Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.
problem Constructing semi-discrete and discrete surfaces explicitly.
method Using Jacobi elliptic functions and τ-functions.
result Explicit constructions and periodicities of semi-discrete and discrete surfaces.
Study classifies semi-discrete linear Weingarten surfaces with Weierstrass-type representations and analyzes their singularities.
problem Characterizing semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities.
method Established properties, classified, and analyzed the singularities of semi-discrete linear Weingarten surfaces in Riemannian and Lorentzian spaceforms.
result Defined and analyzed the singularities of semi-discrete linear Weingarten surfaces, including those with non-zero constant Gaussian curvature, parallel surfaces of minimal and maximal surfaces, and constant mean curvature 1 surfaces in de Sitter 3-space.
Transforms curves to surfaces with constant curvature.
problem Creating surfaces with constant curvature from curves.
method Introducing a Darboux transformation for polarized space curves and studying its properties.
result Semi-discrete isothermic surfaces can be described as sequences of Darboux transforms of polarized curves.
We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or integrable systems principles. Finally, we state the shared properties that those…
New geometric transformations link discrete and continuous curve motions.
problem Establishing a connection between discrete and continuous curve motions.
method Infinitesimal Darboux transformations of smooth curves.
result Alternate geometric interpretation for semi-discrete mKdV equation.
Proves hardness of semi-discrete optimal transport and proposes regularization methods.
problem Computing Wasserstein distance between discrete and non-discrete probability measures.
method Proves hardness, introduces distributionally robust dual optimal transport, regularizes primal objective, uses stochastic gradient descent.
result Regularization schemes and improved convergence guarantees for semi-discrete optimal transport problems.
Stochastic optimization improves semi-discrete OT map estimation with a minimax rate.
problem Empirical success of SGD in semi-discrete OT, but lack of theoretical guarantees.
method Averaged projected SGD with a minimax convergence rate of O(1/√n).
result SGD methods can estimate the OT map with a minimax convergence rate of O(1/√n).
We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the τ τ τ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …
DRAG decreases regularization to accelerate semi-discrete OT convergence.
problem Mitigating bias in semi-discrete OT problems with entropic regularization.
method DRAG: Decreasing Regularization Averaged Gradient, a stochastic gradient descent algorithm.
result DRAG achieves unbiased O ( 1 / t ) \mathcal{O}(1/t) O ( 1/ t ) sample and iteration complexity for OT cost and potential estimation, and O ( 1 / t ) \mathcal{O}(1/\sqrt{t}) O ( 1/ t ) rate for OT map. A new variational inference method using optimal transport.
problem Approximating complex posterior distributions with flexible particle-based methods.
method Introducing a new particle-based variational inference method based on semi-discrete optimal transport.
result The method provides a particle approximation and optimal transportation densities.
Framework uses optimal transport for neural architecture search.
problem Optimizing neural architectures in deep learning.
method Semi-discrete optimization using optimal transport.
result Gradient flow and minimizing movement scheme converge to reaction-diffusion equations.
k-GANs uses an ensemble of GANs with semi-discrete OT for better modeling.
problem Mode collapse in GANs.
method Semi-discrete optimal transport for training an ensemble of GANs.
result k-GANs consistently outperforms baseline GANs in experiments.
Paper addresses stability in multi-asset American option pricing.
problem Stability in multi-asset American option pricing problems.
method Semi-discretization approach followed by full discretization.
result Stability conditions found for numerical solution.
Deep convolutional neural networks have led to breakthrough results in practical feature extraction applications. The mathematical analysis of these networks was pioneered by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on identical semi-discrete wavelet frames in each network layer…
New algorithm improves OT map estimation for semi-discrete settings.
problem Improving estimation of OT maps in semi-discrete settings.
method Stochastic Gradient Descent with adaptive entropic regularization and averaging acceleration.
result Achieves nearly minimax rate of O ( t − 1 ) \mathcal{O}(t^{-1}) O ( t − 1 ) for OT map estimation. AlignFlow improves FGMs by optimizing noise and data alignment.
problem Optimal Transport methods for FGMs are limited by scalability issues.
method Introduces Semi-Discrete Optimal Transport (SDOT) to enhance FGM training.
result AlignFlow scales well to large datasets and model architectures.
Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
New method for flexible tubes and structures, enabling rigid-foldability.
problem Creating flexible tubes with rigid-foldability.
method Discrete, semi-discrete, and smooth construction of surfaces (T-hedra and profile-affine surfaces).
result Unified treatment of continuous flexible structures composed of tubes.
Two new algorithms improve neural architecture search efficiency.
problem Optimizing neural architecture search for faster and more accurate models.
method Introduces NASGD and NASAGD using accelerated gradient descent on a semi-discrete space.
result Achieves comparable accuracy with 40x fewer architectures in 12 hours.
New method preserves positivity in financial model simulations.
problem Preserving positivity in financial model simulations.
method Combining semi discrete technique with split step method.
result Explicit and positivity preserving numerical scheme for Ait-Sahalia model.
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.
New methods create full discretized isothermic tori in Euclidean spaces.
problem Creating full discretized isothermic tori in Euclidean spaces.
method Using Darboux transformations and periodic curvature line systems.
result Discrete and semi-discrete k-dimensional isothermic tori in n-dimensional Euclidean space.
This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semi-discrete systems with non-normal matrices A. By employing the logarithmic sp…
Paper proposes a method to train generative networks with minimized Wasserstein distance.
problem Training generative networks to match target distributions accurately.
method Gradual, semi-discrete approach via explicit Wasserstein minimization.
result The approach minimizes Wasserstein distance to both empirical and population target distributions.
Estimates discontinuous optimal transport maps between a discrete and continuous distribution.
problem Estimating discontinuous optimal transport maps between a discrete and continuous distribution.
method Entropic optimal transport estimator, computationally efficient.
result The estimator converges at the minimax-optimal rate n − 1 / 2 n^{-1/2} n − 1/2 in the semi-discrete setting. Derives Ribaucour coordinates for curves and submanifolds, smoothing curvature line nets.
problem Deriving Ribaucour coordinates for curves and submanifolds.
method Uses Bianchi permutability result and Ribaucour transformations.
result Reduction of ambient dimension for submanifolds proved.
New method for pricing options in stochastic volatility models.
problem Pricing options in models with stochastic volatility.
method Time-adaptive, high-order compact finite difference scheme.
result Extends fourth-order multistep methods to stochastic volatility models.
The paper analyzes rates of convergence for optimal transport map estimators using barycentric projections.
problem Estimating optimal transport maps from data sampled according to two distributions.
method Comprehensive analysis of rates of convergence for plug-in estimators defined via barycentric projections.
result New stability estimate for barycentric projections under minimal smoothness assumptions.
A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parame…
In this paper we want to exploit further the semi-discrete method appeared in Halidias and Stamatiou (2015). We are interested in the numerical solution of mean reverting CEV processes that appear in financial mathematics models and are described as non negative solutions of certain stochastic differential equations wi…
On compact surfaces, a Green-Wasserstein inequality cannot be improved without the sqrt(log n) factor.
problem Can the Green-Wasserstein inequality be improved without the sqrt(log n) factor?
method Contradiction proof using second-moment estimates and semi-discrete random matching asymptotics.
result It is impossible to remove the sqrt(log n) factor in the inequality on any compact connected surface.
SVD-based methods reduce computational cost for stochastic systems.
problem High dimensionality and Monte Carlo runs in stochastic systems.
method Extending SVD-based model reduction to stochastic differential equations.
result Preserving symplectic structures improves accuracy and energy conservation.
New findings on optimal transport gradient for generative models, addressing numerical instabilities.
problem Numerical instabilities in training Wasserstein Generative Adversarial Networks (WGAN).
method Valid differentiation theorem for entropic regularized transport, semi-discrete gradient formulation, and optimization algorithm.
result Existence of optimal transport gradient for generative models under specified conditions.
Optimal transport reformulates multiple quantile hedging problem.
problem Multiple quantile hedging problem in incomplete markets.
method Reformulated as Monge optimal transport problem, introduced Kantorovitch version, proved no duality gap.
result Multiple quantile hedging problem can be seen as semi-discrete optimal transport problem.
A new method learns quantization boundaries in continuous space using tessellation.
problem Mapping between discrete and continuous distributions is difficult.
method Constructs normalizing flows on convex polytopes with exact likelihood evaluations.
result Improves likelihood evaluation and quantization learning across various data modalities.
We solve integrable systems to describe the motion of Kaleidocycles.
problem Existence and motion of Kaleidocycles.
method Elliptic theta functions and integrable systems.
result Existence and motion of Kaleidocycles for any number of tetrahedra greater than five.
WGANs use optimal 1-Wasserstein distance to generate distributions.
problem Characterize geometrical properties of generated distributions.
method Analyze WGANs in finite and asymptotic regimes, focusing on univariate latent space.
result WGANs can approach target distribution with optimal 1-Wasserstein distance as sample size increases.
Paper studies how discrete space curves with constant torsion deform to model linkage motions.
problem Modeling and understanding the motion of discrete space curves with constant torsion.
method Using semi-discrete mKdV equations to describe the motion of discrete space curves.
result The motion of discrete space curves is governed by semi-discrete mKdV equations.
This paper tackles co-design of neural hardware and software to improve efficiency.
problem Designing efficient deep learning systems that consider both hardware and software optimizations together.
method Developed a constrained Bayesian optimization framework to automatically identify profitable design points in the joint hardware/software design space.
result Improved energy-delay product by 18% (ResNet) and 40% (DQN) over hand-tuned systems.
Regularizes optimal transport for easier computation and stability.
problem Computational challenges in optimal transport problems.
method Semi-dual formulations and entropic regularization.
result Smooth variational problems for simpler implementation and stability.
Deep neural networks can approximate any target probability distribution given certain conditions.
problem Approximating complex probability distributions with deep neural networks.
method Proving the existence of a deep neural network mapping that approximates a target distribution under various integral probability metrics.
result Upper bounds on the size of the neural network in terms of dimension and approximation error for different metrics.
Enhances CEV model pricing with high-order scheme and adaptive time stepping.
problem Improving accuracy in pricing American CEV models with irregularities.
method High-order time adapted scheme, local mesh refinement, adaptive time stepping, fifth-order 5(4) Dormand-Prince method.
result Highly accurate solution with reduced computational runtime.
Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.
problem Characterizing and analyzing Gaussian fields on 4D Riemannian manifolds.
method Constructing and analyzing co-biharmonic Gaussian fields with covariance kernels defined by the Paneitz operator.
result Rigorous derivation of quantum Liouville measure for ∣ γ ∣ < 8 |γ|<\sqrt8 ∣ γ ∣ < 8 . Optimal transport simplifies machine learning by comparing probability measures.
problem Comparing and manipulating probability distributions in machine learning.
method Uses optimal transport to compare and manipulate probability distributions, combining statistical and geometric perspectives.
result Optimal transport provides a unified framework for various machine learning tasks.
A scalable algorithm for computing Wasserstein barycenters of streaming data.
problem Aggregating data from different, possibly non-identically distributed sources.
method Parallel, semi-discrete algorithm for continuous input distributions.
result Robust, streaming Wasserstein barycenter estimate that tracks nonstationary distributions.
Wasserstein archetypal analysis finds optimal data summaries using Wasserstein metric.
problem Finding optimal data summaries using Wasserstein metric.
method Alternative formulation of archetypal analysis based on Wasserstein metric, with regularization and gradient-based computational approach.
result Existence and consistency of solutions for the regularized problem.
New scalable methods for unbalanced optimal transport improve efficiency and applicability.
problem Scalable algorithms for unbalanced optimal transport remain underexplored.
method Analysis of semi-dual formulation and adaptive gradient methods.
result SGD methods achieve a convergence rate of O(n/εT) for large-scale applications.
This paper extends Mallat's theory of deep CNNs for feature extraction.
problem Feature extraction in deep CNNs for machine learning tasks.
method Develops a comprehensive theory for general convolutional transforms, non-linearities, and pooling operators.
result Translation invariance and deformation sensitivity of the feature extractor with increasing depth.