Low-rank tensor recovery using sequentially optimal modal projections in iterative hard thresholding (SEMPIHT)

Abstract : Iterative hard thresholding (IHT) is a simple and effective approach to parsimonious data recovery. Its multilinear rank (mrank)-based application to low-rank tensor recovery (LRTR) is especially valuable given the difficulties involved in this problem. In this paper, we propose a novel IHT algorithm for LRTR, choosing sequential per-mode SVD truncation as its thresholding operator. This operator is less costly than those used in existing IHT algorithms for LRTR, and often leads to superior performance. Furthermore, by exploiting the sequential optimality of the employed modal projections, we derive recovery guarantees relying on restricted isometry constants. Though these guarantees are suboptimal, our numerical studies indicate that a quasi-optimal number of Gaussian measurements suffices for perfect data reconstruction. We also investigate a continuation technique which yields a sequence of progressively more complex estimated models until attaining a target mrank. When recovering real-world data, this strategy stabilizes the estimation error and can also accelerate convergence. In tensor completion, in particular, it can cope with nonideal characteristics of the sensed tensors and so is crucial for achieving a satisfactory performance. Extensive numerical experiments are reported, including the completion of hyperspectral imaging data and comparisons with several other existing approaches.
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SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2017, 39 (3), pp.A860-A889. 〈10.1137/16M1062089〉
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José Henrique De Morais Goulart, Gérard Favier. Low-rank tensor recovery using sequentially optimal modal projections in iterative hard thresholding (SEMPIHT). SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2017, 39 (3), pp.A860-A889. 〈10.1137/16M1062089〉. 〈hal-01387529v2〉

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