The case
Appendix H (“Spectral Toy Example — Stable Core Topics Across Refinement”), over four parts: wing, engine, window, door.
- Rank 2. M² = 3·φ_ship φ_shipᵀ + 2·φ_castle φ_castleᵀ, with φ_ship = (0.7, 0.7, 0.1, 0.1) and φ_castle = (0.05, 0.05, 0.7, 0.7).
- Rank 3. M³ = 1.5·φ_wing φ_wingᵀ + 1.2·φ_engine φ_engineᵀ + 2·φ_castle φ_castleᵀ, with φ_wing = (0.9, 0.1, 0.1, 0.1), φ_engine = (0.1, 0.9, 0.1, 0.1), and castle unchanged.
The appendix’s entries were checked here and are right. It concludes: “Stable core: ‘castle’ is preserved exactly across refinement”, and the surrounding chapters read archetypes as “strong eigenmodes” that “persist under refinement (blowup)”.
Computed here with numpy.
| eigenvalues | cosine of castle topic with modes 1, 2, 3 | |
|---|---|---|
| M² | 3.213, 1.757, 0, 0 | 0.488, 0.873, 0.000 |
| M³ | 2.215, 1.191, 0.831, 0 | 0.926, 0.372, 0.057 |
M²’s second eigenvector is (−0.30, −0.30, 0.64, 0.64), a contrast of castle against ship, not castle. M³’s first is (−0.35, −0.27, −0.64, −0.64). The two are about 51° apart (|cosine| 0.63).
What it can show
That “castle persists” is true of the factor the matrices were built from and false of their eigenvectors. The topics overlap, and eigenvectors are orthogonal, so the spectrum cannot return them. CLAIM-148 gives the general form. A factorization is identified only up to a change of latent coordinates, and nonnegativity narrows that only under further conditions (anchors). CASE-042 is the same point for collaborative filtering and topic models. A persistence criterion for naming a category (QUESTION-113) has to be stated for factors recovered under such conditions, not for eigenvectors.
What it cannot show
That spectral methods cannot recover topics. They can, given anchor words or higher moments. The monograph’s other worked example (Part III, 2b) uses orthogonal directions by construction, and there the eigenvectors are the directions it built.