  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.
