arXiv research

Study classifies Morse functions with 4 critical points on immersed 2-spheres.

Model criticism tool evaluates text coherence and structure in generated long-form text.

The paper classifies cosymplectic manifolds with critical metrics in dimension 3.

The study identifies all possible vector field structures on specific 2D shapes.

We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact $3$-manifolds. More precisely, we show that a contact $3$-manifold $(M,α)$ admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb fl…

New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.

We develop a topology data analysis-based method to detect early signs for critical transitions in financial data. From the time-series of multiple stock prices, we build time-dependent correlation networks, which exhibit topological structures. We compute the persistent homology associated to these structures in order…

We study the problem of detecting critical structures using a graph embedding model. Existing graph embedding models lack the ability to precisely detect critical structures that are specific to a task at the global scale. In this paper, we propose a novel graph embedding model, called the Ego-CNNs, that employs the eg…

Paper studies critical points of curvature energies in 4D.

Study uncovers complex critical points in tensor decomposition.

Simplified neural network EFTs reveal a single critical condition.

In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, Δ^m J]=0$. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{cr…

To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functi…

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphism…

In this paper we propose a hybrid architecture of actor-critic algorithms for reinforcement learning in parameterized action space, which consists of multiple parallel sub-actor networks to decompose the structured action space into simpler action spaces along with a critic network to guide the training of all sub-acto…

Method reconstructs financial networks from aggregate data, revealing critical link density.

We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…

Researchers extend Godbillon-Vey functional to almost contact manifolds, finding critical structures.

On asymptotically complex hyperbolic (ACH) Einstein manifolds, we consider a certain variational problem for almost complex structures compatible with the metric, for which the linearized Euler-Lagrange equation at Kähler-Einstein structures is given by the Dolbeault Laplacian acting on $(0,1)$-forms with values in the…

Embedding principle explains loss landscape of deep neural networks.

Hard to approximate critical points for simple nonconvex functions.

Learning the distribution of natural images is one of the hardest and most important problems in machine learning. The problem remains open, because the enormous complexity of the structures in natural images spans all length scales. We break down the complexity of the problem and show that the hierarchy of structures …

Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.

We consider the estimation of the policy gradient in partially observable Markov decision processes (POMDP) with a special class of structured policies that are finite-state controllers. We show that the gradient estimation can be done in the Actor-Critic framework, by making the critic compute a "value" function that …

The exponential map fails to be injective near critical points in sub-Riemannian geometry.

We investigate the structure of a harmonic morphism $F$ from a Riemannian 4-manifold M^4 to a 2-surface $N^2$ near a critical point $m_0$. If $m_0$ is an isolated critical point or if $M^4$ is compact without boundary, we show that $F$ is pseudo-holomorphic w.r.t. an almost Hermitian structure defined in a neighbourhoo…

We prove sharp pointwise decay estimates for critical Dirac equations on $\mathbb{R}^n$ with $n\geq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited state…

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the crit…

Study of critical points for 4D conformally invariant curvature energies.

Model criticism is usually carried out by assessing if replicated data generated under the fitted model looks similar to the observed data, see e.g. Gelman, Carlin, Stern, and Rubin [2004, p. 165]. This paper presents a method for latent variable models by pulling back the data into the space of latent variables, and c…

Stock markets are complex systems exhibiting collective phenomena and particular features such as synchronization, fluctuations distributed as power-laws, non-random structures and similarity to neural networks. Such specific properties suggest that markets operate at a very special point. Financial markets are believe…

Functional for Spin(7) forms defined on compact manifolds.

Nearly $G_2$-structures are unstable under a modified $G_2$-Laplacian co-flow.

Characterizes CR manifolds as critical points of an energy functional.

The aim of this paper is to classify three dimensional compact Riemannian manifolds $(M^{3},g)$ that admits a non-constant solution to the equation $-Δf g+Hess f-fRic=μRic+λg,$ for some special constants $(μ, λ)$, under assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we will…

The study connects group structure to smooth actions on one-manifolds.

Introduces a new $G_2$-Hilbert functional in $G_2$-geometry.

Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the f…

Study on critical Lagrangian phase singularities in mean curvature flow.

The paper studies critical points and flows of a $G_2$-Hilbert functional on manifolds with circle actions.

Enhanced Sampling Scheme improves masked generative modeling.

Proposes a value-based method for continuous control without an actor.

Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.

Study proves rigidity of critical points in hydrophobic capillary systems.

The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which enables us to show that if a critical metric of the volume functional on a connected $…

Stochastic subgradient descent avoids critical points in definable functions.

New separation concepts for Anosov representations help bound Thurston asymmetric metric.

The study characterizes complex structures using calculus of variations.