Paper proves existence of a CMC hypertorus in 4D sphere using numerical methods.
problem Proving the existence of a constant mean curvature (CMC) hypertorus in \(S^4\).
method Employed the round Taylor method with rational arithmetic and the Poincare-Miranda theorem.
result Existence of a constant mean curvature (CMC) hypertorus in \(S^4\).
Paper describes a family of periodic solutions in the three body problem.
problem Three body problem in celestial mechanics.
method 1-dimensional family of initial conditions Σ, Round Taylor Method for error tracking.
result Family Σ contains embedded curves with different symmetries.
New Taylor method proves periodic 3-body problem solution.
problem Existence of periodic solutions in the 3-body problem.
method A small variation of the Taylor method with global error formula.
result Rigorous proof of periodic solution existence.
AdaRound improves post-training quantization of neural networks.
problem Improving the accuracy of quantized weights in neural networks.
method Adaptive rounding mechanism that adapts to data and task loss.
result AdaRound outperforms rounding-to-nearest and achieves state-of-the-art performance.
A framework connects Taylor methods with Fisher-efficient NG.
problem Combining Taylor-based methods with Fisher-efficient NG.
method Constructs a theoretical framework linking Taylor approximation and NG.
result Mathematical justification for combining higher order methods with NG.
Study finds Deep Taylor Decomposition is unreliable for explaining neural networks.
problem Reliability of Deep Taylor Decomposition for explaining neural networks.
method Investigated the theoretical foundations of Deep Taylor Decomposition (DTD) and found it under-constrained.
result DTD is unreliable because its theoretical foundations are under-constrained and roots do not align with input.
Proposes a Taylor framework to unify and analyze attribution methods.
problem Lack of a unified guideline for feature contribution assignment in machine learning models.
method Introduces a Taylor attribution framework to model the attribution problem and reformulates fourteen mainstream methods.
result Empirically validates the Taylor reformulations and reveals a positive correlation between performance and principles followed.
TaylorPODA uses Taylor expansions to improve feature attributions for opaque models.
problem Lack of systematic framework for quantifying feature contributions in opaque models.
method Taylor expansion framework with postulates (precision, federation, zero-discrepancy, adaptation).
result TaylorPODA achieves competitive results and provides principled explanations.
Paper proposes a new Taylor moment expansion for non-linear Gaussian filtering and smoothing.
problem Non-linear Gaussian filtering and smoothing in continuous-discrete state-space models.
method Taylor moment expansion (TME) for moment functions directly and in time variable.
result Significantly outperforms state-of-the-art methods in terms of estimation accuracy and numerical stability.
The Adomian decomposition method is shown to be equivalent to the Taylor series approach.
problem Incorrectly perceived complexity of the Adomian decomposition method.
method Demonstrates the Adomian decomposition method as equivalent to the Taylor series approach.
result The Adomian decomposition method is simpler and more straightforward.
Unified framework for analyzing machine learning model attributions.
problem Lack of a general and theoretical framework for understanding attribution methods.
method Proposes a Taylor attribution framework to unify and analyze seven mainstream attribution methods.
result Established three principles for good attribution and empirically validated the Taylor reformulations.
Kernelized Taylor diagram visualizes data populations with fewer assumptions.
problem Limitations of Taylor diagram in capturing non-linear relationships and sensitivity to outliers.
method Proposes a kernelized version of the Taylor diagram that uses maximum mean discrepancy and kernel mean embedding.
result Kernelized Taylor diagram visualizes data populations with minimal assumptions of data distributions.
Paper provides convergence guarantees for rectifier networks using neural Taylor approximations.
problem Smoothness and convexity issues in modern convolutional networks.
method Neural Taylor approximations and Taylor loss for optimization.
result Guarantees match lower bounds for convex nonsmooth functions and accurately capture optimization dynamics.
Taylor expansions improve reinforcement learning policies.
problem Improving reinforcement learning policy optimization.
method Taylor expansion policy optimization.
result Taylor expansions enhance performance of distributed algorithms.
TEAM uses Taylor expansion to generate adversarial examples.
problem Vulnerability of deep neural networks to adversarial examples.
method Approximates DNN output using Taylor expansion and optimizes with Lagrange multiplier method.
result Improves robustness of DNNs through adversarial training.
An expanding literature articulates the view that Taylor rules are helpful in predicting exchange rates. In a changing world however, Taylor rule parameters may be subject to structural instabilities, for example during the Global Financial Crisis. This paper forecasts exchange rates using such Taylor rules with Time V…
We introduce Taylor expansions that do not require the differentiability. We also provide new solutions to partial differential equations. We apply our methods to finance.
Taylorized training improves neural network training at finite width.
problem Understanding and improving neural network training at finite width.
method Training the k-th order Taylor expansion of the neural network at initialization.
result Taylorized training agrees with full neural network training better as k increases and can significantly close the performance gap.
Proves a special case of the Gaussian kinematic formula using large sphere limits.
problem Proving a special case of the Gaussian kinematic formula.
method Viewing the GKF as the limit of spherical kinematic formulas for large dimension spheres.
result Proves a special case of the Gaussian kinematic formula.
Developed Taylor series for muscle-finger system analysis.
problem Understanding the complex relationship between muscle activity and finger movement.
method Used Dendrite Net to develop Taylor series and construct relation spectrum.
result Found muscle synergy and coupling in hand movement.
We provide new exact Taylor's series with fixed coefficients and without the remainder. We demonstrate the usefulness of this contribution by using it to obtain very simple solutions to (non-linear) PDEs. We also apply the method to the portfolio model.
This paper provides algebraic descriptions for aperiodic monotilings.
problem Tackles the aperiodic monotilings introduced by Taylor and Socolar.
method Gives decoration-free algebraic descriptions and shows how monotilings can be derived from a single equation.
result Algebraic equations can describe aperiodic monotilings without needing decorations.
New resonance theory for Anosov flows connects spectral properties to mixing measures.
problem Defining and analyzing Ruelle-Taylor resonances for Anosov actions.
method Combining microlocal methods and J. Taylor's cohomological theory, defining Ruelle-Taylor resonances and proving Fredholm theory.
result Ruelle-Taylor resonances form a discrete subset of Cκ with λ=0 being a leading resonance. We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the …
A new method explains anomalies in one-class models using deep Taylor decomposition.
problem Understanding why one-class models classify data points as anomalies.
method Recompose one-class SVM as a neural network, perform deep Taylor decomposition.
result The method reliably explains a wide set of data anomalies and outperforms baselines.
New approximations for Asian basket spread options using stochastic Taylor expansions.
problem Pricing Asian basket spread options under the Black-Scholes model.
method Stochastic Taylor expansion applied to a log-normal proxy model.
result Highly accurate approximations for Asian and spread options, without numerical integration.
Paper develops a new algorithm to find shortest paths on surfaces.
problem Finding shortest paths on surfaces with defined metrics.
method Uses Taylor expansion of exponential map for numerical computation.
result Developed a new algorithm to find geodesics efficiently.
Paper explains accrual and mark-to-market valuation for interest rate trades.
problem Understanding the valuation differences between accrual and mark-to-market methods for interest rate trades.
method Comparison of discounted cashflow valuation to spread-based valuation, Taylor series approximation, and deferral concept.
result Simple intuition and mathematical explanation of accrual and mark-to-market adjustments.
Optimized online learning with kernels for large-scale adversarial data.
problem Efficient online learning for large-scale, potentially adversarial datasets.
method Online variations of kernel Ridge regression using approximated basis functions.
result Optimal regret for a wide range of kernels with low per-round complexity.
New learning algorithm for real analytic functions without gradient descent.
problem Learning real analytic functions without gradient descent.
method Taylor approximation and sampling data distribution.
result Nonuniform learning result for real analytic functions.
SOAR improves deep networks' robustness against adversarial examples.
problem Improving deep neural networks' robustness against adversarial examples.
method Formulated adversarial robustness problem under robust optimization framework, approximated loss function using second-order Taylor series expansion.
result SOAR significantly improves robustness of networks against adversarial perturbations.
New insights on pruning deep networks by preserving function locality.
problem Designing effective pruning methods for deep neural networks.
method Revisited loss modeling using first and second order Taylor expansions, emphasizing locality.
result Both first and second order Taylor expansions can achieve similar performance in pruning.
In this article we develop a method for the strong approximation of stochastic differential equations (SDEs) driven by Lévy processes or general semimartingales. The main ingredients of our method is the perturbation of the SDE and the Taylor expansion of the resulting parameterized curve. We apply this method to devel…
Paper develops formulas for shape derivatives in wave scattering.
problem Computing high order shape derivatives for wave scattering is challenging.
method Introduces elegant recurrence formulas using differential forms and Lie derivatives.
result Unified framework for computing high order shape perturbations in scattering problems.
This paper extends AD techniques to Monte Carlo processes for efficient derivative calculation.
problem Obtaining derivatives of expectation values in Monte Carlo processes.
method Two approaches: reweighting and Hamiltonian extension of HMC.
result Hamiltonian approach as a change of variables simplifies variance reduction.
Paper introduces cubature method for stochastic Volterra equations.
problem Solving stochastic Volterra integral equations efficiently.
method Derive stochastic Taylor expansion, introduce cubature measure.
result Cubature method is more efficient than Euler scheme under certain conditions.
Extends Bonnet-Myers theorem with new generalizations.
problem Generalizing Bonnet-Myers theorem.
method Complementary generalization of existing extensions by Calabi and Cheeger-Gromov-Taylor.
result New theorem extending previous extensions of Bonnet-Myers.
Paper presents a new backward deep BSDE method for solving nonlinear FBSDE problems.
problem Nonlinear Forward Backward Stochastic Differential Equations (FBSDE) with terminal conditions.
method Backward deep BSDE method applied to FBSDE with nonlinear generators and random initial conditions.
result Derives exact and Taylor-based approximations for time-stepping nonlinear BSDEs.
Taylor's law found in stock market illiquidity, with varying exponents.
problem Understanding the temporal fluctuation scaling in stock illiquidity.
method Investigation of high-frequency illiquidity data from multiple exchanges and industries.
result Taylor's law holds with varying exponents (b > 2 for A-shares, b < 2 for B-shares) across different markets and sectors.
Iterative tilting fine-tunes diffusion models for reward-tilted distributions.
problem Fine-tuning diffusion models for reward-tilted distributions.
method Decomposes large reward tilts into smaller, tractable tilts via first-order Taylor expansion, avoiding backpropagation.
result Validated on a two-dimensional Gaussian mixture, achieving exact closed-form solutions.
Method finds all cross caps formally isometric to a given one.
problem Identifying cross caps that are formally isometric to a given one.
method Finding cross caps with matching Taylor expansions of first fundamental forms.
result A countable family of intrinsic invariants recognizes formal isometry classes completely.
In this work we consider the Taylor expansion of the exponential map of a submanifold immersed in R^n up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in R^3. Also we compute, by using the Taylor expan…
A method estimates and prunes neural network filters to reduce computation and improve accuracy.
problem Reduction of neural network parameters to save computation and energy.
method Estimates each neuron's contribution to loss using first and second-order Taylor expansions; iteratively removes less important neurons.
result High (>93%) correlation between estimated and true importance; 40% FLOPS reduction with 0.02% top-1 accuracy loss.
New method detects text changes under dependencies, outperforming baselines.
problem Detecting structural changes in m-dependent text data. method Kernel change-point detection under m-dependence. result Consistent and weakly consistent detection of change points in m-dependent text. Analyzes explaining nonlinear model predictions.
problem Understanding the contribution of inputs to outputs in nonlinear models.
method Merges integrated gradient and deep Taylor decomposition methods.
result Provides a natural reference point for model at use.
Paper proves existence of solutions for complex surface diffusion equation.
problem Existence of solutions for anisotropic surface diffusion with elasticity.
method Cahn-Taylor minimizing movement scheme for three-dimensional analysis.
result Proves existence of classical solutions without curvature regularization.
New sampling scheme improves ML accuracy in physics simulations.
problem Improving accuracy of ML models in physics simulations.
method Taylor-based data sampling scheme for DNNs.
result Reduces error in DNN solutions of ODE systems.
TEAM generates more powerful adversarial examples for DNNs.
problem Vulnerability of DNNs to imperceptible adversarial examples.
method TEAM uses Taylor expansion and Lagrangian multiplier method to craft adversarial examples.
result TEAM generates adversarial examples with 100% attack success rate using smaller perturbations.