Properties of a parametric curve in R^3 are often determined by analysis of its piecewise linear (PL) approximation. For Bezier curves, there are standard algorithms, known as subdivision, that recursively create PL curves that converge to the curve in distance . The exterior angles of PL curves under subdivision are s…
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48 results for “convergence determination”
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.
It is shown that in a tower of coverings the regularized determinant of a generalized Laplacian converges to the -determinant. This shows generic nontriviality of analytic torsion or regularized determinants since the -counterparts are easier to compute. We further have an "Euler product expansion" for regula…
Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy measures that provably determine the convergence of a sample to its target dist…
The paper studies how hyperbolic surfaces degenerate along harmonic map rays.
problem The degeneration of hyperbolic surfaces along harmonic map rays.
method Using Teichmüller space and holomorphic quadratic differentials, the authors show convergence of rescaled distance functions to the intersection number with a vertical measured foliation.
result Hyperbolic surfaces along the ray converge to the dual R-tree of the vertical measured foliation in the sense of Gromov-Hausdorff.
The study examines when MAML's objective has a benign landscape.
problem Understanding when MAML's objective landscape is benign.
method Analyzing the landscape of MAML objective on LQR tasks.
result The benign landscape of the MAML objective depends on task similarities.
The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,v_{oct}]. We cons…
Kaczmarz++ accelerates convergence for ill-conditioned systems.
problem Solving ill-conditioned linear systems efficiently.
method Adaptive momentum acceleration, Tikhonov-regularized projections, and memoization.
result Kaczmarz++ converges faster than Krylov methods on ill-conditioned systems.
Stochastic algorithm achieves sublinear convergence for bi-objective optimization.
problem Optimizing two conflicting functions using gradient or subgradient descent.
method Stochastic alternating algorithm with varying steps for each objective.
result Achieves sublinear convergence rate of O(1/T) under strong convexity.
New feature Stein discrepancies improve sampler selection and goodness-of-fit testing.
problem Computational inefficiency in existing Stein discrepancies.
method Feature Stein Discrepancies (ΦSDs) and Random Feature Stein Discrepancies (RΦSDs).
result RΦSDs are orders of magnitude faster while maintaining or improving performance.
A set of control points can determine a Bezier surface and a triangulated surface simultaneously. We prove that the triangulated surface becomes homeomorphic and ambient isotopic to the Bezier surface via subdivision. We also show that the total Gaussian curvature of the triangulated surface converges to the total Gaus…
On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant cu…
Flow of curved surfaces converges to a specific shape over time.
problem Behavior of curved surfaces over time.
method Addressed through the -Gauss curvature flow for .
result Flow converges to a translating soliton determined by initial conditions.
In the present paper, we determine the topologies of three-dimensional closed Alexandrov spaces which converge to lower dimensional spaces in the Gromov-Hausdorff topology.
We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…
For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance. We prove that the exterior angles of control polygons under subdivision converge to 0 at the rate of , where is the number of subdivisions.…
Paper accelerates MM algorithm for faster inference of ranking scores from comparison data.
problem Inference of Bradley-Terry model parameters from comparison data.
method Developed and analyzed MM algorithm for maximum likelihood and Bayesian estimation, proposed an accelerated version.
result Accelerated MM algorithm achieves faster convergence rates compared to classical MM algorithm.
In this paper, we will determine the topological types of hyperbolic 3-anifolds H^3/G such that G is a geometric limit of any algebraically convergent sequence of quasi-Fuchsian groups.
This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on R converges to the exponential utility. Under a …
Guts determine the leading coefficients of -Alexander torsions for 3-manifolds.
problem Determining the leading coefficient of -Alexander torsions for 3-manifolds.
method Using a new criterion for the convergence of Fuglede-Kadison determinants and the work of Agol and Zhang on guts of 3-manifolds.
result The leading coefficient equals the relative -torsion of the guts associated to the cohomology class.
Paper presents a novel orbit determination method for spacecraft clusters.
problem Orbit determination for spacecraft clusters with noisy and non-linear observations.
method Kernel embedding techniques for learning orbits from range-rate observations.
result The method can accurately estimate orbits and identify individual satellites.
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergenc…
Global convergence of an online (stochastic) limited memory version of the Broyden-Fletcher- Goldfarb-Shanno (BFGS) quasi-Newton method for solving optimization problems with stochastic objectives that arise in large scale machine learning is established. Lower and upper bounds on the Hessian eigenvalues of the sample …
Momentum speeds up evolutionary processes in machine learning.
problem Accelerating convergence in evolutionary dynamics.
method Combining momentum from machine learning with evolutionary dynamics using information divergences as Lyapunov functions.
result Momentum accelerates convergence of evolutionary dynamics, including the replicator equation and Euclidean gradient descent.
The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.
problem Conditions for Gromov-Hausdorff convergence of metric quotients.
method Analyzes sufficient conditions for Gromov-Hausdorff convergence of metric quotients of a metric space.
result Concrete examples of sequences of two-dimensional conic-flat spheres converging to spheres with singularities.
Let be a link. We study the Heegaard Floer homology of the branched double-cover of , branched along . When is an alternating link, $\HFa$ of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a …
Unified framework for analyzing convergence of RSAs using Wasserstein divergence.
problem Analyzing convergence of constant stepsize recursive stochastic algorithms (RSAs).
method Lifting RSA into a higher-dimensional space as a Markov chain and studying the distribution's contraction property with respect to Wasserstein divergence.
result RSAs' iterates' distribution converges to an invariant distribution under certain contraction properties.
ES and FD gradients converge as optimization dimension grows.
problem Understanding the relationship between Evolution Strategies and Finite Differences gradients.
method Analyzing the convergence of gradients as the optimization dimension increases.
result ES and FD gradients converge as the dimension of the vector under optimization increases.
New scalable methods for log determinant computations speed up Gaussian process kernel learning.
problem Prohibitive computational cost of log determinant calculations for Gaussian process kernel learning.
method Stochastic approximations based on Chebyshev, Lanczos, and surrogate models.
result Lanczos method is superior for kernel learning, and surrogate models are highly efficient and accurate.
Monge-Ampère Iterationmath.DG
New iteration method for complex and real Monge-Ampere equations converges under certain conditions.
problem Proving convergence of Monge-Ampere iterations for complex and real equations.
method Introduced Monge-Ampere iteration for real equations, established convergence conditions, and provided geometric applications.
result Established sufficient conditions for convergence of Monge-Ampere iteration and provided geometric applications.
Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.
problem Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.
method A perturbed form of gradient descent with arbitrary initialization.
result Gradient descent with noise converges to a unique optimum.
Asymptotic Distributions and Rates of Convergence for Random Forests via Generalized U-statisticsstat.ML
Random forests' performance is analyzed with rates of convergence and asymptotic normality established.
problem Theoretical understanding of random forests' performance and behavior.
method Generalized U-statistics framework to analyze random forest predictions.
result Random forest predictions can remain asymptotically normal for larger subsample sizes.
We consider a discrete-time approximation of paths of an Ornstein--Uhlenbeck process as a mean for estimation of a price of European call option in the model of financial market with stochastic volatility. The Euler--Maruyama approximation scheme is implemented. We determine the estimates for the option price for prede…
Wide neural networks can be closely approximated by Gaussian processes, with rates depending on the activation function's properties.
problem Approximating the behavior of wide neural networks using Gaussian processes.
method Established convergence rates for the central limit theorem in an infinite-dimensional functional space, using a transportation distance metric.
result Explicit convergence rates for neural networks approximated by Gaussian processes, varying based on the activation function's properties.
Researchers compute limits of Kähler-Einstein forms on degenerating manifolds.
problem Understanding limits of Kähler-Einstein forms on degenerating manifolds.
method Hybrid convergence of Kähler-Einstein measures using algebro-geometric limits.
result Limit measure is a weighted sum of Dirac masses at divisorial valuations.
Identifiability and optimal rates of convergence for parameters of multiple types in finite mixturesmath.ST
This paper studies identifiability and convergence behaviors for parameters of multiple types in finite mixtures, and the effects of model fitting with extra mixing components. First, we present a general theory for strong identifiability, which extends from the previous work of Nguyen [2013] and Chen [1995] to address…
OGA and HDAIC improve high-dimensional regression with dependent data.
problem High-dimensional linear regression with dependent observations.
method Orthogonal greedy algorithm (OGA) and high-dimensional Akaike's information criterion (HDAIC).
result OGA and HDAIC achieve optimal convergence rate without knowing sparsity.
We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the…
The paper studies the convergence of harmonic metrics on Higgs bundles.
problem Analyzing the asymptotic behavior of harmonic metrics on Higgs bundles.
method Investigates the convergence of harmonic metrics on stable Higgs bundles of degree 0.
result The sequence of harmonic metrics converges to a decoupled harmonic metric at an exponential rate.
Koopman mode analysis applied to neural networks for training optimization.
problem Optimizing neural network training, identifying issues, and speeding up learning.
method Koopman operator analysis of neural network dynamics.
result Spectral analysis of Koopman operator aids in determining network depth, initialization quality, and training termination.
The paper proposes a test to determine the number of latent classes in ordinal categorical data.
problem Determining the correct number of latent classes in latent class models with ordinal categorical data.
method The test statistic centers the largest singular value of a normalized residual matrix by a simple sample-size adjustment.
result The test statistic converges to zero under the null hypothesis and exceeds a fixed positive constant under an under-fitted alternative.
Study finds upper bounds for exotic options using call prices, converging with more data.
problem Finding consistent upper price bounds for exotic options with limited call price data.
method Model-free approach using martingale property of stock price process, focusing on directionally convex payoffs.
result Upper price bounds converge with more observed call prices, especially for directionally convex payoffs.
Improved subgradient method tackles ill-conditioned composite optimization problems.
problem Slow convergence of subgradient method for composite optimization problems.
method Preconditioned subgradient method with Levenberg-Marquardt approach.
result Linear convergence rate for composite optimization problems under mild conditions.
A new RL paradigm reduces state-action-value function approximation inefficiency.
problem Challenges in state-action-value function approximation for RL.
method State Action Separable Reinforcement Learning (sasRL) decouples action space from value function learning.
result sasRL achieves up to 75% better performance than state-of-the-art MDP-based RL algorithms.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an -gradient flow for Euler's elastic energy.
result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
The accuracy of least squares calibration using option premiums and particle filtering of price data to find model parameters is determined. Derivative models using exponential Lévy processes are calibrated using regularized weighted least squares with respect to the minimal entropy martingale measure. Sequential impor…
Paper develops a new method for game options in local volatility models.
problem Optimal stopping in one-dimensional diffusion with general coefficients.
method Skorokhod embedding to construct recombining tree approximations.
result Construct nearly optimal stopping times with convergence rates.
We study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N>0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of knots with 9 crossings or fewer.