Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
Variational approach to basic manifold structures.
problem Understanding basic differential geometric structures.
method Variational description of geometric structures.
result Variational formulation of manifold structures.
Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
Geometrically describes Jacobi equations for field theories with dissipation.
problem Describing field theories with dissipation geometrically.
method Prolongation of the Lagrangian on a k-cosymplectic formulation to describe Jacobi equations and a modified Lagrangian for variational formulation.
result Variational formulation of field theories with dissipation.
New method identifies vanishing arcs for curve singularities.
problem Characterizing arcs sent to geometric vanishing cycles.
method Introducing geometric variation operator and vanishing arcsets.
result Existence of topological exceptional collections of arcsets.
The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples are provided.
Study introduces indecomposability for varifolds, leading to geometric consequences.
problem Understanding the structure of varifolds and their connectedness properties.
method Introducing indecomposability and related concepts for varifolds.
result Substantial geometric consequences derived from the connectedness properties of varifolds.
Geometric Variational Inference improves efficiency in complex probability distributions.
problem Efficiently accessing information in non-linear and high-dimensional probability distributions.
method Geometric Variational Inference (geoVI) uses Riemannian geometry and the Fisher information metric to construct a coordinate transformation.
result geoVI provides a more efficient variational approximation by a normal distribution, demonstrated on various problems.
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.
Study on geometric variational problems for existence, regularity, and uniqueness of solutions.
problem Geometric variational problems, focusing on existence, regularity, and uniqueness of solutions.
method Formulated in Federer and Fleming's theory of currents, discussed the existence theory, and presented core ideas of the (interior) regularity theory for area-minimizing currents and optimal transport paths. Two original results on generic uniqueness of solutions were presented.
result Generic uniqueness of solutions for both Plateau's problem and optimal branched transport problem.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
Geometric structures help in understanding thermodynamics.
problem Understanding thermodynamic systems using geometric methods.
method Using almost cosymplectic structures and variational arguments.
result Evolution equations are derived and discussed.
In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.
Geometric analysis improves convergence of variational inference.
problem Challenges in analyzing convergence of variational inference due to non-convexity and non-smoothness.
method Exploits exponential family structure and Bregman divergences to geometrically analyze the optimization landscape.
result Establishes non-asymptotic convergence rates for gradient descent algorithms.
BBVI with STL converges geometrically under perfect specification, with quadratic variance bound.
problem Convergence rate of BBVI with STL estimator.
method Proved geometric convergence rate with quadratic variance bound for BBVI with STL estimator.
result BBVI with STL converges geometrically under perfect variational family specification.
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We …
GD-VAEs learn dynamics from observations using geometric and topological information.
problem Learning parsimonious representations of nonlinear dynamics from observations.
method Develops data-driven methods incorporating geometric and topological information using Variational Autoencoders (VAEs).
result GD-VAEs provide methods for learning reduced dimensional representations of nonlinear dynamics.
We prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by applications to families of geometric variational problems lacking regularity, several non-smooth extensions of the result are discussed. A…
Geometric interpretation improves VAE performance and robustness.
problem Improving Variational Autoencoder performance and robustness.
method Introducing a geometric perspective on VAEs, sampling from the Riemannian latent space.
result Improved generation and interpolations with competitive or better performance on benchmark datasets.
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propos…
Geometric framework analyzes bias in variational inference for posterior functionals.
problem Analyzing the bias of posterior functionals under variational approximations.
method Developed a geometric framework to evaluate the bias of posterior functionals using the variational tangent space.
result The leading-order bias of a posterior functional is determined by its component orthogonal to the variational tangent space.
We use neural networks as control variates with geometric integration techniques.
problem Analytic integration of neural network approximations for variance reduction.
method Integration domain subdivision using computational geometry for MLPs with continuous piecewise linear activation functions.
result Neural networks can be used as control variates with geometric integration methods.
We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.
GeoPhy uses geometric gradients to efficiently infer phylogenetic trees from molecular data.
problem Challenges in accurately inferring species relationships from molecular data due to combinatorially vast tree topologies.
method Introduces a novel, fully differentiable formulation of phylogenetic inference using geometric spaces and variational Bayesian methods.
result Significantly outperforms other approximate Bayesian methods in inferring phylogenetic trees.
Geometric integrator preserves coadjoint orbits in dissipative systems.
problem Preserving coadjoint orbits in dissipative mechanical systems.
method Adapted discrete variational integrators for forced Euler-Poincaré and Lie-Poisson systems.
result Preserves coadjoint orbits exactly, improving over general-purpose methods.
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
Study on slow convergence in geometric variational problems.
problem Slow convergence of solutions in geometric variational problems.
method Identifying necessary conditions for slowly converging solutions and characterizing their convergence rate and direction.
result Characterization of the rate and direction of convergence for slowly converging solutions.
New theorems prove uniqueness of solutions to geometric PDEs.
problem Proving uniqueness of solutions to geometric PDEs.
method Analyzing nonlinear elliptic PDEs of divergence form.
result Proved several Moser-Bernstein type theorems.
Study fourth-order geometric flow of shape operator for co-dimension one immersions.
problem Analyzing the geometry of isometric immersions in Riemannian manifolds.
method Introduce a moduli flow to decrease curvature variation energy.
result The flow decreases a natural energy measuring curvature variation.
Proves modular functors for SO(3) have integral Hodge structures.
problem Proving modular functors have integral Hodge structures.
method Based on homological models and geometric identification.
result Geometric construction of Hodge structures on SO(3) modular functors.
We prove an extension of a celebrated equivariant bifurcation result of J. Smoller and A. Wasserman, in an abstract framework for geometric variational problems. With this purpose, we prove a slice theorem for continuous affine actions of a (finite-dimensional) Lie group on Banach manifolds. As an application, we discu…
Geometric Mean Market Makers super-hedge impermanent loss without models.
problem Super-hedging impermanent loss in Geometric Mean Market Makers.
method Model-free rebalancing strategy.
result Loss-versus-rebalancing vanishes due to finite variation exchange rate.
Extends geometrical description of tensor manifolds in tree-based formats.
problem Geometrical description of tensor manifolds in tree-based formats.
method Provided a new geometrical description of manifolds of tensors in tree-based format.
result Geometrical description compatible with Tucker format.
Develops a new geometric framework for non-conservative field theories.
problem Non-conservative field theories in classical physics.
method Multisymplectic and contact geometries, variational field equations, jet bundle description.
result Introduces variational field equations in multicontact manifolds.
Noether theorem applied to variational problems on hyperbolic surfaces.
problem Variational problems on hyperbolic surfaces.
method Noether's theorem on symmetry and conservation laws.
result Application to geometric problems on hyperbolic surfaces.
We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subse…
A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension m. In this minicourse we discuss these problems from a ge…
Paper proposes a closed-form formula for geometric Istanbul call options.
problem Pricing geometric Istanbul call options under the Black-Scholes model.
method Second-order Taylor expansion to derive a closed-form approximation.
result The proposed formula accurately approximates GIC values compared to Monte-Carlo simulations.
Investigates invariant hulls of functionals on manifolds.
problem Understanding meaningful functionals on manifolds through reparameterizations.
method Uses inner-variations to transform arbitrary functionals into invariant realizations.
result Explicit computations for volume functional in N-dimensional manifolds, especially in N=2. Derives energy-momentum tensor from Standard Model, examines energy conditions.
problem Validating energy conditions in the context of the Standard Model.
method Geometric variational problem on globally hyperbolic manifold, deriving energy-momentum tensor.
result Validates various energy conditions in general relativity.
We develop VAE-DLM for dynamics with geometric flows in latent space.
problem Learning latent geometric properties for dynamics in high-dimensional data.
method Riemannian approaches to VAEs with a geometric flow in latent space, reformulating ELBO loss.
result Improved performance and robust learning for external dynamics, reducing OOD error.
A recent algorithmic family for distributed optimization, DIGing's, have been shown to have geometric convergence over time-varying undirected/directed graphs. Nevertheless, an identical step-size for all agents is needed. In this paper, we study the convergence rates of the Adapt-Then-Combine (ATC) variation of the DI…
A deep learning model organizes RNA graphs to reveal folding patterns and properties.
problem Organizing and understanding the complex folding patterns of RNA secondary structures.
method Geometric scattering autoencoder (GSAE) network for learning graph embeddings.
result GSAE accurately reflects bistable RNA structures and can sample new folding trajectories.
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a diff…
A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and Born-Infeld systems.
Theory of space-time currents for geometric evolutions.
problem Analysis of geometric evolutions driven by dislocations.
method Development of space-time integral currents with bounded variation, introduction of Lipschitz deformation distance.
result Agreement of Lipschitz deformation distance with integral Whitney flat metric for boundaryless currents.
Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.
problem Construction of moduli spaces for Higgs bundles with specific properties.
method Elementary geometric and combinatorial techniques, focusing on orbit stability of automorphism groups.
result Explicit geometric models for moduli spaces of parabolic Higgs bundles over Riemann sphere.
A second order variational description of the autoparallel curves of some differential-geometric connection for the third order Mathisson's 'new mechanics' of a relativistic free spinning particle is suggested starting from general requirements of invariance and 'variationality'.