Reduced feedback schemes play a critical role in orthogonal frequency division multiplexing-based cellular systems because they facilitate scheduling and rate adaptation by the base station (BS) while reducing the number of subchannels for which channel state information is fed back by the users. We address the problem of reliable transmission even on subchannels that are fed back by a few users due to feedback constraints. For the practically relevant best-$M$ feedback scheme, in which each user reports only its $M$ strongest subchannels to the BS, we derive a nonlinear constrained minimum mean square error estimator that enables the BS to estimate the signal-to-noise-ratios of all the subchannels of every user. We then propose two lower computational complexity approaches that incur a negligible loss in performance. The novelty of these approaches lies in their exploitation of the structure of the best-$M$ feedback information and the correlation among subchannel gains. Applications to general channel models and to quantized feedback are also shown. In terms of system-level impact, the proposed approaches improve the cell throughput compared to several conventional approaches - without requiring any additional feedback - for uncorrelated and correlated subchannels, and for various schedulers. © 2002-2012 IEEE.