Algorithmic linear dimension reduction in the l_1 norm for sparse vectors
<|reference_start|>Algorithmic linear dimension reduction in the l_1 norm for sparse vectors: This paper develops a new method for recovering m-sparse signals that is simultaneously uniform and quick. We present a reconstruction algorithm whose run time, O(m log^2(m) log^2(d)), is sublinear in the length d of the signal. The reconstruction error is within a logarithmic factor (in m) of the optimal m-term approximation error in l_1. In particular, the algorithm recovers m-sparse signals perfectly and noisy signals are recovered with polylogarithmic distortion. Our algorithm makes O(m log^2 (d)) measurements, which is within a logarithmic factor of optimal. We also present a small-space implementation of the algorithm. These sketching techniques and the corresponding reconstruction algorithms provide an algorithmic dimension reduction in the l_1 norm. In particular, vectors of support m in dimension d can be linearly embedded into O(m log^2 d) dimensions with polylogarithmic distortion. We can reconstruct a vector from its low-dimensional sketch in time O(m log^2(m) log^2(d)). Furthermore, this reconstruction is stable and robust under small perturbations.<|reference_end|>
@article{gilbert2006algorithmic,
    title={Algorithmic linear dimension reduction in the l_1 norm for sparse
  vectors},
    author={A. C. Gilbert, M. J. Strauss, J. A. Tropp, and R. Vershynin},
    journal={arXiv preprint arXiv:cs/0608079},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0608079},
    primaryClass={cs.DS}
}