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Several variations of the Kalman filter algorithm, such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are widely used in science and engineering applications. In this paper, we introduce two algorithms of…
Accurately estimating the point spread function (PSF) of an optical system requires solving free-space wave propagation, which entails evaluating a diffraction integral. This integral is traditionally computed numerically using Fast Fourier…
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.…
Principal Component Analysis (PCA) is a powerful and popular dimensionality reduction technique. However, due to its linear nature, it often fails to capture the complex underlying structure of real-world data. While Kernel PCA (kPCA)…
Kernel-based methods enjoy powerful generalization capabilities in handling a variety of learning tasks. When such methods are provided with sufficient training data, broadly-applicable classes of nonlinear functions can be approximated…
We study efficient mechanisms for differentially private kernel density estimation (DP-KDE). Prior work for the Gaussian kernel described algorithms that run in time exponential in the number of dimensions $d$. This paper breaks the…
In this paper, we apply the Feature Space Decomposition (FSD) method developed in [LS24, GLS25, LSSW26, ALSS26] to obtain, under fairly general conditions, matching upper and lower bounds for the population excess risk of spectral methods…
The ensemble Kalman filter (EnKF) is widely used for data assimilation in high-dimensional systems, but its performance often deteriorates for strongly nonlinear dynamics due to the structural mismatch between the Kalman update and the…
Conventional vision algorithms adopt a single type of feature or a simple concatenation of multiple features, which is always represented in a high-dimensional space. In this paper, we propose a novel unsupervised spectral embedding…
The fields of signal processing and information theory have evolved with the goal of developing formulations to extract intrinsic information from limited amount of data. When one considers the modeling of unpredictably varying processes…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
Kernel density estimation on a finite interval poses an outstanding challenge because of the well-recognized bias at the boundaries of the interval. Motivated by an application in cancer research, we consider a boundary constraint linking…
We extend the diffusion-map formalism to data sets that are induced by asymmetric kernels. Analytical convergence results of the resulting expansion are proved, and an algorithm is proposed to perform the dimensional reduction. In this work…
Modeling dynamical systems with ordinary differential equations implies a mechanistic view of the process underlying the dynamics. However in many cases, this knowledge is not available. To overcome this issue, we introduce a general…
Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation,…
Kernel principal component analysis (kPCA) is a widely studied method to construct a low-dimensional data representation after a nonlinear transformation. The prevailing method to reconstruct the original input signal from kPCA -- an…
Foreground segmentation is an essential task in the field of image understanding. Under unsupervised conditions, different images and instances always have variable expressions, which make it difficult to achieve stable segmentation…
Many scientific problems require identifying a small set of covariates that are associated with a target response and estimating their effects. Often, these effects are nonlinear and include interactions, so linear and additive methods can…
As a promising step, the performance of data analysis and feature learning are able to be improved if certain pattern matching mechanism is available. One of the feasible solutions can refer to the importance estimation of instances, and…
Gaussian process regression is a widely-applied method for function approximation and uncertainty quantification. The technique has gained popularity recently in the machine learning community due to its robustness and interpretability. The…