基于凸规划的高鲁棒性误差校正
Highly robust error correction by convex programming
摘要：
本文讨论了一个程式化的通信问题，其中一个人希望发送一个实值信号x在R^n（一个块的n条信息）到远程接收器。我们问它是否有可能可靠地传输此信息时，传输的码字的一部分被任意的粗差损坏，以及当此外，码字的所有条目被较小的错误（例如量化误差）污染。我们表明，如果一个编码的信息为Ax，其中A是一个合适的m × n的编码矩阵（m >= n），有两个解码方案，允许恢复块的n条信息x几乎相同的精度，如果没有发生重大错误传输（或等效的，如果一个有一个预言提供完美的信息的网站和幅度的重大错误）。此外，这两种解码策略都是非常具体的，只涉及解决简单的凸优化程序，无论是线性规划或二阶锥规划。我们补充我们的研究与数值模拟表明，编码器/解码器对执行非常好。
英文摘要：
This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x in R^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably when a fraction of the transmitted codeword is corrupted by arbitrary gross errors, and when in addition, all the entries of the codeword are contaminated by smaller errors (e.g. quantization errors). We show that if one encodes the information as Ax where A is a suitable m by n coding matrix (m >= n), there are two decoding schemes that allow the recovery of the block of n pieces of information x with nearly the same accuracy as if no gross errors occur upon transmission (or equivalently as if one has an oracle supplying perfect information about the sites and amplitudes of the gross errors). Moreover, both decoding strategies are very concrete and only involve solving simple convex optimization programs, either a linear program or a second-order cone program. We complement our study with numerical simulations showing that the encoder/decoder pair performs remarkably well.
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