New algorithm speeds up spectral clustering for dynamic graphs.
problem Prohibitive complexity of spectral clustering for dynamic graphs.
method Reuse past cluster assignments and use fast Chebyshev graph filtering.
result Achieves clustering quality approximating spectral clustering with significant complexity benefits.
Graphon autoencoder generates graphs with arbitrary sizes using Chebyshev filters.
problem Generating graphs with arbitrary sizes and arbitrary structures.
method Induces graphons from observed graphs, uses Chebyshev filters for latent representation, and learns encoder and decoder to minimize Wasserstein distance.
result Graphon autoencoder provides a new paradigm for graph generation with good generalizability and transferability.
Quantum torus methods enhance understanding of skein algebras and modules.
problem Investigating Kauffman bracket skein algebras and modules using quantum torus methods.
method Two quantum torus methods: embedding into quantum Teichmüller space and filtering to a monomial subalgebra.
result Generalized Chebyshev homomorphism and refined unicity theorem for skein modules.
Euler and Chebyshev worked on map drawing and garment fitting.
problem Drawing geographical maps and fitting garments.
method Analyzing historical works of Euler and Chebyshev.
result Found connections between map drawing and garment fitting.
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…
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.
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…
Filtered conformal ellipsoids for graph-native time series
problem Joint prediction sets for multivariate time series
method Filtered conformal ellipsoids
result Sharper at-target ellipsoids than static-covariance and non-filter baselines
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.
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.
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.
Paper revisits graph-CNNs using Laplace-Beltrami spectral filters and polynomials.
problem Improving spectral graph convolutional neural networks (graph-CNNs).
method Developed Laplace-Beltrami CNN (LB-CNN) by replacing graph Laplacian with LB operator and approximating spectral filters using Chebyshev, Laguerre, and Hermite polynomials.
result Classification accuracy of LB-CNN is not dependent on the type of polynomials or operators.
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%.
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. 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.
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.
Study of skein modules of marked 3-manifolds and Chebyshev-Frobenius homomorphism.
problem Understanding skein modules of marked 3-manifolds and their algebraic properties.
method Extension of Muller's result to marked surfaces with unmarked boundary components, development of surgery theory, and application of Chebyshev homomorphism.
result Extension of Chebyshev-Frobenius homomorphism to skein modules of marked 3-manifolds and characterization of its image.
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…
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.
Two methods using Chebyshev tensors improve accuracy and speed in computing Dynamic Initial Margin.
problem Computing Dynamic Initial Margin (DIM) with high accuracy and speed.
method Two methods based on Chebyshev tensors implemented in Monte Carlo engine.
result Better accuracy, speed, and implementation efforts compared to benchmarks.
The abstract discusses historical papers on geographical map construction.
problem Understanding and proving Chebyshev's result on best geographical maps.
method Review and explanation of historical papers, providing complete proofs.
result A best geographical map has a constant conformal factor on its boundary.
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…
A new method uses Chebyshev polynomials for American option pricing.
problem Pricing American options efficiently and accurately.
method Dynamic Chebyshev interpolation in time-stepping, offline computation of moments, online approximation.
result The method delivers fast convergence and efficiency gains compared to existing methods.
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.
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.
Estimates matrix spectrum from few entries.
problem Recovering spectral properties from partial matrix observations.
method Estimating Schatten norms using an unbiased graph-based estimator, then applying Chebyshev approximation or moment matching.
result Accurately estimates Schatten norms from fewer samples than needed for matrix completion.
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.
A new method for calculating implied volatility using Chebyshev polynomials.
problem Calculating the implied volatility for option pricing and calibration.
method Bivariate interpolation of the implied volatility surface using Chebyshev polynomials.
result The method provides a closed-form approximation with subexponential error decay and high accuracy.
Solves PCR with fewer calls to ridge regression.
problem Principal component regression (PCR) with high accuracy.
method Reduces PCR to ridge regression calls and develops stable recurrence for matrix Chebyshev polynomials.
result Achieves PCR with multiplicative accuracy up to 1+γ using fewer calls. Unbounded primitivity index in free groups linked to Chebyshev function.
problem Analyzing primitivity and simplicity indices in free groups.
method Combining topological, group-theoretic, and number-theoretic approaches, including asymptotic properties of the second Chebyshev function.
result Proved the unboundedness of the primitivity index sequence and its asymptotic behavior.
The Alexander polynomials Δ_{n,3}(t) and Δ_{n,4}(t) are presented as a sum of the Alexander polynomials Δ_{k,2}(t). These polynomials are also expressed in the form of a sum of Chebyshev polynomials of the second kind. These expansions allow one to introduce the "coordinates" in corresponding bases, which are proposed …
Alexander invariant created for doodles, vanishes on unlinked doodles.
problem Creating an Alexander type invariant for doodles.
method Deformation of Tits representation and Chebyshev polynomials of second kind.
result Invariant vanishes on unlinked doodles with more than one component.
This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of 2-bridge knot…
Chebyshev methods reduce risk calculations by orders of magnitude.
problem Efficiently revalue portfolios for risk computation.
method Chebyshev interpolation techniques for exponential convergence.
result Significant decrease in computational effort without loss of accuracy.
Improved Anderson acceleration speeds up nonlinear optimization.
problem Optimizing nonlinear functions efficiently.
method Combining Anderson acceleration with Chebyshev polynomials.
result Achieves optimal convergence rate for nonlinear problems.
Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real-time. Simultaneously we observe an increase in model sophistication on the one hand and growing demands on the quality of risk management on the other. To address the resulting computational challenges, it is nat…