There are a lot of references in the statistic literature to "functional data" (i.e. data that are curves), and in parallel, to "high dimensional data" (i.e. when data are high dimensional vectors). My question is about the difference between the two type of data.
When talking about applied statistic methodologies that apply in case 1 can be understood as a rephrasing of methodologies from case 2 through a projection into a finite dimensional subspace of a space of functions, it can be polynomes, splines, wavelet, Fourier, .... and will translate the functional problem into a finite dimensional vectorial problem (since in applied mathematic everything comes to be finite at some point).
My question is: can we say that any statistical procedure that applies to functional data can also be applied (almost directly) to high dimension data and that any procedure dedicated to high dimensional data can be (almost directly) applied to functional data ?
If the answer is no, can you illustrate ?
EDIT/UPDATE with the help of Simon Byrne's answer:
- sparsity (S-sparse assumption, $l^p$ ball and weak $l^p$ ball for $p<1$) is used as a structural assumption in high dimensional statistical analysis.
- "smoothness" is used as a structural assumption in functional data analysis.
On the other hand, inverse Fourier transform and inverse wavelet transform are transforming sparcity into smoothness, and smoothness is transformed into sparcity by wavelet and fourier transform. This make the critical difference mentionned by Simon not so critical?