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5G mmWave MIMO systems enable accurate estimation of the user position and mapping of the radio environment using a single snapshot when both the base station (BS) and user are equipped with large antenna arrays. However, massive arrays are initially expected only at the BS side, likely leaving users with one or very few antennas. In this paper, we propose a novel method for single-snapshot localization and mapping in the more challenging case of a user equipped with a single-antenna receiver. The joint maximum likelihood (ML) estimation problem is formulated and its solution formally derived. To avoid the burden of a full-dimensional search over the space of the unknown parameters, we present a novel practical approach that exploits the sparsity of mmWave channels to compute an approximate joint ML estimate. A thorough analysis, including the derivation of the Cramér-Rao lower bounds, reveals …