Non-Convex Tensor Recovery from Local Measurements

TL;DR

The paper tackles recovering a low tubal rank tensor from local, per-slice measurements, addressing a practical gap in tensor sensing for distributed or streaming data. It adopts a nonconvex tensor factorization ${\mathcal X}=\mathcal U*\mathcal V$ and develops a two-stage algorithm: Stage I provides a truncated spectral initialization, and Stage II performs a local search via alternating updates for $\mathcal U$ and $\mathcal V$, with two variants. The baseline Alt-PGD-Min achieves $\varepsilon$-accuracy in $\mathcal O(\kappa^2\log(1/\varepsilon))$ iterations with a detailed sample complexity, while Alt-ScalePGD-Min uses a preconditioned gradient to obtain a $\kappa$-independent rate of $\mathcal O(\log(1/\varepsilon))$ and improved sample complexity; this makes the method robust to ill-conditioning. The paper provides rigorous analysis (spectral initialization concentration, contraction of gradient steps, and perturbation bounds) and demonstrates the effectiveness of the approach on synthetic data and video sequences. Overall, the work advances local tensor CS by delivering nonconvex algorithms with near-optimal iteration and sample complexity, enabling scalable recovery in distributed and streaming applications.

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 $ε$-accuracy recovery with $\mathcal O\left( κ^2 \log \frac{1}ε\right)$ iteration complexity and $\mathcal O\left( κ^6rn_3\log n_3 \left( κ^2r\left(n_1 + n_2 \right) + n_1 \log \frac{1}ε\right) \right)$ sample complexity, where $κ$ denotes tensor condition number of $\boldsymbol{\mathcal X}^\star$. To further accelerate the convergence, especially when the tensor is ill-conditioned with large $κ$, 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 $κ$ independent iteration complexity $\mathcal O(\log \frac{1}ε)$ and improves the sample complexity to $\mathcal O\left( κ^4 rn_3 \log n_3 \left( κ^4r(n_1+n_2) + n_1 \log \frac{1}ε\right) \right)$. Experiments validate the effectiveness of the proposed methods.

Figures (9)

• Figure 1: Spectral initialization with $m_0=200,m_c=100$.
• Figure 2: Random initialization with $m_c=120$.
• Figure 3: Visualization of frame-7 in recovered videos.
• Figure 4: Quantitative comparison
• Figure 5: Comparison between LRcCS and Alt-PGD-Min under sufficiently large iterations 50000.

Theorems & Definitions (48)

• Definition 1
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