We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…
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48 results for “higher-order terms”
For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann -invariants, which we call higher-order signatures. The higher-order genera o…
New method improves DAG learning by using large coefficients for higher-order terms.
problem Recovering DAG structures from observational data is challenging due to combinatorial optimization.
method Proposes truncated matrix power iteration to approximate DAG constraints efficiently.
result Empirically outperforms previous methods by a factor of 3 or more in structural Hamming distance.
Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
problem Proving relations between Reshetikhin-Turaev and re-normalized link invariants for links.
method Analyzing higher order terms of re-normalized link invariants for plumbed links.
result Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming u…
We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the…
A new method predicts higher-order interactions in evolving graphs using simplicial complexes.
problem Predicting higher-order interactions in dynamic graphs with theoretical guarantees.
method Capturing higher-order interactions as simplices, modeling neighborhoods with face-vectors, and developing a nonparametric kernel estimator.
result Our method outperforms existing higher-order prediction methods and is theoretically consistent.
The parametric complexity is the key quantity in the minimum description length (MDL) approach to statistical model selection. Rissanen and others have shown that the parametric complexity of a statistical model approaches a simple function of the Fisher information volume of the model as the sample size goes to in…
Unified field theory from higher-order Riemannian geometry.
problem Field-theoretical unification of fundamental forces.
method Exploiting higher-order Riemannian geometry and Einstein-Hilbert action, deriving gauge theories and predicting physical constants.
result Theoretical predictions for Weinberg angle and Coulomb's constant match experimental values.
This work improves online SGD's sample complexity for multi-index models by considering higher-order terms.
problem Suboptimal sample complexity for learning multi-index models using online SGD.
method Focus on both second- and higher-order terms to improve sample complexity.
result Online SGD achieves samples for multi-index models.
Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.
problem Uniqueness of hypersurfaces with constant higher order mean curvature in hyperbolic space.
method Generalization of Bernstein theorem and proof of Bernstein type results for immersed hypersurfaces.
result Rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures.
We prove a lower bound for the -th Steklov eigenvalues in terms of an isoperimetric constant called the -th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…
This work explores higher-order algebroids via vector bundle comorphisms.
problem Generalizing concepts of higher-order tangent bundles and Lie algebroids.
method Introduces a vector bundle comorphism approach to describe higher-order algebroids.
result Establishes a one-to-one correspondence between higher-order Lie algebroids and specific algebraic structures.
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations …
GUIDE detects anomalies in attributed networks by reconstructing node attributes and higher-order structures.
problem Lack of effective mechanisms for detecting anomalies in complex network interactions.
method GUIDE uses attribute and structure autoencoders, graph attention, and reconstruction errors to identify anomalies.
result GUIDE significantly outperforms state-of-the-art methods on multiple real-world datasets.
We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kähler geometry.
GENIE accelerates DDM synthesis with higher-order solvers.
problem Efficiently solving the differential equation for high-quality generation.
method Higher-order Taylor methods, utilizing Jacobian-vector products.
result GENIE significantly accelerates synthesis compared to previous solvers.
H-GAT improves stock selection by capturing complex higher-order stock relations and integrating both technical and fundamental analysis.
problem Stock selection difficulty and lack of comprehensive analysis.
method Higher-order Graph Attention Network (H-GAT) that incorporates both technical and fundamental analysis.
result H-GAT outperforms existing methods in stock selection metrics.
We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the inters…
We consider the higher order buckling eigenvalues of the following Dirichlet poly-Laplacian in the unit sphere with order . We obtain universal bounds on the th eigenvalue in terms of the first th eigenvalues independent of the domains. In particular, for , our result is shar…
New approach approximates c-space geometry of multi-loop linkages.
problem Higher-order mobility analysis of multi-loop linkages.
method Higher-order Taylor series expansion of geometric constraint mapping using joint screws.
result Local approximation of c-space and configurations with certain rank.
We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the -Laplacian. In the case of the closed eigenvalue problem and the Neuma…
Users form information trails as they browse the web, checkin with a geolocation, rate items, or consume media. A common problem is to predict what a user might do next for the purposes of guidance, recommendation, or prefetching. First-order and higher-order Markov chains have been widely used methods to study such se…
The moments of spatial probabilistic systems are often given by an infinite hierarchy of coupled differential equations. Moment closure methods are used to approximate a subset of low order moments by terminating the hierarchy at some order and replacing higher order terms with functions of lower order ones. For a give…
In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically -- showing promising results. The following work investigates GNNs from a theoretical…
HAMD optimizes cubic portfolios without quadratization, achieving better results.
problem Optimizing higher-order portfolio models with reduced distortion.
method Hybrid pipeline combining continuous Hamiltonian search, cardinality-preserving projection, and iterated local search.
result HAMD achieves significantly lower native cubic objective values than classical heuristics.
A new probabilistic BTD method for tensor data.
problem Modeling higher-order tensors with robust inference.
method Probabilistic Block-Term Decomposition using variational Bayesian inference and von-Mises Fisher distribution.
result The proposed pBTD can quantify multi-linear structures robustly.
The paper proves existence and classification of translating solitons in warped product manifolds.
problem Existence and classification of translating solitons in warped product manifolds.
method Proving existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows in warped product manifolds.
result Existence and classification of translating solitons in warped product manifolds.
Paper proves higher-order flow matching preserves optimality in generative modeling.
problem Theoretical guarantees for higher-order flow matching in generative modeling.
method Neural network approximations with controlled depth, width, and sparsity.
result Proves worst case optimality for second-order flow matching.
MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldsstat.ML
MACE uses higher-order messages to create fast, accurate force fields.
problem Creating fast and accurate force fields in computational chemistry and materials science.
method Introducing MACE, an equivariant MPNN model that uses four-body messages.
result MACE reduces the required number of message passing iterations to just two, achieving state-of-the-art accuracy.
The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
problem Establishing higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
method Introducing conformally covariant boundary operators, proving extension theorems, and establishing trace inequalities.
result Generalized CR Sobolev trace inequalities for all γ ∈ (0, n+1) \mathbb{N}.
We study the finite horizon Merton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expansion techniques, we derive approximations for the both the value function and the optimal investment strategy. We also analyze the `implied Sharpe ratio' and derive a series a…
Improved estimation of higher order integrals using shrinkage techniques.
problem Estimating higher order Bochner integrals in non-parametric settings.
method Shrinkage of U-statistic towards a target element, considering kernel degeneracy.
result Consistent shrinkage estimators with fast rates of convergence, even for non-degenerate kernels.
Study optimizes zero-order strongly convex function minimization with higher order smoothness.
problem Optimizing a strongly convex function with noisy evaluations.
method Randomized approximation of projected gradient descent with smoothing kernel.
result Upper bounds and minimax lower bounds for the algorithm, showing near-optimality.
A new HOM model improves forecasting of Indian base metal prices.
problem Improving accuracy in predicting base metal prices in the Indian market.
method A Higher Order Markovian (HOM) model with varying order based on market delay.
result The HOM model consistently outperforms the standard Markovian model in forecasting.
A quantum framework optimizes collateral allocation for derivatives.
problem Legal constraints and operational rules in collateral allocation for derivatives.
method Certified higher-order quantum framework that normalizes margin requirements and builds a bounded neighborhood of actions.
result Quantum framework improves certified sample quality compared to classical methods.
New methods compute Alexander polynomials for complex knots.
problem Efficiently computing higher order Alexander polynomials for complex knots.
method Developed new algorithms to compute the Smith normal form of Alexander matrices.
result Computed Alexander polynomials for knots up to 100 crossings.
The paper proves a new discrete Laplacian for 3D meshes and shows its superiority over primal construction.
problem Developing a more accurate discrete Laplacian for 3D meshes.
method Proves the Euler-Lagrange equation for the Dirichlet energy using the associated discrete Laplacian of the dual construction.
result The associated discrete Laplacian is optimal in compared to the primal construction.
Post-Einsteinian Effects in the General Theory of Relativity from Higher-Order Riemannian Geometrymath.DG
Higher-order geometry modifies Newtonian dynamics and predicts anomalies in spacecraft motion.
problem Observing and understanding higher-order effects in general relativity.
method Generalizing the Einstein-Hilbert action to include higher-order infinitesimals and studying field equations and cosmologies.
result Higher-order corrections predict anomalies like the Pioneer and flyby effects.
Efficiently approximates higher-order derivatives for generative models.
problem Expensive computation of higher-order derivatives in generative models.
method Rewrite SM objective in terms of directional derivatives and use finite difference for efficient approximation.
result Comparable results to gradient-based methods but significantly more computationally efficient.
We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.
SpeqNets improve graph neural networks by scaling and adapting to graph sparsity.
problem Graph neural networks struggle with permutation-equivariant functions and scalability to large graphs.
method Introducing sparsity-aware, permutation-equivariant graph networks with heuristics for graph isomorphism.
result Significantly improved predictive performance and reduced computation times compared to existing methods.
For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.
In this paper, using the method of moving frames, we generalise some of Terracini's results on varieties with tangent defect. In particular, we characterise varieties with higher order osculating defect in terms of Jacobians of higher fundamental forms and moreover we characterise varieties with "small" higher fundamen…
In this article a homotopy co-momentum map (à la Callies-Frégier-Rogers-Zambon) trangressing to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coa…
A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucia…
Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical the…