A Matriz 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 aprendizado de máquina, statistics, and processamento de sinais. 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 produto interno 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 |
Essa matriz é simétrica e semi-definida positiva, o que significa que todos its eigenvalues are non-negative. The Gram Matrix is essential in métodos de kernel usada em aprendizado de máquina, particularly in Máquinas de Vetores de Suporte 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 inteligência artificial e análise de dados.