arXiv research

Proposes integrating global and local entropy for more reliable LLMs.

The paper extends Perelman's theorems on Ricci flow entropy.

We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant $r$, $a$, $b$.

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 …

Local gaps in Ricci shrinkers depend only on dimension.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimate along the Ricci flow. It can be used to show the Gromov-Hausdorff convergence of the Kähler Ricci flow on ea…

Self-attention networks localize when eigenspectrum variance is small.

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

LES optimizes designs by sampling descent sequences, achieving strong sample efficiency.

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenval…

The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses we introduce variational characterizations that naturally suggest a two-step scheme for their optimizatio…

As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy.…

EDLP samples flat modes in discrete spaces using entropy.

Proposes a method to solve deep neural networks' local minimum problem.

We discuss a class of (local and non-local) theories of gravity that share same properties: i) they admit the Einstein spacetime with arbitrary cosmological constant as a solution; ii) the on-shell action of such a theory vanishes and iii) any (cosmological or black hole) horizon in the Einstein spacetime with a positi…

We introduce a novel Entropy-driven Monte Carlo (EdMC) strategy to efficiently sample solutions of random Constraint Satisfaction Problems (CSPs). First, we extend a recent result that, using a large-deviation analysis, shows that the geometry of the space of solutions of the Binary Perceptron Learning Problem (a proto…

Gathering the most information by picking the least amount of data is a common task in experimental design or when exploring an unknown environment in reinforcement learning and robotics. A widely used measure for quantifying the information contained in some distribution of interest is its entropy. Greedily minimizing…

We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also impl…

Proves convergence of gradient Ricci shrinkers with uniform bounds.

Pointwise localization allows more precise localization and accurate interpretability, compared to bounding box, in applications where objects are highly unstructured such as in medical domain. In this work, we focus on weakly supervised localization (WSL) where a model is trained to classify an image and localize regi…

We identify a class of over-parameterized deep neural networks with standard activation functions and cross-entropy loss which provably have no bad local valley, in the sense that from any point in parameter space there exists a continuous path on which the cross-entropy loss is non-increasing and gets arbitrarily clos…

Entropy data replaces classical charts for smooth manifolds.

We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infin…

The entropy of minimal surfaces is minimized in hyperbolic manifolds.

The paper extends entropy formulas to super Ricci flows on metric measure spaces.

The paper develops techniques to study entropy and rigidity in RCD-spaces.

Local Sobolev inequality on Ricci flows with applications.

In this work we derive local gradient and Laplacian estimates of the Aronson-Bénilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discove…

The paper proposes a method to identify high-quality financial patterns using entropy.

Enhances RL by controlling policy stochasticity through trajectory entropy constraints.

The paper establishes bounds for Ricci flows using entropy and heat kernel methods.

In 1870s, L. Boltzmann proved the famous $H$-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 $W$-entropy and proved the $W$-entropy formula for the Ricci flow. This plays a crucial role in…

Ricci flow controls curvature on manifolds with bounds.

Geodesic flows on compact manifolds without conjugate points are shown to have a unique measure of maximal entropy.

Improved loss functions adapt to weight-space anisotropy, outperforming isotropic counterparts.

Ecker's and Huisken's quantities agree for ancient mean curvature flows.

A relation between the conformal anomaly and the logarithmic term in the entanglement entropy is known to exist for CFT's in even dimensions. In odd dimensions the local anomaly and the logarithmic term in the entropy are absent. As was observed recently, there exists a non-trivial integrated anomaly if an odd-dimensio…

Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a …

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists for all time and converges towards a Ricci soliton. If $g$ is not a local maxim…

Calculates local Granger causality for Gaussian and nonlinear systems.

EPSTE: A geometric token and deep learning approach to estimating transfer entropy in neuroimaging time series

Constructs QFT on curved surfaces, proving axioms and calculating entropy.

Novel CE-method variants reduce local minima convergence with fewer function evaluations.

The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.

Develops correlation number for specific potentials and Hitchin representations.

The paper studies empirical processes from nearest neighbors in regression.