Chapter 8: Singular Value Decomposition (SVD). In the chapter mind map this icon labels Randomized SVD: Random Projection. The discussion below is excerpted and lightly edited from § Randomized SVD in Mathematics for AI and Machine Learning.

For large matrices $A \in \mathbb{R}^{M \times N}$ where $M, N$ are very large, computing the full SVD or even a truncated SVD can be computationally expensive (typically $O(MN \min(M,N))$). Randomized SVD is a randomized algorithm that computes an approximate rank-$k$ truncated SVD with significantly lower computational cost, especially when $k \ll \min(M,N)$. It is important to emphasize that randomized SVD is not a new model or a new decomposition—the mathematical meaning of SVD remains unchanged: we still seek $A = U \Sigma V^\top$ where $U$ and $V$ are orthonormal and $\Sigma$ is diagonal. Rather, randomized SVD uses random projections at the algorithmic level to approximate the computation of this same decomposition, making it computationally efficient for large matrices. The output is still a standard SVD decomposition $A \approx U_k \Sigma_k V_k^\top$; only the method of computation differs.