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 proportional to the identity matrix.

Formal definitions

Let $D$ be a distribution over vectors in the vector space $\mathbb^n$.
Then $D$ is in isotropic position if, for vector $v$ sampled from the distribution,
$\mathbb\, vv^\mathsf = \mathrm.$

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 $K$ in $\mathbb^n$ is called isotropic if it has volume $, K, = 1$, center of mass at the origin, and there is a constant $\alpha > 0$ such that
$\int_K \langle x, y \rangle^2 dx = \alpha^2 , y, ^2,$
for all vectors $y$ in $\mathbb^n$; here $, \cdot,$ stands for the standard Euclidean norm.

See also

* Whitening transformation

References

*

Machine learning
Random matrices

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