arXiv research

We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than or equal to the topological entropy for the metric of constant negative curvatu…

News novelty predicts negative stock market returns.

Unified interpretation of softmax cross-entropy and negative sampling for knowledge graph embedding.

We survey several notions of entropy related to a compact manifold of negative curvature, some relations between them, and the rigidity problems.

Sharp inequality in spaces with non-negative Ricci curvature.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

Study on entropy stability in product spaces of negatively curved symmetric spaces.

The paper connects currents and entropy in hyperbolic 3-manifolds.

Liouville entropy increases strictly along Ricci flow on surfaces.

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

The study examines entropy and pressure at infinity in negatively curved manifolds, linking them to strong positive recurrence.

Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.

The entropy of minimal surfaces is minimized in hyperbolic manifolds.

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy i…

Bayesian Markowitz portfolio problem shows entropy regularization is ineffective.

In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In …

In deep neural network, the cross-entropy loss function is commonly used for classification. Minimizing cross-entropy is equivalent to maximizing likelihood under assumptions of uniform feature and class distributions. It belongs to generative training criteria which does not directly discriminate correct class from co…

Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.

Quantum machine learning uses quantum cross entropy to minimize loss, but measurement loss affects this process.

We compute the second variation of the Ricci expander entropy and briefly discuss the linear stability of compact negative Einstein manifolds.

We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow in terms…

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, o…

Geodesics in curved spaces spread evenly over time.

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we a…

Study proves uniqueness of corrugated negatively curved immersions in differential geometry.

Empirical evidence suggests that even the most competitive markets are not strictly efficient. Price histories can be used to predict near future returns with a probability better than random chance. Many markets can be considered as {\it favorable games}, in the sense that there is a small probabilistic edge that smar…

In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and poten…

ELBO converges to a sum of entropies for many generative models.

Study on $k$-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.

New scalable algorithm for non-negative linear regression with entropy-regularized OT loss.

New approach to portfolio optimization shows entropy regularization is ineffective.

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological ent…

Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension

ELBO of VAEs converges to a sum of three entropies.

We empirically investigate the (negative) expected accuracy as an alternative loss function to cross entropy (negative log likelihood) for classification tasks. Coupled with softmax activation, it has small derivatives over most of its domain, and is therefore hard to optimize. A modified, leaky version is evaluated on…

Study non-negative curvature Markov chains, proving entropy contraction.

A new approach improves numerical tabular data imputation by addressing diffusion models' limitations.

In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the ri…

If pricing kernels are assumed non-negative then the inverse problem of finding the pricing kernel is well-posed. The constrained least squares method provides a consistent estimate of the pricing kernel. When the data are limited, a new method is suggested: relaxed maximization of the relative entropy. This estimator …

New method for comparing different mass measures on tree structures using entropy partial transport.

Study geodesic flows on hyperbolic manifolds without conjugate points, proving unique measure of maximal entropy.

We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and pa…

The paper introduces a new system of equations for Hessian-cscK metrics.

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a th…

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

New method uses asymmetric Tsallis relative entropy for better risk assessment in financial portfolios.

FTRL algorithm with negative entropy regularizer achieves best-of-three-world results for linear bandits.

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic…