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.
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.
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…
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…
Locally-verifiable conditions ensure exactness of spline discrete de Rham complex.
problem Ensuring cohomological equivalence of spline discrete complex to continuous de Rham complex.
method Theoretical analysis and locally-verifiable sufficient conditions for exactness.
result Locally-verifiable conditions guarantee exactness of hierarchical B-spline discrete de Rham complex.
SMART combines decision trees and MARS for better regression modeling.
problem High variance in decision trees for continuous relationships, poor performance in MARS for discontinuities.
method SMART uses a decision tree to identify subsets with distinct continuous relationships, then applies MARS to fit these relationships independently.
result SMART improves regression performance over state-of-the-art methods in capturing discontinuities and continuous relationships.
A new knot selection method for GAMs reduces model complexity.
problem Choosing optimal knots for B-spline regression in GAMs.
method Adaptive splines combined with Fellner-Schall tuning for automatic knot selection.
result Comparable performance with P-splines but using fewer knots.
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.
Adaptive algorithm improves nonlinear data assimilation for non-Gaussian systems.
problem Challenges of non-Gaussian statistics in data assimilation.
method Triangular measure transport with P-spline basis functions and an information criterion.
result Automatic selection of parsimonious parametrization for efficient adaptation.
This paper introduces a spline-based method for nonparametric ADVI that handles complex posterior distributions.
problem Learning complex posterior distributions with skewness, multimodality, and bounded support.
method Develops a spline-based nonparametric approximation approach for ADVI.
result Establishes the asymptotic consistency of the derived lower bound for importance weighted autoencoder.
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.
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. 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…
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.
Adaptive RBF-KAN improves KANs by dynamically adjusting kernel parameters.
problem Efficiently approximating multivariate functions using learnable univariate edge functions.
method Integrates LOOCV-based kernel scale estimation with adaptive kernel learning.
result Adaptive RBF-KAN outperforms fixed kernel KANs on various benchmark functions.
Proposes a new method for estimating non-pathwise differentiable functional parameters.
problem Estimating dose-response curves for continuous exposure.
method Targeted Highly Adaptive Lasso (HAL) for non-pathwise differentiable functional parameters.
result The Targeted HAL-MLE achieves dimension-free rates up to log(n) factors and outperforms other methods in simulations.
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.
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…
We present a theoretical and empirical study of the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network fun…
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.
Dropout improves regularization in flexible models for rare features.
problem Understanding theoretical properties of dropout in generalized linear models.
method Theoretical analysis and application to adaptive smoothing with B-splines.
result Dropout prefers rare features in mean and dispersion parameters.
A new modeling framework CSN simplifies and interprets machine learning models.
problem Complexity and interpretability issues in machine learning models.
method Combines spline transformation and cross-network to create CSN.
result CSN is as performant and interpretable as XGBoost and FCNN.
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.
Batch normalization improves deep networks by aligning their decision boundaries with data.
problem Improving the performance and generalization of deep networks.
method Theoretical analysis of batch normalization as a function approximation technique for continuous piecewise affine splines.
result Batch normalization adapts the geometry of a deep network's partition to match the data, improving learning and generalization.
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.
Develops a new method to model overlapping asymmetric datasets effectively.
problem Handling overlapping asymmetric datasets in data science.
method Twice penalized P-Spline approximation method.
result Improves model fit by over 65% in a real-life dataset.
A novel online framework for analyzing multidimensional functional data.
problem Analysis of multidimensional functional data streams poses significant challenges.
method Online functional principal component analysis using tensor product splines on a Stiefel manifold with Riemannian stochastic gradient descent.
result Efficient and scalable modeling of multidimensional functional data.
Neural networks require a careful design in order to perform properly on a given task. In particular, selecting a good activation function (possibly in a data-dependent fashion) is a crucial step, which remains an open problem in the research community. Despite a large amount of investigations, most current implementat…
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 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.
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.
Framework for designing nonlinearities in neural networks with slope constraints.
problem Designing nonlinearities with specific properties for signal processing.
method Variational framework with regularization for slope constraints and optimization of adaptive splines.
result Adaptive nonuniform linear splines achieve global optimum in constrained optimization.
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.
Robust model detects outliers in spatiotemporal epidemic data.
problem Outliers in epidemic data can mislead public health decisions.
method RST-GAM with mean-shift, adaptive Lasso, splines, and proximal algorithm.
result Demonstrates effectiveness in real-world COVID-19 data analysis.
In many classification problems it is desirable to output well-calibrated probabilities on the different classes. We propose a robust, non-parametric method of calibrating probabilities called SplineCalib that utilizes smoothing splines to determine a calibration function. We demonstrate how applying certain transforma…
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.
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…
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 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 …
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.
We use splines and the Sasaki metric to analyze and compare manifold-valued trajectories.
problem Analyzing and comparing trajectories on Riemannian manifolds.
method Riemannian hierarchical model, Bézier splines, Sasaki metric.
result Spline-based approaches outperform state-of-the-art methods in intensity classification of trajectories.
Cubic spline smoothing improves interpolation between irregularly sampled data.
problem Interpolation discontinuity in recurrent neural networks for irregularly sampled sequences.
method Cubic spline smoothing compensation module trained end-to-end with ODE-RNN.
result Improves interpolation between irregularly sampled data points.
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.
Smoothing splines provide a powerful and flexible means for nonparametric estimation and inference. With a cubic time complexity, fitting smoothing spline models to large data is computationally prohibitive. In this paper, we use the theoretical optimal eigenspace to derive a low rank approximation of the smoothing spl…
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 …
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.