This paper analyzes the sample complexity of SPS method for scalar linear regression.
problem Analyzing the sample complexity of the Sign-Perturbed Sums (SPS) identification method.
method The paper provides high probability upper bounds for the sizes of SPS confidence intervals under different sets of assumptions.
result The sizes of SPS confidence intervals shrink at a geometric rate around the true parameter, if observation noises are subgaussian.
The paper reformulates regression in infinite dimensions as an inverse problem, showing it's equivalent to compact inverse problems.
problem Learning a linear operator between Hilbert spaces from empirical observations.
method Reformulates regression as an inverse problem, proving equivalence to compact inverse problems under specific conditions.
result The inverse problem is equivalent to compact inverse problems in terms of spectral properties and regularisation theory.
The paper proposes a gradient-based method for multi-penalty Ridge regression.
problem Optimizing multiple regularization hyperparameters for linear regression.
method Gradient-based optimization through matrix differential calculus.
result The method outperforms traditional regularization techniques like LASSO and Ridge.
A new tree method for tensor data improves regression accuracy.
problem Efficiently modeling tensor data for regression problems.
method Scalar-output regression tree models for scalar-on-tensor problems, and tensor-on-tensor problems using additive tree ensemble approaches.
result The tensor-input tree (TT) method outperforms tensor-input GP models in efficiency and accuracy.
PSLR classifies functional data with scalar covariates using path signatures.
problem Classical functional logistic regression models have limitations in capturing nonlinear and cross-channel dependencies.
method PSLR uses truncated path signatures to create a basis-free representation of functional data.
result PSLR outperforms traditional functional classifiers in accuracy and robustness, especially under non-uniform sampling.
New insights into how large learning rates affect transformer training dynamics.
problem Understanding how large learning rates impact the training of transformer models.
method Analyzing a simplified linear transformer model with a two-factor product map.
result Large learning rates can lead to various training outcomes including cycles, chaos, or divergence.
Overparameterized MLR fits hyper-curves, improving model robustness.
problem Improper predictors degrade model generalizability.
method Parameterizing with a scalar and monomial basis, fitting hyper-curves.
result Hyper-curve approach yields robust predictions for noisy data.
Paper adapts CVS method for TL in functional linear regression.
problem Improving estimation and prediction in scalar-on-function regression.
method Adapts control variates method for transfer learning.
result Establishes theoretical connection between O-TL and CVS-based TL.
The SPS method constructs confidence regions for true parameters with optimal sample complexity.
problem Constructing exact, non-asymptotic confidence regions for true system parameters.
method Sign-Perturbed Sums (SPS) method, generalized to various types of problems.
result High probability upper bounds for SPS confidence regions show optimal shrinkage rate.
Bayesian Additive Distribution Regression (DistBART) predicts distributions from grouped data.
problem Predicting distributions from grouped data with varying characteristics.
method Bayesian nonparametric approach using BART for modeling the regression function.
result Empirical and theoretical evidence supports DistBART's effectiveness in learning from low-dimensional marginals.
Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
Proposes a new model for non-linear regression of multivariate time series data.
problem Regression models for non-scalar variables, especially time series, have limitations.
method Develops a non-linear function-on-function model using neural networks.
result Demonstrates effectiveness through real-world applications.
Functional PLS improves prediction and inference for scalar responses from functional predictors.
problem Estimating scalar responses from functional predictors in an ill-posed inverse problem.
method Functional partial least squares (PLS) estimator with adaptive early stopping and new tests.
result PLS attains nearly minimax-optimal convergence rates and detects local alternatives.
Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending func…
Neural network predicts functional responses from scalar inputs.
problem Regression of functional responses with large scalar predictors and nonlinear relationships.
method Transform functional response to finite dimensions, design feed-forward neural network, modify output via objective functions, apply roughness penalty.
result Proposed neural network outperforms conventional methods in multiple scenarios.
This work extends VQR to non-linear cases and provides scalable solvers.
problem Limitations of VQR in handling non-linear relationships and scalability.
method Extension to non-linear VQR, vector monotone rearrangement, fast solvers.
result Substantial improvement over linear VQR and scalable solvers.
Enhances Gaussian process regression with multi-fidelity models and active subspaces for high-dimensional problems.
problem Data scarcity and high-dimensional input spaces with low intrinsic dimensionality.
method Employ Gaussian processes in a Bayesian setting, augmenting with low-fidelity models, and exploiting active subspaces.
result Improves model accuracy through multi-fidelity Gaussian process regression with active subspaces.
The paper examines scalar fourth-order linear differential operators and their invariants.
problem Equivalence problem of scalar fourth-order linear differential operators.
method Investigation of differential invariants.
result Application of differential invariants to the equivalence problem.
This study examines the relationship between PLS and OLS regression using eigenvalue distributions.
problem Analyzing the difference between PLS and OLS regression in terms of eigenvalue distributions.
method Examined the distance between PLS and OLS regression coefficients using the Mahalanobis distance and eigenvalue distributions of the regressor covariance matrix.
result Provided a bound on the distance between PLS and OLS regression coefficients that depends only on the eigenvalue distribution of the regressor covariance matrix.
Improves neural network regression performance using encoder-decoder approach.
problem Suboptimal regression results from classical methods in neural networks.
method Proposes a new method using an encoder-decoder approach for regression.
result Shows improved performance on various real-world datasets.
New bounds on self-normalized martingales improve online linear regression performance.
problem Improving regret bounds in online linear regression.
method Characterizing scale-invariant bounds on self-normalized martingales.
result For d = 1 d=1 d = 1 , O ( log T ) O(\log T) O ( log T ) doubly-uniform regret is possible; for d > 1 d>1 d > 1 , sublinear doubly-uniform regret is impossible. Solves Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
problem Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
method Fully non-linear elliptic equation
result Solves if datas are strictly convex
We propose a framework for the linear prediction of a multi-way array (i.e., a tensor) from another multi-way array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or…
Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.
problem Limited understanding of self-normalized concentration for vector-valued processes outside sub-Gaussian frameworks.
method Developed concentration inequalities for self-normalized processes with light tails (e.g., Bennett, Bernstein bounds) for vector-valued data.
result Provided new insights and bounds for self-normalized processes with non-sub-Gaussian distributions.
Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.
problem Prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds.
method Reduced to solving a semi-linear partial differential equation with exponential nonlinearity using super and sub-solution method.
result Existence of solution depends on the sign of a constant associated to Gauduchon degree.
Proves effective linear volume growth for 3-manifolds with positive scalar curvature.
problem Volume growth of three-manifolds with positive scalar curvature.
method Utilizes the technique of μ-bubbles and almost-splitting theorem.
result Proves effective linear volume growth for 3-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
ESNs trained with Tikhonov least squares approximate ergodic dynamical systems in L2(μ) norm.
problem Approximating ergodic dynamical systems using ESNs.
method Tikhonov least squares regression on ESNs trained on observations from an ergodic dynamical system.
result ESNs trained with Tikhonov least squares approximate the target function in the L2(μ) norm.
Paper optimizes hyperparameters for high-dimensional regression models.
problem Optimizing robustness radius in high-dimensional linear regression.
method Distributionally robust optimization (DRO) with high-dimensional asymptotic statistics.
result Optimal hyperparameter selection minimizes estimation error efficiently.
We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as x ↦ W N W N − 1 ⋯ W 1 x x \mapsto W_N W_{N-1} \cdots W_1 x x ↦ W N W N − 1 ⋯ W 1 x ) by minimizing the ℓ 2 \ell_2 ℓ 2 loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are…
Functional input neural networks approximate continuous functions on weighted spaces.
problem Approximating continuous functions on infinite-dimensional weighted spaces.
method Additive family mapping, non-linear activation, linear readouts, Stone-Weierstrass theorem.
result Global universal approximation of continuous functions on weighted spaces.
In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the g…
Functional BART adds shape priors to Bayesian tree regression for better curve fitting.
problem Regression with function-on-scalar data and shape constraints.
method Bayesian tree structure with spline representations, customized Bayesian backfitting algorithm, shape priors.
result Improved estimation and prediction accuracy with shape priors.
The paper tackles the trade-off between fairness and accuracy in machine learning models.
problem Ensuring fairness in machine learning often reduces model accuracy.
method The paper introduces formal tools for reconciling the fairness-accuracy tension using Pareto optimality from multi-objective optimization.
result The Chebyshev scalarization scheme is superior for finding Pareto optimal solutions compared to the linear scalarization scheme.
This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order d d d (i.e., a d d d -fold multiway array) in R n 1 × n 2 × ⋯ × n d \mathbb{R}^{n_1 \times n_2 \times \cdots \times n_d} R n 1 × n 2 × ⋯ × n d . It focuses on the task of estimatin…
We simplify complex regression coefficients using linearization and feature comparison.
problem Interpreting high-dimensional regression coefficients from nonlinear responses.
method Developed a linearization method to derive feature coefficients and compare them with regression coefficients.
result Shows how regression coefficients relate to linearized feature coefficients and how they change under regularization.
Corrects GCV for inconsistent risk estimation in finite ensembles of penalized estimators.
problem Inconsistent risk estimation of GCV for finite ensembles of penalized estimators.
method Identifies a correction involving an additional scalar correction based on degrees of freedom adjusted training errors from each ensemble component.
result CGCV maintains computational advantages of GCV and is model-free uniformly consistent for ridge regression.
Efficiently maps indoor magnetic fields with SKI and D-SKI.
problem Computing large-scale magnetic field maps in indoor environments.
method Structured kernel interpolation (SKI) with derivatives (D-SKI) for Gaussian process regression.
result Achieves better accuracy and faster computation than state-of-the-art methods.
Robust learning mixtures of linear regressions improve robustness.
problem Improving robustness in learning mixtures of linear regressions.
method Connecting mixtures of linear regressions and mixtures of Gaussians with thresholding for a quasi-polynomial time algorithm.
result The algorithm has significantly better robustness than previous results.
This paper studies robust regression in the settings of Huber's ε ε ε -contamination models. We consider estimators that are maximizers of multivariate regression depth functions. These estimators are shown to achieve minimax rates in the settings of ε ε ε -contamination models for various regression problems including nonpa…
This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm…
Study on deformation of weighted scalar curvature, proving geometric results and stability.
problem Deformation of weighted scalar curvature and related geometric properties.
method Linearization of weighted scalar curvature, studying kernel of formal adjoint.
result Definition and study of weighted vacuum static spaces, stability results on flat spaces.
Locally adaptive interpretable regression improves linear regression's predictability.
problem Linear regression's predictability is limited; it lacks adaptability.
method Locally adaptive interpretable regression (LoAIR) uses neural networks to predict percentile of a Gaussian distribution for regression coefficients.
result LoAIR achieves comparable or better predictive performance than state-of-the-art baselines.
Study numerical methods for singular FBSDEs with degenerate forward component.
problem Numerical approximation of singular fully coupled FBSDEs with degenerate forward component and non-smooth terminal condition.
method Splitting approach to treat diffusion and transport parts separately.
result The splitting method converges with rate 1/2 under structural condition.
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.
Study on thermodynamic costs of simple linear regression.
problem Understanding thermodynamic costs in machine learning models.
method Approximated thermodynamic lower bounds for exact and stochastic linear regression.
result Derived scaling laws for optimal dataset size based on generalization error.
PARC uses piecewise linear predictors for regression and classification.
problem Multivariate regression and classification problems.
method Alternates between ridge and softmax regression, and cluster assignment based on accuracy and separability.
result Converges to a local minimum in a finite number of steps.
Paper introduces semi-supervised linear extremile regression for high-dimensional data.
problem Challenges in high-dimensional extremile regression due to data sparsity and overfitting.
method Proposes semi-supervised learning for linear extremile regression, achieving n \sqrt{n} n -consistency. result Demonstrates improved estimation efficiency and performance in high-dimensional settings.