Prepare
Define variables, units, missing-data treatment, centering and any justified scaling.
SRAI Book 1 · Chapter 7 · PU-B01-C07
Singular Value Decomposition, PCA, and Dimensionality Reduction
Decompose matrices, verify low-rank approximations and derive PCA from centered data. Quantify what dimensionality reduction preserves and loses before using a compressed representation in a decision.
01 / LEARNING OUTCOMES
Reduce dimensionality without hiding consequential information.
- Decompose a real matrix as A = UΣVT.
- Verify reconstruction, orthogonality and ordered nonnegative singular values.
- Construct and evaluate rank-k approximations.
- Derive PCA from the SVD of centered data.
- Interpret explained variance without turning it into a causal claim.
- Document what a compressed representation preserves and loses.
02 / ANALYTICAL SEQUENCE
Define the use, compute the representation, quantify loss, then decide.
Decompose
Compute SVD and verify reconstruction and orthogonality within a stated tolerance.
Reduce
Choose k using reconstruction evidence and the intended downstream use.
Govern
Record what is removed, affected groups, revision triggers and accountable authority.
03 / SVD, LOW RANK AND PCA
Mathematical optimality under one norm is not universal decision fitness.
Keeping the first k singular directions gives Ak = UkΣkVkT, the best rank-k approximation in Frobenius norm.
For centered data, the rows of VT provide principal directions and UΣ provides scores. Standardization changes the geometry and must be justified.
CORE CONTROLS
| Reconstruction | Measured against an explicit tolerance |
| Explained variance | Squared singular values divided by their total |
| Preprocessing | Fit without leakage and documented |
| Loss review | Residuals, rare events and subgroups checked |
All examples use synthetic instructional data.
Explained variance measures retained variation.
It does not establish importance, causality, fairness or predictive accuracy. Component names must not overstate mathematical association.
Compression is an information-removal decision.
A defensible choice states what is preserved, what is lost, who may reverse it and which evidence requires revision or stopping.
04 / REPRODUCIBILITY AND RELEASE CONTROL
Inspect the evidence, not only the retained percentage.
- The controlled notebook completed all 16 portable validation code cells with saved outputs and no errors.
- The owner verified the portable notebook in Windows VS Code and Google Colab.
- Reconstruction, subgroup loss and rare-event effects must be assessed against the intended use.
- Preprocessing fitted on the complete dataset can create leakage.
- No blanket claim of causal meaning or independent review is made.
CONTROLLED RESOURCES
Read, reproduce, practise and review.
READ
View the complete GitHub production unit ↗
Controlled Chapter 7
Controlled publication edition.
PDF ↗ REPRODUCECanonical notebook
Execution-tested and owner-approved.
IPYNB ↗ PRACTISEExercises and solutions
Progress toward independent application.
PDF ↗ RUNOpen in Google Colab
Run the approved notebook and inspect the saved validation evidence.
COLAB ↗ DOWNLOADComplete release package
Lesson, notebook, exercises, Executive Brief, presentation and validation records.
ZIP ↓ APPLYExecutive brief
Connect analytical controls to decisions.
PDF ↗