low-rank approximation concept
| Definition | The best approximation of a matrix by one of rank k in total squared error keeps the first k terms of its singular value decomposition, and the error is the sum of the squared singular values dropped, the Eckart and Young theorem. It is the identity reader's best approximation, and chapter 4 is about consumers for which it is not the best. Primer L, equation L.18. Also low-rank, Eckart. |
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| Example | Keeping only the root 3 term of the three by two example leaves squared error 1, one quarter of the total. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id low-rank-approximation, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation L.18. |
| Assumptions and scope | none |
| Prior art | none recorded |
| Evidence | none |
| Reviewed | semantic review 2026-09-09; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation L.18.
\[\|A\|_F^{2}=\sum_{i,j}A_{ij}^{2}=\sum_i\sigma_i^{2},\qquad \min_{\operatorname{rank}B\le k}\|A-B\|_F^{2}=\sum_{i>k}\sigma_i^{2}.\]
Conditions
none
Ledger
none
First stated
Primer L section L.8 of Data Mining as Observation, added in draft 0.3 (2026-09-09) for the ECE 514 readers whose first courses are far behind. The idea is standard and TSK Appendix A covers it at length.
Measurements
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Failures and corrections
none
Invariance envelope
none declared
Machine checked
none
Used in
Data Mining as Observation primer L, chapters 6, 11.
Related
singular value decomposition; truncation; explained variance; the flip; identity reader.
See also
none
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.