Performance of low-rank approximations in tensor train format (TT-SVD) for large dense tensors.

There are several factorizations of multi-dimensional tensors into lower-dimensional components, known as `tensor networks'. We consider the popular `tensor-train' (TT) format and ask, how efficiently can we compute a low-rank approximation from a full tensor on current multi-core CPUs. Compared to sparse and dense linear algebra, there are much fewer and less extensive well-optimized kernel libraries for multi-linear algebra. Linear algebra libraries like BLAS and LAPACK may provide the required operations in principle, but often at the cost of additional data movements for rearranging memory layouts. Furthermore, these libraries are typically optimized for the compute-bound case (e.g.\ square matrix operations) whereas low-rank tensor decompositions lead to memory bandwidth limited operations. We propose a `tensor-train singular value decomposition' (TT-SVD) algorithm based on two building blocks: a `Q-less tall-skinny QR' factorization, and a fused tall-skinny matrix-matrix multiplication and reshape operation. We analyze the performance of the resulting TT-SVD algorithm using the Roofline performance model. In addition, we present performance results for different algorithmic variants for shared-memory as well as distributed-memory architectures. Our experiments show that commonly used TT-SVD implementations suffer severe performance penalties. We conclude that a dedicated library for tensor factorization kernels would benefit the community: Computing a low-rank approximation can be as cheap as reading the data twice from main memory. As a consequence, an implementation that achieves realistic performance will move the limit at which one has to resort to randomized methods that only process part of the data.