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Abstract:
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We consider the problem of estimating the low-rank matrice when corrupted by a random matrix when there is a prior structure for the principal components. When the noise matrix has an entry-wise iid Gaussian distribution, the Approximate Message Passing algorithm can achieve the Bayes-optimal accuracy under certain regimes. However, for real-world data, the entry-wise iid Gaussian distribution for the noise matrix assumption may be a rough approximation, suggesting the possibility for further improvement using methods that consider more general noise matrices.
In this work, we study the AMP algorithm for low-rank matrices estimation problems where the data is corrupted by orthogonal rotationally invariant noise. The algorithm extends the base AMP algorithms that require an oracle initialization correlated but independent of the ground-truth spikes and allows for spectral initializations. For sufficiently large signal strengths and strong prior structures, the algorithm can substantially outperform the simple spectral estimates.
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