On vector averaging over the unit hypersphere

Sample averaging is a commonly used way to smooth out irregularities of data and to get rid of random fluctuations in measurements analysis. In adaptive signal processing, where an adaptive system learns its own parameters in order to perform a predefined task, the learnt parameters-pattern may depend on the initial learning state and on the fluctuations of the statistical features of the input signals to the system. In adaptive system learning, averaging may be employed as a method to merge several learnt parameters-patterns in order to get a better representative pattern. Even in the case of scalar parameters, the concept of averaging is not uniquely defined as scalar parameters spaces may exhibit a rich structure to be dealt with. The case of multiple parameter patterns where single parameters are mutually constrained to each other may exhibit an even richer structure. In the present paper, we deal with the case of parameters-patterns belonging to the unit hypersphere and develop an averaging technique based on the differential geometrical structure of such a curved space. Numerical experiments illustrate the behavior of the developed averaging algorithm.