On a Connection Between Fast and Sparse Oblivious Subspace Embeddings

Fast Johnson-Lindenstrauss Transform (FJLT) and Sparse Johnson-Lindenstrauss Transform (SJLT) are two important oblivious subspace embeddings. So far, the developments of these two methods are almost orthogonal. In this work, we propose an iterative algorithm for oblivious subspace embedding which makes a connection between these two methods. The proposed method is built upon an iterative implementation of FJLT and is equipped with several theoretically motivated modifications. One important strategy we adopt is the early stopping strategy. On the one hand, the early stopping strategy makes our algorithm fast. On the other hand, it results in a sparse embedding matrix. As a result, the proposed algorithm is not only faster than the FJLT, but also faster than the SJLT with the same degree of sparsity. We present a general theoretical framework to analyze the embedding property of sparse embedding methods, which is used to prove the embedding property of the proposed method. This framework is also of independent interest. Lastly, we conduct numerical experiments to verify the good performance of the proposed algorithm.