Low noise sensitivity analysis of Lq-minimization in oversampled systems
Low noise sensitivity analysis of Lq-minimization in oversampled systems
复制标题
过采样系统中 Lq 最小化的低噪声灵敏度分析
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
A. Maleki
中科院分区：
文献类型：
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作者：
Haolei Weng;A. Maleki
本文考虑了一类$ellq $-正则化最小二乘（LQLS）来估计$eta在mathbb{R}^p$从它的$n$噪声线性观测$y=Xeta + w$.这些计划的性能进行了研究下的高维渐近设置，其中的维度的信号增长与测量的数量呈线性。在这种渐近设置中，相变（PT）图通常用于比较不同估计器的性能。PT指定某个估计器从其无噪声线性观测恢复结构化信号（例如稀疏信号）所需的最小观测数。尽管PT分析为压缩感知提供了有用的信息，但它忽略了测量噪声的事实不仅限制了其在许多应用领域的适用性，而且可能导致误解。例如，考虑一个线性回归问题，其中$n>p$，信号并不完全稀疏。如果在这样的系统中忽略测量噪声，则正则化技术（如LQLS）似乎无关紧要，因为即使是普通最小二乘（OLS）也会返回精确解。然而，众所周知，如果$n$并不比$p$大得多，那么正则化技术可以提高OLS的性能。针对PT分析的这一局限性，我们考虑了低噪声灵敏度分析。我们表明，这种分析框架（i）揭示了LQLS的优势OLS，（ii）捕捉不同的LQLS估计之间的差异，即使当$n>p$，和（iii）提供了一个公平的比较不同的估计在高信噪比。作为这个框架的应用，我们将表明，在温和的条件下，LASSO优于其他LQLS，即使当信号是密集的。最后，通过一个简单的变换，我们将我们的低噪声灵敏度框架连接到经典的渐近状态，其中$n/p
ightarrow infty$，并描述如何以及何时正则化技术提供普通最小二乘法的改进，以及当样本量很大时，哪个正则化器提供了最大的改进。
The class of $ell _q$-regularized least squares (LQLS) are considered for estimating $eta in mathbb{R}^p$ from its $n$ noisy linear observations $y=Xeta + w$. The performance of these schemes are studied under the high-dimensional asymptotic setting in which the dimension of the signal grows linearly with the number of measurements. In this asymptotic setting, phase transition (PT) diagrams are often used for comparing the performance of different estimators. PT specifies the minimum number of observations required by a certain estimator to recover a structured signal, e.g. a sparse one, from its noiseless linear observations. Although PT analysis is shown to provide useful information for compressed sensing, the fact that it ignores the measurement noise not only limits its applicability in many application areas, but also may lead to misunderstandings. For instance, consider a linear regression problem in which $n>p$ and the signal is not exactly sparse. If the measurement noise is ignored in such systems, regularization techniques, such as LQLS, seem to be irrelevant since even the ordinary least squares (OLS) returns the exact solution. However, it is well known that if $n$ is not much larger than $p$, then the regularization techniques improve the performance of OLS. In response to this limitation of PT analysis, we consider the low-noise sensitivity analysis. We show that this analysis framework (i) reveals the advantage of LQLS over OLS, (ii) captures the difference between different LQLS estimators even when $n>p$, and (iii) provides a fair comparison among different estimators in high signal-to-noise ratios. As an application of this framework, we will show that under mild conditions LASSO outperforms other LQLS even when the signal is dense. Finally, by a simple transformation, we connect our low-noise sensitivity framework to the classical asymptotic regime in which $n/p
ightarrow infty$, and characterize how and when regularization techniques offer improvements over ordinary least squares, and which regularizer gives the most improvement when the sample size is large.
影响因子：
1.6
作者：
Amelunxen, Dennis;Lotz, Martin;Tropp, Joel A.
通讯作者：
Tropp, Joel A.