Английская Википедия:Isotropic position

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Шаблон:Short description 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 <math display="inline">D</math> be a distribution over vectors in the vector space <math display="inline">\mathbb{R}^n</math>. Then <math display="inline">D</math> is in isotropic position if, for vector <math display="inline">v</math> sampled from the distribution, <math display="block">\mathbb{E}\, vv^\mathsf{T} = \mathrm{Id}.</math>

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

See also

• Whitening transformation

References

• Шаблон:Cite journal

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