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We perform a finite sample analysis of the detection levels for sparseprincipal components of a high-dimensional covariance matrix. Our minimaxoptimal test is based on a sparse eigenvalue statistic. Alas, computing thistest is known to be NP-complete in general, and we describe a computationallyefficient alternative test using convex relaxations. Our relaxation is alsoproved to detect sparse principal components at near optimal detection levels,and it performs well on simulated datasets. Moreover, using polynomial timereductions from theoretical computer science, we bring significant evidencethat our results cannot be improved, thus revealing an inherent trade offbetween statistical and computational performance.

Optimal detection of sparse principal components in high dimension