  The Gaia optical reference frame is intrinsically undefined with respect to
global orientation and spin, so it needs to be anchored in the radio-based
International Celestial Reference Frame (ICRF) to provide a referenced and
quasi-inertial celestial coordinate system. The link between the two
fundamental frames is realized through two samples of distant extragalactic
sources, mostly AGNs and quasars, but only the smaller sample of radio-loud
ICRF sources with optical counterparts is available to determine the mutual
orientation. The robustness of this link can be mathematically formulated in
the framework of functional principal component analysis using a set of vector
spherical harmonics to represent the differences in celestial positions of the
common objects. The weakest eigenvectors are computed, which describe the
greatest deficiency of the link. The deficient or poorly determined terms are
specific vector fields on the sphere which carry the largest errors of absolute
astrometry using Gaia in reference to the ICRF. This analysis provides
guidelines to the future development of the ICRF maximizing the accuracy of the
link over the entire celestial sphere. A measure of robustness of a
least-squares solution, which can be applied to any linear model fitting
problem, is introduced to help discriminate between reference frame tie models
of different degrees.
