Paper proves regularity and existence of Riemannian splines.
problem Regularity and existence of Riemannian splines on manifolds.
method Generalization of DuBois-Reymond Lemma for higher-order splines.
result Established existence of minimizers for spline energy.
We extend the adaptive regression spline model by incorporating saturation, the natural requirement that a function extend as a constant outside a certain range. We fit saturating splines to data using a convex optimization problem over a space of measures, which we solve using an efficient algorithm based on the condi…
With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general…
Deep neural networks (DNNs) generate much richer function spaces than shallow networks. Since the function spaces induced by shallow networks have several approximation theoretic drawbacks, this explains, however, not necessarily the success of deep networks. In this article we take another route by comparing the expre…
The paper introduces a spline-based method for calibrating neural networks.
problem Ensuring neural network outputs are reliable for safety-critical applications.
method Approximating the empirical cumulative distribution function using splines to map network outputs to calibrated probabilities.
result The spline-based recalibration consistently outperforms existing methods on calibration measures.
A new spline method for manifold learning using Hessian-based curvature penalties.
problem Learning manifolds with curvature penalties in high dimensions.
method Generalizes thin-plate splines to flat manifolds using Hessian matrices, minimizing square error with curvature constraints.
result Existence and uniqueness of the spline solution, expressed as Green's functions and Hessian approximations.
Bayesian nonparametric LABS model adapts to function smoothness in Besov spaces.
problem Estimating functions with unknown smoothness in Besov spaces.
method Lévy Adaptive B-spline (LABS) regression model with automatic smoothness adaptation.
result LABS posterior contracts around true function in Besov classes at nearly minimax-optimal rates.
Multivariate splines linked to infinitely-wide neural networks with improved numerical performance.
problem Understanding the relationship between multivariate splines and neural networks.
method Showed multivariate splines can be represented as random features in infinitely-wide neural networks with a homogeneous activation function.
result The function space of multivariate splines is a Sobolev space on a Euclidean ball with explicit norm bounds on derivatives.
RST improves environmental time series classification accuracy using randomized B-spline trees.
problem Improving accuracy in classifying complex environmental time series.
method Randomized Spline Trees (RST) integrates randomized functional representations into ensemble learning.
result RST variants outperform standard Random Forests and Gradient Boosting on most environmental time series datasets.
New framework explains deep neural networks using variational spline theory.
problem Understanding functions learned by deep neural networks.
method Developed a variational framework and function space.
result Deep ReLU networks are solutions to regularized data fitting problems over the proposed function space.
This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines …
Gaussian processes are the leading class of distributions on random functions, but they suffer from well known issues including difficulty scaling and inflexibility with respect to certain shape constraints (such as nonnegativity). Here we propose Deep Random Splines, a flexible class of random functions obtained by tr…
This paper presents an efficient algorithm for evolving point cloud data on smooth manifolds using B-Splines.
problem Evolution of point cloud data on smooth manifolds in higher dimensions.
method Lagrangian approach using adaptive B-Spline interpolation.
result Demonstrates the convergence of geometric quantities and the effectiveness of the approach.
We reparametrize ReLU NNs as splines to understand their learning dynamics.
problem Understanding the learning dynamics and inductive bias of neural networks.
method Reparametrize ReLU NNs as continuous piecewise linear splines to study learning dynamics.
result Standard weight initializations yield very flat functions, leading to strength and type of implicit regularization.
Extends gradient-based optimization to spline functions.
problem Limitations of standard differentiable programming methods.
method Derives Jacobian of spline functions and uses it in predictive models.
result Improved performance in various applications.
Gradient descent training of neural networks leads to solutions close to natural cubic splines.
problem Understanding the implicit bias of gradient descent in neural networks.
method Analysis of gradient descent training for wide neural networks, focusing on the curvature penalty and initialization schemes.
result The solutions of gradient descent training are polyharmonic splines for certain initialization schemes.
We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalized least squares criterion, in which the penalty term sums the absolute kth order discrete derivatives over the input points…
We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factoria…
Temporal Functional Circuits explain KAN forecasts with interpretable edge functions.
problem Lack of mechanistic explanations in KAN forecasting.
method Transform KAN edge functions into faithful, temporally grounded explanations using a gated residual KAN.
result Gated KAN achieves lower MSE than linear-only models on regime-switching signals.
Optimizes spectral density estimation for stationary and nonstationary processes.
problem Estimating spectral density of time series with complex structure.
method Optimally adaptive Bayesian spectral density estimation using smoothing spline covariance structure.
result Optimal eigendecomposition provides superior performance compared to alternative covariance functions.
Paper finds maximum curvature of Bézier-spline curves.
problem Finding maximum curvature of Bézier-spline curves.
method Modified B-spline solutions for inverse interpolation problem.
result Determined maximum curvature of Bézier-spline curves.
Researchers found new functions for spherical clothoids using special functions.
problem Developing new mathematical functions for spherical clothoids.
method Used confluent hypergeometric functions and Meixner-Pollaczek polynomials.
result Presented Cartesian coordinate functions and stereographic projections.
We propose a novel method to determine the dissimilarity between subjects for functional data clustering. Spline smoothing or interpolation is common to deal with data of such type. Instead of estimating the best-representing curve for each subject as fixed during clustering, we measure the dissimilarity between subjec…
Cardiac motion modeling using LDDMM and shape splines.
problem Difficulties in probing cardiac function due to shape and deformation interactions.
method LDDMM framework, parallel transport, normalization, shape splines.
result Significant differences in model parameters between pathologies, revealing insights into disease dynamics.
The paper proposes a new method for density estimation using spline quasi-interpolation for clustering.
problem Density estimation and clustering modeling for multivariate data.
method Spline quasi-interpolation for mono-variate approximation, copulas for multivariate modeling.
result The proposed method achieves accurate clustering of data using copulas and spline quasi-interpolation.
We propose to optimize the activation functions of a deep neural network by adding a corresponding functional regularization to the cost function. We justify the use of a second-order total-variation criterion. This allows us to derive a general representer theorem for deep neural networks that makes a direct connectio…
Revisits stochastic collocation with exponential splines for option pricing.
problem Improving the accuracy of option price interpolation using stochastic collocation.
method Uses exponential quadratic splines and optimizes abscissae or parameters of B-splines.
result Shows that fixing abscissae and optimizing parameters leads to better interpolation accuracy.
Improves BN graph learning with splines for scalability.
problem Learning accurate BN graph structures from data.
method Score-and-search approach with MARS for CPD modeling.
result Improves BN graph accuracy and scalability.
A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as th…
Represents neural networks as solutions to inverse problems in Banach spaces.
problem Understanding the function learned by neural networks.
method Variational framework, representer theorem, polynomial ridge splines.
result Neural networks are solutions to inverse problems in Banach spaces.
This work relates the framework of model-based clustering for spatial functional data where the data are surfaces. We first introduce a Bayesian spatial spline regression model with mixed-effects (BSSR) for modeling spatial function data. The BSSR model is based on Nodal basis functions for spatial regression and accom…
Develops flexible non-parametric ACFs using B-spline kernels.
problem Flexible modelling of the autocovariance function (ACF) in time-series, spatial, and spatio-temporal analysis.
method Derives the inverse Fourier transform of B-spline spectral bases to create a general class of non-parametric ACFs.
result Provides a provably dense, flexible, and general class of non-parametric ACFs for various types of processes.
Bayesian method for knot inference in multivariate spline regression.
problem Inference on knot locations in multivariate spline regression due to non-differentiability and varying dimensions.
method Fully Bayesian approach with a new prior on knot number and analytic formula for normal model, extended Bayesian information criterion for non-normal cases, reversible jump Markov chain Monte Carlo.
result Demonstrated superior performance in function fitting with jumping discontinuity.
Improves spline quality and accuracy in computational microscopy.
problem Detecting slender, overlapping structures in microscopy images.
method Differentiable rendering approach for spline refinement.
result Achieves high reliability and sub-pixel accuracy.
Kolmogorov-Arnold Networks achieve optimal convergence rates in nonparametric regression.
problem Nonparametric function approximation in multivariate settings.
method Structured additive and multiplicative KANs using B-splines.
result Achieve minimax-optimal convergence rate O(n−2r/(2r+1)) for Sobolev space functions. Approximates nonlocal curvature of curves using splines.
problem Approximating nonlocal curvature of planar curves.
method Extending nonlocal curvature definition, using incomplete beta function, and linear interpolating spline approximation.
result Nonlocal curvature of a planar curve can be approximated by a spline.
A machine learning method selects optimal orthonormal bases for functional data analysis.
problem Lack of formal criteria for choosing initial orthonormal bases in functional data methods.
method Proposes a machine learning algorithm to learn and place knots for efficient orthogonal spline bases (splinets).
result Demonstrates efficiency, especially for sparse functional data and complex physical systems.
The paper develops a new method for estimating non-parametric regression functions with spatio-temporal dependencies.
problem Estimating non-parametric regression functions with spatio-temporal dependencies.
method Locally Adaptive Regression Splines (LARS) with ADMM algorithm.
result The method shows superior performance compared to existing techniques.
A new model approximates complex functions in parameter space.
problem Complex and nonlinear functional regression problems.
method Mapping-to-Parameter function model with B-spline free knot placement.
result Robust knot placement algorithms improve model performance.
This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.
problem Existing methods for constructing splines on Lie groups have limitations and assumptions that may not reflect actual curves.
method The paper introduces a new approach using solutions of the Poisson equation on Lie groups to construct splines.
result The new method allows for global splines with arbitrary initial conditions, improving curve reconstruction.
Sig-Splines model uses signatures and splines for time series data, achieving universality and convexity.
problem Creating a generative model for multivariate time series data.
method Combines linear transformations and signature transforms into a neural spline flow.
result Achieves universality and introduces convexity in model parameters.
Improves MARS for nonparametric multivariate regression with dimension reduction.
problem High number of basis functions in MARS for high-order interactions.
method Linear combinations of covariates for dimension reduction, facilitating gradient calculation and eigen-analysis for estimation.
result Asymptotic theory and numerical studies show improved performance over MARS.
We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators. Our key result is that a large class of DNs can be written as a composition of max-affine spline operators (MASOs), which provide a powerful portal through which to view and analyze their inner workings. …
HAR regression improves performance on small datasets.
problem Small datasets with complex functions.
method Data-adaptive kernel ridge regression using tensor-product spline basis.
result Achieves n−1/3 convergence rate for right-continuous functions. Sinh-acceleration speeds up B-spline option pricing.
problem Improving efficiency in option pricing calculations.
method Using sinh-acceleration on B-spline probability density projection.
result SINH acceleration technique improves error control and reduces CPU time.
Combines spline interpolation and ARIMA for stock market forecasting.
problem Limited predictive performance of ARIMA in noisy data.
method Integrates cubic spline interpolation and ARIMA for time series forecasting.
result Demonstrates guidance for short-term stock market forecasting.
A new method evolves point clouds using B-splines for smooth surfaces.
problem Evolution of smooth surfaces from discrete point clouds.
method Adaptive Lagrangian B-spline framework for geometric evolution.
result Efficient and accurate reproduction of surface evolution phenomena.
A new nonparametric approach for system identification has been recently proposed where the impulse response is seen as the realization of a zero--mean Gaussian process whose covariance, the so--called stable spline kernel, guarantees that the impulse response is almost surely stable. Maximum entropy properties of the …