A new method is presented for the optimal or near-optimal quantization of memoryless Gaussian data. The basic construction of the codebook is motivated by related ideas in the statistical framework of sparse recovery in linear regression. Similarly, the encoding is performed by a convex-hull iterative algorithm. Preliminary theoretical results establish the optimality of the proposed algorithm for a certain range of the parameter values. Experimental results demonstrate these theoretical findings on simulated data. The performance of the proposed algorithm is compared with that of trellis-coded quantization and of the recently proposed algorithms in and. Depending on the choice of the relevant design parameters, the complexity of the encoding algorithm varies, and is generally polynomial in the data block-length. The present results are, in part, motivated by the analogous channel coding results of.