Bayesian interpretation of kernel regularization

Bayesian interpretation of kernel regularization examines how <a href="/facts/Kernel_methods/fYuURIPk">kernel methods</a> in <a href="/facts/Machine_learning/e0w0XJTu">machine learning</a> can be understood through the lens of <a href="/facts/Bayesian_statistics/9w7b1Bw4">Bayesian statistics</a>, a framework that uses probability to model uncertainty. Kernel methods are founded on the concept of similarity between inputs within a structured space. While techniques like <a href="/facts/Support_vector_machine/XobxpdBG">support vector machines</a> (SVMs) and their <a href="/facts/Regularization_(mathematics)/K601A3y2">regularization</a> (a technique to make a model more generalizable and transferable) were not originally formulated using Bayesian principles, analyzing them from a <a href="/facts/Bayesian_probability/Cvn7WWfG">Bayesian</a> perspective provides valuable insights.
In the Bayesian framework, kernel methods serve as a fundamental component of <a href="/facts/Gaussian_processes/MrBq7kYW">Gaussian processes</a>, where the kernel function operates as a covariance function that defines relationships between inputs. Traditionally, these methods have been applied to <a href="/facts/Supervised_learning/JlNFNPFt">supervised learning</a> problems where inputs are represented as vectors and outputs as scalars. Recent developments have extended kernel methods to handle <a href="/facts/Kernel_methods_for_vector_output/p3gzGg6S">multiple outputs</a>, as seen in <a href="/facts/Multi-task_learning/av9lCQUF">multi-task learning</a>.
The mathematical framework for kernel methods typically involves <a href="/facts/Reproducing_kernel_Hilbert_space/VegkYAXb">reproducing kernel Hilbert spaces</a> (RKHS). Not all kernels form inner product spaces, as they may not always be positive semidefinite (a property ensuring non-negative similarity measures), but they still operate within these more general RKHS. A mathematical equivalence between regularization approaches and Bayesian methods can be established, particularly in cases where the reproducing kernel Hilbert space is finite-dimensional. This equivalence demonstrates how both perspectives converge to essentially the same <a href="/facts/Estimator/CbkjcKvN">estimators</a>, revealing the underlying connection between these seemingly different approaches.

Bayesian interpretation of kernel regularization open-in-new

Bayesian interpretation of kernel regularization