Editing Singular value decomposition (section)

===Truncated SVD===
In many applications the number {{tmath|r}} of the non-zero singular values is large making even the Compact SVD impractical to compute. In such cases, the smallest singular values may need to be truncated to compute only {{tmath|t \ll r}} non-zero singular values. The truncated SVD is no longer an exact decomposition of the original matrix {{tmath|\mathbf M,}} but rather provides the optimal [[#Low-rank matrix approximation|low-rank matrix approximation]] {{tmath|\tilde{\mathbf M} }} by any matrix of a fixed rank {{tmath|t}}

<math display=block>
\tilde{\mathbf{M}} = \mathbf{U}_t \mathbf \Sigma_t \mathbf{V}_t^*,
</math>

where matrix {{tmath|\mathbf U_t}} is {{tmath|m \times t,}} {{tmath|\mathbf \Sigma_t}} is {{tmath|t \times t}} diagonal, and {{tmath|\mathbf V_t^*}} is {{tmath|t \times n.}} Only the {{tmath|t}} column vectors of {{tmath|\mathbf U}} and {{tmath|t}} row vectors of {{tmath|\mathbf V^*}} corresponding to the {{tmath|t}} largest singular values {{tmath|\mathbf \Sigma_t}} are calculated. This can be much quicker and more economical than the compact SVD if {{tmath|t \ll r,}} but requires a completely different toolset of numerical solvers.

In applications that require an approximation to the [[Moore–Penrose inverse]] of the matrix {{tmath|\mathbf M,}} the smallest singular values of {{tmath|\mathbf M}} are of interest, which are more challenging to compute compared to the largest ones.

Truncated SVD is employed in [[latent semantic indexing]].<ref>{{cite journal | last1 = Chicco | first1 = D | last2 = Masseroli | first2 = M | year = 2015 | title = Software suite for gene and protein annotation prediction and similarity search | journal = IEEE/ACM Transactions on Computational Biology and Bioinformatics | volume = 12 | issue = 4 | pages = 837–843 | doi=10.1109/TCBB.2014.2382127 | pmid = 26357324 | hdl = 11311/959408 | s2cid = 14714823 | url = https://doi.org/10.1109/TCBB.2014.2382127 | hdl-access = free }}
</ref>