Counting Real Critical Points of the Distance to Orthogonally Invariant Matrix Sets

Counting Real Critical Points of the Distance to Orthogonally Invariant Matrix Sets

Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and count the real smooth critical points of a data matrix on an orthogonally invariant set of matrices. The technique relies on “transfer principles...

The Euclidean Distance Degree of Orthogonally Invariant Matrix Varieties

We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.

Orthogonally Invariant Estimation of the Skew-symmetric Normal Mean Matrix

A b s t r a c t. The unbiased estimator of risk of the orthogonally invariant es-timator of the skew-symmetric normal mean matrix is obtained, and a class of minimax estimators and their order-preserving modification are proposed. The estimators have applications in paired comparisons model. A Monte Carlo study to compare the risks of the estimators is given.

Counting Lattice Points in Admissible Adelic Sets

On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...