In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations fro…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,695 papers · 148 categories
Trend · papers per month
48 results for “higher order derivatives”
Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic space…
A method to estimate high order derivatives of data distributions from samples.
problem Estimating high order derivatives of data distributions efficiently and accurately.
method Generalizing denoising score matching via Tweedie's formula to estimate higher order derivatives.
result Models trained with the proposed method can approximate second order derivatives more efficiently and accurately than via automatic differentiation.
The paper glosses different forms of an introducing of higher order tangent-like functors, especially functors derived from higher order nonholonomic tangent functors. A special attention is devoted to higher order osculating bundles: their identification with higher order tangent bundles is demonstrated as the main re…
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the -th order spectral variations of singular values.
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate .
Higher-order tangent bundles have geometric structures compatible with their iterated bundle structure.
problem Connection towers and Sasaki metrics on higher-order tangent bundles
method Introduce the notion of a connection tower and study the geometric structures induced by such towers.
result Connection towers determine multiconnections, adapted splittings, and canonical vector bundle structures.
Transversality Conditions for Higher Order Infinite Horizon Discrete Time Optimization Problemsmath.OC
In this paper, we examine higher order difference problems. Using the "squeezing" argument, we derive both Euler's condition and the transversality condition. In order to derive the two conditions, two needed assumptions are identified. A counterexample, in which the transversality condition is not satisfied without th…
State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothn…
This paper deals with the enumeration of the higher order non-trivial compositions of the differential operations and the directional derivative in the space (). We present the recurrences for a counting the higher order non-trivial compositions.
Introduces new connections in higher geometry.
problem Defining connections in higher geometry.
method Develops formal differentiation and integration of maps to derived stacks.
result Establishes new -algebras of higher symmetries.
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 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…
In this paper, we describe a geometric setting for higher-order lagrangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting appli…
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 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…
Improved GAN performance using higher-order Wasserstein moments.
problem Stabilizing and enhancing GANs for better mode coverage and stability.
method Deriving and training a GAN with a modified Wasserstein distance using higher-order moments.
result Training a GAN with higher-order Wasserstein moments improves performance, even with increased computational cost.
New minimal surfaces in 4D space derived from parametric equations.
problem Deriving explicit parametric equations for higher-order Henneberg-type minimal surfaces in .
method Generalized Weierstrass--Enneper representation and differential geometric analysis.
result Explicit parametric equations and differential geometric characteristics of the Henneberg-type minimal surfaces in .
Study improves BN TTA under distribution shift using higher-order asymptotics.
problem Improving BN TTA for changing data distributions.
method Integrates Edgeworth expansion and saddlepoint approximation with one-step M-estimation.
result Derives optimal weighting parameter for minimized mean-squared error.
This paper precisely estimates transformer derivatives for explicit learning guarantees.
problem Computing fully-explicit generalization bounds for transformers with precise higher-order derivative estimates.
method Analyzes and estimates all higher-order derivatives of transformers with multiple attention heads and layer normalization.
result Obtains explicit pathwise generalization bounds for transformers learning from non-i.i.d. samples.
The paper defines and analyzes higher-order Yang-Mills-Higgs functionals and their gradient flows.
problem Analyzing the behavior of higher-order Yang-Mills-Higgs functionals and their gradient flows.
method Gauge fixing technique, -bound of the Higgs field, local -derivative estimates, energy estimates, blow-up analysis.
result Solutions to the gradient flow do not hit finite time singularities under certain conditions.
In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures natura…
Paper introduces a framework for diagnosing Alzheimer's disease using higher-order topological features from fMRI.
problem Diagnosing Alzheimer's disease using brain network topology.
method Persistent homology to extract higher-order features (cycles, cavities) from fMRI data.
result Framework significantly outperforms existing methods in AD classification.
In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-W…
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 …
Closed-form relations and approximations for SE(3) derivatives for robust numerical simulations.
problem Deriving closed-form derivatives and approximations for SE(3) for robust numerical simulations.
method Avoiding block partitioning, deriving higher-order approximations for differential, first and second derivatives, Jacobian, and Hessian.
result Compact and numerically robust closed-form relations for SE(3) derivatives.
We show that solutions to certain higher-order intrinsic geometric flows on a compact manifold, including some flows generated by the ambient obstruction tensor, are unique. With the goal of providing a complete self-contained proof, details surrounding map covariant derivatives and a careful application of the DeTurck…
Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points su…
Higher-Order Ranking and Link Prediction: From Closing Triangles to Closing Higher-Order Motifscs.LG
In this paper, we introduce the notion of motif closure and describe higher-order ranking and link prediction methods based on the notion of closing higher-order network motifs. The methods are fast and efficient for real-time ranking and link prediction-based applications such as web search, online advertising, and re…
Derives formulae for general permutation equivariant layers and presents a second order graph variational encoder.
problem Tackles the limitation of previous equivariant neural networks by considering permutations of matrices.
method Derives formulae for general permutation equivariant layers, including matrix permutations. Presents a second order graph variational encoder.
result Latent distribution of equivariant generative models must be exchangeable.
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…
Combines neural networks and probabilistic graphical models for efficient higher-order inference.
problem Lack of efficient higher-order relational information in graph neural networks and probabilistic graphical models.
method Derives efficient approximate sum-product loopy belief propagation for higher-order PGMs, embeds into neural network, proposes methods for constructing higher-order factors.
result Substantially outperforms state-of-the-art k-order graph neural networks in molecular datasets.
CDF2PDF is a method of PDF estimation by approximating CDF. The original idea of it was previously proposed in [1] called SIC. However, SIC requires additional hyper-parameter tunning, and no algorithms for computing higher order derivative from a trained NN are provided in [1]. CDF2PDF improves SIC by avoiding the tim…
In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher acc…
We consider the local solution to the Calabi flow for C^αinitial metric. We also prove that the Calabi flow on compact Kaehler surfaces can be extended once the metrics along the flow are bounded in L^\infty sense. This can be viewed as obtaining higher order derivative estimates from second order derivatives for a fou…
We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element . Given this, we introduce an infinite sequence of higher brackets on the image of the project…
We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second varia…
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher--order in…
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.
Paper tackles stochastic control with mean and higher-order moments, finding Nash equilibria.
problem Time-inconsistent stochastic control problems with mean and higher-order moments.
method Developed closed-loop and open-loop Nash equilibrium controls using PDEs and maximum principles.
result Identical closed-loop and open-loop Nash equilibria controls, independent of state value and random path.
A Quasilinear Algorithm for Computing Higher-Order Derivatives of Deep Feed-Forward Neural Networkscs.LG
A new algorithm speeds up neural network derivative calculations.
problem Exponential runtime of autodifferentiation for high-order derivatives in neural networks.
method n-TangentProp, a quasilinear algorithm for computing higher-order derivatives.
result Computes exact derivatives in quasilinear time, not exponential.
Confusion of Tagged Perturbations in Forward Automatic Differentiation of Higher-Order Functionscs.SC
Forward Automatic Differentiation (AD) is a technique for augmenting programs to compute derivatives. The essence of Forward AD is to attach perturbations to each number, and propagate these through the computation. When derivatives are nested, the distinct derivative calculations, and their associated perturbations, m…
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
Paper characterizes equilibrium strategies for stochastic control with higher-order moments.
problem Stochastic control problems with higher-order moments.
method Novel characterization of time-consistent control problems, deriving equilibrium conditions via BSDEs.
result Derives sufficient and necessary conditions for an open-loop Nash equilibrium control (ONEC) in a novel way.
Using the reviewed Riemann-Liouville fractional derivative we introduce the fractional osculator Lagrange space of k order and the main structures on it. The results are applied at the k order fractional prolongation of Lagrange, Finsler and Riemann fractional structures.
In this study, we tested the interaction effect of multimodal datasets using a novel method called the kernel method for detecting higher order interactions among biologically relevant mulit-view data. Using a semiparametric method on a reproducing kernel Hilbert space (RKHS), we used a standard mixed-effects linear mo…
New principle for optimal control with higher order differential constraints.
problem Optimal control problems with higher order differential constraints.
method Derivation of the Principle of Minimal Labour and generalization of Pontryagin Maximum Principle.
result Generalized Pontryagin Maximum Principle for higher order constraints.
Defines a bundle map for currents on manifolds using higher covariant derivatives.
problem Defining a bundle map for currents on manifolds.
method Using higher covariant derivatives on a manifold equipped with a torsion-free connection.
result The bundle of generalized Weyl algebras and its properties.