We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent …
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48 results for “second-order decomposition”
Conformal operators on weighted forms; their decomposition and null space on Einstein manifoldsmath.DG
There is a class of Laplacian like conformally invariant differential operators on differential forms which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explic…
We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) a…
The purpose of this present paper is to investigate the geometric structure of regular overdetermined systems of second order with two independent and one dependent variables from the point of view of rank 2 prolongations. Utilizing this notion of prolongations, we characterize the type of these overdetermined systems.…
We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's diameter, we argue that the method performs well when initialized with a second-…
Study on spin random fields using chaos decomposition for cosmic microwave background modeling.
problem Modeling polarization of Cosmic Microwave Background using spin random fields.
method Explicit Wiener-Itô chaos decomposition of area measures of level sets.
result Reveals a clear difference between high frequency regime and zero spin case.
Paper studies matching of samples from two distributions with a Gibbs probability weight.
problem Matching two independent i.i.d. samples from two distributions with a weighted cost.
method Uses chaos decomposition of polynomial functions of empirical distributions to derive asymptotics.
result Convergence of resulting random joint distribution to Schrödinger problem solution as N→∞.
Sensitivity analysis of the utility maximization problem with respect to model perturbationsq-fin.PM
We study the sensitivity of the expected utility maximization problem in a continuous semi-martingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value …
The Geometry of Filteringmath.DG
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …
A new method for computing image curvature efficiently and accurately.
problem Low performance, low accuracy, and requirement of second order differentiability in conventional computation schemes.
method Proposes a novel discrete computation scheme for weighted Gaussian curvature.
result More accurate, computationally more efficient, and does not require second order differentiability.
New bounds show current methods overestimate system parameter errors.
problem Current bounds overestimate parameter errors in system identification.
method Utilized asymptotic normality and second-order decomposition.
result Obtained finite-sample bounds matching optimal rates up to constants.
Study uses G-BSDEs to decompose pricing kernels under robust G-expectation.
problem Long-term decomposition of robust pricing kernels under G-expectation.
method Proposes and analyzes three types of quadratic G-BSDEs to decompose pricing kernels.
result Pricing kernels decomposed into four components: discounting, transitory, symmetric martingale, and volatility uncertainty.
We consider the problem of online subspace tracking of a partially observed high-dimensional data stream corrupted by noise, where we assume that the data lie in a low-dimensional linear subspace. This problem is cast as an online low-rank tensor completion problem. We propose a novel online tensor subspace tracking al…
New method solves stochastic optimization problems with random models.
problem Optimizing stochastic objectives with deterministic constraints.
method Trust-Region Sequential Quadratic Programming with random model.
result Global convergence guarantees for first- and second-order stationary points.
SINGD improves KFAC for memory-efficiency and stability in low-precision training.
problem Memory inefficiency and numerical instability of KFAC in low-precision training.
method Formulated inverse-free KFAC update and imposed structures in Kronecker factors.
result SINGD is memory-efficient and numerically robust, often outperforming AdamW in half precision.
In this note, we focus on smooth nonconvex optimization problems that obey: (1) all local minimizers are also global; and (2) around any saddle point or local maximizer, the objective has a negative directional curvature. Concrete applications such as dictionary learning, generalized phase retrieval, and orthogonal ten…
The Dynamic Mode Decomposition (DMD) extracted dynamic modes are the non-orthogonal eigenvectors of the matrix that best approximates the one-step temporal evolution of the multivariate samples. In the context of dynamical system analysis, the extracted dynamic modes are a generalization of global stability modes. We a…
QMME balances cost and speed in convex optimization.
problem Slow convergence of first-order methods and high cost of second-order methods.
method Minimizing quadratic majorants with fixed curvature at each iteration.
result QMME framework achieves sequential convergence under standard assumptions.
Joint blind source separation (J-BSS) is an emerging data-driven technique for multi-set data-fusion. In this paper, J-BSS is addressed from a tensorial perspective. We show how, by using second-order multi-set statistics in J-BSS, a specific double coupled canonical polyadic decomposition (DC-CPD) problem can be formu…
We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strate…
This paper tackles gauge fixing and regularity for perturbations around spherical backgrounds.
problem Understanding gauge freedom and regularity in perturbation theory for symmetric tensors.
method Analyzing Hodge-type decomposition for axially symmetric and axistationary tensors, showing existence and uniqueness of gauge tensors.
result Stationary and axially symmetric second order perturbations can be rendered in a canonical form with only one degree of differentiability loss near the origin.
Study geometric flows of G2-structures, determining curvature and torsion invariants.
problem Investigate geometric flows of G2-structures and their invariants.
method Explicitly compute differential invariants, decompose curvature and torsion, analyze principal symbols.
result Established short-time existence and uniqueness for geometric flows of G2-structures.
Improved sampling for Diffusion Models by accounting for covariance.
problem Sampling quality degradation in few-step Diffusion Models.
method Covariance-aware sampler using Tweedie's formula and Fourier-space decomposition.
result Consistently superior samples compared to state-of-the-art samplers.
Given a pair of second order diffusion operators, one on the total space of a principle bundle and the other on the base space , intertwined by the projection , if the operator on the base manifold has constant rank, we define a semi-connection on the principal bundle which allows to spl…
This paper studies large-scale dynamical networks where the current state of the system is a linear transformation of the previous state, contaminated by a multivariate Gaussian noise. Examples include stock markets, human brains and gene regulatory networks. We introduce a transition matrix to describe the evolution, …
Tensor Decompositions for Identifying Directed Graph Topologies and Tracking Dynamic Networksstat.ML
Directed networks are pervasive both in nature and engineered systems, often underlying the complex behavior observed in biological systems, microblogs and social interactions over the web, as well as global financial markets. Since their structures are often unobservable, in order to facilitate network analytics, one …
Second-order estimator improves continuous-time policy evaluation.
problem Estimating value surfaces from discrete data with time-inhomogeneous dynamics.
method Moment-matching coefficients for high-order generator regression.
result Second-order estimator consistently outperforms Bellman baseline.
This work analyzes how preconditioning affects generalization in machine learning models.
problem The impact of preconditioning on the generalization of machine learning models.
method An asymptotic bias-variance decomposition of the generalization error for ridgeless regression under various preconditioners.
result The optimal preconditioner depends on label noise, model specification, and signal alignment, with NGD potentially better under certain conditions.
Extensions and Applications of Stein-Weiss Operators to the Study of Traceless Symmetric Tensorsmath.DG
The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.
problem Understanding Stein-Weiss operators on symmetric tensors of arbitrary rank.
method Analyzing the decomposition of tensor spaces into irreducible components and computing Weitzenbock formulas.
result Unified framework for second-order Stein-Weiss operators and tools for geometric analysis.
The study examines numerical aspects of Karhunen-Loève expansions for stochastic processes.
problem Constructing Karhunen-Loève expansions for second-order stochastic processes.
method Spectral decomposition of covariance operator via Fredholm integral equation, discretization, singular value decomposition of weight-scaled sample matrix.
result Consistent solutions for model-based and data-driven KLE construction, characterized by convergence of SVD-based eigenvalue estimates and KL coefficients distributions.
We consider large scale empirical risk minimization (ERM) problems, where both the problem dimension and variable size is large. In these cases, most second order methods are infeasible due to the high cost in both computing the Hessian over all samples and computing its inverse in high dimensions. In this paper, we pr…
Exact and high order discretization schemes for Wishart processes and their affine extensionsmath.PR
This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator, in order to use composition techniques as Ninomiya and Victoir or Alfonsi. Doing so, we have found a remarkable splitting for Wishart proc…
Second-order economic theory considers new variables to improve price volatility predictions.
problem Current economic models focus on first-order variables, missing second-order variables that affect price volatility.
method Introduces second-order economic theory with new variables composed of sums of squares of agents' transactions.
result Second-order economic theory complements first-order variables and introduces new macroeconomic variables.
Latent variable models with hidden binary units appear in various applications. Learning such models, in particular in the presence of noise, is a challenging computational problem. In this paper we propose a novel spectral approach to this problem, based on the eigenvectors of both the second order moment matrix and t…
Second-order optimizers retain residual information after data deletion, affecting machine unlearning.
problem Residual information in second-order optimizers after data deletion.
method Comparison of first-order and second-order learners, eigendecomposition analysis.
result Second-order optimizers retain residual information, not detectable by first-order analysis.
Exact second-order optimization for deep learning reduces computational cost and improves performance.
problem Inadequate use of second-order optimization methods in deep learning due to high computational cost and non-convexity.
method Developed an exact stochastic second-order Newton method that addresses the non-convexity issue and provides an expression for the stochastic Hessian.
result Exact second-order Newton direction formula and its application in deep learning datasets.
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.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
problem Analyzing convexity in differential equations on Lie algebroids.
method Theory development for local convexity of SODEs on Lie algebroids.
result Extensive discussion of homogeneous quadratic SODEs on Lie algebroids.
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.
Negative step sizes improve second-order methods for neural networks.
problem Second-order methods discard negative curvature, limiting their effectiveness.
method Introduce negative step sizes in second-order methods combined with Wolfe line search.
result Negative step sizes lead to global convergence and improved performance.
Simplified argument for second order estimate in quaternionic Calabi-Yau problem.
problem Second order estimates for quaternionic Calabi-Yau problem on hyperkähler manifolds.
method Simplified argument to derive the estimate.
result Simplified derivation of second order estimate.
Paper studies second order tail probabilities in risk models.
problem Analyzing tail probabilities in risk models with constant interest force.
method Asymptotic expansion and weighted Kesten-type inequality for second order subexponential random variables.
result Second order asymptotic formulae for continuous-time renewal risk models are derived.
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.
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.
Paper analyzes Nyström regularization for time series forecasting with sequential sub-sampling.
problem Learning rate analysis of Nyström regularization for -mixing time series.
method Banach-valued Bernstein inequality and integral operator approach for -mixing sequences.
result Almost optimal learning rates for Nyström regularization with sequential sub-sampling.
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 …
In this paper, we study stochastic non-convex optimization with non-convex random functions. Recent studies on non-convex optimization revolve around establishing second-order convergence, i.e., converging to a nearly second-order optimal stationary points. However, existing results on stochastic non-convex optimizatio…