Non-Convex Tensor Recovery from Local Measurements

TL;DR

This paper introduces a novel nonconvex tensor recovery method from local measurements, leveraging low tubal rank structure and proposing algorithms with provable convergence and sample complexity guarantees.

Contribution

It develops a new tensor compressed sensing model and algorithms with theoretical guarantees for recovery from local measurements, improving efficiency and robustness.

Findings

Achieves $ ext{epsilon}$-accuracy with logarithmic iteration complexity.

Provides sample complexity bounds dependent on tensor condition number.

Validates effectiveness through experiments.

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 ${\boldsymbol {\mathcal X}}^\star$ 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 \textsf{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 \textsf{Alt-PGD-Min} achieves $\epsilon$-accuracy recovery with $\mathcal O\left( \kappa^2 \log \frac{1}{\epsilon}\right)$ iteration complexity and $\mathcal O\left( \kappa^6rn_3\log n_3 \left( \kappa^2r\left(n_1 + n_2 \right) + n_1 \log \frac{1}{\epsilon}\right) \right)$ sample complexity, where $\kappa$ denotes tensor condition number of $\boldsymbol{\mathcal X}^\star$. To further accelerate the convergence, especially when the tensor is ill-conditioned with large $\kappa$, we prove \textsf{Alt-ScalePGD-Min} that preconditions the gradient update using an approximate Hessian that can be computed efficiently. We show that \textsf{Alt-ScalePGD-Min} achieves $\kappa$ independent iteration complexity $\mathcal O(\log \frac{1}{\epsilon})$ and improves the sample complexity to $\mathcal O\left( \kappa^4 rn_3 \log n_3 \left( \kappa^4r(n_1+n_2) + n_1 \log \frac{1}{\epsilon}\right) \right)$. Experiments validate the effectiveness of the proposed methods.