Isotropic position
In the fields of machine learning, the theory of computation, and random matrix theory, a probability distribution over vectors is said to be in isotropic position if its covariance matrix is equal to the identity matrix.
Formal definitions
Let be a distribution over vectors in the vector space . Then is in isotropic position if, for vector sampled from the distribution,
A set of vectors is said to be in isotropic position if the uniform distribution over that set is in isotropic position. In particular, every orthonormal set of vectors is isotropic.
As a related definition, a convex body in is in isotropic position if, for all vectors in , we have
See also
References
- Rudelson, M. (1999). "Random Vectors in the Isotropic Position". Journal of Functional Analysis. 164 (1): 60–72. arXiv:math/9608208.
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