The objective of motion capture (mocap) data completion is to recover missing measurement of the body markers from mocap. It becomes increasingly challenging as the missing ratio and duration of mocap data grow. Traditional approaches usually recast this problem as a low-rank matrix approximation problem based on the nuclear norm. However, the nuclear norm defined as the sum of all the singular values of a matrix is not a good approximation to the rank of mocap data. This paper proposes a novel approach to solve mocap data completion problem by adopting a new matrix norm, called truncated nuclear norm. An efficient iterative algorithm is designed to solve this problem based on the augmented Lagrange multiplier. The convergence of the proposed method is proved mathematically under mild conditions. To demonstrate the effectiveness of the proposed method, various comparative experiments are performed on synthetic data and mocap data. Compared to other methods, the proposed method is more efficient and accurate.

Let A be a singular diagonalizable complex matrix with three distinct eigenvalues. We derive all explicit solutions X of the Yang-Baxter-like matrix equation AXA = XAX, by taking advantage of the Jordan form structure of A. The result generates the formula obtained in Chen et al. (2019) and M. Saeed Ibrahim Adam et al. (2019). We give examples to illustrate the validity of the results obtained in this paper.