One of the most fascinating, critical and widely used pre-processing steps in scientific fields is the transformation of a random variable to unit covariance, a procedure commonly called whitening or sphering. The attractiveness of this transformation lies in its rotational invariance, which allows for many possible ways of whitening. Since the seminal paper of Bell and Sejnowski (1997), where a Mahalanobis type-whitening was introduced, this procedure has become increasingly popular in neuroscience and related fields, and a cornerstone of deep learning methods. 

Here an integral approach to whitening is proposed, making this pre-processing step and its optimisation a relevant issue that should be confronted in the context of functional spaces. Using the factorization of the inverse covariance operator under certain boundary conditions, we introduce the concept of whitening operator and functional whitening. The major advantage of this approach is that whitening can be robustly enforced by the smooth topological characteristics of the data thus avoiding the embedding of spurious noise in the transformation.  

We further provide algorithms to compute different whitening transformations in terms of basis systems that are not necessarily orthonormal in the usual sense (B-splines, non-orthogonal wavelets,…). Through simulations, we demonstrate how these transformations can be optimized by investigating the cross-covariance and cross-correlation operator between the functional sphered variable and the original one. Some strategies are also suggested to enhance the practical use of these transformations when data is of high dimensions.  

The interested reader can easily access the R code of the paper at https://github.com/m-vidal/functional-whitening, or visualize the performance of our methods at https://mvidal.shinyapps.io/whitening/