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The Generalized Lasso With Non-Linear Observations

机译：具有非线性观测的广义套索

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摘要

We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that the non-linear observations may be treated as noisy linear observations, and thus, the signal may be estimated using the generalized Lasso. This is appealing because of the abundance of efficient, specialized solvers for this program. Just as noise may be diminished by projecting onto the lower dimensional space, the error from modeling non-linear observations with linear observations will be greatly reduced when using the signal structure in the reconstruction. We allow general signal structure, only assuming that the signal belongs to some set . We consider the single-index model of non-linearity. Our theory allows the non-linearity to be discontinuous, not one-to-one and even unknown. We assume a random Gaussian model for the measurement matrix, but allow the rows to have an unknown covariance matrix. As special cases of our results, we recover near-optimal theory for noisy linear observations, and also give the first theoretical accuracy guarantee for 1-b compressed sensing with unknown covariance matrix of the measurement vectors.
机译：当信号属于掩埋在高维空间中的低维集合时，我们将从非线性观测中研究信号估计问题。在实践中经常使用的粗略启发式假设将非线性观测结果视为有噪声的线性观测值，因此，可以使用广义的套索估计信号。由于该程序有大量高效，专业的求解器，因此具有吸引力。正如可以通过投影到低维空间来减少噪声一样，在重建中使用信号结构时，使用线性观测对非线性观测建模的误差也会大大降低。仅假设信号属于某个集合，我们才允许使用一般的信号结构。我们考虑非线性的单指标模型。我们的理论允许非线性是不连续的，不是一对一的，甚至是未知的。我们假设测量矩阵为随机高斯模型，但允许行具有未知的协方差矩阵。作为结果的特例，我们恢复了有噪线性观测的近似最佳理论，并为测量向量未知协方差矩阵的1-b压缩传感提供了第一个理论精度保证。

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