Shallow neural networks can represent polynomials efficiently.
problem Representing polynomials using shallow neural networks.
method Using shallow neural networks of width 2(R+d)d to represent d-variate polynomials of degree R. result Derives minimax optimal convergence rate for shallow networks to unknown univariate regression functions.
New method finds better loss functions for neural nets.
problem Finding effective loss functions for deep neural networks.
method Optimizes multivariate Taylor polynomial parameterizations using CMA-ES.
result TaylorGLO finds loss functions that outperform existing methods.
Develops AMITE for analyzing neural network nonlinearities.
problem Addressing difficulties in verification, explainability, and security in neural network analysis.
method Analytically modified integral transform expansion (AMITE) for neural network nonlinearities.
result First to provide six mutually exclusive desired expansion properties.
In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor expansion…
A mathematical framework connects neural networks and polynomial regression for better model understanding.
problem Neural networks are black boxes with challenges in dimensioning and prediction error evaluation.
method Developed a mathematical framework using Taylor expansion to relate neural networks and polynomial regression.
result Polynomial approximations from neural networks trained on polynomial data are accurate locally.
New triangulations encode flows with vanishing polynomial.
problem Constructing veering triangulations with vanishing taut polynomial.
method Using connections between veering triangulations and pseudo-Anosov flows.
result Created arbitrarily large veering triangulations with vanishing taut polynomial.
NN2Poly converts deep neural networks into polynomial models for better understanding.
problem Improving neural network interpretability and theoretical understanding.
method Taylor expansion on activation functions, combinatorial properties, and polynomial coefficients calculation.
result NN2Poly accurately represents deep feed-forward neural networks as polynomial models.
By analyzing the affine Taylor expansion of a non-degenerate plane curve, we obtain characterizations of classes of such curves via curvature properties of the gravity curve. The proof is based on an analysis of the degree parity and leading coefficients of polynomials occurring in the expansion.
Developed algorithms to compute three polynomial invariants of veering triangulations.
problem Computing polynomial invariants of veering triangulations.
method Introduced and used algorithms for taut, veering, and Teichmüller polynomials based on upper and lower tracks of veering triangulations.
result Proved that the lower and upper taut polynomials are equal but the veering polynomials can differ.
Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-ma…
The paper proves that Gaussian field critical points have finite moments.
problem Proving the finiteness of moments for Gaussian field critical points.
method General approach not specific to critical points, using Taylor polynomial non-degeneracy.
result The finiteness of moments of the number of critical points of Gaussian fields.
Defines natural tensors for submanifolds of pseudo-Riemannian manifolds.
problem Characterizing tensors for submanifolds of pseudo-Riemannian manifolds.
method Constructs geodesic normal coordinates and expresses metric coefficients as polynomials in curvature and second fundamental form derivatives.
result Natural tensors are linear combinations of contractions of curvature and second fundamental form derivatives.
New algorithm speeds up polynomial kernel approximations.
problem Efficiently approximating polynomial kernels of high degree.
method Oblivious sketching combined with novel sampling.
result Polynomial factor slowdown removed in running time.
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.
CBN improves batch normalization for small mini-batch sizes.
problem Reduced effectiveness of Batch Normalization in small mini-batch sizes.
method CBN uses statistics from multiple recent iterations, compensating for network weight changes via Taylor polynomials.
result CBN outperforms original batch normalization and direct iteration statistics in object detection and image classification.
The taut polynomial equals a twisted Alexander polynomial.
problem Understanding the relationship between taut polynomials and Alexander polynomials.
method Defined taut polynomial of veering triangulations and proved it equals a twisted Alexander polynomial.
result The taut polynomial equals a twisted Alexander polynomial of the underlying manifold.
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.
This paper improves neural network learning by escaping the NTK regime and efficiently learning sparse polynomials.
problem Learning sparse polynomials efficiently using neural networks.
method Spectral analysis of NTK, identifying 'good' directions, and constructing a regularizer.
result Gradient descent on a two-layer neural network can learn sparse polynomials efficiently, improving over the NTK and QuadNTK.
This work improves polynomial approximations for functions with asymmetric behavior.
problem Efficiently approximating functions with asymmetric behavior, especially those growing unbounded on one side.
method Introduces weighted deep polynomial approximants that combine learnable deep polynomials with one-sided weights.
result Weighted deep polynomial approximants outperform existing methods in approximating functions with asymmetric behavior.
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.
Paper improves kernel approximations for better statistical learning.
problem Improving kernel approximations for better statistical learning.
method Taylor series approximations of radial kernel functions.
result Establishes upper bounds for eigenfunctions, leading to better approximations.
Paper approximates XVA for European contingent claims using BSDEs and polynomial expansions.
problem Computing Value Adjustment of European contingent claims with nonlinear features.
method Reduced-form approach, nonlinear Backward Stochastic Differential Equation (BSDE), change of numeraire, Taylor's polynomial expansion.
result Simple first-order approximation can be computationally efficient for CIR intensity model.
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.
We construct an invariant J_M of integral homology spheres M with values in a completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at each root of unity ζgives the the SU(2) Witten-Reshetikhin-Turaev invariant τ_ζ(M) of M at ζ. Thus J_M unifies all the SU(2) Witten-Reshetikhin-Turaev invariants of…
The first aperiodic monotiling, introduced by Taylor, was based on a trapezoidal prototile equipped with 14 distinct decorations. A presentation of the closely related Taylor-Socolar aperiodic monotiling is based on a hexagonal prototile equipped with 7 decorations. This paper gives decoration-free algebraic descriptio…
J.P. Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the μˉ-invariants of a string link with the link as its closure. We give another description…
The paper is concerned with non-linear Gaussian filtering and smoothing in continuous-discrete state-space models, where the dynamic model is formulated as an Itô stochastic differential equation (SDE), and the measurements are obtained at discrete time instants. We propose novel Taylor moment expansion (TME) Gaussian …
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.
In this paper we study the pricing of exchange options under a dynamic described by stochastic correlation with random jumps. In particular, we consider a Ornstein-Uhlenbeck covariance model with Levy Background Noise Process driven by Inverse Gaussian subordinators. We use expansion in terms of Taylor polynomials and …
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.
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.
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.
Modern convolutional networks, incorporating rectifiers and max-pooling, are neither smooth nor convex; standard guarantees therefore do not apply. Nevertheless, methods from convex optimization such as gradient descent and Adam are widely used as building blocks for deep learning algorithms. This paper provides the fi…
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.
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.
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.
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.
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…
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…
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. 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.
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.
In this paper we study the Taylor series of an operator-valued function related to the differential of the exponential map. For a smooth manifold M with a torsion-free affine connection the operator Ep(v) acting on the space TpM is defined to be the composition of the differential …
Examines how central bank policies affect stock markets and asset prices.
problem Understanding the impact of monetary policy on stock markets and asset prices.
method Used Taylor rule equations to analyze data from 1990 to 2020 for US and UK, testing with various econometric methods.
result Monetary policy can explain asset price volatility and output gap better than just inflation rate.
We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which exp…
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 …
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 …