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 § Definition: Truncated SVD in Mathematics for AI and Machine Learning.
One more reduction is possible: keep only the top $t$ singular values with $t < r$ (the book figure
- $U_t \in \mathbb{R}^{M \times t}$
- $\Sigma_t = \mathrm{diag}(\sigma_0,\ldots,\sigma_{t-1})$
- $V_t \in \mathbb{R}^{N \times t}$
Unlike Thin and Compact SVD, this product is no longer exact: it is a low-rank approximation—a truncated SVD. We take up its error analysis later in the chapter (@sec:truncated-svd).
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.