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.
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.
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.
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.
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.
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.
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…
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…
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. 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. 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.
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…
This paper aims to solve a basic problem in distributed statistical inference: how many machines can we use in parallel computing? In kernel ridge regression, we address this question in two important settings: nonparametric estimation and hypothesis testing. Specifically, we find a range for the number of machines und…
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.
Many problems on signal processing reduce to nonparametric function estimation. We propose a new methodology, piecewise convex fitting (PCF), and give a two-stage adaptive estimate. In the first stage, the number and location of the change points is estimated using strong smoothing. In the second stage, a constrained s…
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 lasso variant of MARS for nonparametric regression.
problem Nonparametric regression with MARS.
method Least squares estimation over convex function combinations with a complexity constraint.
result Achieves logarithmic convergence rate in dimensionality.
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.
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.
There are various algorithms and methodologies used for automated screening of cervical cancer by segmenting and classifying cervical cancer cells into different categories. This study presents a critical review of different research papers published that integrated AI methods in screening cervical cancer via different…
This paper characterizes how randomized neural networks generalize well in multi-dimensional tasks.
problem Understanding the generalization of randomized neural networks in multi-dimensional tasks.
method Characterizes RSNs as an IGAM formalized by an optimization problem with a regularization functional and loss.
result RSNs generalize well in multi-dimensional tasks, akin to spline regression under certain conditions.
GTMs model complex multivariate data with varying conditional independencies.
problem Modeling multivariate data with intricate marginals and complex dependency structures.
method Semiparametric approach using penalized splines and lasso regularization.
result GTMs accurately learn complex dependencies and identify conditional independencies.
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.
Kronecker trend filtering improves lattice data smoothing.
problem Estimating smooth functions on lattice data.
method Penalized least squares with Kronecker products of univariate trend filtering penalties.
result Kronecker trend filtering outperforms linear smoothers in high dimensions.
Develops fast approximations for conditional Shapley values in linear and polynomial models.
problem Estimating conditional Shapley values using regression models is computationally expensive.
method A new approximative estimation method for conditional Shapley values using linear and polynomial regression models.
result Our method significantly reduces computation time compared to existing methods.
Prediction of dynamical time series with additive noise using support vector machines or kernel based regression has been proved to be consistent for certain classes of discrete dynamical systems. Consistency implies that these methods are effective at computing the expected value of a point at a future time given the …
Model-based clustering approaches concern the paradigm of exploratory data analysis relying on the finite mixture model to automatically find a latent structure governing observed data. They are one of the most popular and successful approaches in cluster analysis. The mixture density estimation is generally performed …
The paper analyzes the generalizability of linear autoencoders and multivariate linear regression.
problem Limited theoretical understanding of linear autoencoders' performance.
method Proposes a PAC-Bayes bound for multivariate linear regression and shows LAEs as constrained models.
result The proposed PAC-Bayes bound is tight and correlates with practical metrics.
Deep P-Spline automates DNN structure selection for complex regression problems.
problem Challenges in selecting optimal network structures for DNNs.
method Linking neuron selection to knot placement in basis expansion techniques, introducing a difference penalty for automated knot selection.
result Deep P-Spline extends model class and forms a latent variable modeling framework with theoretical guarantees.
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.
Graph poly-Laplacian method improves regression accuracy.
problem Regression with noisy labels on graphs.
method Graph poly-Laplacian regularization for non-parametric regression.
result Rate of convergence matches known results for smoothing splines.
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.
A new kernel-based nonconformity score improves multivariate prediction regions.
problem Tackling the challenge of compressing multivariate residual vectors into scalars while preserving geometric structure.
method Introducing a Multivariate Kernel Score (MKS) that decomposes into an anisotropic MMD, providing finite-sample coverage guarantees and convergence rates.
result The MKS produces prediction regions that explicitly adapt to geometric structure, reducing volume compared to ellipsoidal baselines.
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.
Optimal transport improves multivariate prediction uncertainty quantification.
problem Uncertainty quantification in multivariate learning tasks, especially in regression and classification.
method Introducing a novel Conformal Prediction procedure using optimal transport to handle multivariate score functions and construct flexible prediction regions.
result Ensures finite-sample, distribution-free coverage guarantees for multivariate prediction sets.
Bayesian model clusters brain activity time series.
problem Heterogeneous multivariate time series in brain imaging.
method Group-based Bayesian mixture of smoothing splines with covariate effects.
result Distinct brain activity patterns identified.
FineMorphs models smooth transformations for multivariate regression.
problem Efficiently modeling complex transformations for multivariate regression.
method Optimal control of affine and diffeomorphic transformations using smooth vector fields.
result FineMorphs can reduce dimensionality and adapt to large datasets.
Neural optimal transport improves multivariate conformal prediction.
problem Multivariate quantile regression challenges and existing methods ignore joint distribution geometry.
method Combines neural optimal transport with amortized optimization for efficient training and faster inference.
result Constructs tighter and more informative predictive regions for multivariate conformal prediction.
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…
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.
Develops a new multivariate regression model for complex outcomes.
problem Flexible, heterogeneous, and residual-dependent multivariate regression problems.
method MultiVCBART framework with Graphical Horseshoe priors.
result Empirically outperforms existing models on sparse, high-dimensional datasets.
NQE uses quantile regression for fast SBI with cubic Hermite splines.
problem Efficient Bayesian inference for complex models with limited data.
method Neural Quantile Estimation (NQE) learns quantiles autoregressively and interpolates them using cubic Hermite splines.
result NQE achieves state-of-the-art performance on various benchmark problems.
Develops higher-order Euler-Poincaré field equations for principal G-bundles.
problem Formulating field equations for higher-order jet bundles of principal G-bundles.
method Reduction theory applied to G-invariant Lagrangian field theories on jet bundles, transferring Hamilton's principle to reduced configuration bundles. result Higher-order Euler-Poincaré field equations are equivalent to conservation of Noether current.
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.
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.
EP method speeds up Bayesian probit regression in high dimensions.
problem Computational challenges in high-dimensional Bayesian probit regression.
method Adapting EP approximation to multivariate Gaussian prior and skew-normal distribution.
result EP routine is computationally feasible in high-dimensional settings.
SBAMDT uses adaptive soft splits to model complex decision boundaries.
problem Limited ability of standard decision trees to capture complex decision boundaries.
method Probabilistic additive decision tree model with adaptive soft multivariate splits.
result Demonstrated improved predictive performance on synthetic and real datasets.
Optimizes minimum-volume prediction sets for multivariate regression.
problem Lack of efficient methods for multivariate conformal prediction.
method Optimization-driven framework for minimum-volume covering sets.
result Efficient and informative prediction sets with tight coverage.