A numerical algorithm to decompose an exact low-rank skew-symmetric tensor into a sum of elementary skew-symmetric tensors is introduced. The algorithm uncovers this Grassmann decomposition based on linear relations that are encoded by the kernel of the differential of the natural action of the general linear group on the tensor, following the ideas of Brooksbank et al. [ Detecting sparsity patterns in tensor data , arXiv:2408.17425v3, 2026]. The Grassmann decomposition can be recovered, up to scale, from the diagonalization of a generic element in this kernel. Numerical experiments illustrate that the algorithm is computationally efficient and quite accurate for mathematically low-rank tensors.

We present ATC, a C++ library for advanced Tucker-based lossy compression of dense multidimensional numerical data in a shared-memory parallel setting, based on the sequentially truncated higher-order singular value decomposition (ST-HOSVD) and bit plane truncation. Several techniques are proposed to improve speed, memory usage, error control and compression rate. First, a hybrid truncation scheme is described which combines Tucker rank truncation and TTHRESH quantization. We derive a novel expression to approximate the error of truncated Tucker decompositions in the case of core and factor perturbations. We parallelize the quantization and encoding scheme and adjust this phase to improve error control. Implementation aspects are described, such as an ST-HOSVD procedure using only a single transposition. We also discuss several usability features of ATC, including the presence of multiple interfaces, extensive data type support, and integrated downsampling of the decompressed data. Numerical results show that ATC maintains state-of-the-art Tucker compression rates while providing average speed-up factors of 2.2 to 3.5 and halving memory usage. Our compressor provides precise error control, deviating only 1.4% from the requested error on average. Finally, ATC often achieves higher compression than non-Tucker-based compressors in the high-error domain.

We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We characterize the complexity of our algorithm in terms of an algebraic property of this polynomial system—the multigraded regularity. We prove effective bounds for many tensor formats and ranks, which are of independent interest for overconstrained polynomial system solving. Moreover, we conjecture a general formula for the multigraded regularity, yielding a (parameterized) polynomial time complexity for the tensor rank decomposition problem in the considered setting. Our numerical experiments show that our algorithm can outperform state-of-the-art numerical algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption.