New CFNN architecture approximates functions with machine accuracy.
problem Function approximation with high precision.
method Chebyshev Feature Neural Network (CFNN) with learnable frequencies.
result Achieves machine accuracy in function approximation.
New technique reduces FRTB IMA capital calculation burden by over 90%.
problem Reduction of computational burden in FRTB IMA capital calculation.
method Orthogonal Chebyshev Sliding Technique based on high-dimensional Chebyshev Tensors.
result Reduction of computational burden by more than 90%.
Efficiently calibrates volatility models using Chebyshev Tensors.
problem Calibrating pricing models efficiently.
method Used Chebyshev Tensors to speed up calibration of the rough Bergomi volatility model.
result Chebyshev Tensors can calibrate the rough Bergomi volatility model 40,000 times more efficiently than brute-force methods.
New method improves support estimation for unknown distributions.
problem Estimating the support size of an unknown distribution.
method Regularized Weighted Chebyshev Approximations, joint optimization of bias and variance, linear programming.
result Significant improvements in worst-case risk for synthetic data and accurate bacterial genus estimation for microbiome data.
New method calculates credit exposures for complex options efficiently.
problem Efficient calculation of credit exposures for complex options.
method Dynamic Chebyshev method for closed-form approximation.
result Highly efficient evaluation of credit exposures for large paths.
We solve principal component regression (PCR), up to a multiplicative accuracy 1+γ, by reducing the problem to O~(γ−1) black-box calls of ridge regression. Therefore, our algorithm does not require any explicit construction of the top principal components, and is suitable for large-scale PCR instances. In…
Chebyshev polynomials analyze Czech enterprises' stock dynamics.
problem Analyzing stock dynamics of enterprises not following normal distribution.
method Chebyshev polynomial decomposition of stock time series.
result Allows effective analysis of stock dynamics without variance and correlation.
We introduce a new method to price American options based on Chebyshev interpolation. In each step of a dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. The key advantage of this approach is that it allows to shift the model-dependent computations into an offline phase pri…
A Chebyshev knot is a knot which admits a parametrization of the form x(t)=Ta(t); y(t)=Tb(t); z(t)=Tc(t+φ), where a,b,c are pairwise coprime, Tn(t) is the Chebyshev polynomial of degree n, and $φ\in \RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that the…
New method reduces high-dimensional financial problems using low-rank tensor approximation.
problem High-dimensional financial problems in pricing, calibration, and risk assessment.
method Low-rank tensor approximation for Chebyshev interpolation in tensor train (TT) format.
result Efficiently approximates interpolation coefficients using tensor completion.
Chebyshev steps improve convergence in deep-unfolded gradient descent.
problem Improving convergence speed in iterative algorithms.
method Introducing Chebyshev steps to bound convergence rate of gradient descent.
result Chebyshev steps lead to asymptotically optimal convergence rate.
For node level graph encoding, a recent important state-of-art method is the graph convolutional networks (GCN), which nicely integrate local vertex features and graph topology in the spectral domain. However, current studies suffer from several drawbacks: (1) graph CNNs relies on Chebyshev polynomial approximation whi…
For applications as varied as Bayesian neural networks, determinantal point processes, elliptical graphical models, and kernel learning for Gaussian processes (GPs), one must compute a log determinant of an n×n positive definite matrix, and its derivatives - leading to prohibitive O(n3) computatio…
The paper connects Chebyshev polynomials and Gram determinants on Möbius bands.
problem Exploring the relationship between Chebyshev polynomials and Gram determinants on Möbius bands.
method Analyzing Mersenne numbers and Chebyshev polynomials, proving conjectures, and developing algorithms.
result A factor of the Gram determinant supports a conjecture about its closed formula involving Chebyshev polynomials.
A Chebyshev knot C(a,b,c,φ) is a knot which has a parametrization of the form x(t)=Ta(t);y(t)=Tb(t);z(t)=Tc(t+φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ∈R. We show that any two-bridge knot is a Chebyshev knot with a=3 and also with a=4. For e…
We report on the works of Euler and Chebyshev on the drawing of geographical maps. We point out relations with questions about the fitting of garments that were studied by Chebyshev.
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
A new method for computing Greeks without bias, improving stability.
problem Inaccurate and unstable computation of second order Greeks (like Gamma) in financial instruments.
method Apply Chebyshev interpolation techniques to finite differences for improved stability.
result Improved stability and accuracy in computing spot Greeks without bias.
New sampling method for Heston model reduces complexity.
problem Efficient sampling for Heston model's time integrated variance.
method Series expansion, change of measure, Chebyshev polynomial approximations.
result Strong, efficient sampling scheme established for Heston model.
Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
problem Exploring the Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
method Generalization of splitting homomorphism for stated skein modules of 3-manifolds.
result Existence and properties of Chebyshev-Frobenius homomorphism for 3-manifold skein modules.
Quantum computing speeds up Bermudan option pricing.
problem Efficient pricing of financial derivatives, especially Bermudan options.
method Quantum amplitude estimation combined with Chebyshev interpolation.
result Quadratic speed-up over classical methods.
Exact formulas for volumes of specific knot cone-manifolds.
problem Finding exact volumes of cone-manifolds with two-bridge knots.
method Provided exact integral formulas using Chebyshev polynomials and algebraic equations.
result Exact formulas for hyperbolic and spherical volumes of cone-manifolds.
The study bounds positive bases of skein algebras using Chebyshev polynomials.
problem Finding positive bases in skein algebras of surfaces.
method Using Chebyshev polynomials to establish bounds.
result Normalized Chebyshev polynomials of type one give the only positive basis for the closed torus.
This paper categorifies Chebyshev polynomials using diagrammatic algebra.
problem Categorifying two-variable Chebyshev polynomials of the second kind.
method Using A2 spider and Karoubi envelope of A2 spider, the recursive formula is shown. result A q-deformation of the two-variable Chebyshev polynomials is defined. We solve for functions from their truncated Hilbert transforms using Chebyshev series.
problem Finding functions from their truncated Hilbert transforms.
method Express functions in Chebyshev series and numerically estimate coefficients.
result Numerical methods work well for extrapolating functions from truncated Hilbert transforms.
Paper uses Chebyshev Tensors for accurate dynamic sensitivities and ISDA SIMM computation.
problem Computing dynamic sensitivities and initial margin for financial instruments.
method Uses Chebyshev Tensors in Monte Carlo simulations to compute dynamic sensitivities and ISDA SIMM.
result High accuracy and computational gains for FX swaps and Spread Options.
The implied volatility is a crucial element of any financial toolbox, since it is used for quoting and the hedging of options as well as for model calibration. In contrast to the Black-Scholes formula its inverse, the implied volatility, is not explicitly available and numerical approximation is required. We propose a …
Some results on existence of global Chebyshev coordinates on a Riemannian manifold or, more generally, on Aleksandrov surface are proved. For instance, if the positive and the negative parts of integral curvature of a Riemannian manifold M are less than 2πeach, then there exist global Chebyshev coordinates on M. These …
Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones-Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which em…
In this paper we use Bernstein and Chebyshev polynomials to approximate the price of some basket options under a bivariate Black-Scholes model. The method consists in expanding the price of a univariate related contract after conditioning on the remaining underlying assets and calculating the mixed exponential-power mo…
We introduce a new activation function using Chebyshev-Lagrange polynomials for improved neural network performance.
problem Improving data efficiency and accuracy of neural networks.
method Parameterized piece-wise polynomial activation functions based on Chebyshev nodes and Lagrangian interpolation.
result Significant improvements in model capacity and accuracy, especially in linear extrapolation.
A Chebyshev curve C(a,b,c,φ) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ\in \RR. When C(a,b,c,φ) has no double points, it defines a polynomial knot. We determine all possible knots when a, b and c are given.
Chebyshev Greedy Algorithm is a generalization of the well known Orthogonal Matching Pursuit defined in a Hilbert space to the case of Banach spaces. We apply this algorithm for constructing sparse approximate solutions (with respect to a given dictionary) to convex optimization problems. Rate of convergence results in…
The paper proposes using function approximations to reduce the computational burden in measuring counterparty credit exposure.
problem The need for regular exposure calculations in finance, balancing between computational cost and risk simplification.
method Replacing derivative pricers with function approximations, proving error bounds, and using Chebyshev interpolation for convergence.
result Derives probabilistic and finite sample error bounds, showing significant run-time reductions and asymptotic efficiency gains.
The paper computes groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
problem Computing groups and modules for wheel graphs.
method Utilized Fibonacci and Chebyshev polynomials to compute the Reduced Fox Coloring Group and Alexander-Burau-Fox Module.
result Computed groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x=T3(t),y=Tb(t),z=C(t) where Tk(t) are the Chebyshev polynomials and b+°C=3N. If C(t)=Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots …
New bound improves on weighted majority vote risk estimation.
problem Improving risk estimation for weighted majority vote.
method Novel Chebyshev-Cantelli inequality and PAC-Bayes-Bennett inequality.
result New bounds improve on existing methods.
We show that every rational knot K of crossing number N admits a polynomial parametrization x=Ta(t),y=Tb(t),z=C(t) where Tk(t) are the Chebyshev polynomials, a=3 and b+°C=3N. We show that every rational knot also admits a polynomial parametrization with a=4. If C(t)=Tc(t) is a Chebyshev p…
The paper develops a method to accurately estimate the Bayes misclassification error rate.
problem Estimating the best achievable classifier performance without learning a Bayes-optimal classifier.
method Learning to benchmark using an ensemble of ε-ball estimators and Chebyshev approximation.
result The proposed method achieves an optimal mean squared error rate of O(N^(-1)) under a smoothness assumption.
We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA), a generalization of the Weak Orthogonal Matching Pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setti…
Directly simulates squared Bessel processes efficiently.
problem Simulating squared Bessel processes accurately and efficiently.
method Two-dimensional Chebyshev expansion for non-central chi-square distribution inverse.
result Accurate and efficient simulation for various degrees of freedom.
In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary compone…
Max-convolution is an important problem closely resembling standard convolution; as such, max-convolution occurs frequently across many fields. Here we extend the method with fastest known worst-case runtime, which can be applied to nonnegative vectors by numerically approximating the Chebyshev norm $\| \cdot \|_\infty…
Closed formulas for η-corrections in the once-punctured torus identified.
problem Identifying η-corrections in the Kauffman bracket skein algebra of the once-punctured torus.
method Explicit closed formulas for Chebyshev-threaded families and η-corrections.
result Explicit Chebyshev expansions and coefficients for η-corrections.
Effective dimensionality reduction improves accuracy and reduces costs in estimating option Greeks.
problem Estimating Greeks for barrier and arithmetic average Asian options.
method Global sensitivity analysis, Chebyshev interpolation, conditional pathwise method, randomized Quasi Monte Carlo, Brownian bridge discretization, importance sampling.
result Reduced effective dimensionality enhances convergence rate and accuracy of randomized Quasi Monte Carlo integration.
Chebyshev technique reduces FRTB-IMA equity autocallables computation costs by 90%.
problem Efficient computation of FRTB-IMA capital for equity autocallables.
method Orthogonal Chebyshev Sliding Technique applied to equity autocallables.
result Computational cost reduction of about 90% for equity autocallables.
The paper interprets learned step sizes in deep-unfolded gradient descent.
problem Intuitive interpretation of learned non-constant step sizes in deep-unfolded gradient descent.
method Theoretical analysis and optimization of spectral radius.
result Chebyshev steps achieve the lower bound of convergence rate for first-order methods.
Jones polynomials derived from K-theory of a cluster algebra.
problem Jones polynomials of knots and links.
method K-theory of a cluster C*-algebra of the sphere with two cusps.
result Interplay between Chebyshev and Jones polynomials.