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Mathematics

Some math topics with relation to ML

orthogonal projections of ellipsoid

Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

In the mathematical section of this blog we deal with the geometry of Multivariate Normal Distributions [MVNs]. We found that the contour surfaces of MVNs are multidimensional ellipsoids given by quadratic forms of their constituting vectors. We have identified the variance-covariance matrix of a MVN as the inverse matrix mediating the required quadratic form. We also considered ellipsoidal cores of… Read More »Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

3-dim projections of 4-dim MVN

n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

In the third post of this series we have discussed an idea of I. Rivin (see [1], [2]). Rivin has shown that the surface area of a general (n-1)-dimensional ellipsoid in a n-dimensional Euclidean space can be expressed in terms of an expectation value of a specific function weighted by a multivariate Gaussian probability density [pdf]. In contrast to (n-1)-spheres… Read More »n-dimensional spheres and ellipsoids – IV – numerical check of Rivin’s formula for the surface areas of ellipsoids in 3 dimensions and the perimeters of ellipses

n-dimensional spheres and ellipsoids – III – Surface area of n-dimensional ellipsoid and its relation to MVN-statistics

In the 2nd post of this series we have derived an explicit formula for the volume of a n-dimensional ellipsoid (in an Euclidean space). One reason for the relatively simple derivation was that the determinant of the generating linear transformation could be taken in front of the volume integral. Unfortunately, an analogue sequence of steps is not possible for an… Read More »n-dimensional spheres and ellipsoids – III – Surface area of n-dimensional ellipsoid and its relation to MVN-statistics

Covariance matrix of a cut-off Multivariate Normal Distribution – IV – theoretical prediction for a 3-dimensional MVN-core

In this post series we study ellipsoidal cores of Multivariate Normal Distributions [MVNs]. We defined a “core” as the volume enclosed by a selected contour surface of constant probability density. We constructed cut-off distributions by setting the probability density to zero outside the core. In previous posts we have already discussed volume integrals which would give us a relation between… Read More »Covariance matrix of a cut-off Multivariate Normal Distribution – IV – theoretical prediction for a 3-dimensional MVN-core