  Dimension reduction is crucial in functional data analysis (FDA). The key
tool to reduce the dimension of the data is functional principal component
analysis. Existing approaches for functional principal component analysis
usually involve the diagonalization of the covariance operator. With the
increasing size and complexity of functional datasets, estimating the
covariance operator has become more challenging. Therefore, there is a growing
need for efficient methodologies to estimate the eigencomponents. Using the
duality of the space of observations and the space of functional features, we
propose to use the inner-product between the curves to estimate the
eigenelements of multivariate and multidimensional functional datasets. The
relationship between the eigenelements of the covariance operator and those of
the inner-product matrix is established. We explore the application of these
methodologies in several FDA settings and provide general guidance on their
usability.
