Truncated SVD
Chapter 8: Singular Value Decomposition (SVD) — Truncated SVD
From the book
Chapter 8: Singular Value Decomposition (SVD). In the chapter mind map this icon labels Truncated SVD: Rank-$k$ Approx. The discussion below is excerpted and lightly edited from § Theorem: Energy loss from truncated SVD - Truncated SVD Error in Mathematics for AI and Machine Learning.
The proof is similar to the total energy one. Using the fact that we have
Combined with $\|A\|_F^2 = \sum_{i=0}^{r-1} \sigma_i^2$ from the energy theorem, we obtain the desired result. So we omit the detailed proof.
What this drawing shows
What you see. Shows keeping only the leading singular directions to form a low-rank approximation.
In the mind map. Chapter 8 — Truncated SVD: Rank- Approx. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 8: Singular Value Decomposition (SVD) — see the mind-map node Truncated SVD: Rank- Approx.