3,028 research outputs found

    Subspace Evolution and Transfer (SET) for Low-Rank Matrix Completion

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    We describe a new algorithm, termed subspace evolution and transfer (SET), for solving low-rank matrix completion problems. The algorithm takes as its input a subset of entries of a low-rank matrix, and outputs one low-rank matrix consistent with the given observations. The completion task is accomplished by searching for a column space on the Grassmann manifold that matches the incomplete observations. The SET algorithm consists of two parts -- subspace evolution and subspace transfer. In the evolution part, we use a gradient descent method on the Grassmann manifold to refine our estimate of the column space. Since the gradient descent algorithm is not guaranteed to converge, due to the existence of barriers along the search path, we design a new mechanism for detecting barriers and transferring the estimated column space across the barriers. This mechanism constitutes the core of the transfer step of the algorithm. The SET algorithm exhibits excellent empirical performance for both high and low sampling rate regimes

    Alternating Least-Squares for Low-Rank Matrix Reconstruction

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    For reconstruction of low-rank matrices from undersampled measurements, we develop an iterative algorithm based on least-squares estimation. While the algorithm can be used for any low-rank matrix, it is also capable of exploiting a-priori knowledge of matrix structure. In particular, we consider linearly structured matrices, such as Hankel and Toeplitz, as well as positive semidefinite matrices. The performance of the algorithm, referred to as alternating least-squares (ALS), is evaluated by simulations and compared to the Cram\'er-Rao bounds.Comment: 4 pages, 2 figure
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