在我们投稿之后，收到了revison建议，对下面几篇相关性较强的论文进行了阅读，前两篇可以涵盖在我们的框架中。第二篇我认为算法框架没我们的好且缺失收敛性分析；后三篇着重于理论分析，可以作为我们未来的拓展方向。
Hong, D., Kolda, T. G., & Duersch, J. A. (2020). Generalized canonical polyadic tensor decomposition. SIAM Review, 62(1), 133-163.
This work develop the GCP framework for computing the CP tensor decomposition with an arbitrary elementwise loss function. They derive the gradient for GCP with respect to the model components, along with a straightforward way of handling missing data. Without a convergence analysis, I think it is not solid enough than our paper.
Chi, E. C., & Kolda, T. G. (2012). On tensors, sparsity, and nonnegative factorizations. SIAM Journal on Matrix Analysis and Applications, 33(4), 1272-1299.
They present a new algorithm for Poisson tensor factorization called CP-APR. The negative log likelihood function for Poisson is equal to the KL divergence. The subproblems are solved using a majorization minimization (MM) approach. If the algorithm is restricted to a single inner iteration per subproblem, it reduces to the standard Lee–Seung multiplicative. They show how to prevent the algorithm from converging to non-KKT points and prove convergence of CP-APR under mild conditions.
Xia, D., Yuan, M., & Zhang, C. H. (2017). Statistically optimal and computationally efficient low rank tensor completion from noisy entries. Annals of Statistics, to appear.
This work not only established minimax optimal rates of convergence for noisy tensor completion, but also proposed an efficient algorithm based upon power iteration and a second-order spectral initialization that achieves this rate.
This model with sub-Gaussian noise is quite different from ours, but it used a lot of new techniques for theoretical analysis, which provide tools for the further study of ExpCP.
Han, R., Willett, R., & Zhang, A. (2020). An Optimal Statistical and Computational Framework for Generalized Tensor Estimation. arXiv preprint arXiv:2002.11255.
This paper described a projected gradient descent based tensor estimation framework for many tensor data applications and established both an upper bound on statistical error and the linear convergence rate. The proposal achieves the minimax optimal rate of convergence in estimation error and shows efficacy in real data analysis.
The proposed scheme can be applied to several problems, including sub-Gaussian denoising, tensor regression, Poisson and binomial tensor PCA. Unfortunately, it cannot be directly extended to the generalized tensor completion problem.