spectral embedding instrument

Definition	The coordinates a node receives from its values in the first few nontrivial eigenvectors of the graph Laplacian. Clustering those coordinates is spectral clustering. Chapter 3. Also spectral clustering, row normalization.
Example	A ring of ten nodes embeds on a circle, and the angle between two nodes’ embeddings tracks their distance along the ring.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `spectral-embedding`, kind instrument.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 0.19.
Assumptions and scope	The coordinates a node receives from its values in the first few nontrivial eigenvectors of the graph Laplacian. Row normalization puts every embedded row on the unit sphere, leaves the cosine between rows unchanged, and ignores a positive rescaling of a row, so it is the projection onto the read subspace of the geodesic-rank consumer. Its magnitude converges to degree noise and its angle keeps the geodesics, so the cosine between embedded rows is the right similarity and the Euclidean distance the wrong one. Clustering the normalized coordinates is spectral clustering, and a density reader of the same embedding reads the complementary coordinate. The embedding of equation 0.20 scales each eigenvector entry by the eigenvalue and the degree. Its squared Euclidean distance is the resistance, commute time over volume, exactly for the full spectrum and approximately when only the low modes are kept.
Prior art	none recorded
Evidence	`geometric-observation/chapters/ch09_legibility.md:31-41`, `geometric-observation/chapters/ch03_historical_precursors.md:98-110`, `geometric-observation/chapters/ch09_legibility.md:10-41`, `lean/DataMiningAsObservation/SpectralEmbedding.lean`, `lean/DataMiningAsObservation/Laplacian.lean`
Reviewed	semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page.

A node's values in the low eigenvectors of the Laplacian.

Equation

Book equation 0.19.

\[L=I-D^{-1/2}AD^{-1/2},\qquad D=\operatorname{diag}(d_1,\dots,d_n),\qquad L\,u_k=\lambda_k u_k,\ \ 0=\lambda_1\le\lambda_2\le\cdots.\]

Book equation 3.3.

\[X_i=r_i\,\theta_i,\qquad r_i\ \to\ \frac1{\sqrt{d_i}},\qquad \theta_i=\frac{X_i}{\|X_i\|}\in S^{m-1}.\]

Conditions

The coordinates a node receives from its values in the first few nontrivial eigenvectors of the graph Laplacian. Row normalization puts every embedded row on the unit sphere, leaves the cosine between rows unchanged, and ignores a positive rescaling of a row, so it is the projection onto the read subspace of the geodesic-rank consumer.
Its magnitude converges to degree noise and its angle keeps the geodesics, so the cosine between embedded rows is the right similarity and the Euclidean distance the wrong one. Clustering the normalized coordinates is spectral clustering, and a density reader of the same embedding reads the complementary coordinate.
The embedding of equation 0.20 scales each eigenvector entry by the eigenvalue and the degree. Its squared Euclidean distance is the resistance, commute time over volume, exactly for the full spectrum and approximately when only the low modes are kept.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

Ng, Jordan, and Weiss, on spectral clustering, 2002, as chapter 3 section 3.3 of Data Mining as Observation reads it, with the program’s angular reading in the-angular-observer/theorem.md:110-146.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 3 section 3.3	row normalization is the projection onto the read subspace of the geodesic-rank consumer	`geometric-observation/chapters/ch09_legibility.md:31-41`; `geometric-observation/chapters/ch03_historical_precursors.md:98-110`
chapter 9 section 9.1	the geodesic-rank reader discards the radius, row normalization is the angular projection	`geometric-observation/chapters/ch09_legibility.md:10-41`; `the-angular-observer/theorem.md:110-146`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/SpectralEmbedding.lean, theorems dot_self_nonneg, rowNormalize_unit, dot_rowNormalize, rowNormalize_smul, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

lean/DataMiningAsObservation/Laplacian.lean, theorems quad_eq, quad_nonneg, quad_const, quad_pos_of_edge, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 0, 3, 9.

Laplacian; commute time; geodesic distance; degree; recognizer.

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.

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