  The assumption of independent subvectors arises in many aspects of
multivariate analysis. In most real-world applications, however, we lack prior
knowledge about the number of subvectors and the specific variables within each
subvector. Yet, testing all these combinations is not feasible. For example,
for a data matrix containing 15 variables, there are already 1 382 958 545
possible combinations. Given that zero correlation is a necessary condition for
independence, independent subvectors exhibit a block diagonal covariance
matrix. This paper focuses on the detection of such block diagonal covariance
structures in high-dimensional data and therefore also identifies uncorrelated
subvectors. Our nonparametric approach exploits the fact that the structure of
the covariance matrix is mirrored by the structure of its eigenvectors.
However, the true block diagonal structure is masked by noise in the sample
case. To address this problem, we propose to use sparse approximations of the
sample eigenvectors to reveal the sparse structure of the population
eigenvectors. Notably, the right singular vectors of a data matrix with an
overall mean of zero are identical to the sample eigenvectors of its covariance
matrix. Using sparse approximations of these singular vectors instead of the
eigenvectors makes the estimation of the covariance matrix obsolete. We
demonstrate the performance of our method through simulations and provide real
data examples. Supplementary materials for this article are available online.
