<p>Federated learning is widely adopted in scientific and engineering fields, with tensors forming the core structure for handling massive datasets. Federated tensor decomposition has thus emerged as a critical tool in federated data processing. However, existing approaches remain nascent and face two main limitations: the direct adoption of classical decomposition models that lack adaptability to data characteristics for accuracy enhancement, and the absence of efficient algorithms suitable for large-scale tensor computations. To overcome the restrictions of current federated Tucker decomposition in accuracy and speed, we introduce a novel federated low-multilinear-rank approximation model based on a modified Tucker decomposition. We further develop a joint randomized algorithm to solve the proposed model efficiently. By transforming tensor data into a transformed domain, the algorithm rapidly identifies structural differences across data modalities via sampling, thereby capturing intrinsic feature correlations. Theoretical analysis provides an expected error bound for the algorithm, while numerical experiments on diverse real-world datasets validate its effectiveness.</p>