Phase transition and higher order analysis of Lq regularization under dependence.
Phase transition and higher order analysis of Lq regularization under dependence.
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依赖性下 Lq 正则化的相变和高阶分析。
DOI：
10.1093/imaiai/iaae005
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发表时间：
2024
期刊：
影响因子：
--
通讯作者：
Yang,Qinglong
中科院分区：
文献类型：
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作者：
Huang,Hanwen;Zeng,Peng;Yang,Qinglong
本文研究了在此模型下，由一组噪声观测估计a-稀疏信号的问题，其中观测矩阵的行由分布抽取。我们考虑一类正则化最小二乘（LQLS）的制定，其中表示的范数。在和固定的情况下，我们得到了任意协方差矩阵的渐近风险，推广了标准高斯设计的已有结果，即。结果来自非严格的复制品方法。我们在小误差范围内对LQLS进行了高阶分析，其中第一个主导项可以用来确定LQLS的相变行为。我们的结果表明，在和的情况下，第一个主导项不依赖于的协方差结构，这表明预测变量之间的相关性只影响相变曲线。拉索。为了研究和情形下的协方差结构对LQLS性能的影响，我们导出了渐近风险的小误差展开式中第二个主导项的显式公式。大量的计算实验证实了我们的分析预测与数值结果是一致的。
We study the problem of estimating a-sparse signalfrom a set of noisy observationsunder the model, whereis the measurement matrix the row of which is drawn from distribution. We consider the class of-regularized least squares (LQLS) given by the formulation, wheredenotes the-norm. In the settingwith fixedand, we derive the asymptotic risk offor arbitrary covariance matrixthat generalizes the existing results for standard Gaussian design, i.e.. The results were derived from the non-rigorous replica method. We perform a higher-order analysis for LQLS in the small-error regime in which the first dominant term can be used to determine the phase transition behavior of LQLS. Our results show that the first dominant term does not depend on the covariance structure ofin the casesandwhich indicates that the correlations among predictors only affect the phase transition curve in the casea.k.a. LASSO. To study the influence of the covariance structure ofon the performance of LQLS in the casesand, we derive the explicit formulas for the second dominant term in the expansion of the asymptotic risk in terms of small error. Extensive computational experiments confirm that our analytical predictions are consistent with numerical results.