FastRank: Fast Tensor Rank Approximation based on Spectral Energy

Complex multi-dimensional data are often represented as tensors, analyzed through tensor decompositions. A central challenge is selecting the right number of components for the decomposition. In the Canonical Polyadic Decomposition (CPD), this means determining the canonical rank, which directly impacts decomposition quality. Existing methods typically estimate rank by repeatedly computing CPDs, an expensive process. We introduce $FastRank$, a theoretically grounded method that estimates rank without CPD computation. By applying Singular Value Decomposition (SVD) to a sum-reduced matrix of the tensor and analyzing its eigenspectrum, FastRank achieves over $1000\times$ speedup and surpasses state-of-the-art accuracy. We validate it using both synthetic and real data, including noisy settings, and highlight its scalability in knowledge graph completion, where prior methods fail due to computational limitations.