arXiv research

In this paper, we develop the notion of entropy for uniform hypergraphs via tensor theory. We employ the probability distribution of the generalized singular values, calculated from the higher-order singular value decomposition of the Laplacian tensors, to fit into the Shannon entropy formula. We show that this tensor …

Uniform entropy bound for Ricci shrinkers with bounded curvature.

Uniform bounds derived for fully non-linear equations.

Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.

Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.

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

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is unif…

Study shows uniform-time chaos propagation in mean field Langevin dynamics.

Entropy rigidity for Finsler flows but collapse for Reeb flows.

Proves convergence of gradient Ricci shrinkers with uniform bounds.

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

The paper studies empirical processes from nearest neighbors in regression.

Proves uniform K-stability is open in Kähler cone.

Study Transformer layers under cross-entropy training using mean field control.

Stability result for a popular algorithm in optimal transport.

We prove that, among metrics on a compact quotient of $\left(\mathbb{H}^2\right)^n$ (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.

Self-attention networks localize when eigenspectrum variance is small.

Entropy rigidity proven for 3D and higher convex projective manifolds.

We show that all closed $2$-dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the subspace are both $\leq C\,(1+γ)$ for some universal constant $C$ and genus $γ$. Th…

Uniform Sobolev inequality for Kähler metrics with entropy bound.

Unified estimate for complex Monge-Ampère equations on Kähler manifolds.

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…

A new loss function HUG decouples and generalizes neural collapse.

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…

Uniform bounds prove connection between Kähler metrics and RCD spaces.

New insights into neural network training efficiency.

The study evaluates different probability models for uncertainty visualization using entropy calculations.

In this paper we prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in $C^\infty$-topology to a smooth strictly convex soliton as $t$ approaches to infinity is obtained as a consequence of these estimates together with an earlier result of Andre…

Investigates properties of volume, entropy, and diameter in higher Teichmüller spaces.

New bounds for agnostic learning with average smoothness.

Entropy regularization is used to get improved optimization performance in reinforcement learning tasks. A common form of regularization is to maximize policy entropy to avoid premature convergence and lead to more stochastic policies for exploration through action space. However, this does not ensure exploration in th…

We aim to develop off-policy DRL algorithms that not only exceed state-of-the-art performance but are also simple and minimalistic. For standard continuous control benchmarks, Soft Actor-Critic (SAC), which employs entropy maximization, currently provides state-of-the-art performance. We first demonstrate that the entr…

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

Unified framework for uniform signal recovery in nonlinear GCS with 1-bit/quantized measurements.

Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.

Study ergodic properties of geodesic flows on specific manifolds without conjugate points.

New bounds show empirical EOT adapts to simpler measure.

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…

New inequality links surface orthospectrum to boundary length.

Paper refines Chen-Cheng's estimates for Kähler metrics.

Unified framework for self-supervised learning via latent distribution matching.

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…

Choquet regularization improves exploration in RL.

The paper proves rigidity results for Einstein manifolds with specific geometric constraints.

In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L2-bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.

Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $Γ$ acting on $X$, then either $Γ$ is virtually nilpotent or the algebraic entropy $Ent (Γ) \geq C(n,a)$.

In signal analysis and synthesis, linear approximation theory considers a linear decomposition of any given signal in a set of atoms, collected into a so-called dictionary. Relevant sparse representations are obtained by relaxing the orthogonality condition of the atoms, yielding overcomplete dictionaries with an exten…

Entropy measure quantifies volatility correlation and risk diversity in asset portfolios.