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The groupwise approach to non-rigid image registration, solving the dense correspondence problem, has recently been shown to be a useful tool in many applications, including medical imaging, automatic construction of statistical models of appearance and analysis of facial dynamics. Such an approach overcomes limitations of traditional pairwise methods but at a cost of having to search for the solution (optimal registration) in a space of much higher dimensionality which grows rapidly with the number of examples (images) being registered. Techniques to overcome this dimensionality problem have not been addressed sufficiently in the groupwise registration literature. In this paper, we propose a novel, fast and reliable, fully unsupervised stochastic algorithm to search for optimal groupwise dense correspondence in large sets of unmarked images. The efficiency of our approach stems from novel dimensionality reduction techniques specific to the problem of groupwise image registration and from comparative insensitivity of the adopted optimisation scheme (simultaneous perturbation stochastic approximation (SPSA)) to the high dimensionality of the search space. Additionally, our algorithm is formulated in way readily suited to implementation on graphics processing units (GPU). In evaluation of our method we show a high robustness and success rate, fast convergence on various types of test data, including facial images featuring large degrees of both inter- and intra-person variation, and show considerable improvement in terms of accuracy of solution and speed compared to traditional methods.