Study complex structures with totally real sections, providing integrability equations.
problem Existence and integrability of complex structures with totally real sections.
method Explicit integrability equations derived from fiberwise Taylor expansions.
result Detailed fiberwise Taylor expansions and integrability equations in a geometric case.
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 provides an explicit formula for complex structures in embeddings of manifolds.
problem Finding explicit formulas for complex structures in embeddings of manifolds.
method Recursive expression and fiberwise Taylor expansion of the canonical complex structure.
result Evidence of canonical vanishing of integrability equations in general settings.
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.
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.
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…
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.
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.
We develop closed-form approximations for European put options under stochastic volatility models.
problem Tackling the pricing of European put options under stochastic volatility models with time-dependent parameters.
method Using a second-order Taylor expansion around the mean of the argument, we write the option price as an expectation of a Black-Scholes formula. We then simplify the resulting expectations and derive closed-form pricing formulas under the assumption of piecewise-constant parameters.
result We derive closed-form pricing formulas and bounds on the remainder term generated by the Taylor expansion, showing that the errors are well within acceptable ranges for practical applications.
The paper calculates Bachelier option prices using Taylor expansions and applies it as a variance reduction technique.
problem Calculating Bachelier option prices and variance reduction in correlated cases.
method Taylor expansions and classical Itô calculus to derive option prices, uses negative powers of future mean volatility.
result The paper provides a new method to calculate Bachelier option prices and applies it to reduce variance in Monte Carlo simulations.
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.
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 this paper we propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model, based on Taylor expansions and the calculation of mixed exponential-power moments of a Gaussian distribution. Our numerical results show that a second order expansion provides accurate prices o…
Study exchange option pricing with stochastic volatility and correlation.
problem Pricing exchange options under stochastic volatility and correlation.
method Approximation using a closed-form solution with Taylor expansion.
result Numerical results show the effectiveness of the proposed method.
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.
Expanding the rough Heston model in H
problem Analyzing the dependence of the fractional Riccati equation on the Hurst parameter H method Deriving a Taylor expansion of the Riccati solution in H result Local uniform convergence and analyticity of the fractional Riccati solution
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 …
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.
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 work explores functional expansions to handle path dependence in various fields.
problem Path dependence and infinite-dimensional problems in non-Markovian systems.
method Generalizes Wiener series and functional Taylor expansion to handle static and dynamic functionals.
result Elegant separation of functionals from future trajectories in dynamic cases.
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.
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.
The paper modifies asset pricing models using Taylor series expansions and market-based averages.
problem Improving asset pricing models to better reflect market dynamics.
method Derives new pricing equations using Taylor series expansions and market-based averages.
result New expressions for asset prices and volatilities derived from market data.
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.
Paper finds dropout noise approximation invalid for logistic regression and neural networks.
problem Invalidity of dropout noise approximation for logistic regression and neural networks.
method Derived equivalence between dropout noise injection and L2 regularisation using divergent Taylor expansion. result Approximation approach is invalid for robust constraints and general neural network topologies.
Approximates option prices in Barndorff-Nielsen and Shephard models using Taylor expansion.
problem Approximating option prices in complex stochastic volatility models.
method Taylor expansion and recursive algorithm for closed-form approximations.
result Explicit results for inverse Gaussian and gamma stationary distributions, with favorable comparisons to characteristic function.
Paper studies third order open mapping in sub-Riemannian geometry.
problem Analyzing third order open mapping in sub-Riemannian geometry.
method Third order open mapping results for maps from a Banach space into a finite dimensional manifold. Computing third order term in the Taylor expansion of the end-point map.
result Specialization of abstract theory to study length-minimality of sub-Riemannian strictly singular curves and third order analysis of specific extremal curves.
We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kern…
Study on Einstein deformations of negative Kähler Einstein metrics.
problem Understanding Einstein deformations of Kähler Einstein metrics.
method Relate second order Einstein deformation theory to complex geometry, gauge normalise, and use Taylor expansion.
result Taylor expansion to order two of an Einstein deformation is determined by h12 and the divergence of the Kodaira-Spencer bracket. Paper proposes a closed-form formula for geometric Istanbul call options.
problem Pricing geometric Istanbul call options under the Black-Scholes model.
method Second-order Taylor expansion to derive a closed-form approximation.
result The proposed formula accurately approximates GIC values compared to Monte-Carlo simulations.
Study Bergman kernel metrics on degenerating hyperelliptic surfaces.
problem Asymptotic behavior of Bergman kernels near singularities.
method Taylor expansion for Abelian differentials and period matrices.
result Explicit coefficients in asymptotic formulas for Bergman kernels.
New formulas for pricing Asian and basket options using stochastic expansion.
problem Pricing Asian and basket options under time-dependent parameters.
method Stochastic Taylor expansion around a log-normal proxy model.
result Highly accurate approximations for Asian options and vanilla options with discrete dividends.
We derive asymptotic expansions for the prices of a variety of European and barrier-style claims in a general local-stochastic volatility setting. Our method combines Taylor series expansions of the diffusion coefficients with an expansion in the correlation parameter between the underlying asset and volatility process…
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…
Derives functional Itô formula for non-anticipative maps of rough paths.
problem Functional Itô formula for non-anticipative maps of càdlàg rough paths.
method Approximation properties of the signature and Marcus transformation.
result Functional Taylor expansion for sufficiently regular non-anticipative maps.
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.
Formalizes synthetic differential geometry in Lean.
problem Formalizing synthetic differential geometry in a proof assistant.
method Formalization of synthetic differential geometry with Lean and mathlib.
result Proves a Taylor theorem for functions of several variables.
Geodesic random walks in Riemannian manifolds analyzed for large deviations.
problem Analyzing large deviations for geodesic random walks in Riemannian manifolds.
method Direct proof of Cramér's theorem, exploiting vector space structure, Taylor expansions, and parallel transport.
result Obtained the analogue of Cramér's theorem for geodesic random walks.
New perturbative method improves stochastic gradient descent for binary weights.
problem Improving stochastic gradient descent for binary weights.
method Perturbative expansion around the mean of the sampling distribution, Taylor-corrected estimators, variance reduction techniques.
result Perturbative correction improves convergence of stochastic variational inference.
We consider a financial market with liquidity cost as in Çetin, Jarrow and Protter [2004], where the supply function Sε(s,ν) depends on a parameter ε≥0 with S0(s,ν)=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [2010] of the super-hedging cost of a…
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.
A new DP method for deep learning with faster convergence and better privacy.
problem Challenges in differentially private training of deep neural networks.
method Method of auxiliary coordinates with perturbed Taylor expansion for privacy.
result Empirically shows decent trained model quality with modest privacy budget.
A method to estimate functions of the return using its moments in reinforcement learning.
problem Estimating functions of the return directly using temporal difference methods is challenging.
method Modified temporal difference algorithm to learn moments of the return, then use these moments in a Taylor expansion to approximate functions of the return.
result Functions of the return can be estimated efficiently using the proposed method.
Unified bounds for neural networks incorporating physical laws.
problem Limitations in existing generalization analyses for PINNs and VPINNs.
method Unified framework using Taylor expansion and Koopman-based analysis.
result High-rank networks can generalize well even with differential operators.
Diffeomorphism freedom induces a gauge dependence in the theory of spacetime perturbations. We derive a compact formula for gauge transformations of perturbations of arbitrary order. To this end, we develop the theory of Taylor expansions for one-parameter families (not necessarily groups) of diffeomorphisms. First, we…
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.