Convex Coupled Matrix and Tensor Completion
This work addresses data completion challenges in multi-modal datasets, though it appears incremental as it builds on existing norms and algorithms.
The authors tackled the problem of coupled matrix and tensor completion by proposing convex low rank inducing norms that share information between matrices and tensors, resulting in a globally optimal solution and better excess risk bounds compared to uncoupled methods.
We propose a set of convex low rank inducing norms for a coupled matrices and tensors (hereafter coupled tensors), which shares information between matrices and tensors through common modes. More specifically, we propose a mixture of the overlapped trace norm and the latent norms with the matrix trace norm, and then, we propose a new completion algorithm based on the proposed norms. A key advantage of the proposed norms is that it is convex and can find a globally optimal solution, while existing methods for coupled learning are non-convex. Furthermore, we analyze the excess risk bounds of the completion model regularized by our proposed norms which show that our proposed norms can exploit the low rankness of coupled tensors leading to better bounds compared to uncoupled norms. Through synthetic and real-world data experiments, we show that the proposed completion algorithm compares favorably with existing completion algorithms.