🤖 AI Summary
To address the high computational cost of SVD approximation updates in streaming data and the scalability limitations of existing incremental/truncated SVD methods at large truncation ranks, this paper proposes a low-rank update framework based on Bidirectional Diagonal Decomposition (BDD). Our method enables efficient rank-$r$ updates via three key innovations: (1) a compact Householder transformation reducing memory usage by 50%; (2) Givens rotations enabling $O(r^2)$-complexity rank-$r$ updates; and (3) a hybrid sparse-plus-low-rank separation strategy for accurate and scalable matrix approximation. Experiments on recommendation systems and network subspace tracking demonstrate that our approach significantly outperforms LAPACK SVD and state-of-the-art incremental SVD methods—achieving superior accuracy, real-time performance even at high truncation ranks, and balanced throughput–precision trade-offs.
📝 Abstract
For a datastream, the change over a short interval is often of low rank. For high throughput information arranged in matrix format, recomputing an optimal SVD approximation after each step is typically prohibitive. Instead, incremental and truncated updating strategies are used, which may not scale for large truncation ranks. Therefore, we propose a set of efficient new algorithms that update a bidiagonal factorization, and which are similarly accurate as the SVD methods. In particular, we develop a compact Householder-type algorithm that decouples a sparse part from a low-rank update and has about half the memory requirements of standard bidiagonalization methods. A second algorithm based on Givens rotations has only about 10 flops per rotation and scales quadratically with the problem size, compared to a typical cubic scaling. The algorithm is therefore effective for processing high-throughput updates, as we demonstrate in tracking large subspaces of recommendation systems and networks, and when compared to well known software such as LAPACK or the incremental SVD.