A Gram-Matrix, 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 maschinellem Lernen, statistics, and Signalverarbeitung. 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 inneres Produkt 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 |
Diese Matrix ist symmetrisch und positiv semi-definit, was bedeutet, dass alle its eigenvalues are non-negative. The Gram Matrix is essential in Kernel-Methoden im maschinellen Lernen, particularly in Support-Vektor-Maschinen 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 künstliche Intelligenz und Datenanalyse wesentlich.