Upper bound on Jones polynomials density modulo primes.
problem Density of Jones polynomials modulo prime numbers.
method Derived an upper bound on Jones polynomials density within a large degree range.
result Upper bound on Jones polynomials density modulo primes.
Paper develops polynomial approximations for complex probability densities.
problem Approximating high-dimensional concentrated probability densities.
method Tensor-product spectral polynomials and KR rearrangements.
result Efficient approximation of complex densities using composite maps.
Polynomial density theorem for specific subgroup orbits in quotient spaces.
problem Effective density of orbits in arithmetic quotients of SL2(C) and SL2(R)imesSL2(R). method Use of Margulis function, incidence geometry tools, and spectral gap of ambient space.
result Proved effective density theorems with polynomial error rate.
Quantum computers outperform classical methods in density modeling.
problem Density modeling with quantum computers.
method Quantum-classical separation for density modeling.
result Quantum computers offer a super-polynomial advantage over classical algorithms for density modeling.
New study shows low-degree polynomial algorithms struggle at clause densities close to Fix's.
problem Finding satisfying assignments in random k-SAT formulas at high clause densities.
method Analysis of low-degree polynomial algorithms and a new many-way overlap gap property.
result No efficient algorithms can find satisfying assignments at clause densities close to Fix's.
Study on Jones polynomials and their roots in the unit circle and complex plane.
problem Understanding the roots of Jones polynomials for knots and links.
method Analyzing solutions of the equation JK(t)=1 for double-twist knots and links. result The set of solutions to JKn(t)=1 is dense in the unit circle and complex plane. The paper proposes a new method for probabilistic load forecasting using Bernstein-Polynomial Normalizing Flows.
problem High variability in short-term load forecasting at the low-voltage level due to fluctuating demand and increasing electrification.
method Flexible conditional density forecasting based on Bernstein polynomial normalizing flows with neural network control.
result Density predictions outperform traditional methods for 24h-ahead load forecasting.
Classifies area-minimizing surfaces in R^4 as algebraic.
problem Classifying entire area-minimizing surfaces in R^4.
method Using quadratic area growth and holomorphic polynomials to cut out surfaces.
result Entire 2-dimensional area-minimizing or stable surfaces in R^4 are algebraic.
New method efficiently interpolates nonparametric density estimators.
problem Efficient evaluation of nonparametric density estimators.
method Piecewise multivariate polynomial interpolation scheme.
result New estimator with low space requirements and efficient querying.
We survey the construction and properties of the Yamada polynomial of spatial graphs and present the Yamada polynomial formulae for some classes of graphs. Then we construct an infinite family of spatial graphs for which roots of Yamada polynomials are dense in the complex plane.
Polynomial-time algorithm estimates edge density of random graphs with privacy and robustness.
problem Estimating edge density of random graphs while maintaining privacy and robustness.
method Sum-of-squares algorithm for robust edge density estimation and reduction from privacy to robustness.
result Optimal error rate up to logarithmic factors, matching theoretical lower bounds.
Paper proposes a robust LPR method using similarity kernels.
problem Outliers and high-leverage points affect traditional LPR's accuracy.
method Integrates predictor and response variables in weighting mechanism using a conditional density kernel.
result Lower empirical bias compared to iterative robust LOWESS.
This paper presents a method for efficient density estimation in nonlinear systems.
problem Accurate representation of non-Gaussian distributions in nonlinear dynamical systems is challenging.
method Uses Seminonparametric (SNP) densities with probabilists' Hermite polynomial basis and Monte Carlo approximation for maximum likelihood estimation.
result Demonstrates that the method can accurately capture non-Gaussian density structure and compute quantiles using fewer samples than raw Monte Carlo.
We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also …
Derives a series expansion for Asian option pricing with polynomial jump-diffusion moments.
problem Pricing Asian options with polynomial jump-diffusion processes.
method Uses Hermite polynomials and moments of the underlying process for closed-form computation.
result Explicit computation of Greeks and accurate series expansion for Asian options.
Samplets and multiwavelets constructed from scattered data converge to specific densities in the limit.
problem Constructing data-adapted multiresolution analyses and multiwavelets with flexible vanishing moments.
method Probabilistic framework for samplet construction; convergence to multiwavelets with broken polynomial densities.
result Samplet construction converges to multiwavelets in the infinite data limit.
We accelerate CNF by reducing ODE truncation errors with polynomial regularization.
problem High computation cost of CNF due to large truncation errors in solving ODEs.
method Add polynomial regularization to approximate ODE trajectories with polynomial functions.
result 42.3% to 71.3% reduction of NFE on density estimation, 19.3% to 32.1% on variational auto-encoder.
Study proposes a method to construct copulas using corrected Hermite polynomial expansion for estimating foreign exchange volatility.
problem Estimating cross foreign exchange volatility with complex correlation structures.
method Applying corrections to the finite sum of multivariate Hermite polynomial expansions to construct copulas.
result The proposed copula method accurately reproduces the volatility smile of cross currency pairs.
FNFs model parameter-dependent densities by combining a fixed flow with a polynomial parameter-dependent transformation.
problem Learning a separate flow for every parameter configuration is intractable.
method Factorizable Normalizing Flows (FNFs) represent the parameter-dependent density as a fixed flow for a reference configuration and a learnable polynomial transformation factorized over parameters.
result FNFs enable the recovery of the combined effect of multiple parameters without sampling their joint space, providing a scalable and interpretable solution.
A new method for sampling on manifolds reduces density estimation errors.
problem Sampling on implicitly defined manifolds in various applications.
method Polynomial-Maximization Moment (PMM) estimator replacing local k-nearest-neighbour density estimate.
result Reduces density estimation errors by 22--36% on asymmetric gamma and boundary-spacing regimes.
OPAA estimates probability densities using functional analysis.
problem Estimating probability density functions efficiently and accurately.
method OPAA uses a parallelizable algorithm based on functional analysis to estimate probability distributions.
result OPAA provides an efficient method to estimate probability density functions and normalizing weights.
New sampling method for heavy-tailed distributions using Langevin Algorithm.
problem Sampling from heavy-tailed distributions efficiently.
method Transformed Unadjusted Langevin Algorithm on specific transformations.
result Polynomial-order oracle complexities for certain heavy-tailed densities.
Let L be any infinite biperiodic alternating link. We show that for any sequence of finite links that Folner converges almost everywhere to L, their determinant densities converge to the Mahler measure of the 2-variable characteristic polynomial of the toroidal dimer model on an associated biperiodic graph.
Entire area-minimizing surfaces of density 2 are planar or quadratic
problem Classifying entire area-minimizing surfaces
method Using density and algebraic properties
result All such surfaces are planar or algebraic
A number of fundamental quantities in statistical signal processing and information theory can be expressed as integral functions of two probability density functions. Such quantities are called density functionals as they map density functions onto the real line. For example, information divergence functions measure t…
We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let p be an arbitrary distribution over an interval I which is τ-close (in total variation distance) to an unknown probability distribution $…
Study cohomology spaces of sl(2) acting on n-ary differential operators.
problem Computing cohomology spaces for sl(2) action on n-ary differential operators.
method Analyzes polynomial μ-densities as sl(2) modules and computes cohomological spaces H^2.
result Computed cohomological spaces H^2 of sl(2) on n-ary differential operators.
Local polynomial regression (Fan and Gijbels 1996) is an important class of methods for nonparametric density estimation and regression problems. However, straightforward implementation of local polynomial regression has quadratic time complexity which hinders its applicability in large-scale data analysis. In this pap…
This paper describes a recursive estimation procedure for multivariate binary densities (probability distributions of vectors of Bernoulli random variables) using orthogonal expansions. For d covariates, there are 2d basis coefficients to estimate, which renders conventional approaches computationally prohibitive …
New method uses Hermite polynomials for American option valuation.
problem Valuation of American options with complex jump-diffusion dynamics.
method Hermite polynomial expansions of transition density and early exercise premium.
result Converging approximations to true option prices and exercise boundaries.
HR in 8D encodes unique conformal gravity with negative curvature.
problem Holographic Renormalisation in 8D Einstein Gravity.
method Relating HR to Topological Regularisation and adding the Euler term.
result The unique conformal gravity theory reproduces the polynomial and cancels divergent terms.
The study approximates option prices using Hermite polynomials without assuming a specific distribution.
problem Approximating option prices without assuming a specific distribution of returns.
method Approximating the logarithmic return's density by a linear combination of rescaled Hermite polynomials.
result Empirical results suggest reasonable performance for options with moderate strike prices.
We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existen…
Triangular map is a recent construct in probability theory that allows one to transform any source probability density function to any target density function. Based on triangular maps, we propose a general framework for high-dimensional density estimation, by specifying one-dimensional transformations (equivalently co…
Here we develop an option pricing method based on Legendre series expansion of the density function. The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density fun…
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.
problem Understanding isolation properties of geodesic planes in hyperbolic 3-manifolds.
method Quantitative estimates of geodesic planes in frame bundles, using tight areas and densities.
result Polynomial estimates of isolation properties with degree given by modified critical exponents.
High-dimensional models trained on smooth manifolds achieve optimal rates in Wasserstein metrics.
problem Training score-based generative models on complex, low-dimensional manifolds.
method Proves optimal rates for SGMs on smooth manifolds, separating into noise regimes and using ReLU nearest-projection coordinates.
result Optimal intrinsic Wasserstein rates are achieved, with polynomial ambient dependence for families with controlled geometry and density.
We consider a stochastic volatility model with Lévy jumps for a log-return process Z=(Zt)t≥0 of the form Z=U+X, where U=(Ut)t≥0 is a classical stochastic volatility process and X=(Xt)t≥0 is an independent Lévy process with absolutely continuous Lévy measure ν. Small-time expansio…
Study on manifolds with density using modified Hessians for curvature comparison.
problem Developing comparison geometry on manifolds with density.
method Modified Hessian approach based on weighted sectional curvature framework.
result Derivation of Hessian comparison and shape operator comparison theorems.
Exact causal network discovery is polynomial for sparse networks.
problem Finding the optimal causal Bayesian network from data is computationally hard.
method Pruning the search space using network properties, combined with dynamic programming and shortest-path searches.
result Exact discovery is polynomial for sparse causal Bayesian networks.
Polynomial-time algorithm matches correlated random graphs with non-vanishing correlation.
problem Matching correlated random graphs with non-vanishing edge correlation.
method Iterative algorithm for polynomial-time recovery of latent matching.
result Algorithm succeeds in recovering latent matching as long as edge correlation is non-vanishing.
We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold $(M,\rg)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on T∗M and of differential operators on tensor densities over $M…
Introduction 1. The two-eigenvalue problem 2. Hecke algebra representations of braid groups 3. Duality of Jones-Wenzl representations 4. Closed images of Jones-Wenzl sectors 5. Distribution of evaluations of Jones polynomials 6. Fibonacci representations
Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we in…
Detects dense subhypergraphs in random hypergraphs using low-degree polynomials.
problem Detecting a planted dense subhypergraph in a random hypergraph model.
method Degree-n^o(1) polynomials of adjacency tensor entries.
result Thresholds for detection in different density regimes.
We present a constructive approach to surface comparison realizable by a polynomial-time algorithm. We determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solut…
Lower bounds show density estimation requires linear samples or query time.
problem Statistical-computational trade-offs in density estimation.
method Lower bound analysis on data structures.
result Lower bounds demonstrate statistical-computational trade-offs for density estimation.