Ping Li

Chenglin Fan, Ping Li, Hanyu Peng

The densest subgraph of a large graph usually refers to some subgraph with the highest average degree, which has been extended to the family of $p$-means dense subgraph objectives by~\citet{veldt2021generalized}. The $p$-mean densest subgraph problem seeks a subgraph with the highest average $p$-th-power degree, whereas the standard densest subgraph problem seeks a subgraph with a simple highest average degree. It was shown that the standard peeling algorithm can perform arbitrarily poorly on generalized objective when $p>1$ but uncertain when $0<p<1$. In this paper, we are the first to show that a standard peeling algorithm can still yield $2^{1/p}$-approximation for the case $0<p < 1$. (Veldt 2021) proposed a new generalized peeling algorithm (GENPEEL), which for $p \geq 1$ has an approximation guarantee ratio $(p+1)^{1/p}$, and time complexity $O(mn)$, where $m$ and $n$ denote the number of edges and nodes in graph respectively. In terms of algorithmic contributions, we propose a new and faster generalized peeling algorithm (called GENPEEL++ in this paper), which for $p \in [1, +\infty)$ has an approximation guarantee ratio $(2(p+1))^{1/p}$, and time complexity $O(m(\log n))$, where $m$ and $n$ denote the number of edges and nodes in graph, respectively. This approximation ratio converges to 1 as $p \rightarrow \infty$.

Yunfeng Cai, Ping Li

In recent years, low-rank tensor completion (LRTC) has received considerable attention due to its applications in image/video inpainting, hyperspectral data recovery, etc. With different notions of tensor rank (e.g., CP, Tucker, tensor train/ring, etc.), various optimization based numerical methods are proposed to LRTC. However, tensor network based methods have not been proposed yet. In this paper, we propose to solve LRTC via tensor networks with a Tucker wrapper. Here by "Tucker wrapper" we mean that the outermost factor matrices of the tensor network are all orthonormal. We formulate LRTC as a problem of solving a system of nonlinear equations, rather than a constrained optimization problem. A two-level alternative least square method is then employed to update the unknown factors. The computation of the method is dominated by tensor matrix multiplications and can be efficiently performed. Also, under proper assumptions, it is shown that with high probability, the method converges to the exact solution at a linear rate. Numerical simulations show that the proposed algorithm is comparable with state-of-the-art methods.