# Convex Coupled Matrix and Tensor Completion

## TL;DR

This paper introduces convex low-rank norms for coupled matrix and tensor completion, enabling globally optimal solutions and improved theoretical bounds, with demonstrated effectiveness on synthetic and real data.

## Contribution

It proposes a novel convex norm for coupled tensors that combines overlapped and latent norms, providing a globally optimal completion algorithm with better risk bounds.

## Key findings

- The proposed norms outperform existing methods in synthetic data experiments.
- The completion algorithm achieves superior accuracy on real-world datasets.
- Theoretical analysis shows improved excess risk bounds for coupled tensor completion.

## Abstract

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.