From Event: SPIE Optical Engineering + Applications, 2017
We study matrix completion with non-uniform, deterministic sampling patterns. We introduce a computable parameter, which is a function of the sampling pattern, and show that if this parameter is small, then we may recover missing entries of the matrix, with appropriate weights. We theoretically analyze a simple and well-known recovery method, which simply projects the (zero-padded) subsampled matrix onto the set of low-rank matrices. We show that under non-uniform deterministic sampling, this method yields a biased solution, and we propose an algorithm to de-bias it. Numerical simulations demonstrate that de-biasing significantly improves the estimate. However, when the observations are noisy, the error of this method can be sub-optimal when the sampling is highly non-uniform. To remedy this, we suggest an alternative which is based on projection onto the max-norm ball whose robustness to noise tolerates arbitrarily non-uniform sampling. Finally, we analyze convex optimization in this framework.
Simon Foucart, Deanna Needell, Yaniv Plan, and Mary Wootters, "De-biasing low-rank projection for matrix completion," Proc. SPIE 10394, Wavelets and Sparsity XVII, 1039417 (Presented at SPIE Optical Engineering + Applications: August 08, 2017; Published: 24 August 2017); https://doi.org/10.1117/12.2275004.
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