Seminar Details
2019-10-01 (14:00) : Two 20-minute talks
At Euler building (room A.002)
Organized by Mathematical Engineering
Section 1: Alternating minimization algorithm for graph regularized tensor completion
Speaker :
Guan, Yu
Abstract :
In this talk, we consider low-rank tensor completion problem which aims to exactly recover a low-rank tensor from an
incomplete observation. It plays an important role in many applications such as signal processing, computer vision, machine learning, and
neuroscience. A widely used convex relaxation of low-rank tensor completion problem is to minimize the sum of the nuclear norm of its
unfolding matrices. We build our low-rank completion model based on the CANDECOMP/PARAFAC decomposition approach. In order
to improve the recovery quality, the graph information such as correlations between the column/row entities from the side matrices are exploited. In this paper, we propose efficient alternating minimization algorithms combined with conjugate gradient method or alternating
direction method of multiplier method dealing with the subproblems. Furthermore, based on the Kurdyka- ojasiewicz property, we show
that the sequence generated by the alternating minimization globally converges to a critical point of the objective function since our model
is a Kurdyka- ojasiewicz function. Moreover, the complexity and convergence rate of proposed algorithms are also derived. In addition,
the regularizer terms in the model could be reformulated as a generalized weighted nuclear norm. As a result, the statistical consistency
is guaranteed for our completion problem. Extensive numerical experiments including synthetic data, real data indicate the proposed
algorithms are effective.
Section 2: Optimization Problems with Orthogonality Constraints -- from Feasible to Infeasible
Speaker :
Gao, Bin
Abstract :
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthogonalization procedure. However, such demand is particularly huge in some application domains such as material computation. In this talk, we introduce several efficient algorithms including feasible and infeasible methods. Such methods have much lower computational complexity and can also benefit from parallel computing since they are full of BLAS3 operations. In the infeasible algorithm based on a modified augmented Lagrange method, the orthogonalization procedure is only invoked once as the last step. Consequently, the main parts of the proposed algorithms can be parallelized naturally. We establish global subsequence convergence results for our proposed algorithms. Worst-case complexity and local convergence rate are also studied under some mild assumptions. Numerical experiments, including tests under parallel environment, illustrate that our new algorithms attain good performances and high scalability in solving discretized Kohn-Sham total energy minimization problems.