$N$-Dimensional Tensor Completion for Nuclear Magnetic Resonance Relaxometry

Abstract

This paper deals with tensor completion for the solution of multidimensional inverse problems arising in nuclear magnetic resonance (NMR) relaxometry. We study the problem of reconstructing an approximately low-rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, nonuniform sampling strategies, and parameter selection methods are developed in this context. In particular, we derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property-based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by nonuniform sampling from a Parseval tight frame. The proposed algorithm is then applied to the setting of nuclear magnetic resonance relaxometry, for both simulated and experimental data. We compare our results with basis pursuit as well as with the state-of-the-art nonsubsampled data acquisition and reconstruction approach. Our experiments indicate that tensor recovery promises to significantly accelerate $N$-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments to be performed. Our methods could also be applied to other similar inverse problems arising in machine learning, signal and image processing, and computer vision.