Abstract
We study the meta-learning for support recovery (i.e., non-zero coordinates of the eigenvectors) in high-dimensional Principal Component Analysis. We reduce the sufficient sample complexity in a novel task, with the information that is learned from auxiliary tasks, where a task is defined as a random Principal Component (PC) matrix with its own support. We pool data from all the tasks to execute an improper estimation of a single PC matrix, by maximising the ℓ_1-regularised predictive covariance. With m tasks for p-variate sub-Gaussian random vectors, we establish the sufficient sample complexity for each task to be of the order O(√(m^-1log p)), with high probability. This is very relevant for meta-learning where there are many tasks m = O(log p), each with very few samples, i.e., n = O(1), in an scenario where multi-task learning fails. For a novel task, we prove that the sufficient sample complexity of successful support recovery can be reduced to O(log |J|), under an additional constraint that the support of the novel task is a subset of the estimated support union (J) from the auxiliary tasks. This reduces the original sample complexity of O(log p) for learning a single task. Theoretical claims are validated with numerical simulations and the problem of true covariance estimation in brain-imaging and cancer genetics data sets are considered to validate the proposed methodology.