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Recovering low-rank matrices from binary measurements

  • * Corresponding author: Simon Foucart

    * Corresponding author: Simon Foucart 
The first author is partially supported by NSF grants DMS-1622134 and DMS-1664803.
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  • This article studies the approximate recovery of low-rank matrices acquired through binary measurements. Two types of recovery algorithms are considered, one based on hard singular value thresholding and the other one based on semidefinite programming. In case no thresholds are introduced before binary quantization, it is first shown that the direction of the low-rank matrices can be well approximated. Then, in case nonadaptive thresholds are incorporated, it is shown that both the direction and the magnitude can be well approximated. Finally, by allowing the thresholds to be chosen adaptively, we exhibit a recovery procedure for which low-rank matrices are fully approximated with error decaying exponentially with the number of binary measurements. In all cases, the procedures are robust to prequantization error. The underlying arguments are essentially deterministic: they rely only on an unusual restricted isometry property of the measurement process, which is established once and for all for Gaussian measurement processes.

    Mathematics Subject Classification: 94A12, 65F10, 90C22.


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  • Figure 1.  Comparison of the procedures (25) and (39) when the dimension $ n $ varies

    Figure 2.  Recovery error $ \varepsilon: = \left\| {\mathbf X} - \dfrac{ {\mathbf X}^{\rm ht}}{\| {\mathbf X}^{\rm ht}\|_F} \right\|_F $ vs magnitude $ \mu: = \| {\mathbf e}\|_1 $ of prequantization error

    Figure 3.  Recovery error $ \varepsilon: = \| {\mathbf X} - \widehat{ {\mathbf X}}\|_F $ vs oversampling factor $ \lambda $ for the procedure (62)

    Figure 4.  Recovery error $ \varepsilon: = \| {\mathbf X} - \widehat{ {\mathbf X}}\|_F $ vs oversampling factor $ \lambda $ for the procedure (73)-(74)-(75)

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