Fast and Guaranteed Tensor Decomposition via Sketching
This work addresses the problem of efficient tensor decomposition for applications in statistical learning and data mining, representing an incremental advancement through novel sketching techniques.
The paper tackles the computational challenge of tensor CP decomposition by proposing fast randomized algorithms based on sketching, achieving significant speed improvements on both sparse and dense tensors while maintaining approximation quality independent of tensor properties like sparsity.
Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results.