  We introduce a new method for sparse principal component analysis, based on
the aggregation of eigenvector information from carefully-selected axis-aligned
random projections of the sample covariance matrix. Unlike most alternative
approaches, our algorithm is non-iterative, so is not vulnerable to a bad
choice of initialisation. We provide theoretical guarantees under which our
principal subspace estimator can attain the minimax optimal rate of convergence
in polynomial time. In addition, our theory provides a more refined
understanding of the statistical and computational trade-off in the problem of
sparse principal component estimation, revealing a subtle interplay between the
effective sample size and the number of random projections that are required to
achieve the minimax optimal rate. Numerical studies provide further insight
into the procedure and confirm its highly competitive finite-sample
performance.
