  The exponential mechanism is a fundamental tool of Differential Privacy (DP)
due to its strong privacy guarantees and flexibility. We study its extension to
settings with summaries based on infinite dimensional outputs such as with
functional data analysis, shape analysis, and nonparametric statistics. We show
that one can design the mechanism with respect to a specific base measure over
the output space, such as a Guassian process. We provide a positive result that
establishes a Central Limit Theorem for the exponential mechanism quite
broadly. We also provide an apparent negative result, showing that the
magnitude of the noise introduced for privacy is asymptotically non-negligible
relative to the statistical estimation error. We develop an \ep-DP mechanism
for functional principal component analysis, applicable in separable Hilbert
spaces. We demonstrate its performance via simulations and applications to two
datasets.
