Measurement Manipulation of the Matrix Sensing Problem to Improve Optimization Landscape
Keywords: non-convex optimization, low-rank matrix optimization, matrix sensing, preconditioning algorithm
Abstract: This work studies the matrix sensing (MS) problem through the lens of the Restricted Isometry Property (RIP). It has been shown in several recent papers that two different techniques of convex relaxations and local search methods for the MS problem both require the RIP constant to be less than 0.5 while most real-world problems have their RIPs close to 1. The existing literature guarantees a small RIP constant only for sensing operators having an i.i.d. Gaussian distribution, and it is well-known that the MS problem could have a complicated landscape when the RIP is greater than 0.5. In this work, we address this issue and improve the optimization landscape by developing two results. First, we show that any sensing operator with a model not too distant from i.i.d. Gaussian has a slightly higher RIP than i.i.d. Gaussian, and that its RIP constant can be reduced to match the RIP constant of an i.i.d. Gaussian via slightly increasing the number of measurements. Second, we show that if the sensing operator has an arbitrary distribution, it can be modified in such a way that the resulting operator will act as a perturbed Gaussian with a lower RIP constant. Our approach is a preconditioning technique that replaces each sensing matrix with a weighted sum of all sensing matrices. We numerically demonstrate that the RIP constants for different distributions can be reduced from almost 1 to less than 0.5 via the preconditioning of the sensing operator.
Primary Area: optimization
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Submission Number: 6103
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