Neural processes amortize Gaussian process inference, replacing the exact $O(n^3)$ posterior with a learned $O(n)$ map from context sets to predictive distributions. For a class of latent neural processes, we bound the Kullback--Leibler (KL) divergence between the GP and LNP predictives, decomposing it into three interpretable sources, namely label contamination as the neural process uses label values to estimate a quantity that is label-independent in the exact GP, an information bottleneck because the finite-dimensional representation cannot resolve the full context geometry, and amortization error from a single encoder network shared across all contexts. The bottleneck truncation term decays in the representation dimension $d$ as $O(e^{-cd^{2/d_x}})$ for squared-exponential kernels on $\mathbb{R}^{d_x}$ where $c > 0$ is a kernel-dependent constant and as $O(d^{-2ν/d_x})$ for Matérn-$ν$ kernels, directly linking architecture sizing to kernel smoothness and input dimension. The label contamination term is $O(1)$ in general, with only the observation-noise component decaying as $O(1/n)$, identifying a persistent cost of routing uncertainty estimation through a label-dependent representation. These results characterize the costs of amortization within the analyzed class and yield architectural recommendations to predict variance from context locations alone in the GP-amortization regime, and replace mean aggregation with second-order pooling to close the dominant amortization gap.

Feature learning is widely regarded as the key mechanism distinguishing neural networks from fixed-kernel methods, yet its impact on the induced function space remains poorly understood. In this work, we precisely characterize how the function space spanned by the features of a two-layer neural network evolves during gradient descent training. We prove that, in the high-dimensional proportional regime, after a large gradient step the post-update feature distribution is well approximated by a target-dependent spiked Gaussian covariance. This induces a data-adaptive kernel that reshapes the function space and modifies its spectral structure. Our analysis reveals that feature learning can be interpreted as a distributional transformation in either parameter space or input space, equivalently as the introduction of a target-dependent kernel. In particular, it selectively amplifies eigenvalues aligned with the target direction and mixes leading eigenfunctions, coupling the top radial mode with a target-aligned quadratic harmonic. Overall, our results provide a precise function-space perspective on early-stage feature learning: rather than just rescaling a fixed kernel, gradient descent induces a data-adaptive deformation that preferentially enhances directions aligned with the signal in the data.

Recent studies have reported $\textit{saturation effects}$ and $\textit{multiple descent behavior}$ in large dimensional kernel ridge regression (KRR). However, these findings are predominantly derived under restrictive settings, such as inner product kernels on sphere or strong eigenfunction assumptions like hypercontractivity. Whether such behaviors hold for other kernels remains an open question. In this paper, we establish a broad, new family of large dimensional kernels and derive the corresponding convergence rates of the generalization error. As a result, we recover key phenomena previously associated with inner product kernels on sphere, including: $i)$ the $\textit{minimax optimality}$ when the source condition $s\le 1$; $ii)$ the $\textit{saturation effect}$ when $s>1$; $iii)$ a $\textit{periodic plateau phenomenon}$ in the convergence rate and a $\textit {multiple-descent behavior}$ with respect to the sample size $n$.

Tree ensembles such as random forests (RFs) and gradient boosting machines (GBMs) are among the most widely used supervised learners, yet their theoretical properties remain incompletely understood. We adopt a spectral perspective on these algorithms, with two main contributions. First, we derive minimax-optimal convergence for RF regression, showing that, under mild regularity conditions on tree growth, the eigenvalue decay of the induced kernel operator governs the statistical rate. Second, we exploit this spectral viewpoint to develop compression schemes for tree ensembles. For RFs, leading eigenfunctions of the kernel operator capture the dominant predictive directions; for GBMs, leading singular vectors of the smoother matrix play an analogous role. Learning nonlinear maps for these spectral representations yields distilled models that are orders of magnitude smaller than the originals while maintaining competitive predictive performance. Our methods compare favorably to state of the art algorithms for forest pruning and rule extraction, with applications to resource constrained computing.

Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.

Advanced manufacturing technologies allow for the production of intricate parts featuring high shape complexity and spatially-varying material composition. Data fusion of point clouds with chromatic attributes provides 4D point clouds, a compact and informative representation that encodes both shape and material information. In this paper, we present a registration-free framework for Simultaneous Monitoring of shApe and Color (SMAC) via 4D point clouds. The proposed framework leverages Laplace-Beltrami operator spectral properties to capture and monitor geometric features and the relationship between shape and surface color. A combined monitoring scheme is proposed to effectively detect shape deformations and color anomalies, along with a spatially-aware post-signal diagnostic procedure to determine the source of change and localize color anomalies. Importantly, neither component relies on registration or mesh reconstruction, eliminating error-prone and computationally expensive preprocessing steps. A Monte Carlo simulation study and a case study on functionally graded materials demonstrate that SMAC achieves effective detection performance, particularly for subtle defects, while providing diagnostic capabilities to identify the source and location of anomalies.

Existing guarantees for misspecified kernelized bandit optimization pay for misspecification through kernel complexity: in generic offline bounds, the misspecification level $\varepsilon$ is multiplied by $\sqrt{d_\mathrm{eff}}$, where $d_\mathrm{eff}$ is the kernel effective dimension, while in online regret bounds, the corresponding penalty is $\sqrt{γ_n}\,n\varepsilon$, where $γ_n$ is the maximum information gain after $n$ rounds of interaction. In this work, we show that, for a large class of kernels, the misspecification amplification can be reduced to logarithmic or polylogarithmic growth. In the offline setting, we first prove high-probability simple-regret bounds whose misspecification term is governed by a spectral Lebesgue constant. This yields logarithmic amplification for one-dimensional monotone spectra and polylogarithmic amplification for multivariate Fourier-diagonal product kernels. In the online setting, we modify a domain-splitting algorithm and prove a cumulative regret bound of $\widetilde{\mathcal O}(\sqrt{γ_n n}+n\varepsilon)$ under mild localized eigendecay assumptions, removing the extra $\sqrt{γ_n}$ factor from the misspecification term. The common principle is localization: spectral localization controls the Lebesgue constant of the offline approximation operator, while domain splitting implements the spatial analogue of this mechanism in the online setting, preventing local misspecification errors from being amplified globally.

In this section, we present some preliminary results that will be useful in proving Theorem 3.2, Theorem 3.3 and Proposition 3.4. We draw upon existing theory on properties of random kernel matrices and extend these properties to cluster-correlated data. Specifically, we show the convergence of eigenvalues and eigenvectors of an empirical kernel matrix based on clustered data. Let (X,F,P) be a probability space and H be a Hilbert space over (X,F,P) with a symmetric kernel function k: X X R. Let H be a compact operator on H, defined by Hg(x) = Z Equivalently, Hn can be viewed as an n nreal matrix whose (i,j)-th entry is {Hn}i,j = 1 n k(Xi,Xj). This is the empirical kernel matrix scaled by a factor of 1/n. Here we restrict our discussion to a reproducing kernel Hilbert space (RKHS) H, where the kernel function k is positive semi-definite. We also assume that the operator H is Hilbert-Schmidt, with E[k2(X,X0)] < . Let λ(T) denote the spectrum of a compact, symmetric operator T. Then λ(H) and λ(Hn) are the sets of eigenvalues for H and Hn, respectively.