各向同性随机正交矩阵：LASSO的性能和最小圆锥奇异值
Isotropically Random Orthogonal Matrices: Performance of LASSO and Minimum Conic Singular Values
摘要：
最近，广义LASSO算法从压缩噪声测量中恢复结构化信号的精确性能，通过i.i.d.高斯矩阵，已被特征化。该分析是基于一个框架引入Stojnic和严重依赖于使用戈登的高斯最小-最大定理（GMT），高斯过程的比较原则。其结果是，相应的表征测量矩阵的其他合奏尚未开发。在这项工作中，我们分析了各向同性随机正交（i.r.o.）测量.我们考虑广义LASSO的约束版本，并推导出其归一化平方误差在大系统极限的尖锐表征。当它的高斯对应相比，我们的结果分析证实了在性能的i.r.o.合奏。我们的第二个结果，得出了一个渐近下界的最小锥奇异值的i.r.o.矩阵这个界大于高斯矩阵的相应界。为了证明我们的结果，我们表示i.r.o.矩阵的高斯和显示，经过一些修改，GMT框架仍然适用。
英文摘要：
Recently, the precise performance of the Generalized LASSO algorithm for recovering structured signals from compressed noisy measurements, obtained via i.i.d. Gaussian matrices, has been characterized. The analysis is based on a framework introduced by Stojnic and heavily relies on the use of Gordon's Gaussian min-max theorem (GMT), a comparison principle on Gaussian processes. As a result, corresponding characterizations for other ensembles of measurement matrices have not been developed. In this work, we analyze the corresponding performance of the ensemble of isotropically random orthogonal (i.r.o.) measurements. We consider the constrained version of the Generalized LASSO and derive a sharp characterization of its normalized squared error in the large-system limit. When compared to its Gaussian counterpart, our result analytically confirms the superiority in performance of the i.r.o. ensemble. Our second result, derives an asymptotic lower bound on the minimum conic singular values of i.r.o. matrices. This bound is larger than the corresponding bound on Gaussian matrices. To prove our results we express i.r.o. matrices in terms of Gaussians and show that, with some modifications, the GMT framework is still applicable.
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