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We propose a new algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We describe the complexity of our algorithm in terms of the multigraded regularity of a multihomogeneous ideal. We prove effective bounds for many formats and ranks and conjecture a general formula. These bounds are of independent interest for overconstrained polynomial system solving. Our experiments show that our algorithm can outperform state-of-the-art algebraic algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption.
In this paper we study the problem of recovering a tensor network decomposition of a given tensor $mathcal{T}$ in which the tensors at the vertices of the network are orthogonally decomposable. Specifically, we consider tensor networks in the form of
Quaternion matrix approximation problems construct the approximated matrix via the quaternion singular value decomposition (SVD) by selecting some singular value decomposition (SVD) triplets of quaternion matrices. In applications such as color image
Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. Ho
The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and prove the low
We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we show