Effective algorithms for tensor train decomposition via the UTV framework

The tensor-train (TT) decomposition is widely used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the well-known TT-SVD method, which performs singular value decompositions (SVDs) on the successive matrices sequentially. But in practical applications, it is often unnecessary to compute full SVDs. In this article, we propose a new method called the TT-UTV. It utilizes the virtues of rank-revealing UTV decomposition to compute the TT format for a large-scale tensor, resulting in lower computational cost. We analyze the error bounds on the accuracy of these algorithms in both the URV and ULV cases and then recommend different sweep patterns for these two cases. Based on the theoretical analysis, we also formulate the rank-adaptive algorithms with prescribed accuracy. Numerical experiments on various applications, including magnetic resonance imaging data completion, are performed to illustrate their good performance in practice.