  Cosine similarity is an established similarity metric for computing
associations on vectors, and it is commonly used to identify related samples
from biological perturbational data. The distribution of cosine similarity
changes with the covariance of the data, and this in turn affects the
statistical power to identify related signals. The relationship between the
mean and covariance of the distribution of the data and the distribution of
cosine similarity is poorly understood. In this work, we derive the asymptotic
moments of cosine similarity as a function of the data and identify the
criteria of the data covariance matrix that minimize the variance of cosine
similarity. We find that the variance of cosine similarity is minimized when
the eigenvalues of the covariance matrix are equal for centered data. One
immediate application of this work is characterizing the null distribution of
cosine similarity over a dataset with non-zero covariance structure.
Furthermore, this result can be used to optimize over a set of transformations
or representations on a dataset to maximize power, recall, or other
discriminative metrics, with direct application to noisy biological data. While
we consider the specific biological domain of perturbational data analysis, our
result has potential application for any use of cosine similarity or Pearson's
correlation on data with covariance structure.
