A matrice de noyau, also known as a Matrice de Gram, is a mathematical representation that captures the inner products between pairs of data points in a transformed espace de caractéristiques. In apprentissage automatique, particularly in algorithms like Machines à vecteurs de support (SVM) and kernelized versions of other algorithms, the kernel matrix is essential for enabling the application of linear methods to non-linear problems.
Lorsque les données sont transformées dans un espace de dimension supérieure, elles peuvent devenir linéairement séparables, allowing for better classification. The kernel function computes the inner product of two vectors in this higher-dimensional space without explicitly performing the transformation, making calculations more efficient. Common kernel functions include the linear kernel, polynomial kernel, and radial basis function (RBF) kernel.
The kernel matrix is symmetric and positive semi-definite, meaning its eigenvalues are non-negative, which is crucial for ensuring stability in optimization problems. Each entry in the kernel matrix corresponds to the similarity between two data points, providing a représentation compacte that simplifies computations in various machine learning tasks, such as classification, regression, and clustering.
In summary, the kernel matrix is a powerful tool in machine learning that enables the application of linear algorithms to complex, non-linear datasets by encapsulating the relationships between data points in a mathematically manageable form.