Difeng Cai
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View article: Posterior Covariance Structures in Gaussian Processes
Posterior Covariance Structures in Gaussian Processes Open
In this paper, we present a comprehensive analysis of the posterior covariance field in Gaussian processes, with applications to the posterior covariance matrix. The analysis is based on the Gaussian prior covariance but the approach also …
View article: Data-Driven Linear Complexity Low-Rank Approximation of General Kernel Matrices: A Geometric Approach
Data-Driven Linear Complexity Low-Rank Approximation of General Kernel Matrices: A Geometric Approach Open
A general, {\em rectangular} kernel matrix may be defined as $K_{ij} = κ(x_i,y_j)$ where $κ(x,y)$ is a kernel function and where $X=\{x_i\}_{i=1}^m$ and $Y=\{y_i\}_{i=1}^n$ are two sets of points. In this paper, we seek a low-rank approxim…
View article: AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows
AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows Open
Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting fun…
View article: Data-driven Construction of Hierarchical Matrices with Nested Bases
Data-driven Construction of Hierarchical Matrices with Nested Bases Open
Hierarchical matrices provide a powerful representation for significantly reducing the computational complexity associated with dense kernel matrices. For general kernel functions, interpolation-based methods are widely used for the effici…
View article: Hybrid A Posteriori Error Estimators for Conforming Finite Element Approximations to Stationary Convection-Diffusion-Reaction equations
Hybrid A Posteriori Error Estimators for Conforming Finite Element Approximations to Stationary Convection-Diffusion-Reaction equations Open
We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. We present an explicit hybrid estimator, which, in each regime, is proved to b…
View article: Fast deterministic approximation of symmetric indefinite kernel matrices with high dimensional datasets
Fast deterministic approximation of symmetric indefinite kernel matrices with high dimensional datasets Open
Kernel methods are used frequently in various applications of machine learning. For large-scale high dimensional applications, the success of kernel methods hinges on the ability to operate certain large dense kernel matrix K. An enormous …
View article: Fast and stable deterministic approximation of general symmetric kernel matrices in high dimensions.
Fast and stable deterministic approximation of general symmetric kernel matrices in high dimensions. Open
Kernel methods are used frequently in various applications of machine learning. For large-scale applications, the success of kernel methods hinges on the ability to operate certain large dense kernel matrix K. To reduce the computational c…
View article: A stable matrix version of the fast multipole method: stabilization strategies and examples
A stable matrix version of the fast multipole method: stabilization strategies and examples Open
The fast multipole method (FMM) is an efficient method for evaluating matrix-vector products related to certain discretized kernel functions. The method involves an underlying FMM matrix given by a sequence of smaller matrices (called gene…
View article: Eigenvalue Problems for Exponential-Type Kernels
Eigenvalue Problems for Exponential-Type Kernels Open
We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate | λ k - λ k , h | ≤ C k h 2 {\lvert\lambda_{k}-\lambda_{k,h}\rvert\l…
View article: SIMULATION OF FISH MIGRATION AT DIFFERENT WATER DEPTHS BASED ON BACKPROPAGATION NEURAL NETWORK
SIMULATION OF FISH MIGRATION AT DIFFERENT WATER DEPTHS BASED ON BACKPROPAGATION NEURAL NETWORK Open
This paper aims to disclose the law of fish migration trajectories at different water depths.For this purpose, the grass carps in a reservoir in southwestern China were taken as the targets, outdoor experiments were performed to monitor th…
View article: ROBUST AND EXPLICIT A POSTERIORI ERROR ESTIMATION TECHNIQUES IN ADAPTIVE FINITE ELEMENT METHOD
ROBUST AND EXPLICIT A POSTERIORI ERROR ESTIMATION TECHNIQUES IN ADAPTIVE FINITE ELEMENT METHOD Open
The thesis presents a comprehensive study of a posteriori error estimation in the adaptive solution to some classical elliptic partial differential equations. Several new error estimators are proposed for diffusion problems with discontinu…
View article: SMASH: Structured matrix approximation by separation and hierarchy
SMASH: Structured matrix approximation by separation and hierarchy Open
This paper presents an efficient method to perform Structured Matrix Approximation by Separation and Hierarchy (SMASH), when the original dense matrix is associated with a kernel function. Given points in a domain, a tree structure is firs…