Zhiyu Liu

ZHiyu Liu, Haobo Geng, Xudong Wang et al.

We study the problem of recovering a low-tubal-rank tensor $\mathcal{X}\_\star\in \mathbb{R}^{n \times n \times k}$ from noisy linear measurements under the t-product framework. A widely adopted strategy involves factorizing the optimization variable as $\mathcal{U} * \mathcal{U}^\top$, where $\mathcal{U} \in \mathbb{R}^{n \times R \times k}$, followed by applying factorized gradient descent (FGD) to solve the resulting optimization problem. Since the tubal-rank $r$ of the underlying tensor $\mathcal{X}_\star$ is typically unknown, this method often assumes $r < R \le n$, a regime known as over-parameterization. However, when the measurements are corrupted by some dense noise (e.g., Gaussian noise), FGD with the commonly used spectral initialization yields a recovery error that grows linearly with the over-estimated tubal-rank $R$. To address this issue, we show that using a small initialization enables FGD to achieve a nearly minimax optimal recovery error, even when the tubal-rank $R$ is significantly overestimated. Using a four-stage analytic framework, we analyze this phenomenon and establish the sharpest known error bound to date, which is independent of the overestimated tubal-rank $R$. Furthermore, we provide a theoretical guarantee showing that an easy-to-use early stopping strategy can achieve the best known result in practice. All these theoretical findings are validated through a series of simulations and real-data experiments.

Zhiyu Liu, Zhi Han, Yandong Tang et al.

We consider the noisy matrix sensing problem in the over-parameterization setting, where the estimated rank $r$ is larger than the true rank $r_\star$ of the target matrix $X_\star$. Specifically, our main objective is to recover a matrix $ X_\star \in \mathbb{R}^{n_1 \times n_2} $ with rank $ r_\star $ from noisy measurements using an over-parameterized factorization $ LR^\top $, where $ L \in \mathbb{R}^{n_1 \times r}, \, R \in \mathbb{R}^{n_2 \times r} $ and $ \min\{n_1, n_2\} \ge r > r_\star $, with $ r_\star $ being unknown. Recently, preconditioning methods have been proposed to accelerate the convergence of matrix sensing problem compared to vanilla gradient descent, incorporating preconditioning terms $ (L^\top L + λI)^{-1} $ and $ (R^\top R + λI)^{-1} $ into the original gradient. However, these methods require careful tuning of the damping parameter $λ$ and are sensitive to step size. To address these limitations, we propose the alternating preconditioned gradient descent (APGD) algorithm, which alternately updates the two factor matrices, eliminating the need for the damping parameter $λ$ and enabling faster convergence with larger step sizes. We theoretically prove that APGD convergences to a near-optimal error at a linear rate. We further show that APGD can be extended to deal with other low-rank matrix estimation tasks, also with a theoretical guarantee of linear convergence. To validate the effectiveness and scalability of the proposed APGD, we conduct simulated and real-world experiments on a wide range of low-rank estimation problems, including noisy matrix sensing, weighted PCA, 1-bit matrix completion, and matrix completion. The extensive results demonstrate that APGD consistently achieves the fastest convergence and the lowest computation time compared to the existing alternatives.

Zhiyu Liu, Bo Wu, Jin Dai et al.

In future intelligent transportation systems, networked vehicles coordinate with each other to achieve safe operations based on an assumption that communications among vehicles and infrastructure are reliable. Traditional methods usually deal with the design of control systems and communication networks in a separated manner. However, control and communication systems are tightly coupled as the motions of vehicles will affect the overall communication quality. Hence, we are motivated to study the co-design of both control and communication systems. In particular, we propose a control theoretical framework for distributed motion planning for multi-agent systems which satisfies complex and high-level spatial and temporal specifications while accounting for communication quality at the same time. Towards this end, desired motion specifications and communication performances are formulated as signal temporal logic (STL) and spatial-temporal logic (SpaTeL) formulas, respectively. The specifications are encoded as constraints on system and environment state variables of mixed integer linear programs (MILP), and upon which control strategies satisfying both STL and SpaTeL specifications are generated for each agent by employing a distributed model predictive control (MPC) framework. Effectiveness of the proposed framework is validated by a simulation of distributed communication-aware motion planning for multi-agent systems.