Sequential randomized matrix factorization for Gaussian processes

Fast approximate truncated SVD

“…For higher levels, parallelization must rely on parallel implementation of matrix multiplication. Another factor that contributes to possible parallelization is simplicity of the method. Performance on streaming data: In many applications (e.g., Gaussian Processes and other correlation analysis), it is typical that the entries of the matrix X are not available all at once, but come one column a time. In such a case, our scheme allows one to process data as they come, effectively scanning the binary tree bottom‐to‐top along the branches and releasing intermediate results of factorization each time when the number of columns reaches 2 i q , i =1,….…”

“…Lemma 1. Let the parameters r j be defined by (8), (9), and q ≥ r. Then, the total computational cost of the algorithm is as following.…”

“…Step II, the cost of forming of the matrix P i,j and its factorization is (8). Thus, the total cost for one submatrix does not exceed…”

“…Moreover, the method has several attractive features: it is one‐pass (i.e., each entry of the matrix is used just once) and is readily parallelizable . In the case of streaming data, when the matrix columns or rows are obtained successively over time, the factorization can be computed as new data become available, similar to the model used in other works . The total asymptotical cost is the same as if all the data are available from the beginning.…”

“…In the case of streaming data, when the matrix columns or rows are obtained successively over time, the factorization can be computed as new data become available, similar to the model used in other works. 7,8 The total asymptotical cost is the same as if all the data are available from the beginning. The general scheme of the proposed method is similar to the one used in the papers.…”