Integrating Kernel Methods Research Articles

Integrating Graph Structures into Kernel Regression Models

Kernel methods are highly flexible and powerful tools for capturing complex, nonlinear relationships in data. In this paper, we propose a substantial extension of existing network-based regression models by integrating kernel methods with graph-theoretic constraints. Our approach builds upon the foundational work of Li et al. [1], who incorporated network cohesion into generalized linear models (GLMs). We extend their framework by introducing a kernelized regression model that allows for the modeling of nonlinear interactions while leveraging network data. This kernelized framework relaxes the linearity constraints of GLMs, making the model more versatile and capable of capturing complex patterns in high-dimensional spaces. We demonstrate the effectiveness of our approach using simulated and real-world datasets, such as the Teenage Friends and Lifestyle Study. Results show that our kernelized network regression model not only outperforms traditional linear and generalized linear models in predictive accuracy but also scales efficiently to larger datasets. Our work represents a significant advancement in the modeling of network-linked data, providing a robust, scalable, and interpretable framework that extends the application of kernel methods beyond their traditional constraints. Future directions include exploring more complex graph structures, such as weighted and directed graphs, and developing optimized algorithms for even larger datasets.

Integration of Data-Level Fusion Model and Kernel Methods for Degradation Modeling and Prognostic Analysis

To prevent unexpected failures of complex engineering systems, multiple sensors have been widely used to simultaneously monitor the degradation process and make inference about the remaining useful life in real time. As each of the sensor signals often contains partial and dependent information, data-level fusion techniques have been developed that aim to construct a health index via the combination of multiple sensor signals. While the existing data-level fusion approaches have shown a promise for degradation modeling and prognostics, they are limited by only considering a linear fusion function. Such a linear assumption is usually insufficient to accurately characterize the complicated relations between multiple sensor signals and the underlying degradation process in practice, especially for complex engineering systems considered in this study. To address this issue, this study fills the literature gap by integrating kernel methods into the data-level fusion approaches to construct a health index for better characterizing the degradation process of the system. Through selecting a proper kernel function, the nonlinear relation between multiple sensor signals and the underlying degradation process can be captured. As a result, the constructed health index is expected to perform better in prognosis than existing data-level fusion methods that are based on the linear assumption. In fact, the existing data-level fusion models turn out to be only a special case of the proposed method. A case study based on the degradation signals of aircraft gas turbine engines is conducted and finally shows the developed health index by using the proposed method is insensitive for missing data and leads to an improved prognostic performance.

Online learning control using adaptive critic designs with sparse kernel machines.

In the past decade, adaptive critic designs (ACDs), including heuristic dynamic programming (HDP), dual heuristic programming (DHP), and their action-dependent ones, have been widely studied to realize online learning control of dynamical systems. However, because neural networks with manually designed features are commonly used to deal with continuous state and action spaces, the generalization capability and learning efficiency of previous ACDs still need to be improved. In this paper, a novel framework of ACDs with sparse kernel machines is presented by integrating kernel methods into the critic of ACDs. To improve the generalization capability as well as the computational efficiency of kernel machines, a sparsification method based on the approximately linear dependence analysis is used. Using the sparse kernel machines, two kernel-based ACD algorithms, that is, kernel HDP (KHDP) and kernel DHP (KDHP), are proposed and their performance is analyzed both theoretically and empirically. Because of the representation learning and generalization capability of sparse kernel machines, KHDP and KDHP can obtain much better performance than previous HDP and DHP with manually designed neural networks. Simulation and experimental results of two nonlinear control problems, that is, a continuous-action inverted pendulum problem and a ball and plate control problem, demonstrate the effectiveness of the proposed kernel ACD methods.

Weighted Lp estimates for the ∂-equation on convex domains of finite type

We prove non-isotropic Lp (1 ≤ p ≤ ∞) estimates with weights for solutions of the Cauchy-Riemann equation on bounded convex domains of finite type in C n using the integral kernel method. We also give an example which guarantees the optimum of the estimates.

Optimal Hölder and Lp estimates for [formula omitted] on boundaries of convex domains of finite type

We prove optimal Hölder and L p estimates for solutions of the tangential Cauchy–Riemann equation on the boundary of a bounded convex domain of finite type in C n using the integral kernel method.

Optimal Sobolev estimates for �? on convex domains of finite type

We prove that solutions for ¯∂ get 1/M-derivatives more than the data in L p -Sobolev spaces on a bounded convex domain of finite type M by means of the integral kernel method. Also we prove that the Bergman projection is invariant under the L p -Sobolev spaces of fractional orders by different methods from McNeal-Stein's. By using these results, we can get L p -Sobolev estimates of order 1/M for the canonical solution for ¯∂.

Optimal Hoelder and L p Estimates for ∂   b   on the Boundaries of Real Ellipsoids in C n

Let D be a real ellipsoid in Cn, n > 3, with defining function p(Z) = En=,(x ,k + yk2k)1, Zk = Xk + iYk, where nk, mk E N. In this paper we study the sharp Holder and Lp estimates for the solutions of the tangential Cauchy-Riemann equations 0b on the boundary OD of D using the integral kernel method. In particular, we proved that if a E L(o j)(OD) such that Oba = 0 on AD in the distribution sense, then there exists a u E A112m(OD) satisfying ObU = a and IIUIIA,/2(9D) 0 independent of a, where Al/2m(OD) is the Lipschitz space with exponent 21 and 2m = maxl<k<f min(2nk, 2mk) is the type of the m << k domain D.