envelope construction

This paper addresses the finite-sample estimation of L₁ calibration error for binary classifiers. Existing approaches rely either on strong distributional assumptions or asymptotic analysis, limiting their practical applicability. To overcome these limitations, we propose a distribution-free, non-asymptotic theoretical framework. First, we derive a tight upper bound on the L₁ calibration error for calibration functions of bounded variation—a novel result in calibration theory. Second, we design a general-purpose, post-hoc correction method that controls calibration error without requiring model retraining or compromising original predictive performance. Our approach integrates bounded-variation function analysis, nonparametric calibration modeling, and distribution-free probabilistic inequalities. Experiments across multiple benchmark datasets demonstrate that the framework enables highly reliable calibration assessment with low computational overhead, significantly improving both accuracy and robustness of L₁ calibration error estimation—particularly in small-sample regimes.

For interpolation of large-scale scattered data using Matérn-type radial basis functions (RBFs), this paper proposes an efficient algorithm based on multilevel residual correction and samplet coordinate representation. The method constructs a hierarchical system by incorporating samplets—local discrete signed measures with vanishing moments—into the Matérn multiresolution framework, ensuring uniform boundedness of the condition number and reducing overall complexity to (O(N log^2 N)). Key components include multilevel residual decomposition, samplet coordinate transformation, diagonal scaling preconditioning, and sparse approximation of the generalized Vandermonde matrix. Theoretical analysis establishes both the well-conditionedness of the resulting system and controllable interpolation error. Numerical experiments in two and three dimensions confirm near-linear-logarithmic scaling in both matrix assembly and solver time, significantly outperforming conventional RBF approaches.

This work addresses the problem of continuous interpolation and approximation of multivariate scattered data under optional monotonicity constraints. The authors propose a training-free, Lipschitz-continuous approximation method that constructs tight upper and lower bounds, integrating local interpolation with smoothing strategies to ensure strict adherence to prescribed monotonicity while avoiding discontinuities. The key contribution lies in achieving shape-preserving, continuous, and computationally efficient instance-wise approximation. By leveraging GPU-based parallelization, the method achieves substantial performance gains. The proposed algorithms have been implemented in the open-source Python package LipFit, making them readily applicable to large-scale constrained scattered data fitting tasks.

This study addresses the fundamental challenge of achieving optimal function approximation under limited evaluation data—a central problem in numerical analysis and machine learning. From the perspective of information-based complexity, the work systematically investigates function recovery under generalized sampling by integrating information-theoretic analysis, optimal recovery theory, and nonlinear, adaptive, and randomized sampling mechanisms. It uncovers intrinsic connections among diverse sampling strategies, characterizes the information-theoretic limits of function approximation given finite data, and proposes efficient algorithms and sampling schemes that approach these limits. The results provide foundational insights and a unified framework for optimal sampling theory.

Scientific machine learning faces a fundamental verification challenge: statistical methods rely on strong assumptions, while classical interpolation—though possessing rigorous error bounds—is computationally intractable. This work introduces the first verifiable modeling framework that integrates adaptive interpolation with tight, computationally feasible error upper bounds, enabling pointwise mathematically provable falsifiability of predictions. Our approach innovatively couples radial basis function interpolation, interval analysis, Lipschitz constant estimation, and uncertainty propagation modeling to formulate an error-aware training paradigm, augmented by a constraint-aware loss function. Evaluated on partial differential equation surrogate modeling tasks, the framework achieves a 99.2% error-bound coverage rate and reduces verification latency by three orders of magnitude. These advances significantly enhance trustworthiness and enable reliable closed-loop decision-making in scientific simulation workflows.

This study addresses the problem of efficiently estimating the excess risk of large-scale empirical risk minimization models when only black-box access is available. To this end, the authors propose an interleaved resampling-and-refitting algorithm that constructs pseudo-responses from a single training dataset and iteratively generates small synthetic subsets, enabling accurate prediction error estimation without requiring additional validation data or full model retraining. This approach constitutes the first black-box excess risk estimator that operates solely on one dataset and avoids costly full-scale retraining, substantially reducing both computational and data overhead. Theoretical analysis—leveraging randomized residual symmetrization, empirical process theory, and tensor concentration inequalities—establishes high-probability upper bounds on excess risk under both fixed and random design settings, thereby confirming the method’s validity and reliability.

This work addresses the exponential decay of signal-to-noise ratio (SNR) in classical polynomial Stein discrepancies when increasing the polynomial order, which severely undermines the statistical power of goodness-of-fit tests due to uncontrolled variance. For the first time, the construction of Stein discrepancies is explicitly formulated as an SNR² maximization problem. The authors propose the λ-PSD method, which integrates covariance-aware reweighting, low-dimensional subspace approximation, and Rayleigh quotient–based optimization of Stein eigenfunctions. Under Gaussian assumptions, this approach effectively prevents SNR collapse induced by high-order polynomials while retaining linear time complexity. Empirical results demonstrate substantially improved test power, highlighting the critical role of SNR-aware design in scalable Stein discrepancy methods.

This study addresses the challenge of overfitting in exact interpolation under noisy observations, which compromises the generalization capability of multivariate surface modeling. The authors establish a unified slice-based training/testing protocol, implementing Clough–Tocher cubic and Multiquadric radial basis function (RBF) interpolations using SciPy/NumPy. To ensure rigor and reproducibility, they incorporate fixed random seeds, repeated data splits, and Bootstrap-based uncertainty quantification into a standardized evaluation framework. Experimental results demonstrate that both methods achieve high accuracy in noise-free settings, yet exhibit overfitting when noise is present. Notably, cubic interpolation consistently outperforms RBF across RMSE, MAE, and R² metrics, showing greater stability. The findings highlight that structured interpolation can effectively recover physically meaningful process behavior even when anomalous measurements are retained, offering a promising approach for modeling noisy data.

This work investigates the worst-case degrees of freedom of isotonic regression with binary responses and its theoretical guarantees for probability calibration. By analyzing the number of distinct fitted values and leveraging tools from analytic number theory and statistical learning theory, the authors establish—for the first time—a tight asymptotic upper bound on the degrees of freedom, with leading term $\frac{3}{(4\pi^2)^{1/3}} n^{2/3}$. Building on this result, they derive the first nontrivial upper bound on the expected calibration error (ECE) that requires no assumptions on the underlying model or data distribution, other than the response variable $Y$ being binary ($Y \in \{0,1\}$). This provides rigorous theoretical support for the use of isotonic regression in calibration tasks.

Existing subsampling methods based on Determinantal Point Processes (DPPs) struggle to construct continuous DPPs that simultaneously achieve favorable variance reduction properties and lack efficient, structure-preserving discretization schemes. This work proposes a novel wavelet-based continuous DPP and introduces a general discretization framework that converts continuous kernels into low-rank discrete kernels while preserving their variance decay characteristics. The approach is the first to enable DPP-based subsampling for target functions with arbitrarily low regularity and provides explicit convergence rates that depend on the function’s smoothness. The proposed wavelet DPP outperforms existing methods both theoretically and empirically in terms of accuracy and efficiency, substantially broadening the applicability and effectiveness of DPPs in machine learning subsampling tasks.