F-transform, dimensionality reduction, Laplacian eigenmaps, fuzzy partition, market volatility, Black-Scholes

Two distinguished properties of the F-transform: the best approximation in a local sense and the reduction in dimension imply the fact that the F-transform has many successful applications. We show that the technique of F-transform fully agrees with the technique of dimensionality reduction, based on Laplacian eigenmaps. To justify this claim, we characterize the processed by the F-transform data in terms of the adjacency graph that reflects their (data) intrinsic geometry. In the second part, we give an overview of the F-transform applications to mathematical finance: we discussed the estimation of market volatility and the numerical scheme and solution to the one-factor Black-Scholes partial differential equation.