GS02 — PCA and Subspace Tracking
| Field | Value |
|---|---|
| Domain | Grassmannian Systems |
| System | Top-k eigenspace of n×n data matrix |
| Group | U(n) acting on Gr(k, n) |
| H^k tier | H⁰ |
| ISA | Origami (β → ∞) |
| Status | Validated |
| Opcodes | ORBIT · SPLIT · SPLAT · LABEL |
| Papers | Paper 574, Paper 596 |
Physical system
Principal Component Analysis (PCA) finds the k-dimensional subspace V* ∈ Gr(k, n) that captures maximum variance in a data matrix X ∈ ℝ^{n×m}. The solution is the top-k eigenspace of the covariance matrix Σ = XX†/m — the dominant k-plane. PCA is an ORBIT to a fixed point on Gr(k, n).
The subspace V* is the tropical argmax of the variance function f(V) = tr(V† Σ V): the unique global maximum on the compact manifold Gr(k, n). Power iteration (repeatedly multiply by Σ and orthogonalise) is the discrete geodesic flow on Gr(k, n) converging to V*. Oja’s rule (online PCA) is the continuous-time limit — a gradient ORBIT on Gr(k, n) driven by the natural gradient of the Fisher-Rao metric of the data distribution.
Connection to alchemi/chemistry: the NOON spectrum {σ₁²,…,σₖ²} of the C/T skeleton (C06) is exactly the output of PCA applied to the natural orbital occupation matrix. CASSCF orbital optimisation (Paper 596) is PCA on the electronic wavefunction Grassmannian — the OPU (P01) running power iteration.
Target category
Gr(k, n) — the Grassmannian as a Riemannian symmetric space U(n)/(U(k)×U(n−k)), with the Fubini-Study (round) metric. Objects: k-planes V ∈ Gr(k,n). Morphisms: orthogonal projections P_V: ℝⁿ → V. The variance function f(V) = tr(P_V Σ P_V) is the LABEL eigenvalue; its gradient flow is the ORBIT; the global maximum V* is the tropical fixed point (the Schubert cell of the top-k eigenspace).
Interpretation functor
F: C → Gr(k, n) defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Power iteration step: V → orth(ΣV); geodesic on Gr(k,n) toward top-k eigenspace V* |
| SPLIT | SVD: X = UΣV† (thin); extract left singular vectors U[:,1:k] = the k-plane |
| SPLAT | Reconstruction: X̂ = UΣV† restricted to top-k; low-rank approximation |
| LABEL | Eigenvalue σ₁² ≥ σ₂² ≥ … ≥ σₖ²: the NOON spectrum; encodes fraction of variance explained |
ISA programme
DATA: LABEL[Sigma = X X† / m | sample covariance] -- compute covariance
SPLIT: SPLIT[X = U Sigma V† | SVD] -- singular value decomp
NOON: LABEL[{sigma_i^2} | top-k singular values] -- NOON spectrum
ORBIT: ORBIT[V -> orth(Sigma V) | power iteration] -- geodesic to V*
GAP?: LABEL[sigma_k^2 / sigma_{k+1}^2 > gap_threshold?] -- Schubert variety crossing
SNAP: LABEL[snap if gap >> 1 | beta* condition] -- top-k subspace stable
SPLAT: SPLAT[X_hat = U_k Sigma_k V_k†] -- reconstruct from V*
OUTPUT: LABEL[V* in Gr(k,n) | top-k eigenspace] -- the PCA solution
Computable output
- Top-k eigenspace V* ∈ Gr(k, n): the ORBIT fixed point. Computable to machine precision via SVD in O(n²k) time.
- Explained variance Σᵢ σᵢ² / tr(Σ): the LABEL output. The “elbow” in the scree plot is the β* snap — the spectral gap σₖ²/σₖ₊₁² that determines the natural dimension k.
- Wedin perturbation bound: if Σ̃ = Σ + E is a perturbed covariance, ‖sin Θ(V, Ṽ)‖ ≤ ‖E‖ / (σₖ² − σₖ₊₁²) — the perturbation in the subspace (measured by principal angles) is bounded by the noise divided by the spectral gap. The gap σₖ² − σₖ₊₁² is the ISA β* snap threshold: when the gap is large, the ORBIT converges fast; when it vanishes, the subspace is degenerate (BIND obstruction at H²).
- Principal angles θᵢ(V₁, V₂) ∈ [0, π/2]: the ORBIT distance between two k-planes in Gr(k,n). The chordal distance d_c = √(Σᵢ sin²θᵢ) is the Grassmannian metric. Equivalent to the θ_G angle in the alchemi C/T skeleton (C06) when applied to the orbital subspace.
The NOON spectrum connection
The NOON spectrum of the alchemi C/T skeleton (C06) is the PCA output on the molecular orbital Grassmannian:
| PCA concept | Chemistry concept | ISA |
|---|---|---|
| Data matrix X | CI coefficient matrix | State vector |
| Covariance Σ = XX†/m | 1-RDM (one-body density matrix) | Reduced density |
| Top-k eigenvalue σₖ² | NOON (natural orbital occupation) | LABEL eigenvalue |
| Gap σₖ²/σₖ₊₁² | C/T threshold (C-box vs T-arrow) | β* snap condition |
| V* ∈ Gr(k,n) | Natural orbital basis | ORBIT fixed point |
| Schubert cell of V* | Active space | Positroid cell |
The CASSCF algorithm (P01, Paper 596) is PCA on the electronic wavefunction Grassmannian, run iteratively. The orbital gradient is the Riemannian gradient of the energy functional on Gr(k,n); the orbital Hessian is the Fisher-Rao metric; the convergence criterion (σ₁² crossing 0.88 threshold in x596c) is the Schubert variety crossing = the PCA gap condition.
Oja’s rule and natural gradient
Oja’s online PCA rule: dV/dt = (I − VV†)ΣV is the Riemannian gradient of f(V) = tr(V†ΣV) on Gr(k,n), using the round metric. It converges to V* at rate proportional to σₖ² − σₖ₊₁² — the spectral gap, the ISA β*. Oja’s rule is the natural gradient ORBIT on the statistical manifold of data distributions parametrised by their principal subspace — the intersection of PCA (GS02) and information geometry (IG01).
Validation
- SVD algorithm: Golub & Reinsch (1970). Computes top-k eigenspace to machine precision in O(nk²) time. Implemented in LAPACK, NumPy, PyTorch.
- Oja’s rule: Oja (1982). Convergence proved; stationary points = top-k eigenspace.
- Wedin’s theorem: Wedin (1972). Perturbation bound for eigenspaces; exact.
- Alchemi / NOON spectrum: Papers 588/596; x596a–e PASS 7/7 confirming CASSCF convergence = Schubert variety crossing.