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We investigate the problem of sparse modeling for predictive coding and introduce an efficient algorithm for computing sparse stereo linear predictors for lossless audio compression. Sparse linear predictive coding offers both improved compression and reduction of decoding complexity compared with non-sparse linear predictive coding. The modeling part amounts to finding the optimal structure of a sparse linear predictor using a fully implementable minimum description length (MDL) approach. The MDL criterion, simplified conveniently under realistic assumptions, is approximately minimized by a greedy algorithm which solves sequentially least squares partial problems, where the LDLT factorization ensures numerically stable solutions and facilitates a quasi-optimal quantization of the parameter vector. The overall compression system built around this modeling tool is shown to achieve the main goals: improved compression and, even more importantly, faster decoding speeds than the state of the art lossless audio compression methods. The optimal MDL sparse predictors are shown to provide parametric spectra that constitute new alternative spectral descriptors, capturing important regularities missed by the optimal MDL non-sparse predictors.