A Single-Mode Quasi Riemannian Gradient Descent Algorithm for Low-Rank Tensor Recovery

TL;DR

This paper focuses on recovering a low-rank tensor from its incomplete measurements by exploiting the benefits of both fixed-rank matrix tangent space projection in Riemannian gradient descent and sequentially truncated high-order singular value decomposition (ST-HOSVD).

Abstract

This paper focuses on recovering a low-rank tensor from its incomplete measurements. We propose a novel algorithm termed the Single Mode Quasi Riemannian Gradient Descent (SM-QRGD). By exploiting the benefits of both fixed-rank matrix tangent space projection in Riemannian gradient descent and sequentially truncated high-order singular value decomposition (ST-HOSVD), SM-QRGD achieves a much faster convergence speed than existing state-of-the-art algorithms. Theoretically, we establish the convergence of SM-QRGD through the Tensor Restricted Isometry Property (TRIP) and the geometry of the fixed-rank matrix manifold. Numerically, extensive experiments are conducted, affirming the accuracy and efficacy of the proposed algorithm.

Figures (5)

• Figure 1: Illustration for the SM-QRGD Algorithm.
• Figure 1: (a): The phase transition plot for varying rank $r$ and sampling rate $\rho$ when $n = 100$. (b): The CPU times of SM-QRGD and SeMPIHT v.s. relative error under different signal-to-noise ratios when $n = 100, r = 3$ and $\rho = 0.3$.
• Figure 2: Results for different modes tangent space projection with $n = 100, r_1 = 10, r_2 = 20, r_3 = 30$ and $\rho = 0.3$.
• Figure 3: Results of SM-QRGD under different condition number
• Figure 4: Comparisons with other algorithms

Theorems & Definitions (20)

• Definition 2.1: Quasi-projection property of low-multilinear-rank tensor map $\hat{\mathscr{P}}_{\Theta}$
• Proposition 2.2: Quasi-projection property of T-HOSVD and ST-HOSVD
• Lemma 3.1: Projection onto Substitution Tangent Space
• Proof 1
• Definition 3.2: Tensor First-mode Restricted Isometry Property
• Remark 3.3
• Theorem 3.4: Recovery guarantee with $\alpha = 1$
• Theorem 3.5: Recovery guarantee with normalized step size
• Remark 3.6
• Lemma 6.1