arXiv research

SOAR improves deep networks' robustness against adversarial examples.

Neural networks learn higher-order derivatives for physics problems.

Improved Frank-Wolfe method reduces dependence on data size for empirical risk minimization.

Developed Taylor series for muscle-finger system analysis.

Derives functional Itô formula for non-anticipative maps of rough paths.

The Adomian decomposition method is shown to be equivalent to the Taylor series approach.

New insights on pruning deep networks by preserving function locality.

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 on Einstein deformations of negative Kähler Einstein metrics.

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 $\mathcal{M}$ with a torsion-free affine connection the operator $\mathcal{E}_p(v)$ acting on the space $T_p\mathcal{M}$ is defined to be the composition of the differential …

Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local-stochastic volatility setting. Our price approximations require only a normal CDF and our implied volatility approximations are fully explicit (ie, they require no spec…

Let M and N be smooth manifolds. For an open V of M let emb(V,N) be the space of embeddings from V to N. By results of Goodwillie and Goodwillie-Klein, the cofunctor V |--> emb(V,N) is analytic if dim(N)-dim(M) > 2. We deduce that its Taylor series converges to it. For details about the Taylor series, see Part I.

The paper modifies asset pricing models using Taylor series expansions and market-based averages.

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.

We consider closed-form approximations for European put option prices within the Heston and GARCH diffusion stochastic volatility models with time-dependent parameters. Our methodology involves writing the put option price as an expectation of a Black-Scholes formula and performing a second-order Taylor expansion aroun…

Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M,N) should come from an analysis of the cofunctor V |--> emb(V,N) from the poset O of open subsets of M to spaces. We therefore abstract some of the properties of this cofunctor, and …

In every point of a Kähler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kähler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and…

This paper extends AD techniques to Monte Carlo processes for efficient derivative calculation.

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.

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…

Explicit Taylor series for the volume of tubes in Lie groups

Classifies almost-toric systems in four dimensions.

Paper proposes a closed-form formula for geometric Istanbul call options.

Efficient method classifies locally stationary time series based on second-order characteristics.

Formalizes synthetic differential geometry in Lean.

Proves and tests methods for learning time-series with breaks.

Agents learn state ambiguity from non-linear sensor data using Gaussian approximations.

This paper provides intuition on the relationship of accrual and mark-to-market valuation for cash and forward interest rate trades. Discounted cashflow valuation is compared to spread-based valuation for forward trades, which explains the trader's view on valuation. This is followed by Taylor series approximation for …

Paper improves kernel approximations for better statistical learning.

CO2 algorithm creates coresets for generic smooth divergences efficiently.

TEAM generates more powerful adversarial examples for DNNs.

New flow for G2-structures helps find torsion-free structures.

Unified framework for imputation and prediction in healthcare time series.

In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately ch…

After the torch of Anders Kock [Taylor series calculus for ring objects of line type, Journal of Pure and Applied Algebra, 12 (1978), 271-293], we will establish the Baker-Campbell-Hausdorff formula as well as the Zassenhaus formula in the theory of Lie groups.

Structural pruning of neural network parameters reduces computation, energy, and memory transfer costs during inference. We propose a novel method that estimates the contribution of a neuron (filter) to the final loss and iteratively removes those with smaller scores. We describe two variations of our method using the …

Paper proposes a new method for efficient second-order neural network training.

Kernelized Taylor diagram visualizes data populations with fewer assumptions.

Ever since the proof of asymptotic normality of maximum likelihood estimator by Cramer (1946), it has been understood that a basic technique of the Taylor series expansion suffices for asymptotics of $M$-estimators with smooth/differentiable loss function. Although the Taylor series expansion is a purely deterministic …

Paper extracts features from time series to improve forecasting accuracy.

Expanding the rough Heston model in $H$

We report a general technique to study a given experimental time series with superstatistics. Crucial for the applicability of the superstatistics concept is the existence of a parameter $β$ that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration. The propose…

Taylor expansions improve reinforcement learning policies.

This work explores functional expansions to handle path dependence in various fields.

Taylorized training improves neural network training at finite width.

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…

AdaRound improves post-training quantization of neural networks.

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focu…