Xiangyu Yang

Hao Wang, Ye Wang, Xiangyu Yang

This paper considers the problem of minimizing the sum of a smooth function and the Schatten-$p$ norm of the matrix. Our contribution involves proposing accelerated iteratively reweighted nuclear norm methods designed for solving the nonconvex low-rank minimization problem. Two major novelties characterize our approach. Firstly, the proposed method possesses a rank identification property, enabling the provable identification of the "correct" rank of the stationary point within a finite number of iterations. Secondly, we introduce an adaptive updating strategy for smoothing parameters. This strategy automatically fixes parameters associated with zero singular values as constants upon detecting the "correct" rank while quickly driving the rest of the parameters to zero. This adaptive behavior transforms the algorithm into one that effectively solves smooth problems after a few iterations, setting our work apart from existing iteratively reweighted methods for low-rank optimization. We prove the global convergence of the proposed algorithm, guaranteeing that every limit point of the iterates is a critical point. Furthermore, a local convergence rate analysis is provided under the Kurdyka-Łojasiewicz property. We conduct numerical experiments using both synthetic and real data to showcase our algorithm's efficiency and superiority over existing methods.

Xiangyu Yang, Jiashan Wang, Hao Wang

This paper primarily focuses on computing the Euclidean projection of a vector onto the $\ell_{p}$ ball in which $p\in(0,1)$. Such a problem emerges as the core building block in statistical machine learning and signal processing tasks because of its ability to promote the sparsity of the desired solution. However, efficient numerical algorithms for finding the projections are still not available, particularly in large-scale optimization. To meet this challenge, we first derive the first-order necessary optimality conditions of this problem. Based on this characterization, we develop a novel numerical approach for computing the stationary point by solving a sequence of projections onto the reweighted $\ell_{1}$-balls. This method is practically simple to implement and computationally efficient. Moreover, the proposed algorithm is shown to converge uniquely under mild conditions and has a worst-case $O(1/\sqrt{k})$ convergence rate. Numerical experiments demonstrate the efficiency of our proposed algorithm.

Xiangyu Yang, Sheng Hua, Yuanming Shi et al.

With the rapid upsurge of deep learning tasks at the network edge, effective edge artificial intelligence (AI) inference becomes critical to provide low-latency intelligent services for mobile users via leveraging the edge computing capability. In such scenarios, energy efficiency becomes a primary concern. In this paper, we present a joint inference task selection and downlink beamforming strategy to achieve energy-efficient edge AI inference through minimizing the overall power consumption consisting of both computation and transmission power consumption, yielding a mixed combinatorial optimization problem. By exploiting the inherent connections between the set of task selection and group sparsity structural transmit beamforming vector, we reformulate the optimization as a group sparse beamforming problem. To solve this challenging problem, we propose a log-sum function based three-stage approach. By adopting the log-sum function to enhance the group sparsity, a proximal iteratively reweighted algorithm is developed. Furthermore, we establish the global convergence analysis and provide the ergodic worst-case convergence rate for this algorithm. Simulation results will demonstrate the effectiveness of the proposed approach for improving energy efficiency in edge AI inference systems.