random fourier feature mapping

This work addresses the computational and memory bottlenecks of traditional Grassmannian kernel methods, which require constructing full Gram matrices and thus struggle with high-dimensional subspace data. To overcome these limitations, the authors propose a scalable kernel approximation framework based on random rank-one projections combined with bounded nonlinear transformations—either periodic or binary—that yield compact one-bit subspace feature representations. This approach enables continuous interpolation between the inverse Binet–Cauchy kernel and Gaussian-like kernels while effectively preserving the intrinsic geometry of subspaces. The method substantially reduces computational, memory, and storage costs. Experimental results on synthetic data and the ETH-80 classification benchmark demonstrate that the proposed technique accurately maintains Grassmannian geometric relationships with high fidelity, confirming its efficiency and practical utility.

Conventional Random Fourier Features (RFF) fail for non-separable Laplacian kernels and their Matérn and powered-exponential generalizations, due to their heavy-tailed and non-separable spectral distributions. Method: We propose a novel, implementable random feature mapping: for the first time, we design heavy-tailed random weight matrices with weak coupling structures tailored to these three kernel classes, integrating spectral analysis with structured sampling to achieve efficient, low-bias feature approximation—without explicitly computing high-dimensional kernel matrices. Contribution/Results: Our method preserves theoretical approximation accuracy while substantially reducing computational cost. Empirical evaluation across multiple real-world datasets demonstrates that it matches the predictive performance of exact kernel methods on classification and regression tasks, yet accelerates training by one to two orders of magnitude—effectively overcoming the longstanding modeling bottleneck for non-separable, heavy-tailed kernel functions.

Existing scalable kernel methods struggle to effectively model nonstationary processes—those exhibiting complex patterns whose statistical properties vary with input location. This work proposes the Random Wavelet Features (RWF) framework, which extends random feature methods to nonstationary settings for the first time. By sampling from wavelet families to construct explicit feature maps, RWF leverages the localization and multiresolution properties of wavelets to enable efficient and scalable approximation of nonstationary kernels. Theoretical analysis guarantees that the resulting kernels are positive definite, unbiased, and uniformly convergent. Empirical evaluations demonstrate that RWF outperforms conventional stationary random feature approaches across multiple synthetic and real-world datasets, achieving a superior trade-off between accuracy and computational efficiency compared to more complex models such as deep Gaussian processes.

This work addresses the limitation of Random Fourier Features (RFF) in approximating non-Gaussian isotropic shift-invariant kernels. We propose a unified scale-mixture representation of spectral distributions: for the first time, we rigorously prove that the spectral distribution of any isotropic positive-definite shift-invariant kernel admits a representation as an α-stable random vector scaled by a scalar factor, whose distribution is analytically determined by the kernel function. Building on this characterization, we derive a general spectral sampling formula applicable to a broad class of kernels—including exponential-power, generalized Matérn, generalized Cauchy, and Beta/Kummer/Tricomi families—without modifying existing Gaussian RFF implementations. The approach seamlessly extends RFF-based algorithms such as SVM, kernel ridge regression, and Gaussian processes to diverse kernel families, significantly enhancing their compatibility and flexibility while preserving computational efficiency.

Standard multilayer perceptrons (MLPs) suffer from spectral bias, hindering accurate modeling of high-frequency complex-valued optical field perturbations. To address this, we propose Fourier Feature Networks (FFNs), which map inputs into a perturbation-dependent Fourier basis space, transforming nonlinear learning into linear combination of precomputed basis functions. FFNs enable end-to-end learning of the complex-valued transmission matrix under multimode fiber compression. This approach significantly reduces model complexity while enhancing generalization. Experiments demonstrate that FFN achieves one-order-of-magnitude lower prediction error than standard MLPs, attains an average complex correlation coefficient of 0.995 for both amplitude and phase, and reduces parameter count by 85%. By explicitly encoding high-frequency priors via Fourier features, FFN effectively overcomes the representational bottleneck of conventional neural networks in optical high-frequency modeling.

本文证明了随机特征方法在多维目标中的谱收敛性，并通过分析不同规则性的频率分布，为椭圆边界值和特征值问题提供了解决方案。

This work addresses the ambiguity inherent in traditional scalar representations of angular data, which fail to distinguish between nearby angles differing by more than π due to periodicity. To resolve this, the authors propose a high-dimensional, real-valued distributed representation based on Fourier embeddings, integrated with spatial semantic pointers to enable neurally interpretable encoding of periodic signals. They formalize the Dirichlet kernel and the periodic Gaussian kernel within this framework, allowing flexible and theoretically grounded control over angular similarity measures. The resulting approach provides unambiguous representations for arbitrarily close angles and establishes a principled design framework for similarity functions with customizable kernel shapes and provable theoretical guarantees.

This study addresses the limitations of Fourier Neural Operators (FNOs) in learning high-frequency components due to frequency truncation, as well as the challenges in coordinate encoding design. To this end, we propose the CAFE+FNO framework, which introduces a novel CAFE+ mechanism that fuses Fourier-Chebyshev features via the Hadamard product to enhance frequency-domain interactions. By integrating implicit neural representations, parallel affine branches, and shared-kernel MLPs, the method generates Fourier kernels through explicit feature composition, enabling efficient PDE solving with model parameters independent of the number of modes. Experiments across five PDE benchmarks demonstrate that the proposed approach significantly outperforms existing FNO variants, fully validating its advantages in bandwidth learnability.

This work addresses the lack of effective random feature methods for Bernstein–Schur kernels—products of finite-dimensional feature maps and completely monotone translation-invariant kernels—which fall outside the scope of Bochner’s theorem and are incompatible with polynomial sketching. To overcome this, the authors propose a dual randomization strategy: applying matrix sketching to compress the finite-dimensional modulation component and combining Bernstein–Widder scale sampling with Gaussian random Fourier features for the radial completely monotone part. This approach yields the first unified, unbiased random feature construction for this kernel class, preserving the exact limiting behavior of the modulation while providing an explicit variance expression and operator norm bounds based on intrinsic dimensionality. Crucially, it decouples sketching error from radial sampling error. Under theoretical guarantees, only \(D_m\) features—far fewer than the \(O(d^2)\) required for exact modulation—suffice for high-accuracy approximation; for the YAT kernel instance, single-scale, single-frequency sampling achieves variance optimality under a fixed radial budget.

本文提出了带符号随机傅里叶特征(SRFF)技术，解决了在使用不定核进行大规模数据集的核密度估计(KDE)时计算成本高的问题。