design partitions of unity

This work proposes a unified functional analytic framework that interprets both supervised and unsupervised learning as variational optimization problems within a function space induced by the data distribution. The central insight is that the fundamental distinction between these learning paradigms arises from the choice of the functional being optimized, rather than from differences in the underlying function space itself. Data structure is characterized via operators induced by the distribution, and target functions are estimated in the eigenbasis of these operators. This framework systematically integrates classical algorithms—including kernel methods, spectral clustering, and manifold learning—revealing their intrinsic coherence and underscoring the foundational role of function spaces and associated operators in modern machine learning.

Neural network function approximation suffers from low training efficiency, poor generalization—especially in extrapolation—and non-transferable parameters. To address these limitations, we propose a reusable initialization framework based on basis-function pretraining. Our method (1) performs unsupervised pretraining of network weights using polynomial basis functions to construct a domain-agnostic parameter prior; (2) introduces an input-domain mapping mechanism that enables adaptive alignment of pretrained parameters to arbitrary function domains; and (3) supports modular training and cross-task parameter transfer. Extensive experiments on one- and two-dimensional function approximation tasks demonstrate that our approach achieves an average 2.3× speedup in training convergence, improves extrapolation accuracy by 37%–61% (measured by error reduction), and enhances model stability. This work establishes a scalable, composable paradigm for function modeling in scientific computing and machine learning.

This work addresses the challenge of reconstructing analytic ordinary differential equation (ODE) vector fields from limited discrete trajectory data. Methodologically, it introduces a novel approximation framework centered on the push-forward operator—employed here for the first time as the core modeling tool—combined with the Fourier–Borel transform and Fock space theory to construct finite-dimensional operator approximations within a local analytic functional space. Theoretically, it establishes rigorous convergence guarantees with explicit rates, proving that truncated least-squares polynomials achieve superior approximation both inside and outside their support domain. Experimentally, the method accurately recovers vector fields induced by analytic flow maps, exhibits strong extrapolation capability, and maintains numerical stability. Overall, it provides a new paradigm for analytic dynamical system modeling from sparse data.

This work addresses the limitation of the Tanimoto kernel (Jaccard index), which is restricted to binary sets or nonnegative real-valued functions. We propose the first generalized Tanimoto kernel for **arbitrary real-valued functions**. Our method decomposes each function into signed magnitude components, mapping it to a pair of signed sets; this yields a rigorously defined set-based representation, from which we derive an explicit feature map and the associated reproducing kernel Hilbert space (RKHS) structure. Building on general kernel design principles, we further provide a piecewise-linear analytic formulation and a differentiable smooth approximation. The resulting framework unifies similarity modeling for real-valued functions, combining theoretical soundness with computational tractability. Empirically, it significantly improves generalization performance in function regression and similarity learning tasks.

This study systematically characterizes the approximation capacity of tree tensor networks (TTNs) for multivariate functions, addressing two central problems: the approximation rates of TTNs for classical smooth function classes, and the intrinsic structure of their attainable approximation classes. Methodologically, it integrates tensor network theory, approximation theory, and function space embedding analysis. The contributions are threefold: (i) it establishes that TTNs achieve near-optimal *h*-uniform and *h*-adaptive approximation rates; (ii) it identifies the TTN approximation class as a quasi-Banach space strictly containing—yet not contained in—the classical isotropic, anisotropic, and mixed smoothness spaces; and (iii) it constructs a rigorous theoretical framework for universal approximation by TTNs, demonstrating expressive power comparable to deep ReLU networks and establishing continuous embeddings from multiple smoothness spaces into the TTN approximation class.

This work investigates low-degree sandwiching polynomial approximations for geometric function classes with low intrinsic dimension—such as intersections of $k$ halfspaces—under the Gaussian distribution. The authors propose a novel approach that directly leverages the smoothness of the target function’s boundary to construct sandwiching Lipschitz functions, thereby circumventing the technical complexities of traditional FT-mollification techniques. By integrating tools from high-dimensional approximation theory, they achieve the first construction of sandwiching polynomials for intersections of $k$ halfspaces with degree $\mathrm{poly}(k)$, improving exponentially upon the previous best-known bound of $2^{O(k)}$. Furthermore, for low-dimensional polynomial threshold functions, their method yields a doubly exponential improvement in the degree dependence.

This work addresses optimization problems defined over products of simplices, such as low-rank learning of discrete multivariate probability distributions and function data registration based on the Square-Root Velocity Function (SRVF) representation. To tackle the inherent constraints, the authors propose a smooth reparameterization that is strictly convex element-wise, transforming the constrained problem into an unconstrained optimization over a Riemannian manifold. The resulting problem is solved via Riemannian gradient descent (RGD). Theoretical analysis shows that this reparameterization maps second-order KKT points on the manifold to weak second-order KKT points of the original problem, ensuring theoretical soundness while enhancing computational efficiency. Experiments demonstrate that RGD significantly outperforms projected gradient descent (PGD), achieving more accurate shape-preserving registration in functional data and efficiently solving probability tensor decomposition tasks.

This work addresses the absence of readily available Gaussian quadrature rules for nonclassical weight functions by proposing a general framework that constructs such rules for arbitrary weights via the method of moments and the Stieltjes procedure. Innovatively integrating type-generic programming with adaptive high-precision arithmetic, the approach effectively controls round-off errors and, for the first time, systematically introduces tailored Gaussian quadrature methods to the statistics community. Implemented in Julia as the CustomGaussQuadrature package—accessible from R through JuliaConnectoR—the resulting quadrature rules achieve exact integration of polynomials up to degree \(2n-1\) while substantially reducing the number of function evaluations, thereby offering both high accuracy and computational efficiency.

This work investigates the expressive limitations of infinitely wide neural networks—specifically Barron functions—in variational problems involving physical phenomena with intricate local geometry, such as bending and folding of elastic shells. By integrating techniques from the calculus of variations, functional analysis, and neural network approximation theory, the authors construct explicit counterexamples and perform energy-approximation comparisons. They rigorously establish, for the first time, that while Barron functions exhibit no energy gap relative to Lipschitz functions for a broad class of first-order integral functionals, they fundamentally fail to approximate energy-minimizing curved-fold configurations in certain elastic shell models, being restricted to straight-fold solutions. This reveals an intrinsic limitation of Barron spaces in scientific machine learning and demonstrates a depth separation phenomenon rooted in geometric expressivity.

This study addresses the universal approximation of continuous functionals defined on compact subsets of Hilbert space products. It establishes that architectures comprising finite continuous linear measurements, scalar nonlinear activations, and a fusion step can uniformly approximate any such functional. The result is further extended to Banach space–valued mappings. To the best of our knowledge, this work provides the first rigorous theoretical guarantee for this widely adopted network structure in operator learning and imaging, specifically on compact sets. By proving a universal approximation theorem for continuous functionals over compact domains, the study validates the theoretical soundness of deep operator network designs commonly used in practice.