Abstract/Summary

We propose that the square root of a probability density function can be represented as a linear combination of Gaussian kernels. It is shown that if the Gaussian kernel centres and kernel width are known, then the maximum likelihood parameter estimator can be formulated as a Riemannian optimisation problem on sphere manifold. The first order Riemannian geometry of the sphere manifold and vector transport are initially explored, then the well-known Riemannian conjugate gradient algorithm is used to estimate the model parameters. For completeness, the k-means clustering algorithm and a grid search are employed to determine the centres and kernel width, respectively. Simulated examples are employed to demonstrate that the proposed approach is effective in constructing the estimate of the square root of probability density function.