New methods using natural gradient for structured optimization.
problem Structured optimization problems.
method Structured second-order methods via natural gradient descent.
result Efficiency demonstrated on non-convex and deep learning problems.
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48 results for “second-order structures”
In this paper we define th order Hessian structures on manifolds and study them. In particular, when , we make a detailed study and establish a one-to-one correspondence between {\it third-order Hessian structures} and a {\it certain class of connections} on the second-order tangent bundle of a manifold. Furt…
Using a model for the bundle of semi-holonomic second order frames of a manifold as an extension of the bundle of holonomic second order frames of , we introduce in a principal bundle structure over , the structure group being the add…
New superintegrable systems derived from Frobenius structures.
problem Constructing second-order superintegrable systems.
method Using conification and direct product construction, applying to semi-simple and nilpotent algebras.
result Explicitly constructed second-order superintegrable systems in three dimensions.
Paper generalizes connections between Lie groups and affine connections.
problem Exploring properties of infinitesimal groups and affine connections.
method Introducing second-order infinitesimal groups and using them to define Lie brackets and connections.
result Generalized correspondence between symmetric and non-symmetric affine connections.
The geometry of systems of third order differential equations induced by second order Lagrangiansmath.DG
A dynamical system on the total space of the fibre bundle of second order accelerations, , is defined as a third order vector field on , called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that …
Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.
problem Exploring properties of second order symmetric parallel tensors in generalized f.pk-space forms.
method Analyzes the properties of second order symmetric parallel tensors and deduces the existence or non-existence of certain tensors and hypersurfaces.
result There does not exist second order skew-symmetric parallel tensor in f.pk-space form. There is no parallel hypersurface in a generalized f.pk-space form but there is semi-parallel hypersurface.
Superintegrable systems on curved manifolds found to have Hessian structures.
problem Characterizing superintegrable systems on curved manifolds.
method Identifying and computing Hessian coordinates for superintegrable systems.
result Examples of superintegrable systems in 2D and 3D have natural Hessian coordinates.
Superintegrable systems on surfaces are classified geometrically.
problem Classifying superintegrable systems on conformal surfaces.
method Geometric structures on conformal surfaces, conformal covariant structural equations.
result Explicit set of algebraic equations defining superintegrable systems on all constant curvature surfaces.
Paper uses second-order differential geometry to study stochastic mechanics.
problem Stochastic differential equations and their symmetries.
method Develops second-order differential geometry to study symmetries of SDEs and constructs stochastic mechanics.
result Establishes stochastic Lagrangian and Hamiltonian mechanics and their relations with HJB equations.
Study second-order obstruction to nearly structure deformations.
problem Proper nearly structure rigidity on Aloff-Wallach space.
method Second-order obstruction analysis, building on Alexandrov and Semmelmann work.
result Proves rigidity for nearly structure on .
Equivalence of second order differential operators in vector bundles studied.
problem Equivalence problem for second order linear differential operators in vector bundles.
method Description of rational invariants of symbols, finding connections associated with differential operators.
result Solving problems of local and global equivalency of differential operators.
In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of …
Study on 4D PDEs with half-flat conformal structure leading to Monge-Ampere equations.
problem Characterizing second-order PDEs in 4D with specific conformal structures.
method Analysis of Monge-Ampere property and dispersionless Lax pairs.
result All known scalar second-order integrable dispersionless PDEs in 4D are of Monge-Ampere type.
Study reveals geometric context of second-order superintegrable systems.
problem Understanding second-order superintegrable systems and their Weylian geometry.
method Re-examined second-order maximally conformally superintegrable Hamiltonian systems, revealing their Weyl structure.
result Extended conformal superintegrability to Weyl structures, interpreting systems as semi-Weyl structures.
New insights into 3D PDEs via Einstein-Weyl geometry.
problem Understanding second-order PDEs in 3D with Einstein-Weyl conformal structure.
method Analyzing solutions of second-order dispersionless integrable PDEs in 3D, relating them to Einstein-Weyl geometry.
result The covector w can be expressed in terms of the equation for generic second-order PDEs, providing a dispersionless integrability test.
SOLBP extends efficient inference to uncertain Bayesian networks.
problem Inference in uncertain Bayesian networks with second-order probabilities.
method Extends Loopy Belief Propagation to second-order Bayesian networks.
result Generates inferences consistent with sum-product networks, more efficient and scalable.
The paper constructs new structures for manifolds using connections and combinations.
problem Understanding smooth manifolds with precise infinitesimal affine structures.
method Constructing new infinitesimal structures for higher-order neighbourhoods of the diagonal.
result Any symmetric affine connection on a manifold extends to a second-order infinitesimally affine structure.
Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.
problem Capturing higher-order tail behavior and dependence effects in risk measures.
method Second-order asymptotic expansions using extreme value theory and regular variation theory.
result Second-order approximations reduce approximation errors, especially at extreme confidence levels.
Develops second order infinitesimal structures on Teichmüller space.
problem Understand the infinitesimal structures of Teichmüller space.
method Formulated second order infinitesimal structures over Teichmüller space.
result Affirmative answers to two folklore problems on Teichmüller space.
The paper studies connections in superintegrable systems, revealing geometric insights.
problem Understanding non- and semi-degenerate superintegrable systems.
method Analyzes two torsion-free connections associated with superintegrable systems.
result Semi-degenerate secondary structure tensor is the Ricci curvature of a natural torsion-free connection.
Proves and tests methods for learning time-series with breaks.
problem Learning time-series with structural breaks.
method Complete proofs and experimental validation of a regularized loss function.
result Experimental results support the validity of the techniques.
Global invariant for path structures and differential equations defined on torus.
problem Global invariant for path structures and differential equations.
method Computed as a secondary invariant from a Cartan connection on a canonical bundle.
result Formula for global invariant of second order differential equations on torus.
A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-rec…
In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.
We give a summary of recent results on the explicit local form of the second-order symmetric Lorentzian manifolds in arbitrary dimension, and its global version. These spacetimes turn out to be essentially a specific subclass of plane waves.
We give the definition of angles on a Gromov-Hausdorff limit space of a sequence of complete n-dimensional Riemannian manifolds with a lower Ricci curvature bound. We apply this to prove there is a weakly second order differential structure on these spaces and prove there is a unique Levi-Civita connection allowing us …
For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in…
This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting denote the configuration fiber bundle, we show that both the multisymplectic structure on as well…
The paper studies properties of a second-order tangent bundle with a deformed metric.
problem Characterizing geometric properties of a second-order tangent bundle with a deformed metric.
method Computing Levi-Civita connection, Riemannian curvature tensor, and conditions for semi-symmetry and pluriholomorphic B-manifold.
result Conditions for to be semi-symmetric and pluriholomorphic B-manifold.
We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution assoc…
New theory allows ICA without assuming non-Gaussian sources.
problem Traditional ICA struggles with Gaussian sources.
method Developed identifiability theory based on second-order statistics and sparsity.
result Identifiability theory and estimation methods validated experimentally.
The paper connects a second order ODE to Sasakian structures and bi-Hamiltonian systems.
problem Defining and analyzing Sasakian structures associated with second order ODEs.
method Defining contact metric structures and Poisson structures, showing compatibility with bi-Hamiltonian systems.
result A compatible bi-Hamiltonian structure for the Reeb vector field is found, and conditions for the vanishing of the first Chern class are derived.
New pruning method captures global correlations for efficient neural network inference.
problem Efficiently pruning neural networks for faster inference and reduced memory usage.
method Second-order structured pruning (SOSP-H) with innovative saliency-based approaches.
result SOSP-H scales to large-scale vision tasks and improves accuracy without compromising efficiency.
Study second order integrability of Einstein deformations on Riemannian and Kähler manifolds.
problem Integrability condition for second order infinitesimal Einstein deformations.
method New expression for integrability condition, Koiso obstruction simplification.
result Complete description of integrable deformations on complex 2-plane Grassmannian.
Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat mediamath.DG
A Lie groupoid, called \textit{second-order non-holonomic material Lie groupoid}, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference crystal. The corresponding Lie algebroid, called \textit{second-order non-holonomic m…
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geome…
The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well de…
The paper classifies second-order superintegrable systems with torsion and semi-degeneracy.
problem Classifying second-order superintegrable systems with torsion and semi-degeneracy.
method Information-geometric structure and geometric conditions for non-degeneracy.
result A -parameter potential is non-degenerate if a certain trace-free tensor field vanishes.
A new method for faster optimization on statistical manifolds.
problem Slow convergence of first-order methods in manifold optimization.
method Dual Riemannian Newton method on manifolds with dual connections.
result Local quadratic convergence of the dual Riemannian Newton method.
PCA outperforms random projections in retaining second order signals from latent groups.
problem Preserving second order structure in latent groups under unsupervised linear projections.
method Theoretical framework and quasi-exhaustive enumeration of projections.
result PCA outperforms random projections in retaining second order signals across a broad range of data-generating parameters.
New MOSC clusters networks by considering both second- and third-order structures.
problem Limited consideration of higher-order structures in spectral clustering.
method Mixed-Order Spectral Clustering (MOSC) combining GL and RW for second- and third-order structures.
result MOSC outperforms existing SC methods on real-world networks.
We study the fillability (or embeddability) of 3-dimensional structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the initial structure is fillable, then it keeps having the same property as long as …
Ambrose, Palais and Singer \cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath \cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Frechet manifolds. It is…
We briefly review the notion of second order constrained (continuous) system (SOCS) and then propose a discrete time counterpart of it, which we naturally call discrete second order constrained system (DSOCS). To illustrate and test numerically our model, we construct certain integrators that simulate the evolution of …
Structural Classification of Locally Stationary Time Series Based on Second-order Characteristicsstat.ME
Efficient method classifies locally stationary time series based on second-order characteristics.
problem Classifying locally stationary time series for various applications.
method Autoregressive approximation, ensemble aggregation, distance-based threshold.
result Zero misclassification error rate asymptotically for mildly differing second-order characteristics.
The aim of this paper is to geometrize time dependent Lagrangian mechanics in a way that the framework of second order tangent bundles plays an essential role. To this end, we first introduce the concepts of time dependent connections and time dependent semisprays on a manifold and their induced vector bundle struc…
These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…