Non-Convex Tensor Recovery from Local Measurements

Abstract Motivated by the settings where sensing the entire tensor is infeasible, this paper proposes a novel tensor compressed sensing model, where measurements are only obtained from sensing each lateral slice via mutually independent matrices. Leveraging the low tubal rank structure, we reparameterize the unknown tensor ?* using two compact tensor factors and formulate the recovery problem as a nonconvex minimization problem. To solve the problem, we first propose an alternating minimization algorithm, termed Alt-PGD-Min, that iteratively optimizes the two factors using a projected gradient descent and an exact minimization step, respectively. Despite nonconvexity, we prove that Alt-PGD-Min achieves ϵ-accuracy recovery with ?(?²log1/?) iteration complexity and ?(?⁶rn₃logn₃(?²r(n₁+n₂)+n₁log1/ε)) sample complexity, where ? denotes tensor condition number of ?*. To further accelerate the convergence, especially when the tensor is ill-conditioned with large ?, we prove Alt-ScalePGD-Min that preconditions the gradient update using an approximate Hessian that can be computed efficiently. We show that Alt-ScalePGD-Min achieves ? independent iteration complexity ?(log1/ε) and improves the sample complexity to ?(?⁴rn₃log n₃(?⁴ r(n₁ + n₂)+n₁log 1/ε)). Experiments validate the effectiveness of the proposed methods.