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首页> 外文期刊>Information Theory, IEEE Transactions on >Exact Recoverability From Dense Corrupted Observations via $ell _{1}$-Minimization
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Exact Recoverability From Dense Corrupted Observations via $ell _{1}$-Minimization

机译：通过$ ell _ {1} $-最小化，从密集的损坏观测中获得精确的可恢复性

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摘要

This paper confirms a surprising phenomenon first observed by Wright under a different setting: given $m$ highly corrupted measurements $y = A_{Omega bullet } x^{star } + e^{star }$, where $A_{Omega bullet }$ is a submatrix whose rows are selected uniformly at random from rows of an orthogonal matrix $A$ and $e^{star }$ is an unknown sparse error vector whose nonzero entries may be unbounded, we show that with high probability, $ell _{1}$-minimization can recover the sparse signal of interest $x^{star }$ exactly from only $m = C mu ^{2} k (log n)^{2}$, where $k$ is the number of nonzero components of $x^{star }$ and $mu = n max _{ij} A_{ij}^{2}$, even if a significant fraction of the measurements are corrupted. We further guarantee that stable recovery is possible when measurements are polluted by both gross sparse and small dense errors: $y = A_{Omega bullet } x^{star } + e^{star }+ nu $, where $nu $ is the small dense noise with bounded energy. Numerous simulation results under various settings are also presented to verify the validity of the theory as well as to illustrate the promising pot- ntial of the proposed framework.
机译：本文证实了赖特首先在不同设置下观察到的令人惊讶的现象：给定$ m $高度损坏的测量值$ y = A_ {Omega bullet} x ^ {star} + e ^ {star} $，其中$ A_ {Omega bullet} $是一个子矩阵，其行是从正交矩阵$ A $的行中随机选择的，而$ e ^ {star} $是未知的稀疏误差向量，其非零项可能是无界的，我们证明了，很有可能$ ell _ {1} $最小化可以仅从$ m = C mu ^ {2} k（log n）^ {2} $中准确地恢复稀疏的感兴趣信号$ x ^ {star} $，其中$ k $是$ x ^ {star} $和$ mu = n max _ {ij} A_ {ij} ^ {2} $的非零分量的数量，即使很大一部分测量结果已损坏。我们进一步保证，当测量值同时受到总的稀疏误差和小的密集误差的污染时，可以实现稳定的恢复：$ y = A_ {Omega bullet} x ^ {star} + e ^ {star} + nu $，其中$ nu $是有限能量的小密集噪声。还提供了各种设置下的大量仿真结果，以验证该理论的有效性以及说明所提出框架的潜力。

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