Optimal interpolation schemes minimize quasiconformal distortion of projective transformations.
problem Minimizing distortion in projective transformations and discrete conformal maps.
method Analysis of quasiconformal dilatation and comparison of interpolation schemes.
result Angle bisector preserving interpolation is optimal in terms of dilatation.
Paper proposes a new numerical scheme for solving BSDEs.
problem Solving backward stochastic differential equations (BSDEs).
method Uses Lagrange interpolation to approximate derivatives and changes sample point distributions for different stability and convergence.
result Guarantees convergence of the scheme under certain conditions on sample point distributions.
The paper analyzes how modern machine learning models can achieve zero training error and robust generalization.
problem Understanding why modern machine learning models achieve strong generalization despite achieving zero training error.
method The paper analyzes local interpolating schemes including geometric simplicial interpolation and singularly weighted k-nearest neighbor methods.
result The nearest neighbor schemes exhibit optimal rates under standard statistical assumptions and provide insights into adversarial examples.
We introduce a new wavelet transform suitable for analyzing functions on point clouds and graphs. Our construction is based on a generalization of the average interpolating refinement scheme of Donoho. The most important ingredient of the original scheme that needs to be altered is the choice of the interpolant. Here, …
Compact scheme solves American put options with regime-switching using finite differences and Hermite interpolation.
problem Pricing American put options with regime-switching model.
method Logarithmic transformation, compact finite difference scheme, Hermite interpolation.
result The scheme provides an accurate and fast solution compared to other methods.
Interpolating noisy data in linear regression leads to zero training error.
problem Understanding why deep neural networks generalize well with noisy data.
method Investigated overparameterized linear regression, analyzing generalization error and proposing a hybrid scheme.
result Interpolating solutions in noisy data can generalize well, with error decaying to zero with more features.
DSoftKI scales GP regression with full derivative observations.
problem Efficiently fitting and predicting full derivative observations in Gaussian Processes.
method Extends SoftKI by using local temperature vectors for interpolation, enabling encoding of local directional sensitivity.
result DSoftKI achieves accurate predictions and scales to larger datasets with full derivative observations.
The study examines deep convolutional neural networks and their learning ability.
problem Understanding the learning ability of deep convolutional neural networks (DCNNs).
method Examines DCNNs under both underparameterized and overparameterized settings, using a novel network deepening scheme.
result Establishes the first learning rates of underparameterized DCNNs and shows how adding layers can create interpolating DCNNs with good learning rates.
Spatial interpolation improves VA valuation efficiency.
problem Efficiently valuing large portfolios of Variable Annuities (VA) products.
method Spatial interpolation framework for large portfolios of VA products.
result Spatial interpolation outperforms nested Monte Carlo simulations in terms of computational efficiency and accuracy.
New methods boost first-order optimization with faster rates.
problem Designing efficient first-order methods for convex problems.
method Shifted objective function with interpolation condition.
result New schemes achieve faster convergence rates.
Interpolatron accelerates deep neural network optimization faster than existing methods.
problem Accelerating nonconvex optimization for deep neural networks.
method Proposes Interpolatron, a new interpolation scheme to accelerate nonconvex optimization.
result Interpolatron converges much faster than state-of-the-art methods on DNNs of great depths.
Algorithm solves American options with regime-switching using multigrid and compact finite difference.
problem Pricing American put options with regime-switching.
method Multigrid iterative algorithm based on compact finite difference schemes and Hermite interpolation.
result The algorithm provides a fast and efficient tool for pricing American put options with regime-switching.
Interpolation improves nearest neighbor algorithms' performance.
problem Improving nearest neighbor algorithms' performance.
method Considered a class of interpolated weighting schemes and characterized their asymptotic performances.
result Mild degree of data interpolation strictly improves prediction accuracy and statistical stability.
SKI accelerates GP inference with sparse grids to handle higher dimensions.
problem SKI scales poorly in high dimensions due to dense grid size.
method Sparse grids within SKI framework, novel matrix-vector multiplication algorithm.
result SKI can be scaled to higher dimensions while maintaining accuracy.
This paper constructs PH spline curves with prescribed arc lengths.
problem Interpolating points, tangent directions, and curvatures with prescribed arc-length.
method Local construction of G2 planar PH biarc curves of degree 7. result Prescribed arc-length can be satisfied for any data and any chosen ratio between boundary tangents.
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.
New algorithm beats traditional methods for seismic data interpolation.
problem Efficiently filling in missing seismic data volumes.
method Primal-dual alternating approach using matrix factors and block-coordinate algorithm.
result Successfully interpolated a large 5D seismic data volume from a 3D model.
Interpolation improves performance in nearest neighbor algorithms without over-parametrization.
problem Achieving zero training error in deep learning without over-parametrization.
method Introduced a class of interpolated weighting schemes in nearest neighbor algorithms.
result Mild data interpolation strictly improves prediction performance and statistical stability.
Market makers use a simplified approach for options trading.
problem Optimal control of a high-dimensional portfolio of options.
method Approximating portfolio vega, using a low-dimensional functional equation, and numerical methods.
result The problem of an option market maker is reduced to a tractable, low-dimensional problem.
Proposes a new metric learning method using Lie group geodesics.
problem Improving distance metrics for k-NN classification.
method Geodesic interpolation on Lie transformation group to calculate velocities and produce a diffeomorphic global transformation.
result Effective in synthetic and real datasets, improving k-NN classification.
Bayesian hyperprior stabilizes image restoration for noisy and missing data.
problem Stability and adaptability in image restoration for noisy and missing data.
method Proposes a hyperprior approach to stabilize Bayesian image restoration.
result Effective restoration of high dynamic range images from a single sensor.
IIC provides a PAC-Bayes bound for interpolating models, revealing factors affecting generalization.
problem Theoretical challenges in understanding overparameterized models and their performance.
method PAC-Bayesian perspective applied to the Interpolating Information Criterion (IIC).
result Test error for overparameterized models achieving zero training error depends on various factors.
New insights into neural network training show some interpolating methods can generalize well, while others fail catastrophically.
problem Understanding why neural networks trained to interpolate can still generalize well or fail catastrophically.
method Analyzing empirical risk minimization (ERM) over large hypotheses classes, focusing on interpolating methods.
result Some interpolating ERM-like methods for large hypotheses classes provide good statistical guarantees, while others fail catastrophically.
For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpol…
Current state-of-the-art discrete optimization methods struggle behind when it comes to challenging contrast-enhancing discrete energies (i.e., favoring different labels for neighboring variables). This work suggests a multiscale approach for these challenging problems. Deriving an algebraic representation allows us to…
Interpolated-MLPs control inductive bias for better performance in low-compute tasks.
problem Low-compute performance gap between MLPs and CNNs.
method Introduced Interpolated MLP (I-MLP) approach to control inductive bias incrementally.
result Continuous logarithmic relationship between inductive bias and performance in low-compute tasks.
Neural network improves VA valuation efficiency.
problem Efficiently valuing large portfolios of complex VA products.
method Neural network approach for spatial interpolation.
result Neural network provides accurate, efficient, and granular valuation.
A method for predicting survival using neural networks for both continuous and discrete time.
problem Survival prediction for both continuous and discrete time data.
method Proposes a scheme for discretizing continuous-time data and two interpolation schemes for continuous-time survival estimates.
result The hazard rate parametrization of neural networks yields better performance than the parametrization of the probability mass function.
Neural networks improve ocean temperature forecasting and data interpolation.
problem Forecasting and reconstructing sea surface temperature from satellite data.
method Patch-level neural network representations that mimic numerical integration schemes.
result Neural networks outperform other data-driven models in forecasting and missing data interpolation.
This work proposes a novel method for interpolating ROMs without solving FEM models.
problem Interpolating ROMs for unseen parameter values without solving FEM models.
method Non-intrusive Space-Time POD interpolation on compact Stiefel manifolds.
result Robust ROMs derived for unseen parameter values with strong correlations to high-fidelity simulations.
New method generates clean data from corrupted observations.
problem Generating clean data from corrupted observations.
method Iterative update of a transport map using black-box corruption channel access.
result Converges to a self-consistent transport map that effectively inverts the corruption channel.
High-probability bound for distributed stochastic approximation tracking error.
problem Analyzing the convergence of distributed stochastic approximation schemes.
method Analysis using ODE approach to stochastic approximation.
result High probability bound for tracking error between iterates and limiting differential equation.
Typically options with a path dependent payoff, such as Target Accumulation Redemption Note (TARN), are evaluated by a Monte Carlo method. This paper describes a finite difference scheme for pricing a TARN option. Key steps in the proposed scheme involve tracking of multiple one-dimensional finite difference solutions,…
Deep learning accelerates Monte Carlo SDE simulations with large time steps.
problem Accurate simulation of SDEs with large time steps.
method Polynomial chaos expansion with neural network learned stochastic collocation points.
result Data-driven scheme achieves strong convergence in Monte Carlo simulations.
Paper explores optimizing GANs' latent space without adversarial training.
problem Understanding the role of saddle point optimization and deep neural networks in GANs.
method Introduces Generative Latent Optimization (GLO) to train deep convolutional generators using simple reconstruction losses.
result GLO achieves desirable GAN properties (synthesizing samples, interpolating, arithmetic with noise) without adversarial training.
The study optimizes Gaussian process approximations for finite-rank models.
problem Posterior behavior of finite-rank approximations differs from parent GP priors.
method Locally supported basis expansions with dependent Gaussian coefficients.
result Finite-rank expansions inherit the same posterior contraction rate as parent GP priors.
This work improves sampling of graph signals with universal bounds and greedy methods.
problem Sampling graph signals is hard due to irregularity and noise.
method Derives universal performance bounds and near-optimal guarantees for greedy sampling.
result Explicit bounds on approximate supermodularity show greedy search can be optimized with worst-case guarantees.
The paper addresses frequency-dependent distortions in massive MIMO systems and proposes a method to recover covariance matrices.
problem Frequency-dependent distortions in the covariance matrix of massive MIMO systems.
method Proposes a novel UL-DL covariance interpolation technique under a mild reciprocity condition.
result The proposed method can recover the covariance matrix in the DL from an estimate in the UL, especially in FDD massive MIMO systems.
Adaptive quadrature improves Bayesian inference through active learning.
problem Efficiently estimating posterior densities in Bayesian inference.
method Sequential node selection using acquisition functions, combining interpolative surrogate models and quadrature rules.
result Positive estimation of marginal likelihood with improved accuracy.
Study finds weak solutions for complex map flows with optimal lifespan.
problem Existence of weak solutions for two-phase matrix-valued harmonic map flows.
method Modified minimizing movement scheme, discretizing time and interpolating solutions.
result Existence of weak solutions with optimal lifespan for the limiting system.
New deformation of link homology for colored diagrams.
problem Understanding colored Khovanov-Rozansky homology.
method Introducing a multi-parameter deformation and extending to braids.
result Link splitting properties and invariants of colored Hopf links.
New geometry for optimal transport cost based on Bregman divergences.
problem Optimal transport cost calculation with Bregman divergences.
method Established properties, defined interpolations, constructed dualistic geometry.
result Derived generalized Pythagorean inequality and Bregman-Wasserstein barycenters.
Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds
problem Convergent approximation of Gaussian Whittle-Matern fields on Riemannian manifolds
method Finite Element approximation of SPDEs
result Universal approximation of precision and covariance matrices
A new method for pricing options with flexible volatility shapes.
problem Parameterizing risk-neutral distributions for accurate option pricing.
method Parsimonious and interpretable parameters for direct control over implied volatility curves.
result Accurate calibration across a large dataset of option curves.
This paper shows how forward rate interpolations are equivalent to discount factor interpolations in yield curve construction.
problem The challenge of choosing between different interpolation methods for yield curve construction.
method Demonstrates the equivalence between forward rate interpolations and discount factor interpolations.
result Some popular interpolation methods on forward rates are equivalent to classical interpolation methods on discount factors.
Paper optimizes neural networks for Bermudan option pricing with faster convergence and risk management tools.
problem Efficiently pricing Bermudan options with static hedging and risk management.
method Monte-Carlo-based artificial neural network framework with novel optimisation algorithm.
result The proposed neural network accelerates convergence and provides improved risk management tools.
New algorithm improves convergence rates for convex optimization problems.
problem Convex optimization problems with noisy stochastic data.
method Stochastic proximal point algorithm with weak linear regularity condition.
result Achieves $\mathcal{O}\left(\frac{1}{k}
ight)$ convergence rate for SPP.
Proposes CDTD, a diffusion model for mixed-type tabular data.
problem Adapting diffusion models to mixed-type tabular data.
method Score matching and score interpolation for continuous features, adaptive noise schedules for categorical features.
result Consistently outperforms state-of-the-art models in mixed-type tabular data.