The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.
problem Curvature-dimension conditions and related inequalities on Riemannian manifolds.
method Information-theoretic approach to study curvature-dimension condition, rigidity theorems, and entropy differential inequalities.
result Equivalence of curvature-dimension condition and entropy differential inequalities on Riemannian manifolds.
We introduce on any smooth oriented minimal surface in Euclidean 3-space a meromorphic quadratic differential, P, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of particular interest is the realization of its real part as a conservation law for a natur…
The paper corrects biases in estimating intrinsic dimension and differential entropy.
problem Systematic bias in estimating intrinsic dimension and differential entropy.
method A bias-corrected estimator for both measures is proposed, highlighting shared steps and useful consequences.
result Simultaneous estimation of differential entropy and intrinsic dimension provides complementary perspectives on underlying manifolds.
Paper develops a new probabilistic method for American options using entropy regularization.
problem Finding optimal stopping times for American options with entropy regularization.
method Entropy-regularized penalization scheme based on Doob-Meyer-Mertens decomposition and reflected backward stochastic differential equations.
result Explicit convergence rates and policy improvement algorithm for American options.
This paper bounds min-entropy leakage for Blowfish privacy using graph symmetries.
problem Bounding min-entropy leakage for Blowfish privacy mechanisms.
method Organizing analysis over symmetrical partitions corresponding to orbits of graph automorphism groups.
result Demonstrates a construction meeting the bound with asymptotic equality, showing tightness.
We approximate differential entropy for efficient Bayesian experimental design.
problem Efficiently estimating expected information gain in large-scale inference problems.
method Approximate differential entropy using Monte Carlo or quasi-Monte Carlo surrogates.
result Our approach achieves comparable or better convergence rates than state-of-the-art methods.
The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
Differentiable PF via entropy-regularized OT for better inference.
problem Non-differentiability of traditional PF resampling methods.
method Entropy-regularized optimal transport for differentiable resampling.
result Convergent differentiable PF method with improved gradient estimates.
This paper optimizes trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
problem Optimizing trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
method Relative entropy-regularized robust optimal control problem, modeled as a stochastic differential game.
result Analytical expressions for optimal strategy and trajectory are derived under specific assumptions.
This paper improves entropy bounds for ranking time-series complexity.
problem Ranking the complexity of time series processes.
method Building on information theoretic bounds, the paper improves the upper bound of conditional differential entropy using Hadamard's inequality and covariance matrix properties.
result The improved bounds can be used to rank the complexity of time series processes.
In this paper, we prove Perelman type W-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.
In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequali…
DE-QT detects optimal Q-learning stopping points.
problem Information loss in Q-learning during prolonged training.
method Introducing DE-QT to detect entropy changes in Q-tables.
result DE-QT identifies the best stopping point for Q-learning.
Study on convergence of SDEs using entropy methods.
problem Analyzing convergence of stochastic differential equations.
method Applied Lyapunov method to Fokker-Planck equation with weighted relative Fisher information.
result Exponential convergence of probability density function to invariant distribution in L1 distance. In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(−K,m)-condition, where m∈[n,∞) and K≥0 are two constants. Moreover, we introduce the W-entropy and prove the W-ent…
A new differential entropy estimator for neural networks training.
problem Lack of effective differential entropy estimators for neural network training.
method KNIFE: a fully parameterized, differentiable kernel-based estimator of differential entropy.
result KNIFE effectively estimates differential entropy and improves neural network training.
Entropy of critical points generalizes Morse theory.
problem Extending Morse theory to group actions.
method Entropy of homologically detectable critical points.
result Entropy lower bound on critical points.
MGD combines maximum entropy and diffusion methods for efficient sampling.
problem Generating samples from limited information in high dimensions.
method Moment Guided Diffusion (MGD) using stochastic differential equations.
result MGD efficiently samples maximum entropy distributions in finite time.
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.
Density destructors simplify complex PDFs to maximize entropy, linking to information theory.
problem Complex multivariate PDFs are hard to analyze.
method Invertible transforms that progressively remove structure from PDFs.
result Density destructors can improve estimates of information theoretic quantities.
In this article we prove two formulas for the topological entropy of an F-optical Hamiltonian flow induced by a C^{\infty} Hamiltonian, where F is a Lagrangian distribution. In these formulas, we calculate the topological entropy as the exponential growth rate of the average of the determinant of the differential of th…
Insider trading is reduced when penalized, affecting expected penalties in a non-monotone way.
problem Reducing insider trading behavior when insiders face legal penalties.
method Characterized via a backward stochastic differential equation (BSDE) with a non-linear operator.
result The insider's expected penalties are non-monotone in the fee structure and determined by relative entropy.
We study the cross-entropy method (CEM) for the non-convex optimization of a continuous and parameterized objective function and introduce a differentiable variant that enables us to differentiate the output of CEM with respect to the objective function's parameters. In the machine learning setting this brings CEM insi…
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.
In this paper, we prove the characterization of the (K,∞)-super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dim…
Study proves uniqueness of corrugated negatively curved immersions in differential geometry.
problem Negatively curved immersions in differential geometry.
method Relative entropy method applied to Gauss-Codazzi system.
result Uniqueness of smooth isometric immersions within corrugated class.
LogDet estimator improves entropy estimation in neural networks.
problem Inconsistent observations and diversified interpretation in neural networks.
method Proposes LogDet estimator for reliable entropy approximation.
result LogDet estimator overcomes distributional diversity issues.
In this paper we aim to find a measure for the diversity of cash flows between agents in an economy. We argue that cash flows can be linked to probabilities of finding a currency unit in a given cash flow. We then use the information entropy as a natural measure of diversity. This leads to a hirarchical inequality meas…
We analyze the problem of learning a single user's preferences in an active learning setting, sequentially and adaptively querying the user over a finite time horizon. Learning is conducted via choice-based queries, where the user selects her preferred option among a small subset of offered alternatives. These queries …
In this note we determine the first two derivatives of the classical Boltzmann-Shannon entropy of the conjugate heat equation on general evolving manifolds. Based on the second derivative of the Boltzmann-Shannon entropy, we construct Perelman's F and W entropy in abstract geometric flows. Monotonicity of the entropies…
The paper explores tail diversification in financial markets using entropy and mutual information.
problem Tail diversification in financial time series.
method Statistical independence through differential entropy and mutual information, using moments as contrast functions.
result Tail covariance matrix is a key driver of tail diversification.
Paper finds a new principle for optimizing consumption and wealth using Tsallis entropy.
problem Optimal consumption-investment problem with recursive utility.
method Established connection to quadratic BSDE, derived stochastic maximum principle.
result Proved existence of optimal strategy and analyzed coupled system.
Compactifies a component by studying metric degeneration.
problem Compactify the SO(2,3)-Hitchin component.
method Study metric degeneration on a surface in pseudo-hyperbolic space.
result Establish closure in projectivized geodesic currents space.
Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.
problem Understanding the behavior of curves with curvature and torsion under curve shortening flow.
method Defined curvature-torsion entropy to analyze the flow of twisted curves.
result Curved curves under curve shortening flow either develop inflection points or exhibit highly irregular singularities.
In this paper we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure theoretic entropy is upper semicontinuous when there is no loss of mass. In case we are losing mass, the critical exponents of parabolic subgroups of the fundamental group have a…
We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable property, which is not invariant under homeomorphisms. We also formulate an obstructi…
Entropy regularized OT test assesses independence between samples.
problem Testing independence between two samples.
method Entropy regularized optimal transport.
result Non-asymptotic bounds for test statistic established.
REMEDI improves neural entropy estimation across various tasks.
problem Challenges in estimating information theoretic quantities in high-dimensional data.
method Combines minimization of cross-entropy with estimation of deviation from data density.
result Improves accuracy in entropy estimation on synthetic and natural data.
We propose a novel information-theoretic approach for Bayesian optimization called Predictive Entropy Search (PES). At each iteration, PES selects the next evaluation point that maximizes the expected information gained with respect to the global maximum. PES codifies this intractable acquisition function in terms of t…
A pricing principle is introduced for non-attainable claims in incomplete markets.
problem Pricing non-attainable contingent claims in incomplete markets.
method Distorted Radon-Nikodym derivative and Tsallis relative entropy over a family of equivalent martingale measures.
result The pricing principle is closely related to backward stochastic differential equations and is arbitrage-free and time-consistent.
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.
State-of-the-art neural networks are vulnerable to adversarial examples; they can easily misclassify inputs that are imperceptibly different than their training and test data. In this work, we establish that the use of cross-entropy loss function and the low-rank features of the training data have responsibility for th…
Analytical results bound the approach to oligarchy in a modified asset exchange model.
problem Analyzing economic inequality in a modified asset exchange model.
method Analytical results using Gini coefficient and differential inequalities.
result The Gini coefficient is bounded by a first-order differential inequality.
Paper proposes AXE loss for non-autoregressive machine translation, improving performance.
problem Challenges in training non-autoregressive models due to lack of autoregressive factors and cross entropy loss penalties.
method Proposes aligned cross entropy (AXE) loss function using a differentiable dynamic program for better word order alignment.
result AXE-based training improves performance on major WMT benchmarks and sets a new state of the art for non-autoregressive models.
We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound's p…
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 …
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.
Choquet regularization improves exploration in RL.
problem Improving exploration in reinforcement learning.
method Introducing Choquet regularizers to measure and manage exploration, reformulating RL problems and deriving explicit solutions.
result Explicit optimal distributions and Choquet regularizers for various exploratory samplers.