Oblivious subspace embeddings for compressed Tucker decompositions

Abstract: Emphasis in the tensor literature on random embeddings (tools for low-distortion dimension reduction) for the canonical polyadic (CP) tensor decomposition has left analogous results for the more expressive Tucker decomposition comparatively lacking. This work establishes general Johnson-Lindenstrauss (JL) type guarantees for the estimation of Tucker decompositions when an oblivious random embedding is applied along each mode. When these embeddings are drawn from a JL-optimal family, the decomposition can be estimated within $\varepsilon$ relative error under restrictions on the embedding dimension that are in line with recent CP results. We implement a higher-order orthogonal iteration (HOOI) decomposition algorithm with random embeddings to demonstrate the practical benefits of this approach and its potential to improve the accessibility of otherwise prohibitive tensor analyses. On moderately large face image and fMRI neuroimaging datasets, empirical results show that substantial dimension reduction is possible with minimal increase in reconstruction error relative to traditional HOOI ($\leq$5% larger error, 50%-60% lower computation time for large models with 50% dimension reduction along each mode). Especially for large tensors, our method outperforms traditional higher-order singular value decomposition (HOSVD) and recently proposed TensorSketch methods.