In this paper we consider a simple estimation problem on the special orthogonal group SO(n) and indicate a possible way to construct approximate filters which is much in the same spirit of the “wide sense” approach to linear filtering theory. Our interest is mainly motivated by applications to computer vision.

We consider group-covariant positive operator valued measures (POVMs) on a finite dimensional quantum system. Following Neumark’s theorem a POVM can be implemented by an orthogonal measurement on a larger system. Accordingly, our goal is to find an implementation of a given group-covariant POVM by a quantum circuit using its symmetry. Based on representation theory of the symmetry group we deve...

An elementary non-technical introduction to group representations and orthogonal polynomials is given. Orthogonality relations for the spherical functions for the rotation groups in Euclidean space (ultraspherical polynomials), and the matrix elements of SU(2) (Jacobi polynomials) are discussed. A general theory for finite groups acting on graphs, giving a finite set of discrete orthogonal poly...

The little Grothendieck problem (a special case of Boolean quadratic optimization) consists of maximizing ∑ ij Cijxixj over binary variables xi ∈ {±1}, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ Rdn×dn a positive semidefinite matrix, the objective is to maximi...

In this paper we derive an explicit formula for the HirzebruchMumford volume of an even lattice L of signature (2, n). If Γ ⊂ O(L) is an arithmetic subgroup, then an application of Hirzebruch-Mumford proportionality allows us to determine the leading term of the growth of the dimension of the spaces Sk(Γ) of cusp forms of weight k, as k goes to infinity. We compute this in a number of examples,...

Transform coding has been the focus of the research for image compression. In previous research, the Amplitude and Group Partitioning (AGP) coding scheme is proved to be a low complexity algorithm with high performance , clearly one of the state-of-art transform coding techniques. However, the previous AGP is used along with the Discrete Cosine Transform (DCT) and the discrete wavelet transform...

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This paper describes the calculation of the character table of the 8-dimensional orthogonal group of maximal index over the field with 3 elements. The group is of interest as it is a subgroup of relatively small index in the sporadic simple group M(23) defined by B. Fischer [2]. The group also has an outer automorphism group of order 24, isomorphic to the symmetric group on 4 symbols.

This paper derives a new algorithm that performs independent component analysis (ICA) by optimizing the contrast function of the RADICAL algorithm. The core idea of the proposed optimization method is to combine the global search of a good initial condition with a gradient-descent algorithm. This new ICA algorithm performs faster than the RADICAL algorithm (based on Jacobi rotations) while stil...

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where W is the symmetric group (the An case) and P is a maximal parabolic subgroup. This generalization o...