Abstract
In [2], Blum et al. demonstrated the first sub-exponential algorithm for learning the parity function in the presence of noise. They solved the length-n parity problem in time 2O(n/logn) but it required the availability of 2O(n/logn) labeled examples. As an open problem, they asked whether there exists a 2o(n) algorithm for the length-n parity problem that uses only poly(n) labeled examples. In this work, we provide a positive answer to this question. We show that there is an algorithm that solves the length-n parity problem in time 2O(n/loglogn) using n 1 + ε labeled examples. This result immediately gives us a sub-exponential algorithm for decoding n × n 1 + ε random binary linear codes (i.e. codes where the messages are n bits and the codewords are n 1 + ε bits) in the presence of random noise. We are also able to extend the same techniques to provide a sub-exponential algorithm for dense instances of the random subset sum problem.
Research supported in part by NSF grant CCR-0093029.
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Lyubashevsky, V. (2005). The Parity Problem in the Presence of Noise, Decoding Random Linear Codes, and the Subset Sum Problem. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_32
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DOI: https://doi.org/10.1007/11538462_32
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