We present a novel approach for recovering a sparse signal from quadratic measurements corresponding to a rank-one tensorization of the data vector. Such quadratic measurements, referred to as interferometric or cross-correlated data, naturally arise in many fields such as remote sensing, spectroscopy, holography and seismology. Compared to the sparse signal recovery problem that uses linear measurements, the unknown in this case is a matrix formed by the cross correlations of the sought signal. This creates a bottleneck for the inversion since the number of unknowns grows quadratically with the dimension of the signal. The main idea of the proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal of the unknown matrix, whose dimension grows linearly with the size of the signal, and use an efficient Noise Collector to absorb the cross-correlated data that come from the off-diagonal elements of this matrix. These elements do not carry extra information about the support of the signal, but significantly contribute to these data. With this strategy, we recover the unknown matrix by solving a convex linear problem whose cost is similar to the one that uses linear measurements. Our theory shows that the proposed approach provides exact support recovery when the data is not too noisy, and that there are no false positives for any level of noise. It also demonstrates that the level of sparsity that can be recovered scales almost linearly with the number of data. The numerical experiments presented in the paper corroborate these findings.

中文翻译：

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从二次测量中快速恢复信号


我们提出了一种新颖的方法，用于从与数据向量的一阶张量化相对应的二次测量中恢复稀疏信号。这种二次测量，称为干涉或互相关数据，自然出现在许多领域，例如遥感、光谱学、全息术和地震学。与使用线性测量的稀疏信号恢复问题相比，这种情况下的未知数是由所寻找信号的互相关形成的矩阵。这造成了反演的瓶颈，因为未知数的数量随着信号的维度呈二次方增长。该方法的主要思想是通过仅恢复未知矩阵的对角线来降低问题的维数，其维数随着信号的大小线性增长，并使用高效的噪声收集器来吸收互相关数据来自该矩阵的非对角元素。这些元素不携带有关信号支持的额外信息，但对这些数据有显着贡献。通过这种策略，我们通过解决凸线性问题来恢复未知矩阵，其成本与使用线性测量的问题相似。我们的理论表明，当数据噪声不太大时，所提出的方法可以提供精确的支持恢复，并且对于任何级别的噪声都不会出现误报。它还表明，可以恢复的稀疏性水平几乎与数据数量成线性比例。论文中提出的数值实验证实了这些发现。