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RIS-aided millimeter wave wireless systems benefit from robustness to blockage and enhanced coverage. In this paper, we study the ability of RIS to also provide enhanced localization capabilities as a by-product of communication. We consider sparse reconstruction algorithms to obtain high resolution channel estimates that are mapped to position information. In RIS-aided mmWave systems, the complexity of sparse recovery becomes a bottleneck, given the large number of elements of the RIS and the large communication arrays. We propose to exploit a multidimensional orthogonal matching pursuit strategy for compressive channel estimation in a RIS-aided millimeter wave system. We show how this algorithm, based on computing the projections on a set of independent dictionaries instead of a single large dictionary, enables high accuracy channel estimation at reduced complexity. We also combine this strategy with a localization approach which does not rely on the absolute time of arrival of the LoS path. Localization results in a realistic 3D indoor scenario show that RIS-aided wireless system can also benefit from a significant improvement in localization accuracy.