New Fourier analysis method for non-uniform Boolean hypercube.
problem Non-uniform probability measures on the Boolean hypercube.
method ANOVA-based decomposition, explicit basis, least squares problem.
result Generalization of Fourier analysis for arbitrary probability measures.
Paper proves L2 regression can learn k-juntas without distributional assumptions.
problem Learning k-juntas using L2 regression without distributional restrictions.
method L2 polynomial regression and minimum mean square estimation (MMSE).
result Agnostic PAC learning of k-juntas using L2 polynomial regression.
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
We show a connection between the Fourier spectrum of Boolean functions and the REINFORCE gradient estimator for binary latent variable models. We show that REINFORCE estimates (up to a factor) the degree-1 Fourier coefficients of a Boolean function. Using this connection we offer a new perspective on variance reduction…
Random SNNs are stable and simple, with low-frequency Fourier spectra.
problem Stability and robustness of spiking neural networks.
method Boolean function analysis and Fourier spectrum concentration.
result Random LIF-SNNs are stable and biased towards simple functions.
We develop Fourier methods to expand translation-invariant kernels.
problem Constructing orthonormal expansions for translation-invariant kernels.
method Fourier analytic technique to derive explicit expansions.
result Explicit expansions for various kernels (Matérn, Cauchy, Gaussian).
GETF efficiently decomposes large-scale Boolean tensors.
problem Efficiently factorizing large-scale Boolean tensors.
method Geometric Expansion for all-order Tensor Factorization (GETF).
result GETF significantly improves reconstruction accuracy and efficiency.
For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.
Due to the isotropy d-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the R-radius hyperboloid model of d-dimensional hyperbolic geometry with R>0 and d≥2, we compute azimuthal Fourier expansions for a fundamental so…
The paper defines and assesses the quality of datasets using a novel expected diameter metric.
problem Lack of rigorous methods to assess data quality.
method Formal definition of data quality, expected diameter metric, Fourier analysis, algebraic methods, probabilistic analysis.
result The expected diameter metric provides theoretical guarantees and practical solutions for data quality assessment.
For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geo…
New algebra counts components of arborescent knots and links.
problem Counting components of arborescent knots and links.
method Developed a new algebra called the crossing algebra.
result The crossing algebra counts the number of components for arborescent knots and links.
New method for European option pricing faster and more robust.
problem Pricing European options efficiently and accurately.
method Fourier cosine series expansions for models with known characteristic functions.
result More robust and faster than the original COS method.
Boolean matrix has been used to represent digital information in many fields, including bank transaction, crime records, natural language processing, protein-protein interaction, etc. Boolean matrix factorization (BMF) aims to find an approximation of a binary matrix as the Boolean product of two low rank Boolean matri…
A new Fourier model improves ODE prediction.
problem Improving the accuracy of ODE solutions, especially for periodic functions.
method Constructing a Fourier state space model and a hybrid model combining Taylor and Fourier methods.
result The hybrid model can predict ODE solutions more accurately, especially for periodic functions.
The COS method proposed in Fang and Oosterlee (2008), although highly efficient, may lack robustness for a number of cases. In this paper, we present a Stable pricing of call options based on Fourier cosine series expansion. The Stability of the pricing methods is demonstrated by error analysis, as well as by a series …
The staircase property aids deep learning by guiding hierarchical feature learning.
problem Understanding how hierarchical structure influences deep learning performance.
method Defined and proved the staircase property for Boolean hypercube functions, and showed its learnability by layerwise stochastic coordinate descent.
result Staircase functions can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks.
This paper analyzes SHAP values using Fourier expansions for model interpretability.
problem Understanding and interpreting SHAP values in complex models.
method Developed a spectral framework using Fourier expansions for SHAP values in various model regimes.
result SHAP values are Lipschitz continuous in the deterministic regime and converge to Gaussian process values in the probabilistic regime.
The degree-d Chow parameters of a Boolean function f:{−1,1}n→R are its degree at most d Fourier coefficients. It is well-known that degree-d Chow parameters uniquely characterize degree-d polynomial threshold functions (PTFs) within the space of all bounded functions. In this paper, we prove …
The theory of learning under the uniform distribution is rich and deep, with connections to cryptography, computational complexity, and the analysis of boolean functions to name a few areas. This theory however is very limited due to the fact that the uniform distribution and the corresponding Fourier basis are rarely …
CodNN uses error-correcting codes to make neural networks more resilient to noise.
problem Neural networks are sensitive to noise, especially in critical applications.
method Construct robust neural networks by coding data or internal layers with error-correcting codes.
result Parity codes can guarantee robustness for a wide range of neural networks, including binarized networks.
Let X be a compact connected strongly pseudoconvex CR manifold of dimension 2n+1,n≥1 with a transversal CR S1 action on X. We establish an asymptotic expansion for the m-th Fourier component of the Szegő kernel function as m→∞, where the expansion involves a contribution in terms of a d…
New bounds for learning polynomial surrogates with L∞ guarantees.
problem Learning polynomial surrogates for bounded binary functions with L∞ error guarantees. method Characterized minimax sample complexity for two classes of polynomials under subgaussian noise.
result Sample complexity rates differ from noiseless case, scaling as nd+1 for degree d polynomials and ns2 for sparse polynomials. A new method integrates Fourier basis expansion and mapping for improved time series forecasting.
problem Inconsistent starting cycles and series length issues in Fourier-based methods.
method Fourier Basis Mapping (FBM) method that integrates time-frequency features through Fourier basis expansion and mapping.
result FBM addresses inconsistencies and preserves temporal characteristics, achieving SOTA performance.
A new NUFFT method speeds up option pricing for various strikes.
problem Efficiently pricing many options of the same maturity but different strikes.
method Non-uniform fast Fourier transform (NUFFT) applied to the COS method.
result Significantly faster computation of option prices.
The paper prices weather contracts using a complex temperature model.
problem Accurate pricing of weather contracts under temperature dynamics.
method Time-changed Levy model with mean-reverting dynamics, Fourier expansion, Esscher transform.
result An accurate approximation of weather contract prices.
We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential Lévy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent Lévy measure. Using techniques from regular perturba…
The paper proposes and proves asymptotic expansions for quantum invariants.
problem Quantum invariants and their expansions under varying metrics.
method Asymptotic expansion conjectures for relative Reshetikhin-Turaev, Turaev-Viro invariants and quantum 6j-symbols.
result Proved asymptotic expansions for special cases, showing geometric dependence on metrics.
FNN approximates functions and solves PDEs with periodic BCs.
problem Approximating and solving periodic functions and PDEs.
method Fourier neural network architecture with activation and loss functions.
result FNN can solve PDEs with periodic BCs and is interpretable.
Let X be a compact strongly pseudoconvex CR manifold with a transversal CR S1-action. In this paper, we establish the asymptotic expansion of Szegő kernels of positive Fourier components and by using the asymptotics, we show that X can be equivariant CR embedded into some CN equipped with a simple $S^…
Boolean logic used for neural network training and inference, with convergence analysis.
problem Discrete optimization in neural networks with Boolean logic.
method Boolean logic backpropagation with convergence analysis.
result First convergence analysis for Boolean logic in neural networks.
Using the Fourier expansion of Markov traces for Ariki-Koike algebras over Q(q,u1,...,ue), we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple …
SLEIPNIR improves Gaussian process regression with derivatives, scaling up efficiently and accurately.
problem Scaling Gaussian process regression with derivatives for large datasets.
method Quadrature Fourier features for feature expansion, proving error bounds.
result Deterministic, non-asymptotic, exponentially fast decaying error bounds for approximated kernel and posterior.
Paper introduces a new method for efficient portfolio risk quantification.
problem Efficiently quantify risk in large portfolios with many trades and few dominant risk factors.
method Combines Fourier-cosine series with tensor decomposition techniques for dimension reduction.
result Achieves relative errors below 0.1% with significant runtime improvement.
Quantization and reduction studied for CR manifolds with group actions.
problem Quantization and reduction for CR manifolds with group actions.
method Consider a compact torsion free CR manifold X with a G-equivariant rigid CR line bundle L. The high tensor powers of L are studied, and a weighted G-invariant Fourier-Szegő operator projects onto the space of G-invariant CR sections. result Quantization commutes with reduction for sufficiently high tensor powers of the line bundle.
Polynomial Chaos Expansion improves operator learning for PDEs.
problem Approximating mappings between infinite-dimensional functional spaces.
method Polynomial Chaos Expansion (PCE) for operator learning.
result PCE achieves strong performance in operator learning and uncertainty quantification.
We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accura…
Multivariate splines linked to infinitely-wide neural networks with improved numerical performance.
problem Understanding the relationship between multivariate splines and neural networks.
method Showed multivariate splines can be represented as random features in infinitely-wide neural networks with a homogeneous activation function.
result The function space of multivariate splines is a Sobolev space on a Euclidean ball with explicit norm bounds on derivatives.
This paper analyzes DONE, an online optimization algorithm that iteratively minimizes an unknown function based on costly and noisy measurements. The algorithm maintains a surrogate of the unknown function in the form of a random Fourier expansion (RFE). The surrogate is updated whenever a new measurement is available,…
Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.
problem Understanding the space of functions computed by deep-layered machines.
method Investigation of Boolean functions on random-layered machines, including neural networks and Boolean circuits.
result The space of functions computed at large depth limit is characterized and the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with depth.
We fit the volatility fluctuations of the S&P 500 index well by a Chi distribution, and the distribution of log-returns by a corresponding superposition of Gaussian distributions. The Fourier transform of this is, remarkably, of the Tsallis type. An option pricing formula is derived from the same superposition of Black…
We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique o…
A new model for pricing ultra-short-term options with complex volatility patterns.
problem Complex pricing of ultra-short-term options due to oscillations in implied volatility.
method Edgeworth++ model with nonparametric stochastic volatility and deterministic shift extension.
result Fast and accurate closed-form option pricing for ultra-short-term options.
Proves existence of Yamabe metrics on conical manifolds with conical points and links.
problem Existence of Yamabe metrics on singular manifolds with conical points and links.
method Derives a counterpart of Aubin's result, uses conical links and Fourier analysis, adds lower-order correction to standard bubbles.
result Derives asymptotic expansions on the Yamabe quotient for generic type metrics.
A new deep learning method using Boolean logic reduces training and inference energy.
problem High computational and energy costs in deep learning training and inference.
method Introduces Boolean weights and inputs for efficient training using Boolean logic.
result Achieves full-precision accuracy in ImageNet classification and surpasses state-of-the-art results in semantic segmentation.
Boolean matrix factorization and Boolean matrix completion from noisy observations are desirable unsupervised data-analysis methods due to their interpretability, but hard to perform due to their NP-hardness. We treat these problems as maximum a posteriori inference problems in a graphical model and present a message p…
New Boolean algebra method shows knot unknotting number is (c+1)/2.
problem Finding the minimum number of region crossing changes to unknot a knot.
method Boolean algebra applied to region crossing changes.
result Region unknotting number is (c+1)/2 for any knot with crossing number c.
This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distri…