Provable Near-Optimal Low-Multilinear-Rank Tensor Recovery
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We consider the problem of recovering a low-multilinear-rank tensor from a small amount of linear measurements. We show that the Riemannian gradient algorithm initialized by one step of iterative hard thresholding can reconstruct an order-d tensor of size n×…× n and multilinear rank (r,…,r) with high probability from only O(nr^2 + r^d+1) measurements, assuming d is a constant. This sampling complexity is optimal in n, compared to existing results whose sampling complexities are all unnecessarily large in n. The analysis relies on the tensor restricted isometry property (TRIP) and the geometry of the manifold of all tensors with a fixed multilinear rank. High computational efficiency of our algorithm is also achieved by doing higher order singular value decomposition on intermediate small tensors of size only 2r×…× 2r rather than on tensors of size n×…× n as usual.
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