A locally Adaptive Normal Distribution

The fundamental building block in many learning models is the distance measure that is used. Usually, the linear distance is used for simplicity. Replacing this stiff distance measure with a flexible one could potentially give a better representation of the actual distance between two points. I will present how the normal distribution changes if the distance measure respects the underlying structure of the data. In particular, a Riemannian manifold will be learned based on observations. The geodesic curve can then be computed—a length-minimizing curve under the Riemannian measure. With this flexible distance measure we get a normal distribution that locally adapts to the data. A maximum likelihood estimation scheme is provided for inference of the parameters mean and covariance, and also, a systematic way to choose the parameter defining the Riemannian manifold. Results on synthetic and real world data demonstrate the efficiency of the proposed model to fit non-trivial probability distributions.