We present a fast algorithm for approximate Canonical Correlation Analysis (CCA). Given a pair of tall-and-thin matrices, the proposed algorithm first employs a randomized dimensionality reduction transform to reduce the size of the input matrices, and then applies any CCA algorithm to the new pair of matrices. The algorithm computes an approximate CCA to the original pair of matrices with provable guarantees, while requiring asymptotically less operations than the state-of-the-art exact algorithms.
suggest a two-stage approach which involves first solving a least-squares problem, and then using the solution to reduce the problem size @cite_26 . However, their technique involves explicitly factoring one of the two matrices, which takes cubic time. Therefore, their method is especially effective when one of the two matrices has significantly less columns than the other. When the two matrices have about the same number of columns, there is no asymptotic performance gain. In contrast, our method is sub-cubic in any case.
@cite_26
2020111956
Dimensionality reduction plays an important role in many data mining applications involving high-dimensional data. Many existing dimensionality reduction techniques can be formulated as a generalized eigenvalue problem, which does not scale to large-size problems. Prior work transforms the generalized eigenvalue problem into an equivalent least squares formulation, which can then be solved efficiently. However, the equivalence relationship only holds under certain assumptions without regularization, which severely limits their applicability in practice. In this paper, an efficient two-stage approach is proposed to solve a class of dimensionality reduction techniques, including Canonical Correlation Analysis, Orthonormal Partial Least Squares, linear Discriminant Analysis, and Hypergraph Spectral Learning. The proposed two-stage approach scales linearly in terms of both the sample size and data dimensionality. The main contributions of this paper include (1) we rigorously establish the equivalence relationship between the proposed two-stage approach and the original formulation without any assumption; and (2) we show that the equivalence relationship still holds in the regularization setting. We have conducted extensive experiments using both synthetic and real-world data sets. Our experimental results confirm the equivalence relationship established in this paper. Results also demonstrate the scalability of the proposed two-stage approach.