PCA and representation geometry
Use pca and representation geometry to move the unsupervised & retrieval production brief toward a defensible release.
Novel equipment failures appear before reliable labels exist.
This lesson isolates pca and representation geometry as one decision inside that system. The people affected are product users and operators; the learning data must carry event time, availability time, ownership, and version; and the operating envelope is a declared latency, cost, and operator-capacity budget.
Choose whether and how to use PCA at a declared prediction cutoff.
Measure decision utility alongside calibration, slice reliability, and system latency—not model score alone.
Novel equipment failures appear before reliable labels exist. An unsafe release must degrade to a named baseline or the last known-good version.
Build the mental model before the machinery.
The core move is to treat pca and representation geometry as a contract between data, a computation, and an action. Version encoder and index together; monitor cluster stability, alert yield, exact-sample recall, latency, and freshness. The implementation becomes easier to debug once you can state which inputs exist, which state is learned, what output means, and what must remain invariant after serialization.
PCA
Define it in a hand-checkable form and name the prediction-time inputs.
explained variance
Connect it to the production metric and identify what it cannot guarantee.
k-means
Stress it with a slice, a temporal boundary, and a failure-safe alternative.
DBSCAN
Stress it with a slice, a temporal boundary, and a failure-safe alternative.
Data & features, Classical baselines
Name every symbol. Check every shape.
Cosine similarity is the central invariant for this lesson. The formula is useful only when its inputs match the production cutoff and its output maps to an action.
Cosine similarity
| Symbol | Meaning / shape / unit |
|---|---|
u, v | embedding vectors |
uᵀv | dot product |
∥·∥₂ | vector magnitude |
Open derivation and numerical substitution
Start from the production quantity being optimized, substitute the observed values with their declared units, then isolate the model-controlled term. Preserve shape annotations at each step so broadcasting or aggregation cannot silently change the result.
- Write the named inputs: u, v, uᵀv, ∥·∥₂.
- Substitute one small, hand-checkable batch before vectorizing.
- Calculate an independent reference value and compare within a declared tolerance.
# equation → code contract
inputs = validate_shapes_and_units(batch)
value = compute_c06(inputs)
assert is_finite(value)Calculate it small. Shape it realistically. Break it on purpose.
A result you can reproduce on paper
Project correlated measurements onto one axis, cluster them, then retrieve the nearest normalized vector.
- Write every input and unit.
- Substitute values into the cosine similarity equation above.
- Compare the result to one simple baseline and explain the direction of the difference.
The same reasoning under real constraints
Find novel device failures and retrieve similar prior incidents for engineering triage.
The production record includes the data snapshot, transformation state, artifact identity, cutoff, score, decision, and the version of the policy that consumed it.
The attractive result you should reject
A stale ANN index never retrieves new inventory; a reranker cannot recover what retrieval omitted.
Diagnostic: replay the smallest failing slice from immutable inputs, then compare each boundary rather than retuning the model.
Change one assumption and make the tradeoff visible.
This lab runs predefined TypeScript only. It never executes learner code. Use the slider, numeric input, reset, live text, or table—the computation is the same.
ANN recall–latency lab
Increase search effort and expose the recall/latency frontier.
Assumption: HNSW-like curve for a fixed one-million-vector index.
Open nonvisual data table
| Item | Computed state | Interpretation |
|---|---|---|
| query 1 | 82.7% | 4.7 ms |
| query 2 | 82.0% | 4.9 ms |
| query 3 | 81.2% | 5.1 ms |
| query 4 | 80.4% | 5.3 ms |
| query 5 | 79.5% | 5.5 ms |
Trace the complete operating path.
- 01
Validate and version PCA.
- 02
Compute pca and representation geometry from prediction-time-safe inputs.
- 03
Persist model, feature, and configuration identities together.
- 04
Serve or materialize behind explicit a declared latency, cost, and operator-capacity budget.
- 05
Join telemetry to mature outcomes and retain a rollback path.
Observability
Join service health, input quality, prediction distributions, slice behavior, and mature outcomes by exact version.
Cost
Measure storage, preprocessing, compute, queueing, and human review under a representative arrival pattern.
Failure modes
A stale ANN index never retrieves new inventory; a reranker cannot recover what retrieval omitted. Add a detector, owner, mitigation, and stop condition for this class of failure.
Alternatives
Compare a rule, a simpler statistical baseline, and a different system boundary before adding model complexity.
Explain the contract, not just the vocabulary.
Incident discovery and retrieval system
Combine PCA, clustering, anomaly triage, embeddings, and ANN search with human review.
- Representation analysis
- Cluster/anomaly validation
- Embedding evaluation
- ANN recall/latency audit
Read primary material with a purpose.
Return to the opening failure.
Version encoder and index together; monitor cluster stability, alert yield, exact-sample recall, latency, and freshness.
For this lesson, the release evidence is a hand-checked formal result, deterministic simulation output, a ≥80% checkpoint, the production rubric, and a named fallback. The course resolves when the system can produce a pca, clustering, anomaly, and nearest-neighbor retrieval investigation.