant to node relabeling.
New CVs preserve transition rates in molecular dynamics.
problem Designing CVs that accurately capture rare events in high-dimensional systems.
method Integrating manifold learning and group-invariant featurization to construct neural network-based CVs that satisfy orthogonality conditions.
result Achieved a CV for butane that reproduces the anti-gauche transition rate with less than ten percent relative error.
We propose a novel node embedding of directed graphs to statistical manifolds, which is based on a global minimization of pairwise relative entropy and graph geodesics in a non-linear way. Each node is encoded with a probability density function over a measurable space. Furthermore, we analyze the connection between th…
Bayesian neural networks learn efficiently at infinite width, matching polynomial-width performance.
problem Understanding the inductive bias of infinite-width neural networks.
method Analyzing the reduced entropy and using subsampling techniques.
result The Bayesian mean-field learner generalizes exactly on polynomially-bounded targets.
We obtain a compactness result for Fano manifolds and Kähler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the Kähler Ricci flows. One assumption is the boundedness of the Ricci potential and the ot…
We study geometric properties of the Lagrangian self-shrinking tori in R4. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conf…
Top-H decoding improves text generation by balancing creativity and coherence.
problem Balancing creativity and coherence in text generation.
method Entropy-constrained minimum divergence problem, NP-hard ECMM problem, top-H decoding.
result Top-H decoding outperforms min-p sampling by up to 25.63% on creative writing benchmarks.
Unified geometric flows improve deep learning efficiency and simplify neural network topologies.
problem Improving deep learning performance and simplifying neural network structures.
method Proposes a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, enabling automated singularity resolution and providing entanglement entropy bounds.
result Demonstrates 2.1× convergence acceleration and 63% topological simplification while maintaining O(NlogN) complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. Develops neural networks that follow thermodynamics principles.
problem Learning physical systems from data while respecting thermodynamics.
method Uses feedforward neural networks and the GENERIC formalism to enforce metriplectic structure.
result Predictions comply with first and second principles of thermodynamics.