Polynomial distribution can be applied to dynamical systems in certain situations. Macroeconomic systems characterized by economic variables such as income and wealth can be modelled similarly using polynomials. We extend our previous work to data regarding income from a more diversified pool of countries, which contai…
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48 results for “polynomial distributions”
Sample- and computationally-efficient distribution estimation is a fundamental tenet in statistics and machine learning. We present SURF, an algorithm for approximating distributions by piecewise polynomials. SURF is: simple, replacing prior complex optimization techniques by straight-forward {empirical probability} ap…
Algorithm samples from Bingham distribution efficiently.
problem Sampling from the Bingham distribution on a sphere.
method Rejection sampling with polynomial approximation.
result Exact samples from Bingham distribution in polynomial time.
Non-negative -approximating polynomials for Gaussian distributions are proven for certain classes of sets.
problem Existence of non-negative -approximating polynomials for Gaussian distributions.
method Proving the existence of degree- non-negative polynomials that approximate indicator functions of sets with Gaussian surface area in -norm.
result Proves the existence of non-negative -approximating polynomials for certain classes of sets with Gaussian surface area.
Study on Alexander polynomials in braids, linking number theory and topology.
problem Distribution of Alexander polynomials in specific families of braids.
method Exploration of arithmetic invariants and analogies with number theory.
result New directions in arithmetic topology and statistics.
The study reveals the efficiency of sampling from tilted distributions.
problem Sampling from a tilted distribution of an unknown underlying distribution.
method Self-normalized importance sampling to characterize accuracy.
result Polynomial vs super-polynomial sample complexity for bounded vs unbounded distributions.
Polynomial-time algorithm learns high-dimensional halfspaces without labels.
problem Learning high-dimensional halfspaces with margins in polynomial time.
method Contrastive moments and polynomial-time algorithm.
result Establishes the unique and efficient identifiability of the hidden halfspace.
Algorithm learns polynomial transformations of Gaussian distributions.
problem Learning high-dimensional polynomial transformations of Gaussian distributions.
method Polynomial-time algorithms for smoothed settings, tensor ring decomposition.
result First end-to-end guarantees for learning pushforwards under neural networks.
The paper generalizes polynomial functions on Lie groups and their properties.
problem Generalizing polynomial functions on Lie groups and understanding their properties.
method Generalizing the notion of polynomial functions and horizontally affine maps on Lie groups.
result S-polynomial functions are equivalent to polynomial functions in the sense of Leibman in connected nilpotent Lie groups.
Algorithm learns halfspaces with Tsybakov noise in polynomial time.
problem PAC learning halfspaces with adversarial noise.
method Reduction to certifying non-optimality, iterative process, warm-start algorithm.
result First polynomial-time algorithm for learning halfspaces with Tsybakov noise.
This is an extended abstract of the talk given at the Oberwolfach Workshop "Algebraic Structures in Low-Dimensional Topology", 25 May -- 31 May 2014. My goal was to describe progress in distributive homology from the previous Oberwolfach Workshop June 3 - June 9, 2012, in particular my work on Yang-Baxter homology; how…
Income and wealth distribution affect stability of a society to a large extent and high inequality affects it negatively. Moreover, in the case of developed countries, recently has been proven that inequality is closely related to all negative phenomena affecting society. So far, Econophysics papers tried to analyse in…
Hermite polynomials improve private data generation by reducing feature count.
problem Infinite-dimensional features in kernel mean embedding are impractical for private data generation.
method Replace random features with Hermite polynomial features, leveraging their ordered nature.
result Hermite polynomial features yield a more accurate approximation of kernel mean embedding with fewer features.
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
The study optimizes polynomial regression for learning under Gaussian distributions.
problem Agnostic learning of Boolean and real-valued functions under Gaussian distributions.
method LP duality and polynomial degree analysis for -regression.
result Optimal SQ lower bounds for various function classes.
Constructive Approximation under Carleman's Condition, with Applications to Smoothed Analysismath.PR
Quantifies polynomial approximation rates for smooth functions under various distributions.
problem Approximating smooth functions with polynomials under different distributional constraints.
method Develops a quantitative analogue of Carleman's theorem using complex analysis.
result Establishes superexponential rates of approximation for certain function classes over general distributions.
In many applications (in particular information systems, such as pattern recognition, machine learning, cheminformatics, bioinformatics to name but a few) the assessment of uncertainty is essential - i.e., the estimation of the underlying probability distribution function. More often than not, the form of this function…
Efficiently estimates binary product distributions with privacy.
problem Estimating means of binary product distributions privately and accurately.
method Polynomial time, pure differential privacy approach.
result Optimal sample complexity with polylogarithmic factors.
New polynomial convergence guarantees for SGM on general data distributions.
problem Efficient guarantees for multimodal and non-smooth distributions in SGM.
method Polynomial convergence guarantees for denoising diffusion models on general data distributions, with no assumptions on functional inequalities or smoothness.
result Wasserstein distance guarantees for distributions of bounded support or decaying tails, and TV guarantees for further smoothness assumptions.
New algorithm learns halfspaces with noise using Forster decomposition.
problem Learning halfspaces in noisy data.
method Forster decomposition and efficient mixture of distributions.
result First polynomial-time algorithm with strongly polynomial sample complexity.
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.
The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.
problem Recovering tensors with low symmetric rank from symmetric rank-one measurements.
method Covering numbers argument, Carbery-Wright inequality, orthogonal polynomials, Fano's inequality.
result Near-optimal sample complexity bounds for log-concave distributions.
Polynomial-time algorithm for learning halfspaces with Gaussian-distributed data and adversarial noise.
problem Learning halfspaces in the presence of adversarial label noise.
method Iterative soft localization technique enhanced with appropriate testers.
result Output a halfspace with misclassification error $O(\opt)+\eps$.
We study proper losses for discrete generative models without knowing the target distribution.
problem Evaluating generative models in the discrete setting without direct access to the target distribution.
method Define and construct black-box proper losses using statistical estimation theory.
result Black-box proper losses must be of polynomial form and involve more samples than the polynomial degree.
Estimation is the computational task of recovering a hidden parameter associated with a distribution , given a measurement sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems ca…
In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and…
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.
Polynomial convergence proved for SGM, improving over previous methods.
problem Learning probability distributions from data and generating samples efficiently.
method Proved polynomial convergence for SGM using accurate score estimates.
result First polynomial convergence guarantees for SGM, independent of dimensionality.
New estimator adapts to various error distributions.
problem Adapting to different error distributions in nonparametric regression.
method Introduces outrigger local polynomial estimator with modified weighted least squares.
result Minimax optimal over Hölder classes with multiplicative factor.
We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let be an arbitrary distribution over an interval which is -close (in total variation distance) to an unknown probability distribution $…
New algorithm learns decision trees faster than before.
problem Properly learning decision trees in polynomial time.
method Membership query algorithm with time complexity.
result Achieved faster learning time for decision trees.
Score matching offers efficient estimation for certain distributions.
problem Estimating probability distributions with intractable constants.
method Score matching as an alternative to maximum likelihood.
result Score matching is computationally and statistically efficient for certain distributions.
The paper develops AMP theory for sparse and robust regression with polynomial iterations.
problem Challenges in high-dimensional statistical estimation due to asymptotic theory breakdown.
method Non-asymptotic distributional theory of AMP for sparse and robust regression.
result First finite-sample non-asymptotic distributional theory of AMP for polynomial iterations.
Robustly estimates multivariate polynomials in noisy data.
problem Estimating multivariate polynomials in noisy data with outliers.
method Generalizes robust multivariate polynomial regression to n-variate setting.
result Achieves optimal sample complexity and approximation error.
Finite element approximation of the Levi-Civita connection and its curvature in two dimensionsmath.NA
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
Lower bounds on MALA and HMC for well-conditioned distributions.
problem Understanding the performance limits of Metropolized sampling methods.
method Analyzing the Metropolis-adjusted Langevin algorithm (MALA) and multi-step Hamiltonian Monte Carlo (HMC) with a leapfrog integrator.
result Nearly-tight lower bound of on the mixing time of MALA from an exponentially warm start.
The paper connects knot theory and cluster algebras via dimer face polynomials.
problem Understanding the relationship between knot theory and cluster algebras.
method Analyzing dimer face polynomials and their connections to Alexander polynomials and cluster algebras.
result Dimer face polynomials are multivariate generalizations of Alexander polynomials and -polynomials in cluster algebras.
Thompson Sampling For Combinatorial Bandits: Polynomial Regret and Mismatched Sampling Paradoxstat.ML
Thompson Sampling shows polynomial regret for combinatorial semi-bandits with subgaussian rewards.
problem Finding optimal solutions in combinatorial semi-bandits with suboptimal sampling.
method Proposes Thompson Sampling with polynomial regret for linear combinatorial semi-bandits.
result Demonstrates 'mismatched sampling paradox' where knowing distributions can lead to worse performance.
New methods test discrete distributions faster with local privacy constraints.
problem Testing discrete distributions under local differential privacy constraints.
method Efficient randomized algorithms and test procedures, both non-interactive and interactive.
result Faster separation rates in interactive privacy mechanisms.
New algorithm approximates distributions with near-linear time and optimal sample efficiency.
problem Approximating distributions from samples efficiently and accurately.
method Near-linear-time estimator for distributions using universal polynomial approximation.
result Establishes for all , achieving optimal approximation.
Given a fibered link, consider the characteristic polynomial of the monodromy restricted to first homology. This generalizes the notion of the Alexander polynomial of a knot. We define a construction, called iterated plumbing, to create a sequence of fibered links from a given one. The resulting sequence of characteris…
Classifies connected shelves up to order six.
problem Classifying finite right-distributive binary algebraic structures called shelves.
method Symbolic computations with Python to classify shelves up to isomorphism, exploring group structure, and defining shelf polynomials.
result Classified all connected shelves with order less than six up to isomorphism.
The paper equidistributes zeros of random polynomials and sections on manifolds.
problem Equidistribution of zeros of random polynomials and sections on manifolds.
method Weighted pluripotential theory, asymptotic Bernstein-Markov measures, variance estimation.
result Equidistribution holds for non-i.i.d. random coefficients and non-homogeneous manifolds.
The paper investigates polynomial alternatives to softmax in transformer models.
problem The effectiveness of softmax attention in transformers is questioned.
method The authors explore polynomial activations as alternatives to softmax, focusing on their ability to regularize the attention matrix.
result Certain polynomials can serve as effective substitutes for softmax in transformer applications, achieving strong performance.
We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized s…
Paper improves likelihood estimation for discrete distributions.
problem Computing profile maximum likelihood for discrete distributions.
method New bounds on Bethe and Sinkhorn permanents for low rank matrices.
result Achieves an approximation factor of exp(-O(sqrt(n) log n)) in polynomial time.
In list-decodable subspace recovery, the input is a collection of points (for some ) of which are drawn i.i.d. from a distribution with a isotropic rank covariance (the \emph{inliers}) and the rest are arbitrary, potential adversarial outliers. The goal is to recover a $O(1/α)…
Study reveals an equivalence principle for the spectrum of random inner-product kernel matrices in polynomial scaling.
problem Understanding the spectrum of random kernel matrices in polynomial scaling regimes.
method Investigates random matrices with nonlinear kernel functions applied to inner products of uniformly distributed vectors.
result The spectrum of the random kernel matrix is asymptotically equivalent to a simpler matrix model through free additive convolution.