A Matrice de Gram, often denoted as G, is a square matrix that is used to represent the inner products of a set of vectors in a vector space. It plays a crucial role in various fields, including apprentissage automatique, statistics, and traitement du signal. In essence, the Gram Matrix captures the angles and lengths between vectors, providing insights into their relationships.
To construct a Gram Matrix from a set of vectors, each entry G(i, j) is computed as the produit interne of the i-th and j-th vectors. For example, if you have vectors v1, v2, and v3, the Gram Matrix G would be:
G = | v1·v1 v1·v2 v1·v3 |
| v2·v1 v2·v2 v2·v3 |
| v3·v1 v3·v2 v3·v3 |
Cette matrice est symétrique et semi-définie positive, ce qui signifie que toutes its eigenvalues are non-negative. The Gram Matrix is essential in méthodes à noyau utilisé en apprentissage automatique, particularly in machines à vecteurs de support and Gaussian processes, where it helps to transform the input space into a higher-dimensional feature space.
In summary, the Gram Matrix is a powerful mathematical concept that provides a compact representation of the relationships between vectors, which is instrumental in various applications of intelligence artificielle et l’analyse de données.