The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset of the Euclidean space and for eve…
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48 results for “higher-order approximations”
Proposes a method to enhance multi-view learning by maximizing higher order correlations.
problem Losing intrinsic interconnections among multiple views in pairwise correlation maximization.
method Formulates multi-view data as a low rank approximation problem using higher order correlation tensor and solves it with the generating polynomial method.
result Consistently outperforms prior methods on real multi-view data.
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.
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…
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.
Cross validation (CV) and the bootstrap are ubiquitous model-agnostic tools for assessing the error or variability of machine learning and statistical estimators. However, these methods require repeatedly re-fitting the model with different weighted versions of the original dataset, which can be prohibitively time-cons…
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…
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…
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.
Several Approximation Algorithms for Sparse Best Rank-1 Approximation to Higher-Order Tensorsmath.NA
Four algorithms improve sparse tensor BR1Approx with theoretical guarantees.
problem Sparse tensor best rank-1 approximation.
method Four approximation algorithms exploiting multilinearity and sparsity.
result Theoretical worst-case approximation lower bounds for all algorithms.
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.
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 present an extension of the Kolmogorov-Smirnov (KS) two-sample test, which can be more sensitive to differences in the tails. Our test statistic is an integral probability metric (IPM) defined over a higher-order total variation ball, recovering the original KS test as its simplest case. We give an exact representer…
We provide improved convergence rates for various \emph{non-smooth} optimization problems via higher-order accelerated methods. In the case of regression, we achieves an iteration complexity, breaking the barrier so far present for previous methods. We arrive at a similar rate fo…
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.
Expectation Propagation (EP) provides a framework for approximate inference. When the model under consideration is over a latent Gaussian field, with the approximation being Gaussian, we show how these approximations can systematically be corrected. A perturbative expansion is made of the exact but intractable correcti…
Enhances stock movement prediction using Higher Order Transformers for multimodal time-series data.
problem Predicting stock movements in financial markets with complex dynamics.
method Introduced Higher Order Transformers, extending self-attention and transformer architecture to capture complex market dynamics. Employed low-rank tensor decomposition and kernel attention to manage computational complexity. Integrated technical and fundamental analysis from historical prices and tweets.
result Demonstrated effectiveness of the method on the Stocknet dataset, improving stock movement prediction.
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…
We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with ar…
Many nonlinear extensions of the Kalman filter, e.g., the extended and the unscented Kalman filter, reduce the state densities to Gaussian densities. This approximation gives sufficient results in many cases. However, this filters only estimate states that are correlated with the observation. Therefore, sequential esti…
This paper constructs metrics with constant fractional higher order curvature on punctured spheres.
problem Constructing complete metrics with constant fractional higher order curvature on punctured spheres.
method The approach involves constructing singular solutions for a conformally invariant integro-differential equation, reducing the problem to solving an infinite-dimensional Toda-type system.
result Unified approach for fractional and higher order cases, proving Fredholm properties for the linearized operator.
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…
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.
We analyze analytic approximation formulae for pricing zero-coupon bonds in the case when the short-term interest rate is driven by a one-factor mean-reverting process with a volatility nonlinearly depending on the interest rate itself. We derive the order of accuracy of the analytical approximation due to Choi and Wir…
We study differential forms and their higher-order generalizations by interpreting them as functions on map spaces. We get a series of approximations of "generalized manifolds" (i.e. of sheaves and stacks) somewhat akin to Taylor series.
Study uses random matrix theory to improve tensor approximation accuracy.
problem Improving tensor approximation accuracy in the presence of noise.
method Random matrix theory applied to tensor unfoldings.
result Characterizes spectral behavior of tensor unfoldings and predicts reconstruction performance.
We provide a simple method and relevant theoretical analysis for efficiently estimating higher-order lp distances. While the analysis mainly focuses on l4, our methodology extends naturally to p = 6,8,10..., (i.e., when p is even). Distance-based methods are popular in machine learning. In large-scale applications, sto…
CSD improves goodness-of-fit testing for higher-order dependence.
problem Insensitivity of standard KSDs to higher-order dependence features like tail dependence.
method Introduces Copula-Stein Discrepancy (CSD) that targets dependence geometry directly on copula density.
result CSD is sensitive to differences in tail dependence coefficients and metrizes weak convergence of copula distributions.
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.
Generalizes SMCI to improve Boltzmann machine learning accuracy.
problem Limitation in applying higher-order SMCI to dense systems.
method Generalized SMCI (GSMCI) and new PBM learning method.
result GSMCI allows higher-order approximations for dense systems.
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.
New method bypasses assumptions for unbiased estimation of complex system interactions.
problem Inferring pair-wise and higher-order interactions from observational data.
method Cross-disciplinary approach using Targeted Learning for unbiased estimation.
result Universal estimator of all-order symmetric interactions without parametric assumptions.
Framework predicts nonlinear system responses using GFDT and generative models.
problem Predicting higher-order moments of nonlinear stochastic systems to small perturbations.
method Combining GFDT with generative modeling to estimate score function directly from data.
result Accurately captures nonlinear and non-Gaussian features of system responses.
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.
Approximates discounted moments for financial products using polynomial expansions.
problem Approximating discounted moments of stochastic processes for financial applications.
method High-order power series expansion of the infinitesimal generator.
result Error decreases to around 10 to 100 times machine precision for higher orders.
Extends RRR to capture nonlinear interactions in multi-response regression.
problem Complex relationships in real-world data cannot be adequately modeled by linear interactions.
method Introduces Higher Order Reduced Rank Regression (HORRR) using tensor representations and Tucker decomposition.
result HORRR can capture nonlinear interactions in multi-response regression.
Paper accelerates diffusion models without retraining, reducing evaluations.
problem Approximating target data distributions efficiently.
method Training-free sampling algorithm using high-order Lagrange interpolation.
result Requires fewer score function evaluations than previous methods.
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.
Sketching is a randomized dimensionality-reduction method that aims to preserve relevant information in large-scale datasets. Count sketch is a simple popular sketch which uses a randomized hash function to achieve compression. In this paper, we propose a novel extension known as Higher-order Count Sketch (HCS). While …
Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive Marginal Quantization of the Euler scheme has recen…
Paper provides Edgeworth expansions for network moments, improving accuracy of sampling distributions.
problem Accurate descriptions of sampling distributions of network moment statistics.
method Edgeworth expansion applied to studentized network moment statistics.
result Higher-order accurate approximation to sampling CDF of network moment statistics.
Transformers can approximate Newton's method for logistic regression.
problem Implementing higher order optimization methods in Transformers.
method Linear attention Transformers with ReLU layers approximating second order optimization algorithms.
result Transformers can implement a single step of Newton's iteration for matrix inversion.
Parametric model reduction of mean-field and stochastic systems via higher-order action matchingstat.ML
This work learns models for population dynamics using variational methods and higher-order quadrature.
problem Modeling population dynamics of physical systems with stochastic and mean-field effects.
method Variational problem to infer gradient fields, combining Monte Carlo sampling with higher-order quadrature rules.
result Accurate prediction of population dynamics over a wide range of parameters.
This technical report constructs a theoretical framework to relate standard Taylor approximation based optimisation methods with Natural Gradient (NG), a method which is Fisher efficient with probabilistic models. Such a framework will be shown to also provide mathematical justification to combine higher order methods …
Paper tackles non-convex optimization for higher moments in portfolio management.
problem Complexity of higher moments in optimization problems.
method Method of successive convex approximation.
result Solves mean-variance-skewness problem using non-convex optimization.
We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetriza…
Low rank tensor learning, such as tensor completion and multilinear multitask learning, has received much attention in recent years. In this paper, we propose higher order matching pursuit for low rank tensor learning problems with a convex or a nonconvex cost function, which is a generalization of the matching pursuit…
The convolution method for the numerical solution of forward-backward stochastic differential equations (FBSDEs), introduced in [21], uses a uniform space grid. In this paper we utilize a tree-like spatial discretization that approximates the BSDE on the tree, so that no spatial interpolation procedure is necessary. In…