Randomized SVD has become an extremely successful approach for efficiently computing a low-rank approximation of matrices. In particular the paper by Halko, Martinsson (who is speaking twice this week), and Tropp (SIREV 2011) contains extensive analysis, and made it a very popular method.
The complexity for $m\times n$ matrices is $O(Nr+(m+n)r^2)$ where $N$ is the cost of a (fast) matrix-vector multiplication; which becomes $O(mn\log n+(m+n)r^2)$ for dense matrices. This work uses classical results in numerical linear algebra to reduce the computational cost to $O(Nr)$ without sacrificing numerical stability. The cost is essentially optimal for many classes of matrices, including $O(mn\log n)$ for dense matrices. The method can also be adapted for updating, downdating and perturbing the matrix, and is especially efficient relative to previous algorithms for such purposes.
- Numerical Analysis Group Internal Seminar