arXiv research

Extended characterization of RAAGs with zero minimal volume entropy.

We show vanishing results about the infimum of the topological entropy of the geodesic flow of homogeneous smooth four manifolds. We prove that any closed oriented geometric four manifold has zero minimal entropy if and only if it has zero simplicial volume. We also show that if a four manifold M admits a geometric dec…

Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.

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…

We determine the minimal entropy martingale measure for a general class of stochastic volatility models where both price process and volatility process contain jump terms which are correlated. This generalizes previous studies which have treated either the geometric Lévy case or continuous price processes with an ortho…

In this short note, we analyze geometric properties of orbit spaces of certain involutions in dimensions four, five, and six. We consider constructions of $\mathcal{F}$-structures on manifolds of dimension at least four that allows us to study minimal entropy, minimal volume, collapse with bounded curvature, and sign o…

In this paper, we propose a general framework to learn a robust large-margin binary classifier when corrupt measurements, called anomalies, caused by sensor failure might be present in the training set. The goal is to minimize the generalization error of the classifier on non-corrupted measurements while controlling th…

The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy…

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

We study the problem of existence of F-structures on compact complex surfaces, giving a complete classification modulo the gap in the classification of surfaces of class VII. We then use these results to study the minimal entropy problem for compact complex surfaces. For instance we prove that compact Kahler surfaces o…

Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic Kähler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound i…

HCLM framework uses entropy regularization for open learning systems.

A new method for generative modeling of discrete data using geometric latent subspaces.

In this paper, we propose a general framework to learn a robust large-margin binary classifier when corrupt measurements, called anomalies, caused by sensor failure might be present in the training set. The goal is to minimize the generalization error of the classifier on non-corrupted measurements while controlling th…

The relaxed maximum entropy problem is concerned with finding a probability distribution on a finite set that minimizes the relative entropy to a given prior distribution, while satisfying relaxed max-norm constraints with respect to a third observed multinomial distribution. We study the entire relaxation path for thi…

Modified Perelman entropy proves RG-2 flow monotonicity.

Researchers calculate the minimal entropy of 3-manifolds, proving it's additive.

Minimal volume entropy vanishes for mapping tori over 3-manifolds.

Study shows rigidity for entropy minimizers in non-monotone cases.

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

The paper explores how neural networks avoid overfitting by learning from flat minima.

Proposes MEDM to balance entropy minimization and diversity maximization for better domain adaptation.

We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of foliated manifold with leafwise Randers structure. In this case the estimate…

Entropy-minimal measure calculated for a stochastic volatility model.

Generalizes entropy-drift inequality for specific geometric spaces.

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…

Constructs entropy-minimizing pseudo-Anosov diffeomorphisms on K3 surfaces.

The entropy of minimal surfaces is minimized in hyperbolic manifolds.

Data-driven anomaly detection methods suffer from the drawback of detecting all instances that are statistically rare, irrespective of whether the detected instances have real-world significance or not. In this paper, we are interested in the problem of specifically detecting anomalous instances that are known to have …

Study minimal surfaces in complex hyperbolic space, linking entropy and volume.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

The study examines conditions for minimal volume entropy of simplicial complexes.

New geometric quantities help classify manifolds and relate to entropy.

Geometric inequalities for equipotential curves derived from convex entropy.

Study on convergence of exponential probability measures with applications to maximum entropy models and SGLD.

We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable 3-manifold with zero simplicial volume, then the minimal entropy can be solved for M…

Entropy corrections improve GBM's predictive accuracy for non-log-normal distributions.

Paper compares two entropy concepts for finite presentation groups.

Defines a new geometric quantity for hyperbolic manifolds, showing it's well-defined and invariant.

The paper studies minimal surface entropy on hyperbolic 3-manifolds and compares it to the hyperbolic case.

New method uses entropy dissipation to prove isoperimetric inequalities.

We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as …

We propose the use of the functional determinant of geometric operators in constructing an entropy functional associated to geometric flows. Our approach is based on the direct computation of the partition function, with a well-defined set of microstates and macrostates in the canonical ensemble. The approach is motiva…

We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of their fundamental groups.

The study bounds entanglement entropy for coherent states on Kähler manifolds.

Quantizes Kähler-Ricci flow for Fano manifolds.

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

The paper applies math and physics to language models, introducing entropy and geometric concepts.