On trimming tensor-structured measurements and efficient low-rank tensor recovery

In this paper, we take a step towards developing efficient hard thresholding methods for low-rank tensor recovery from memory-efficient linear measurements with tensorial structure. Theoretical guarantees for many standard iterative low-rank recovery methods, such as iterative hard thresholding (IHT), are based on model assumptions on the measurement operator, like the restricted isometry property (RIP). However, tensor-structured random linear maps -- while memory-efficient and convenient to apply -- lack good restricted isometry properties; that is, they do not preserve the norms of low-rank tensors sufficiently well. To address this, we propose local trimming techniques that provably restore point-wise geometry-preservation properties of tensor-structured maps, making them comparable to those of unstructured sub-Gaussian measurements. Then, we propose two novel versions of tensor IHT algorithms: an adaptive gradient trimming algorithm and a randomized Kaczmarz-based IHT algorithm, that efficiently recover low-rank tensors from linear measurements. We provide initial theoretical guarantees for the proposed methods and present numerical experiments on real and synthetic data, highlighting their efficiency over the original TensorIHT for low HOSVD and CP-rank tensors.