Kernel Method

Classical spectral descriptors such as the Heat Kernel Signature and Wave Kernel Signature are widely used for non-rigid 3D shape retrieval, yet their failure modes remain poorly understood. We present a frequency-scale saliency framework that audits these descriptors by quantifying the retrieval-level contribution of each descriptor scale interval through ablation. We introduce class spectral fingerprints to characterize category-level scale dependence, and show that descriptor similarity between class pairs is substantially correlated with retrieval failure, with a Spearman correlation of 0.479. Experiments on SHREC'11 demonstrate that short scales dominate retrieval performance while long scales are harmful, that HKS and WKS exhibit distinct scale dependence patterns, and that saliency-weighted retrieval improves mAP on hard categories by 0.156, with cross-fold and random-weight controls confirming that the gain is stable and not due to arbitrary reweighting.

TorchKM is an open-source library for kernel machines, including support vector machines, kernel logistic regression, and kernel quantile regression, with GPU acceleration. The library features a scikit-learn-style API and is designed to exploit GPU-friendly linear algebra, accelerating the full training and model-selection pipeline through intelligent reuse of matrix operations. Benchmarks show competitive predictive performance with substantial speedups over standard baselines. The efficiency and programmable design also make TorchKM a kernel-learning component for AI-driven workflows. Code and documentation are available at https://github.com/YikaiZhang95/torchkm, and the package can be easily installed via PyPI.

Generating realistic financial time series is challenging as training data is often limited to a single historical path. With such scarce data, overfitting is hard to avoid, especially under adversarial training where a trained discriminator can memorize the training samples. To mitigate this, recent approaches train generators to minimize the discrepancy between untrained feature representations of real and generated time series. In these works, the feature maps are based on path signatures, which can fail to capture relevant time series properties at tractable truncation depths. In this work, we instead train generators by matching random convolutional features of real and generated time series. Existing random convolutional feature maps, such as Rocket and Hydra, have been shown to provide informative representations of real-world time series, but cannot supervise generative models because they are non-differentiable. We introduce SOCK (SOft Competing Kernels), a fully differentiable random convolutional feature map, suited to train generative time series models. We show that generators trained by matching random SOCK features consistently outperform signature and diffusion baselines across a wide range of small-sample financial datasets. We further demonstrate SOCK's expressiveness on two-sample hypothesis testing and time series classification tasks, where SOCK matches or outperforms existing unsupervised feature maps.

Vision-language foundation models such as CLIP and SigLIP provide widely used representations for multimodal learning systems. While these models are typically compared through downstream performance, such evaluations often do not explain how their representations differ structurally. In this work, we study this problem through the task of Contrastive Embedding Clustering: identifying sample subsets that are weakly clustered under one representation but strongly clustered under another. We propose \emph{Kernel Optimization for Discrepancy Analysis (KODA)}, a kernel-based framework for contrastive representation comparison and alignment. KODA constructs unified multimodal kernels through modality-wise kernel composition and formulates discrepancy discovery as a constrained optimization problem that searches for coherent structures in one representation while suppressing coherence in a reference representation. This yields interpretable discrepancy directions associated with specific sample subsets and modality interactions. To scale KODA to large vision-language datasets, we develop randomized low-dimensional approximations of joint kernels using random projections, including Random Fourier Features for shift-invariant kernels. Empirically, KODA identifies consistent and interpretable discrepancy structures across vision-language representations and provides sample subsets for representation alignment. The code is available at https://github.com/yokiwuuu/KODA.

Learning a Markov transition model is not merely conditional density estimation: the learned object must be a valid transition kernel before it is iterated in downstream dynamics. This paper introduces a Doeblin-anchored contrastive chart, a statistical-to-dynamical coordinate framework for learning transition kernels from contrastive objectives. Given a restart law and an anchor strength, the chart mixes the target transition with the restart law. The resulting anchored kernel is simultaneously a Doeblin-minorized Markov kernel, the positive conditional law in a binary contrastive experiment, and an explicitly invertible coordinate for the original transition law. We prove that the anchored contrastive risk identifies the anchored transition density and calibrates excess risk to density error. Since inversion of a learned score may produce a signed or unnormalized object, we introduce a measurable Markovization operator that restores kernel validity while preserving integrated $L^1$ accuracy up to a constant factor. Oracle inequalities and Hölder--ReLU approximation bounds yield nonparametric rates for independent transition pairs. For stationary geometrically $β$-mixing trajectories, a conservative thinning-and-coupling extension yields the same reconstruction interface with an effective sample size. Occupancy-weighted perturbation bounds transfer one-step kernel error to finite-horizon marginal, path-law, and occupation-measure errors under explicit coverage.

The Softmax Attention operation in Transformer language models has a quadratic complexity in the sequence length and a growing state size in the form of KV cache, which becomes a bottleneck in long context scenarios. To overcome this limitation, alternative architectures with linear complexity and finite state size have been introduced, such as State-Space Models (SSMs), Linear Attention (LA), and Attention with Bounded-memory Control (ABC). Though linear models achieve similar language perplexity as Transformers, they are still behind in tasks which require retrieval or recall of specific information. In this work, we introduce Blurry Window Attention (BLA) a novel ABC method inspired by SSMs. BLA stores a frequency window from which a blurry KV history is reconstructed via interpolation using Dirichlet kernels. BLA can be understood as a generalization of Sliding Window Attention (SWA) depending on the Dirichlet kernels resolution or as a special case of the Gated Slot Attention (GSA), where the decay factor is implemented with Dirichlet kernels. We describe in details the theory and efficient implementation of BLA. On the Multi-Query Associate Recall (MQAR) synthetic task, we show that the state efficiency of BLA is 8$\times$ better than SWA and is competitive with popular linear attention models, and in the RegBench synthetic task, only BLA and SWA improve their performance as the state size grows among the linear models we tested.

Modeling spatial dependencies is central to spatiotemporal data analysis using Graph Neural Networks (GNNs). Traditional methods rely on distance-based kernels with predefined parameters, which restricts model capacity. Although generic adaptive mechanisms (e.g., Graph Attention Networks) offer flexibility, they often fail to capture the underlying geometric structure, performing worse than distance-based models in data-sparse scenarios. Addressing this, we revisit the kernel parameterization problem and theoretically prove that misspecified kernel parameters introduce unavoidable approximation errors in GNNs. To overcome this, we propose AdaKernel, a simple yet effective approach that learns adaptive kernel parameters within the neural network. Unlike methods that learn graph structures from scratch, AdaKernel adopts a structure-preserving strategy that optimizes the scale of physical interactions rather than discarding them. Extensive experiments on Kriging, Imputation, and Forecasting demonstrate that AdaKernel consistently improves various GNN architectures and outperforms model-agnostic adaptive baselines, validating that accurately learned kernel parameters are superior to both fixed priors and fully latent graph structures.

Median-of-means (MoM) is a powerful technique that theoretically enables near sub-Gaussian finite-sample rate for parameter estimation when the underlying data distribution is heavy-tailed (e.g., assumed to have only two first finite moments). A recent work has extrapolated this technique to median-of-\textit{randomized}-U-Statistics (MoRU) and median-of-\textit{incomplete}-U-Statistics (MoIU) for estimating expectations of heavy-tailed pairwise kernels. In \citet{pmlr-v97-clemencon19a}, a concentration rate that scales like $O(n^{-1/2})$ with sample size has been proven for MoRU. However, despite the computational advantage of the latter, the analysis of finite-sample bound for MoIU remains a significant theoretical challenge. As noted by the authors, a straightforward application of McDiarmid's inequality yields a loose bound of order $O(n^{-1/4})$. In this work, we prove a finite-sample concentration bound for the MoIU estimator that scales as $O(n^{-1/2})$ with respect to the sample size using a delicate convex decomposition approach. Furthermore, we show that our proof can be seamlessly extended to geometric median in multivariate settings. Using a Serfling-type argument, we extrapolate our results into a regime where data pairs are selected without replacement across blocks, breaking the usual block-wise independence condition. Then, using a Bernstein-type treatment for U-Statistics, we tighten the dependency of our bounds on the margin $τ$ from $O(τ^{-3/2})$ achieved in the previous work to $O(τ^{-1})$. Finally, we proved an anti-concentration inequality that is applicable for all median estimators presented in this work to demonstrate that $M\le O(n)$ is an intrinsic restriction on block sizes.

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). Such equivalents make theoretical predictions tractable by reducing a complex model to a simpler one with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful for classification of high-dimensional nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a one-layer feedforward NN, under a canonical nonlinearly separable dataset for the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent of the CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. We identify regimes in which these knobs move the CK beyond the linear equivalent and produce BBP-type transitions to label-aligned outlier eigenspaces. Our analysis helps bring deterministic-equivalence tools from RMT to bear on problems of practical relevance in ML.

Probabilistic smoothing is a standard tool for global optimization, but existing methods rely on Gaussian kernels and specific transforms, often resulting in strong hyperparameter sensitivity and limited robustness. We propose a general smoothing framework that combines flexible symmetric unimodal kernels with monotonic ratio-based transformations. Under mild conditions, we show that the smoothed objective preserves the global maximizer and that all stationary points concentrate near the true optimum for sufficiently large amplification, without requiring a decreasing smoothing schedule. We further provide explicit complexity bounds for stochastic gradient ascent and show that a leave-one-out baseline provably reduces variance. Experiments on high-dimensional benchmarks and black-box adversarial attacks demonstrate improved robustness and competitive performance.

Collaborative analysis of decentralized confidential datasets is important, but direct sharing of original datasets is often restricted by privacy and institutional constraints. Data collaboration (DC) analysis transforms each dataset into privacy-preserving intermediate representations via party-specific obfuscation functions and integrates them into common collaboration representations using an anchor dataset. However, many existing DC analysis methods rely on linear transformations for data obfuscation and integration, which may increase reconstruction risk. Although nonlinear dimensionality reduction can mitigate this risk, conventional linear integration methods cannot accurately align intermediate representations produced by nonlinear transformations. Moreover, existing integration methods mainly minimize discrepancies among parties and do not explicitly incorporate geometric or target-variable information useful for downstream analysis. To overcome these limitations, we first formulate linear kernel integration (LKI) as a linear integration method and then kernelize it to obtain nonlinear kernel integration (NKI). NKI admits a globally optimal solution via kernel ridge regression and an eigenvalue problem. We also introduce graph regularization and a centering constraint so that the target representation can capture geometric and target-variable information useful for downstream analysis. Experiments on image classification tasks demonstrate that NKI improves classification accuracy over existing linear integration methods under nonlinear dimensionality reduction, with further gains from target-variable-aware graph regularization and centering. The results also show that dimensionality reduction choices substantially affect both classification accuracy and reconstruction risk.

A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both k-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for k-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic k-NN neighborhoods, severely biasing the estimator. To correct this, we introduce SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{é}r-Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6%, while consistently matching or outperforming LeJEPA on standard benchmarks-including a +1.8% linear probing gain on ImageNet-1K (ViT-B/14).

Gaussian processes (GPs) provide a principled Bayesian framework for uncertainty estimation, but their computational complexity severely limits scalability to large datasets. We propose SIKA-GP, which accelerates GP inference using sparse inducing kernel approximations based on a dyadic ordered template basis, incurring only ${O}(\log M)$ complexity dependence on the number of inducing points. Our approach constructs compact and expressive kernel representations from sparsely activated bases, enabling efficient tensorized GPU computation and seamless integration with modern large-scale models. SIKA-GP can be naturally embedded into Bayesian neural networks (BNNs) with sparse activations, yielding significant speedups in both training and inference without sacrificing predictive performance. The method naturally extends to deep feature learning, addressing the scalability challenges introduced by deep architectures and high-dimensional feature representations. Empirical results on vision and transformer-based language benchmarks demonstrate that our approach consistently delivers fast and accurate GP models, providing a principled path toward scalable kernel learning.

Conditionally positive definite (CPD) kernels are defined with respect to a function class $\mathcal{F}$. It is well known that such a kernel $K$ is associated with its native space (defined analogously to an RKHS), which in turn gives rise to a learning method -- called conditional kernel ridge regression (conditional KRR) due to its analogy with KRR -- where the estimated regression function is penalized by the square of its native space norm. This method is of interest because it can be viewed as classical linear regression, with features specified by $\mathcal{F}$, followed by the application of standard KRR to the residual (unexplained) component of the target variable. Methods of this type have recently attracted increasing attention. We study the statistical properties of this method by reducing its behavior to that of KRR with another fixed kernel, called the residual kernel. Our main theoretical result shows that such a reduction is indeed possible, at the cost of an additional term in the expected test risk, bounded by $\mathcal{O}(1/\sqrt{N})$, where $N$ is the sample size and the hidden constant depends on the class $\mathcal{F}$ and the input distribution. This reduction enables us to analyze conditional KRR in the case where $K$ is positive definite and $\mathcal{F}$ is given by the first $k$ principal eigenfunctions in the Mercer decomposition of $K$. We also consider the setting where $\mathcal{F}$ consists of $k$ random features from a random feature representation of $K$. It turns out that these two settings are closely related. Both our theoretical analysis and experiments confirm that conditional KRR outperforms standard KRR in these cases whenever the $\mathcal{F}$-component of the regression function is more pronounced than the residual part.

We develop a branched signature kernel solver for linear and nonlinear ordinary differential equations driven by a \emph{single observed trajectory} of a possibly rough forcing signal--a setting common within earthquake engineering, finance, biology, and structural health monitoring, where only one forcing realization is available, and the solver must respect the underlying physical law without an ensemble of realizations. We first introduce a count-sampling construction method to turn the single observation into a hierarchical family of $N+1$ nested training paths on which the branched signature kernel can be evaluated; this allows the signature kernel machinery, originally designed for multi-realization regression problems, to operate on a single-trajectory observation. Then we build a kernel-collocation framework, which places the ansatz either on the highest-order derivative of the solution or on the solution itself. We prove a universal approximation theorem for the branched signature kernel, leveraging the Hairer--Kelly morphism to express branched signature evaluations through geometric signatures of time-extended paths. The offline solver is extended to a streaming Test/Train/Retrain protocol with optional closed-form online updates in both linear and nonlinear cases. Numerical experiments on six benchmarks show accurate, stable predictions across all regimes.

We present a differentiable framework to automatically learn view-dependent 2D kernels in a splatting-based pipeline to improve reconstruction quality and representation efficiency for novel 3D view synthesis. Our volumetric primitive is defined as a bounding ellipsoid and a 3D-kernel latent vector. We first learn a projection network to output a 2D-kernel latent, taking the attributes of the ellipsoid and the 3D-kernel latent as input. Next, the result is sent to a decoder to produce a radially symmetric 2D kernel in terms of Mahalanobis distance, bounded by the projected ellipsoid. The neural networks along with per-primitive attributes are jointly optimized. The effectiveness of our approach is demonstrated on standard benchmarks, comparing favorably against state-of-the-art techniques on both analytical and learned kernels. Finally, we extend the idea to learn general 2D kernels for 2D splatting as well as image representation.

Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.

Fast linear algebra in deep learning usually comes with a choice: fixed geometry and exact computation, as in the Fourier transform, or adaptive geometry paid for by dense parameters, random features, or low-rank surrogates. To move beyond this trade-off, we introduce LAPLEX, a class of exact, trainable (phased) Laplace-kernel operators. A LAPLEX layer is a typically full-rank dense matrix, implicitly defined by learnable coordinate anchors, with FFT-like scaling. Consequently, it supports trainable matrix--vector operations at vector dimensions up to $10^9$ on modern GPUs. As a neural layer, it yields compact projections and classification heads interpretable as soft, trainable routing models. The same primitive also serves as an efficient Gram operator, enabling high-dimensional covariance models on flattened images of dimension $3 \cdot 10^6$ that preserve visible spatial structure without imposing convolutional bias. These applications reflect a single principle: dense geometry can be learned without storing a dense matrix, which enables data-adaptive global interactions in regimes where ordinary dense layers are out of reach. In this sense, LAPLEX separates expressivity from storage cost: it behaves like a dense trainable matrix, but is represented and applied through a small structured set of parameters.

Major depressive disorder (MDD) is a common neuropsychiatric condition whose accurate diagnosis from resting-state functional magnetic resonance imaging (rs-fMRI) remains difficult. Dynamic functional connectivity (DFC) captures time-varying interactions among brain regions and provides rich spatio-temporal information, yet current DFC-based methods face three limitations: sliding-window Pearson correlation yields noisy estimates sensitive to window length and motion artifacts; correlation-derived node features do not fully exploit frequency-domain properties of blood-oxygen-level-dependent (BOLD) signals; and most spatio-temporal graph models handle spatial structure and temporal dynamics in separate stages, restricting their ability to represent coupled brain network evolution. To overcome these issues, we reformulate DFC learning as joint spatio-temporal graph representation learning under a Hawkes-process-inspired temporal dependency prior and propose HWSTCL, a two-stage framework built on a reliability-refined joint spatio-temporal graph with a kernel-weighted pretraining objective. Within each temporal window, BOLD signals are encoded as spectral node descriptors and functional edges are refined by an exponential distance-decay prior that down-weights less reliable long-range connections. The joint graph is then formed by linking each region to itself across future windows through a Hawkes-inspired exponential kernel, allowing spatial and temporal information to be propagated together during message passing. A kernel-weighted contrastive objective further promotes temporal consistency for each region across windows while reducing redundant similarity between different regions. Experiments on a benchmark rs-fMRI dataset show that HWSTCL outperforms recent baselines and yields coherent spatio-temporal representations for MDD diagnosis.

Kernel methods are typically formulated under the assumption of exact, noise-free access to the Gram matrix. However, in emerging settings such as quantum machine learning, each kernel entry must be inferred from noisy observations, and its accuracy depends on how a limited measurement budget is allocated. Despite this, existing approaches overwhelmingly rely on uniform allocation, which equalizes estimator variance but ignores the highly non-uniform dependence of kernelized classifiers on the Gram matrix. In this work, we introduce an adaptive measurement-allocation strategy for learning kernelized Support Vector Machines (SVMs) from noisy Bernoulli observations. Our approach combines two complementary principles: (i) geometric sensitivity, capturing how perturbations of individual kernel entries affect the classifier margin, and (ii) active-set instability, quantifying the probability of discrete changes in support-vector membership induced by measurement noise. These signals define a task-aware allocation scheme that concentrates measurements on the most decision-critical regions of the kernel matrix. We provide a theoretical analysis showing that the benefit of adaptive allocation is governed by the heterogeneity of the induced kernel importance structure, leading to distinct regimes in which adaptive or uniform strategies are preferable. Empirical evaluations on synthetic datasets demonstrate that adaptive allocation significantly improves support-vector recovery, margin estimation, and decision-function accuracy under fixed measurement budgets. A dual-coefficient stability criterion further enables early stopping, achieving near-optimal performance while using only a fraction of the measurement cost. Additional experiments on quantum kernels derived from real-world data reveal a regime-dependent behavior aligned with known phenomena such as kernel concentration. Together...

Neural operators have achieved significant success in modern scientific computing due to their flexibility and strong generalization capabilities. Existing models, however, primarily rely on first-order kernel integral approximations, which severely limit their expressivity. To address this, we propose the Infinite-order Kernel Neural Operator (IKNO), which constructs neural operators via infinite-order kernel integrals and admits an elegant closed-form finite approximation. We develop two complementary infinite-order neural operator constructions: IKNO-Vanilla, which applies the full-kernel resolvent on the product grid via Kronecker eigendecomposition, and IKNO-TP, an alternative tensor-product operator that composes per-axis resolvents. Furthermore, we develop fast computation schemes for both variants of IKNO, which achieve outstanding global information aggregation while maintaining high computational efficiency. Empirically, we evaluate our IKNO on both time-dependent and time-independent benchmarks with arbitrary input shapes, including large-scale industrial datasets. Extensive experiments demonstrate that the IKNO method consistently achieves the SOTA accuracy with significant improvements on nearly all benchmark datasets while maintaining scalability to very large point clouds.

We study fixed-cardinality maximization of the inverse-matrix Solow--Polasky diversity, equivalently finite metric magnitude for the exponential kernel, on one-dimensional and ordered metric sets. The analysis starts from the known finite-line gap formula for the exponential kernel, which writes the excess inverse-matrix diversity as a sum of functions of consecutive gaps. Building on this formula, the main interval theorem proves that, for every $k\geq 2$, the unique maximizing $k$-point subset of $[0,1]$ is the equally spaced set. Thus the objective selects a uniform gap representation on the real line. A converse kernel proposition shows that, among normalized non-increasing distance kernels, requiring the corresponding adjacent-gap additive structure forces the exponential family. Further results transfer the interval theorem to ordered $\ell_1$ (L1, or Manhattan) curves by isometry: the maximizing sets are uniform in accumulated $\ell_1$ length. As a consequence, monotone biobjective Pareto fronts admit Solow--Polasky optimal finite approximations that are uniformly spaced in accumulated objective-space change, a natural representation when all parts of a continuous front should be covered. Examples, including a dense connected front and a finite disconnected ZDT3 front, illustrate how the continuous uniform-gap result appears on discrete candidate sets. Solow-Polasky diversity; diversity measures; finite metric magnitude; L1 distance; uniform spacing; Pareto-front approximation; multiobjective optimization; fixed-cardinality subset selection

Kolmogorov-Arnold Networks (KANs) approximate multivariate functions using learnable univariate edge functions, typically parameterized by B-spline bases. Although effective, spline-based implementations can be computationally expensive. A modified version of KANs, called FastKAN, improves efficiency by replacing splines with Gaussian radial basis functions (RBFs), but it relies on a fixed kernel and shape parameter. In this work, we extend the RBF-based KAN framework by introducing a broader family of radial basis kernels and by initializing the kernel shape parameter using leave-one-out cross-validation (LOOCV). To the best of our knowledge, this is the first study that integrates LOOCV-based kernel scale estimation with deep KAN training. We also introduce Matérn and Wendland kernels into the KAN framework for the first time, enabling more flexible basis representations beyond the Gaussian kernel used in FastKAN. The LOOCV estimate provides a data-driven initialization of the kernel scale, which is subsequently refined during network training. The proposed adaptive RBF-KAN is evaluated on several two-dimensional benchmark functions. The results highlight the importance of kernel selection and adaptive shape parameters, with different kernels showing advantages for smooth functions, discontinuities, and oscillatory patterns. Overall, combining LOOCV-based initialization with adaptive kernel learning provides a practical strategy for improving RBF-based KAN models.

While 3D Gaussian Splatting (3DGS) has revolutionized real-time photorealistic view synthesis, its fundamental reliance on symmetric Gaussian distributions introduces visual artifacts that hinder accurate spatial data exploration. Specifically, symmetric kernels struggle to capture shape and color discontinuities , which cause blurriness and primitive redundancy that mislead human perception during visual analysis. To address these visualization barriers, we introduce 3D Skew Gaussian Splatting (3DSGS), a novel framework that significantly enhances the structural fidelity and compactness of explicit scene representations. Our key insight lies in extending the standard primitive to a general Skew Gaussian counterpart. This generalized primitive inherits the highly efficient rasterization properties of standard Gaussians while gaining intrinsic asymmetric modeling capabilities. We couple this with an enhanced opacity representation to better handle complex transparency, alongside a depth-aware densification strategy that intelligently manages primitive allocation. Furthermore, to make these advancements actionable for real-world visual analytics, we re-derive the CUDA rasterization pipeline to universally support both symmetric and skew Gaussians, integrating it into a decoupled, free-camera interactive visualization engine. Extensive experiments demonstrate that 3DSGS achieves superior rendering quality and structural compactness, particularly in regions with intricate details, while maintaining the real-time frame rates necessary for fluid interactive exploration. Supplementary derivations and visual results are available at \textbf{\textit{https://3d-skew-gs.github.io/}}.

Gaussian Process (GP) kernels are central to Bayesian optimization (BO), yet designing effective kernels for high-dimensional problems still relies on extensive manual engineering. Existing automated approaches struggle in high dimensions for two bottlenecks: their kernel search space is limited to additions and multiplications of base kernels, and LLM-based approaches require conditioning on raw observations, which becomes infeasible due to context-length limits and the difficulty of extracting meaningful patterns. We introduce \textbf{Kernel Discovery}, a LLM-driven evolutionary framework for high-dimensional BO that searches a broader kernel space beyond predefined composition rules and does not require conditioning on observations. Motivated by the observation that directly prompting an LLM to generate kernel code yields syntactically varied but functionally identical kernels, we adopt a two-stage approach: an LLM first proposes novel mathematical forms, then a second LLM call converts each form into validated, executable code. We also propose a leave-one-out continuous ranked probability score (LOO-CRPS) as a selection criterion that penalizes overfitted kernels. On five high-dimensional BO benchmarks, our method achieves an average rank of \textbf{1.2 out of 17}, outperforming competitive baselines. We further analyze the discovered kernels to identify which kernels lead to improvements in high-dimensional BO.

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.

Tabular data generation considers a large table with multiple columns -- each column comprised of numerical, categorical, or sometimes ordinal values. The goal is to produce new rows for the table that replicate the distribution of rows from the original data -- without just copying those initial rows. The last 4 years have seen enormous progress on this problem, mostly using computational expensive methods that employ one-hot encoding, VAEs, and diffusion. This paper describes a new approach to the problem of tabular data generation. By employing copula transformations and modeling the distribution as a kernel density estimate we can nearly match the accuracy and leakage-avoidance achievements of the previous methods, but with almost no training time. Our method is very scalable, and can be run on data sets orders of magnitude larger than prior state-of-the-art on a simple laptop. Moreover, because we employ kernel density estimates, we can store the model as a coreset of the original data -- we believe the first for generative modeling -- and as a result, require significantly less space as well. Our code is available here: \url{https://github.com/tabkde/tabkde-main}

Chunked prefill has become a widely adopted serving strategy for long-context large language models, but efficient attention computation in this regime remains challenging. Existing sparse attention methods are primarily designed for one-shot prefill and do not translate efficiently to chunked prefill: block-sparse kernels lose efficiency when the query length is limited by the chunk size, while fine-grained pattern search becomes costly when repeated over the accumulated KV cache at every chunk. QUOKA, a recent method that directly targets chunked prefill, avoids sparse-kernel overhead but relies on query-subsampled, token-level KV selection, which can miss query-specific KV entries and introduce explicit KV-copy overhead. To address these limitations, we propose CompactAttention, a chunked-prefill attention mechanism based on Block-Union KV Selection. CompactAttention treats 2D block-sparse masks as KV-selection signals rather than direct sparse-kernel execution plans, and converts them into GQA-aware per-group KV block tables through Q-block union and intra-group union. This construction produces the minimal block tables that preserve all KV blocks selected by the input masks under paged execution constraints, enabling selected KV blocks to be accessed in place without explicit KV compaction. On LLaMA-3.1-8B-Instruct, CompactAttention maintains accuracy close to dense attention on the RULER benchmark while delivering up to 2.72$\times$ attention speedup at 128K context length under chunked prefill.

We study a prototypical situation when a learned predictor can discover useful low-dimensional structure in data, while using fewer samples than are needed for accurate prediction. Specifically, we consider the problem of recovering a multi-index polynomial $f^*(x)=h(Ux)$, with $U\in\mathbb{R}^{r\times d}$ and $r\ll d$, from finitely many data/label pairs. Importantly, the target function depends on input $x$ only through the projection onto an unknown $r$-dimensional central subspace. The algorithm we analyze is appealingly simple: fit kernel ridge regression (KRR) to the data and compute the Average Gradient Outer Product (AGOP) from the fitted predictor. Our main results show that under reasonable assumptions the top $r$-dimensional eigenspace of AGOP provably recovers the central subspace, even in regimes when the prediction error remains large. Specifically, if the target function $f^*$ has degree $p^*$, it is known that $n\asymp d^{p^*}$ samples are necessary for KRR to achieve accurate prediction. In contrast, we show that if a low degree $p$ component of $f^*$ already carries all relevant directions for prediction, subspace recovery occurs in the much lower sample regime $n\asymp d^{p+δ}$ for any $δ\in(0,1)$. Our results thus demonstrate a separation between prediction and representation, and provide an explanation for why iterative kernel methods such as Recursive Feature Machines (RFM) can be sample-efficient in practice.

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 $n$.

We present a physics-informed framework for system identification based on randomized stable atomic features. Impulse responses are represented as random superpositions of stable atoms, namely damped complex exponentials associated with poles sampled inside a prescribed disk. Identification is then cast as a convex regularized least-squares problem with optional linear, second-order-cone, and KYP constraints. The approach generalizes random Fourier and random Laplace features to the damped, nonstationary regime relevant to engineering systems while retaining modal interpretability and scalable finite-dimensional computation. The main analytic point is an operator-theoretic Disk-Bochner viewpoint: positive measures over stable poles generate positive-definite kernels with a radius-dependent shift defect, while a converse scalar disk moment representation for an arbitrary kernel is characterized by subnormality of the canonical shift. We prove this statement, establish an RKHS-to-l1 embedding, show that sampled poles induce a valid finite atomic gauge, discuss random-feature convergence, and state sparse-recovery guarantees conditionally on the restricted-eigenvalue properties of the realized disk-Vandermonde or input-output design matrix. We also connect the normalized transfer function problem to Nevanlinna-Pick interpolation and LFT set-membership. The framework directly encodes stability margins, modal localization, DC-gain bounds, monotonicity, passivity, relative degree, settling-time targets, and time/frequency-domain error bounds. Numerical comparisons illustrate how physically meaningful priors can compensate for poor excitation and improve constrained impulse-response recovery in an under-informative data setting.

Generative models are often conditioned on a small set of examples via cross-attention. Under the Gaussian optimal-transport path, we show that the exact velocity field induced by a finite support set is a Nadaraya--Watson kernel smoother whose bandwidth decreases with flow time, from broad averaging at early steps to nearest-neighbor at late steps. A single Gaussian-kernel attention head exactly computes this field, connecting cross-attention conditioning to classical kernel theory. The theory predicts three failure regimes: nearest-neighbor collapse of the kernel at high dimension, mismatch between the isotropic kernel and the data geometry, and insufficient support for nonparametric estimation. Experiments on Gaussian mixtures, spherical shells, and DINOv2 ImageNet features confirm that learned conditioning improves in precisely these regimes, and that IP-Adapter's cross-attention implements approximate NW smoothing in practice.

Modern SMILES-based chemical language models obtain strong MoleculeNet performance by treating SMILES as generic text and compensating with multi-million-molecule self-supervised pretraining. We ask: when a domain carries structural priors as rich as chemistry's, does it warrant a domain-native transformer rather than a generic one rescued by scale? We answer affirmatively with \textbf{GM-Net} (Geometric Measure Network), a transformer family in which every module is replaced by a sphere-native counterpart, and instantiate it as \textbf{Chem-GMNet}. Three blocks follow: SH-Embedding (tokens as learnable directions on $S^{k-1}$ lifted through a Gegenbauer feature map); DualSKA (a per-head fusion of a linear-time gated Sphere-Flow recurrence whose persistent state we prove is the truncated multipole expansion of the input distribution, and a softmax Sphere-Kernel branch over the same Schoenberg-valid kernel); and SH-FFN (sphere projection $\to$ Gegenbauer lift $\to$ moment readout). On canonical DeepChem scaffold splits, against same-shape ChemBERTa-2 baselines under the chemberta3-faithful protocol: (i) random-initialised, Chem-GMNet wins on 7 of 10 MoleculeNet endpoints at $\sim\!35\%$ fewer parameters; (ii) pretrained on the same 10M-SMILES ZINC corpus as ChemBERTa-2 MLM-10M, it matches or beats the public release on 6 of 8 shared endpoints (5/7 excluding a known ClinTox release anomaly). A $(k,L)$ ablation shows that increasing the sphere dimension from $k\!=\!8$ to $k\!=\!10$ at fixed $L\!=\!3$ lowers ESOL RMSE to $0.938$ at scratch, beating pretrained ChemBERTa-2 MLM-10M on this endpoint without any pretraining at all.

Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

We propose Sobolev-regularized Maximum Mean Discrepancy (SrMMD) gradient flow, a regularized variant of maximum mean discrepancy (MMD) gradient flow based on a gradient penalty on the witness function. The proposed regularization mitigates the non-convexity of the MMD objective and yields provable \emph{global} convergence guarantees in MMD in both continuous and discrete time. A more surprising appeal is that our convergence analysis does not rely on isoperimetric assumptions on the target distribution. Instead, it is based on a regularity condition on the difference between kernel mean embeddings. A key highlight of the proposed flow is that it is applicable in both sampling (from an unnormalized target distribution) -- using Stein kernels -- and generative modeling settings, unlike previous works, where a gradient flow is suitable for only generative modeling or sampling but not both. The effectiveness of the proposed flow is empirically verified on a broad range of tasks in both generative modelling and sampling.

Data-free continual learning (DFCIL) relies on model inversion to synthesize pseudo-samples and mitigate catastrophic forgetting. However, existing inversion methods are fundamentally limited by a simplifying assumption: they model feature distributions using diagonal covariance, effectively ignoring correlations that define the geometry of learned representations. As a result, synthesized samples often lack fidelity, limiting knowledge retention. In this work, we show that modeling feature dependencies is a key ingredient for effective DFCIL. We introduce REMIX, a structured covariance modeling framework that enables scalable full-covariance modeling without the prohibitive cost of dense matrix inversion and log-determinant computation. By leveraging a Laplace kernel parameterization, REMIX captures structured feature dependencies using memory that scales linearly with the feature dimensionality, while requiring only an additional logarithmic factor in computation. Modeling these correlations produces more coherent synthetic samples and consistently improves performance across standard DFCIL benchmarks. Our results demonstrate that moving beyond diagonal assumptions is essential for effective and scalable data-free continual learning. Our code is available at https://github. com/pkrukowski1/REMIX-Model-Inversion-via-Laplace-Kernel.

Many three-dimensional spatial fields are anisotropic, with directions of rapid and slow variation that need not align with the coordinate axes. Standard Gaussian process kernels with Automatic Relevance Determination (ARD) capture only axis-aligned anisotropy, while generic full symmetric positive definite (SPD) metrics can represent rotated anisotropy but do not parameterise principal length-scales and directions directly. We introduce an interpretable rotationally anisotropic GP kernel that parameterises a three-dimensional SPD covariance metric using three principal length-scales and an explicit SO(3) rotation. The rotation is represented by an axis-angle vector and mapped to SO(3) via the Lie-algebra exponential map, giving unconstrained Euclidean coordinates for inference while always inducing a valid SPD metric. The construction spans the same family of three-dimensional SPD covariance metrics as a generic full-SPD parameterisation, but exposes the geometry differently: length-scales and orientation are explicit, interpretable, and directly available for prior specification and posterior summaries. We perform Bayesian inference on these quantities using Markov Chain Monte Carlo (MCMC), and characterise the resulting symmetries and weakly identified regimes. On synthetic data with rotated anisotropy, the posterior recovers the generating metric and improves prediction relative to an axis-aligned ARD baseline, while matching the predictive performance of a generic full SPD baseline. When the ground truth is axis-aligned, posterior mass concentrates near the identity rotation and predictive performance matches ARD. On a material-density dataset from a laboratory-fabricated nano-brick, the inferred metric reveals rotated anisotropy that is not captured by axis-aligned kernels.

We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because it enables fast, high-quality generation without distilling a large pretrained diffusion model, but its theory is currently understood mainly for Gaussian kernels, where the drift coincides with smoothed score matching and is identifiable. Our gradient-based reformulation exposes this score-based structure for general kernels: the resulting drift is the score difference between kernel-smoothed data and model distributions, yielding identifiability for characteristic kernels and a smoothed-KL descent interpretation of the drifting dynamics. Since kernel gradients are intrinsic tangent vectors, the same construction extends naturally to Riemannian manifolds and to discrete data via the Fisher-Rao geometry of the probability simplex. Across spherical geospatial data, promoter DNA and molecule generation, kernel-gradient drifting enables state-of-the-art one-step generation beyond the Euclidean setting without distillation.

Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.

We present a theoretically grounded Gaussian process framework that leverages neural feature maps to construct expressive kernels. We show that the learned feature map can be interpreted as an optimal low-rank approximation to a Gram matrix derived from an implied RKHS, from which we establish consistency of the GP posterior. We further analyse the spectral properties of the induced kernels and introduce product feature-map kernels to address oversmoothing. This simple yet powerful approach enables fast, scalable, and accurate exact GP inference with minimal upfront work. The flexibility of kernel design supports seamless application to both regression and classification tasks across diverse data modalities, including tabular inputs and structured domains such as images. On benchmark datasets, this approach surpasses pre-existing methods in terms of accuracy and training and prediction efficiency.

Reliable short horizon forecasting of solar and wind generation is a structural prerequisite of any modern power system yet most published forecasters are tuned and evaluated on a single climatic regime and most algorithmic novelty has been concentrated either on classical recurrent networks or on monolithic foundation models that combine forecasting and explanation We develop a four stage hybrid framework that separates these concerns The first stage acquires hourly generation irradiance and surface weather records through public application programming interfaces The second stage trains three classical baselines autoregressive integrated moving average gradient boosted regression trees and a two layer long short term memory network and produces a strong point forecast together with a residual error series The third stage corrects the residual through a quantum inspired variational kernel built on a six qubit hardware efficient ansatz with three repeated entangling layers The fourth stage uses generative artificial intelligence strictly as an explainability layer that reads the measured benchmark numbers and produces a structured natural language interpretation Across three regions drawn from open public archives Iberian solar North Sea wind and a mixed Texas trace the proposed configuration stays within one percentage point of the strongest classical baseline on the in domain forecasting task and the quantum inspired kernel separates calm and stormy weather regimes with a Fisher discriminant ratio approximately fifteen fold higher than a tuned radial basis kernel

We introduce CUTS-GPR, a new method for performing numerically exact Gaussian process regression (GPR) in high-dimensional settings. The key component of CUTS-GPR is an extremely fast kernel matrix-vector product, which exhibits near-linear or even linear scaling with the amount of training data, $N$, and low-order polynomial scaling with dimensionality, $D$. This is obtained by combining an additive kernel with an incomplete grid and exploiting the resulting structure of the kernel matrix. We demonstrate the scalability of the matrix-vector product by running benchmarks with billions of data points and thousands of dimensions. Full GPR calculations, including hyperparameter optimization, are completed in a matter of hours for $N = 447 265$ and $D = 24$. We demonstrate that our CUTS-GPR enables Bayesian modeling of high-dimensional potential energy surfaces - a longstanding challenge in computational chemistry.

Distributional treatment effects can be invisible to means: a treatment may preserve average outcomes while changing tails, modes, dispersion, or rare-event probabilities. Kernel tests can detect discrepancies between interventional outcome laws, but global tests do not reveal where the laws differ. We propose DR-ME, to our knowledge the first semiparametrically efficient finite-location test for interpretable distributional treatment effects. DR-ME evaluates an interventional kernel witness at learned outcome locations, returning causal-discrepancy coordinates rather than only a global rejection. From observational data, we derive orthogonal doubly robust kernel features whose centered oracle form is the canonical gradient of this finite witness. For fixed locations, we characterize the local testing limit: DR-ME is chi-square calibrated under the null, has noncentral chi-square local power, and uses the covariance whitening that optimizes local signal-to-noise for discrepancies visible through the selected coordinates. This efficient local-power geometry yields a principled location-learning criterion, with sample splitting preserving post-selection validity. Experiments show near-nominal type-I error, competitive power against global doubly robust kernel tests, and interpretable learned locations that localize distributional effects in a semi-synthetic medical-imaging study.

The Maximum Mean Discrepancy (MMD) is a cornerstone statistic for nonparametric two-sample testing, but its test power is dictated entirely by the chosen kernel. Because any fixed kernel inherently fails to distinguish certain distributions, the kernel must be dynamically optimized. However, data-driven optimization violates the foundational i.i.d. assumption, forcing a strict trade-off in existing frameworks. Ratio criteria ignore this dependence, inducing overfitting and variance collapse on rich kernel classes. Conversely, aggregation methods bypass the dependence using finite grids, but this strategy cannot scale to continuous search spaces like deep kernels. To break this dichotomy, we establish data-driven kernel selection as a model selection problem. We propose Complexity-Penalized MMD (CP-MMD), a criterion derived by applying the two-sample uniform concentration inequality of preceding works to the post-optimization MMD problem. The resulting penalty bounds the empirical MMD by the complexity of the kernel search space, mathematically absorbing the cost of optimization, so that CP-MMD enables direct, grid-free maximization over continuous parametric classes, including scalar bandwidths, polynomial feature bandwidths, and deep network parameters. By formally accounting for optimization complexity, we prove that CP-MMD maximizes true test power while ensuring unconditional Type-I validity. Consequently, CP-MMD enables grid-free kernel selection across linear, polynomial-feature, and deep regimes, matching or exceeding state-of-the-art test power.

Modern datasets across many disciplines increasingly consist of time-evolving, potentially infinite-dimensional random objects, such as dynamic functional data, which are naturally modeled in Hilbert spaces. In these settings, characterizing probability measures, for example, through densities, can be ill-defined or technically challenging. Motivated by clustering applications, we propose a Gaussian mixture framework for Hilbert-space-valued data based on kernel mean embeddings and develop efficient optimization algorithms for estimation. We establish theoretical guarantees showing that the proposed algorithm is well defined and that the model yields a dense class of approximations in infinite-dimensional spaces. We evaluate the framework through extensive experiments on diverse structures and data geometries, including $L^2$-functional data and random graphs in Laplacian spaces arising in modern medical applications.

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.

We study the efficient computation of Shapley values for \emph{product games} -- cooperative games in which the coalition value factorizes as a product of per-player terms. Such games arise in machine learning explainability whenever the value function inherits a multiplicative structure from the underlying model, as in kernel methods with product kernels and tree-based models. Our key result is that the Shapley value of each player in a product game admits an exact one-dimensional integral representation: the weighted sum over exponentially many feature coalitions collapses to the integral of a degree-$(d-1)$ polynomial over $[0,1]$, where $d$ is the total number of features. This yields a Gauss--Legendre quadrature scheme that is \emph{provably exact} whenever the number of nodes satisfies $m_q \geq \lceil d/2 \rceil$, and otherwise provides a \emph{near-exact} approximation with error provably decaying geometrically in $m_q$. In practice, a few hundred nodes can achieve highly precise estimates even with thousands of features. Building on this formulation, we derive a numerically stable implementation via log-space evaluation, together with an efficient parallel implementation based on associative scan primitives that achieves $O(d\,m_q)$ total work and $O(\log d)$ parallel time. Experiments show that \textsc{QuadraSHAP} is the fastest numerically stable method across all tested configurations.

We introduce PALACE (Persistence Adaptive-Landmark Analytic Classification Engine), the data-adaptive companion to PLACE, paying a small cross-validation tier on three knobs (budget, radii, bandwidth; $\leq 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yields four closed-form guarantees. (i) A structural lower distortion bound $λ(τ;ν)$ on $\mathcal{D}_n$ under cross-diagram non-interference, with a $(D/L)^2$ budget reduction over the uniform grid when diagrams concentrate. (ii) Equal weights $w_k = K^{-1/2}$ maximizing $λ$, and farthest-point-sampling positions $2$-approximating the optimal $k$-center covering radius; both derived from training labels alone, no gradient training. (iii) A kernel-RKHS classification rate $O((k-1)\sqrt{K}/(γ\sqrt{m_{\min}}))$ with binary necessity threshold $m = Ω(\sqrt K/γ)$ from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin $\hatρ_{\mathrm{Mah}}$ is the strongest closed-form ranker across the chemical-graph pool (mean Spearman $ρ\approx +0.60$); the isotropic surrogate $\hatγ/\sqrt{K}$ admits a selection-consistency rate, and $\widehatλ$ from (i) provides an independent data-level signal (positive on COX2 and PTC). (iv) A per-prediction certificate, in non-asymptotic Pinelis and asymptotic Gaussian forms, with no calibration split. Empirically, PALACE is the strongest closed-form diagram-based method on Orbit5k ($91.3 \pm 1.0\%$, matching Persformer), leads every diagram-based competitor on COX2 and MUTAG, and is competitive on DHFR (within 1 pp of ECP). At $8\times$ domain inflation, adaptive placement maintains $94\%$ while the uniform grid collapses to chance ($25\%$ on 4-class data).

Deep neural networks dominate modern machine learning, while alternative function approximators remain comparatively underexplored at scale. In this work, we revisit kernel methods as drop-in components for standard deep learning pipelines. We introduce \emph{Sparse Kernels} (SKs), a differentiable, localized, and lazy variant of kernel ridge regression (KRR) that defers training to inference time and reduces to the solution of small local systems. We integrate SKs into PyTorch as modular layers that preserve end-to-end trainability, and we show that they expose three distinct sets of parameters -- feature representations, target values, and evaluation points -- each of which can be fixed or learned. This decomposition broadens the design space available to practitioners, enabling, in particular, training-free transfer, nonlinear probing, and hybrid kernel-neural models. Across convolutional networks, vision transformers, and reinforcement learning, SK-based modules serve two complementary roles: in some settings, they match the performance of trained neural readouts with substantially less training; in others, they augment existing models and improve their performance when used as additional components. Our results suggest that kernel methods, once made scalable and differentiable, can be readily integrated with deep learning rather than treated as a separate paradigm.

In this paper, we propose Complex Diffusion Maps (CDM), a novel diffusion mapping framework that aims to reveal the dominant complex harmonics of high-dimensional data. Inspired by the local Gaussian kernel relevant to the heat equation and the nonlocal Schrödinger kernel relevant to the Schrödinger equation, we propose a unified family of $ω$-parameterized complex-valued kernels for the trade-off between local and nonlocal connections. We establish the theoretical foundation based on the operator spectrum theory, where the corresponding diffusion operator, diffusion distance, and complex harmonic maps are well-defined. An optimization-based interpretation of the maps is also developed, aiming to preserve angular structure in the complex diffusion space rather than relying solely on real-valued magnitude. We extensively evaluate CDM on both synthetic and real-world datasets. The complex-valued kernel amplifies differences among easily confusable samples, improving discriminative power over both linear and nonlinear methods based on real-valued kernels. CDM remains robust in high-noise settings, yielding a clearer eigengap that enhances spectral separation. For resting-state fMRI data, CDM captures more strongly correlated and nonlocal spatiotemporal dynamics. Without task-specific tuning, CDM achieves competitive performance on a public EEG sleep dataset, while maintaining high computational efficiency compared with both traditional machine learning and deep neural network approaches, highlighting its generality and practical value.

We study query time bounds for the fundamental problem of estimating the kernel mean $\frac1{|X|}\sum_{x\in X}\mathbf{k}(x,y)$ of a query $y$ in a finite dataset $X\subset\mathbb{R}^d$ up to a prescribed additive error $\varepsilon$. The best known bounds for the Gaussian kernel are $O(d/\varepsilon^2)$, $\widetilde O(d+1/\varepsilon^4)$, and $\widetilde O(d+Δ^2/\varepsilon^2)$, where $Δ$ is the diameter of a region containing the points. We prove the new bound $\tilde O(d+\varepsilonΔ^2+1/\varepsilon^3)$, which improves over the previous ones in regimes with small error $\varepsilon$ and intermediate diameter $Δ$. At the center of our proof is a new fast spherical embedding theorem in the sense introduced by Bartal, Recht and Schulman (2011), which limits the embedded data diameter while preserving local Euclidean distances and avoiding ``distance collapse'' at larger scales. This fast embedding theorem may be of independent interest.

Double-machine-learning pipelines for the Average Dose-Response Function rely on kernel-weighted local-linear smoothers, which inherit unbounded functional influence: a single outlier within a kernel window biases the curve across the entire window. We introduce SHIFT (Self-calibrated Heavy-tail Inlier-Fit with Tempering), a robust DML estimator combining cross-fit nuisance orthogonalization with a kernel-local Welsch-loss second stage optimized by Graduated Non-Convexity, and -- the principal design choice -- a defensive OLS refit whose inlier cutoff is scaled by post-GNC residual MAD rather than the raw-outcome MAD. On a localized-contamination stress test at $p=0.25$ this design choice drops level-RMSE from 1.03 to 0.33 while leaving clean and uniformly-contaminated runs unchanged. Across 1,400 main-sweep fits, SHIFT has competitive worst-case shape recovery (RMSE $0.325$ at $p=0.25$, second to Huber-DML's $0.276$); among the three methods with worst-case RMSE below $0.35$, only SHIFT emits a non-uniform per-sample weight vector, recovering the ground-truth outlier mask at mean $F_1 \approx 0.96$ (range $0.945$--$0.968$) on Gaussian-jump DGPs. We pair the estimator with a six-technique Extreme Value Theory diagnostic suite (Hill, GPD-MLE/PWM, GEV, Mean Excess, parameter stability, causal tail coefficient) that lets a practitioner distinguish Frechet from Weibull regimes and choose between SHIFT and L1 alternatives on empirical grounds. Extensions to binary-treatment CATE (Huber pseudo-outcome X-Learner) and time-series ADRF (block-CV + rolling MAD) are included. A counter-intuitive ablation: linear nuisance models (Ridge, Lasso) outperform gradient-boosted nuisances for robust DML under uniform contamination, inverting the usual more-flexible-is-better heuristic.

Heterogeneous graphs with heterophily have emerged as a powerful abstraction for modeling complex real-world systems, where nodes of different types and labels interact in diverse and often non-homophilous ways. Despite recent advances, robust representation learning for such graphs remains largely unexplored, particularly in the presence of noisy or misleading connectivity. In this work, we investigate this problem and identify structural noise as a critical challenge that significantly degrades model performance. To address this issue, we propose a unified framework, Heterogeneous Graph Unified Learning (HGUL), which jointly handles heterophily and noisy graph structures. The framework consists of three complementary modules: a kNN-based graph construction module that recovers reliable local neighborhoods, a graph structure learning module that adaptively refines the adjacency by filtering noisy edges, and a heterogeneous affinity learning module that captures class-level relationships via an extended affinity matrix derived from a polynomial graph kernel. Extensive experiments on multiple datasets demonstrate that HGUL consistently outperforms existing methods on clean graphs and maintains strong robustness under varying levels of structural noise. The results further underscore the importance of jointly modeling heterophily and noise in heterogeneous graph learning.

Uncertainty estimation is essential for robust decision-making in the presence of ambiguous or out-of-distribution inputs. Gaussian Processes (GPs) are classical kernel-based models that offer principled uncertainty quantification and perform well on small- to medium-scale datasets. Alternatively, formulating the weight space learning problem under tensor network assumptions yields scalable tensor network kernel machines. However, these assumptions break Gaussianity, complicating standard probabilistic inference. This raises a fundamental question: how can tensor network kernel machines provide principled uncertainty estimates? We propose a novel Bayesian Tensor Network Kernel Machine (LA-TNKM) that employs a (linearized) Laplace approximation for Bayesian inference. A comprehensive set of numerical experiments shows that the proposed method consistently matches or surpasses Gaussian Processes and Bayesian Neural Networks (BNNs) across diverse UCI regression benchmarks, highlighting both its effectiveness and practical relevance.

Multimodal recommendation improves user modeling by integrating collaborative signals with heterogeneous item content. In real applications, user interests evolve over time and exhibit nonstationary dynamics, where different preference factors change at different rates. This challenge is amplified in multimodal settings because visual and textual cues can dominate decisions under different temporal regimes. Despite strong progress, most multimodal recommenders still rely on static interaction graphs or coarse temporal heuristics, which limits their ability to model continuous preference evolution with fine-grained temporal adaptation. To address these limitations, we propose TimeMM, a time-conditioned spectral filtering framework for dynamic multimodal recommendation. TimeMM instantiates Time-as-Operator by mapping interaction recency to a family of parametric temporal kernels that reweight edges on the user--item graph, producing component-specific representations without explicit eigendecomposition. To capture non-stationary interests, we introduce Adaptive Spectral Filtering that mixes the operator bank according to temporal context, yielding prediction-specific effective spectral responses. To account for modality-specific temporal sensitivity, we further propose Spectral-Aware Modality Routing that calibrates visual and textual contributions conditioned on the same temporal context. Finally, a ranking-space Spectral Diversity Regularization encourages complementary expert behaviors and prevents filter-bank collapse. Extensive experiments on real-world benchmarks demonstrate that TimeMM consistently outperforms state-of-the-art multimodal recommenders while maintaining linear-time scalability.

Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD we show how to perform gradient-based optimization over the learnable kernel parameters, and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky PDE, showcasing the method's different strengths.

We study online regression with the square loss in a reproducing kernel Hilbert space under a dynamic regret criterion. The learner is compared with a time-varying comparator sequence, and the bounds depend on its path length in the RKHS norm. The proposed method transfers the finite-dimensional discounted Vovk--Azoury--Warmuth approach of Jacobsen & Cutkosky (2024) to the RKHS setting by means of finite-dimensional subspace approximations. For a fixed subspace, we run a VAW-based ensemble of discounted VAW forecasters over a geometric grid of discount factors. The additional approximation error is controlled by the uniform projection error of kernel sections. We then introduce a general orthogonal truncation method: starting from a feature expansion of the kernel, we construct the associated RKHS by introducing an inner product that makes the feature functions orthonormal, and then use the spans of the first basis functions as finite-dimensional approximation spaces. The resulting subspace reduction is applied to several approximation schemes. Explicit feature expansions yield fast-regime bounds for Gaussian and analytic dot-product kernels. Mercer truncations provide a spectral approximation method and lead to dynamic regret bounds in fast and slow regimes, depending on the eigenvalue decay. Finally, we study subspaces spanned by kernel sections and apply this construction to Matérn kernels.

This paper studies the identifiability and stability of drifting fields in the framework of Generative Modeling via Drifting. The motivating question is whether a zero-drift equilibrium identifies the target distribution and whether an approximately vanishing drift implies weak distributional convergence. Since the original drifting model employs the Laplace kernel by default, we first analyze why Gaussian score-based arguments fail to apply. This analysis motivates the introduction of companion-elliptic kernel families, which are characterized by a companion potential satisfying an elliptic closure relation. We show that this class naturally contains the Laplace kernel and consists precisely of Gaussian and Matérn kernels with smoothness parameter $ν>0$. Within this class, we establish field identifiability for arbitrary Borel probability measures on $R^d$: if the drifting field between two such measures vanishes identically, then they must coincide. For stability, we demonstrate that convergence of the field alone does not guarantee weak convergence, since mass may escape to infinity while remaining invisible to the field. Although tightness directly removes this obstruction and restores weak stability, we prove that, even without tightness, every $C_0$-vague cluster point lies exactly on the defect ray $\{cp:0\le c\le1\}$. Consequently, a single scalar $C_0$ observable suffices to detect the missing mass and recover weak convergence.

Gaussian basis functions provide an efficient and flexible alternative to spline activations in KANs. In this work, we introduce the partition-of-unity Gaussian KAN (PU-GKAN), a Shepard-type normalized Gaussian KAN in which the Gaussian basis values on each edge are divided by their local sum over fixed centers. This produces a partition-of-unity feature map with trainable coefficients, while preserving the standard edge-based KAN structure. The normalized construction gives exact constant reproduction at the edge level and admits an explicit finite-feature kernel interpretation. We formulate both the standard Gaussian KAN (GKAN) and PU-GKAN from a finite-feature and additive-kernel viewpoint, making the induced layer kernels and empirical feature matrices explicit. Using the first-layer feature matrix as the reference object, we adopt a practical scale-selection interval for $ε$, with the lower endpoint determined by adjacent-center overlap and the upper endpoint determined by a conservative conditioning threshold. Numerical experiments show that PU-GKAN reduces sensitivity to $ε$, improves validation accuracy for most smooth and moderately non-smooth targets, and gives more stable training behavior. The benefit persists across sample-size and center-number sweeps, higher-dimensional architectures, Matérn RBF bases, and physics-informed examples involving Helmholtz and wave equations. These results indicate that Shepard-type partition-of-unity normalization is a simple and effective stabilization mechanism for RBF-based KANs.

An approach to construct explicit integral representations for two-layer ReLU networks is presented, which provides relatively simple representations for any multivariate polynomial. Quantitative bounds are provided for a particular, sharpened ReLU integral representation, which involves a harmonic extension and a projection. The bounds demonstrate that functions can be approximated with $L^{2}(\mathcal{D})$ errors that do not depend explicitly on dimension or degree, but rather the coefficients of their monomial expansions and the distribution $\mathcal{D}$. We also present a connection to the RKHS of the exponential kernel $K(x,y)=\exp\left(\left\langle x,y\right\rangle \right)$, and a very simple integral representation involving additionally multiplication via a fixed function which has better quantitative bounds.