This paper explores dual relations between line congruences and surfaces in 3D and 4D.
problem Understanding the duality between line congruences in R3 and surfaces in R4. method Analyzes the correspondence between principal lines and asymptotic lines, ridge curves and flat ridge curves, and subparabolic curves.
result Discusses the behavior of subparabolic curves at the discriminant curve of the line congruence and the parabolic curve of the dual surface.
Proposes ridge regression on Riemannian manifolds for time-series prediction.
problem Time-series prediction on Riemannian manifolds.
method Combines Riemannian least-squares fitting via Bézier curves, empirical covariance on manifolds, and Mahalanobis distance regularization.
result Significant error reduction in synthetic spherical experiments and hurricane forecasting.
Theory and method for reducing prediction variance in noisy feature-subsampled ridge ensembles.
problem Reduction of prediction variance in noisy data with feature bagging.
method Developed analytical learning curves for noisy ridge ensembles, introduced heterogeneous feature ensembling.
result Subsampling shifts the double-descent peak, leading to improved performance over a single linear predictor.
New kernel methods estimate complex causal relationships.
problem Estimating nonparametric causal functions like dose-response curves.
method Kernel ridge regression with decomposition property.
result Uniform consistency with finite sample rates proved.
Study shows minimizing the norm of the ERM solution stabilizes kernel ridge-less regression.
problem Stability of kernel ridge-less regression.
method Minimizing the norm of the ERM solution to minimize CV stability.
result Interpolating solution with minimum norm minimizes CV stability.
Research on manifold learning within a density ridge estimation framework has shown great potential in recent work for both estimation and de-noising of manifolds, building on the intuitive and well-defined notion of principal curves and surfaces. However, the problem of unwrapping or unfolding manifolds has received r…
The paper analyzes learning curves for kernel ridge regression with dot-product kernels.
problem Understanding the learning curves for different scaling regimes of data and model.
method Precise formulas for mean test error, bias, and variance in the mo∞ with m/dr constant regime. result A peak in the learning curve at m≈dr/r! for any integer r. Framework for transferring discount curve estimates across fixed-income product classes.
problem Challenges in estimating discount curves from sparse or noisy data.
method Proposes a vector-valued kernel ridge regression (KR) framework with economic regularization.
result Transfer learning tightens confidence intervals and improves extrapolation performance.
Proposes estimators for complex dose-response curves using kernel methods.
problem Estimating complex dose-response curves with continuous treatments, mediators, and covariates.
method Kernel ridge regression with sequential kernel embedding technique.
result Simple estimators for mediated and time-varying dose response curves with nonasymptotic uniform rates.
Kernel ridge regression for causal inference with missing data.
problem Estimating treatment effects with missing data in selected samples.
method Kernel ridge regression estimators for nonparametric dose response curves and semiparametric treatment effects.
result Uniform consistency and finite sample rates for continuous treatment, root-n consistency for discrete treatment.
Unified method visualizes curvature on curves and surfaces.
problem Visualizing curvature on planar curves and surfaces of revolution.
method Tangential angle parameterization.
result Clear, consistent visualizations without arbitrary parameter tuning.
Study on ridge regression in convolutional models shows double descent error behavior.
problem Understanding generalization and estimation error in over-parameterized convolutional models.
method Analysis of ridge estimators for convolutional linear models, derivation of exact error formulae.
result Ridge estimators exhibit double descent error behavior in high-dimensional convolutional models.
The paper analyzes a simple neural network model with algebraic methods.
problem Finding minima of a ridge-regularized mean squared error for ReLU perceptrons.
method Developed a Divide-Enumerate-Merge strategy using computational algebra.
result Identifies both isolated and connected minima of the RR-MSE.
Students replicated classical ML learning curves on 18 datasets, finding power laws with tree ensembles dominating.
problem Scaling laws for classical machine learning on tabular data.
method Distributed replication of 11,536 training runs on 18 datasets using 6 model families, fitting power-law curves.
result Tree ensembles dominate, and power laws fit well (R^2 > 0.8 on 77.7% of cells).
This study presents a rapid multiple incremental and decremental mechanism based on Weight-Error Curves (WECs) for support-vector analysis. Recursion-free computation is proposed for predicting the Lagrangian multipliers of new samples. This study examines Ridge Support Vector Models, subsequently devising a recursion-…
Study the geometry of bifurcation sets for specific types of functions.
problem Understanding the structure of bifurcation sets for specific types of functions.
method Using blow-ups and parametrization, investigate the Gaussian curvature, principal curvatures, and curve behavior.
result Bifurcation sets of D4±-functions can be parametrized as surfaces in R3. Study derives error decay rates for kernel classification under source and capacity conditions.
problem Understanding prediction error decay rates for real data sets.
method Derived decay rates for misclassification error under Gaussian design for SVM and ridge classification.
result Rates accurately describe learning curves for data sets satisfying source and capacity conditions.
Two new ridge solutions improve BLS on added nodes, achieving better accuracy.
problem Improving the Broad Learning System (BLS) for new nodes.
method Proposed two ridge solutions for BLS output weights, updating efficiently.
result Proposed ridge solutions achieve better testing accuracy than original BLS.
Two new algorithms recover ridge lines from point clouds with convergence guarantees.
problem Extracting filamentary structure from point clouds.
method Proposes two novel algorithms with convergence guarantees.
result The algorithms can asymptotically recover the full ridge set.
The paper studies multiple descent in multi-component prediction models.
problem Understanding the risk curves in multi-component prediction models.
method Investigates a 'double random feature model' and 'multiple random feature model' in ridge regression.
result Risk curves of multi-component prediction models can exhibit multiple descents.
New method stabilizes machine learning for physics-informed inverse problems.
problem Reconstructing physical quantities from PDE-compliant measurements.
method Physics-informed learning with smooth inductive bias.
result PDE operators stabilize variance and prevent overfitting in fixed dimensions.
Ridge regularization simplifies model complexity in data science.
problem Overfitting in statistical models.
method Adding a penalty on the magnitude of coefficients.
result Effective in reducing model complexity and improving generalization.
We study the problem of estimating the ridges of a density function. Ridge estimation is an extension of mode finding and is useful for understanding the structure of a density. It can also be used to find hidden structure in point cloud data. We show that, under mild regularity conditions, the ridges of the kernel den…
Improved ridge estimators avoid tuning parameters for high-dimensional data.
problem Difficulty in calibrating tuning parameters for ridge estimators.
method Developed modified ridge estimators that eliminate tuning parameters.
result Modified ridge estimators outperform standard methods in prediction accuracy.
Neural network model improves robustness of mortgage bond yield curve estimation.
problem Overfitting and instability in traditional yield curve estimation methods for small mortgage bond markets.
method Neural network framework with a new loss function for smoothness and stability.
result Empirical results show more robust and stable yield curve estimates compared to existing methods.
Soft diamond regularizers improve deep learning performance and sparsity.
problem Improving deep learning performance and sparsity of trained weights.
method New soft diamond synaptic weight priors based on thick-tailed symmetric alpha stable probability curves.
result Soft diamond regularizers outperform state-of-the-art methods in deep learning tasks.
New equivalences found between subsampling and ridge regularization methods.
problem Establishing precise structural and risk equivalences between subsampling and ridge regularization.
method Proved structural and risk equivalences between subsample ridge estimators and different ridge regularization levels and subsample aspect ratios.
result Optimally tuned ridge regression exhibits a monotonic prediction risk in the data aspect ratio.
Study ridge ensembles in proportional feature-to-sample size regime, proving risk equivalence and GCV consistency.
problem Characterizing and optimizing ridge ensembles in proportional feature-to-sample size regimes.
method Proportional asymptotics analysis, GCV for tuning, proving risk equivalence.
result Risk of optimal full ridgeless ensemble matches optimal ridge predictor's risk.
A conventional wisdom in statistical learning is that large models require strong regularization to prevent overfitting. Here we show that this rule can be violated by linear regression in the underdetermined n≪p situation under realistic conditions. Using simulations and real-life high-dimensional data sets, we d…
We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matri…
Boosting ridge regression for high-dimensional data classification reduces computational cost and improves learning time.
problem High computational demand of inverting regularised covariance matrix in ridge regression for high-dimensional problems.
method Train an ensemble of ridge regressors in randomly projected subspaces, then combine them using adaptive boosting.
result Effective in terms of learning time and improved predictive performance in some cases.
Paper analyzes SGD in kernel regression, showing it outperforms offline methods.
problem Performance of SGD in kernel regression compared to offline methods.
method Analyzes Stochastic Gradient Descent (SGD) in kernel regression under misspecified models.
result SGD achieves min-max optimal rates up to constants, avoiding saturation.
Study shows double descent curve in high-dimensional linear regression with random projections.
problem Understanding the generalization performance in high-dimensional settings with random projections.
method Fixed prediction problem, ridge regression estimator, minimum norm least-squares fit, random matrix theory, asymptotic equivalents.
result Exhibit a double descent curve for high-dimensional linear regression with random projections.
Short proof shows how ridge regression works with random data.
problem Understanding prediction error in ridge regression with random design.
method Combination of exchangeability arguments, matrix perturbation, and operator convexity.
result Elementary proof of prediction error without complex inequalities.
The paper examines how nonlinear transformations affect ridge sets in manifold learning.
problem Understanding the impact of nonlinear transformations on ridge sets in manifold learning.
method Examined the effects of nonlinear transformations on ridge sets using mathematical proofs and numerical experiments.
result The inclusion relationship $\cR(f\circ p)\subseteq \cR(p)$ holds for strictly increasing and concave transformations, and the Hausdorff distance between transformed and non-transformed ridge sets is smaller.
HARFE approximates sparse additive functions using random features and ridge regression.
problem Approximating high-dimensional sparse additive functions.
method Hard-ridge random feature expansion with sparse ridge regression and hard-thresholding pursuit.
result HARFE method converges with a given error bound and achieves lower error than other algorithms.
Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner …
New insights into how neural networks learn features, especially when they are very wide.
problem Understanding how gradient flow in wide neural networks selects solutions, especially in the feature-learning regime.
method Axiomatizing the canonical regularizer as a function-space energy and lift, and deriving geodesic ridge for the feature-learning regime.
result Gradient flow in feature-learning networks biases towards ridge regularization, distorting the inductive bias and damaging pretrained networks.
Unified study of ridge regression structure, cross-validation, and acceleration.
problem Understanding and optimizing ridge regression in large-data settings.
method Unified large-data linear model analysis, cross-validation bias correction, sketching accuracy study.
result Unified understanding and improved methods for ridge regression.
MGD with early stopping tends to ridge regularization in least squares regression.
problem Characterizing the implicit regularization of MGD with early stopping.
method Continuous-time view of MGD (momentum gradient flow) and comparison with explicit ridge regularization.
result Under optimal tuning, the risk of MGF is no more than 1.54 times that of ridge.
Kernel balancing weights are generalized as KRRR, providing better confidence intervals for treatment effects.
problem Lack of generalization error, correct feature specification, and limited to average effects.
method Interpreting kernel balancing weights as KRRR, relaxing feature specification, and extending Gaussian approximation.
result KRRR provides strong generalization properties and justifies confidence sets for causal functions.
Kernel ridge regression imputation with consistent variance estimation for handling missing data.
problem Handling missing data in statistical analysis.
method Kernel ridge regression imputation combined with entropy method for variance estimation.
result Root-n consistency of the imputation estimator in a Sobolev space setting.
A new method for high-dimensional functional regression reduces multicollinearity and improves interpretability.
problem Multicollinearity, overfitting, and interpretability in high-dimensional functional linear models.
method Partition-based functional ridge regression framework.
result Improved numerical stability and enhanced interpretability without explicit variable selection.
Novel algorithm identifies nonlinear Granger causal relationships using kernel ridge regression.
problem Identification of nonlinear Granger causal relationships.
method Flexible plug-in architecture with kernel ridge regression using radial basis function.
result Kernel ridge regression in mlcausality achieves competitive AUC scores and more finely calibrated p-values.
A new method corrects bias in high-dimensional ridge regression.
problem Inherent bias in ridge regression limits statistical efficiency and scalability.
method Iterative bias correction strategy for p<n and Ridge-Screening method for p>n. result Valid inferences and asymptotic properties established for de-biased ridge estimators.
Estimates modes and ridges in mixed Euclidean and directional spaces.
problem Estimating local modes and density ridges in product spaces combining Euclidean and directional metrics.
method Extends mean shift algorithm to product spaces, addressing challenges in generalization.
result Established convergence of the proposed methods and demonstrated effectiveness on real-world datasets.
The paper analyzes the generalization error of random features regression, revealing a double descent curve.
problem Analyzing the generalization error of random features regression.
method Performing ridge regression on N random features of the form σ(wa^T x), where wa are random weights.
result The test error follows a double descent curve, with a global minimum above the interpolation threshold.
Ridge regression linked to Poisson resetting in statistical physics.
problem Understanding and extending ridge regularization in machine learning.
method Connecting stochastic resetting from statistical physics with ridge regularization in machine learning, using renewal processes.
result Exact filter identities for ridge regularization in various reset laws, including exponential and non-exponential.