Operator-based Analysis of Numerical Projection Perturbations in Grassmann Manifold Optimization

<title>Abstract</title> <p>This paper develops a complete perturbation analysis for orthogonal projectors under tangent-structured perturbations arising in Grassmann manifold optimization. For a tangent perturbation E of an exact rank-p orthogonal projector P, the perturbed projector P ̃=P+E satisfies the exact identity P ̃^2-P ̃=E^2, demonstrating that the idempotency defect is intrinsically quadratic. The sharp constant bound ∥P ̃^2-P ̃∥_F≤ϵ^2/√2 is proved, with equality if and only if the reduced perturbation has rank one. The closed-form spectrum of P ̃ is derived: eigenvalues 1/2(1±√(1+4σ_i^2 )) for each singular value σ_i of the reduced perturbation, plus 0 and 1. The residual satisfies ∥P ̃^2-P ̃∥_F=√(2∑σ_i^4 ) with sharp two-sided bounds ϵ^2/√2r≤⋅≤ϵ^2/√2, where ϵ=∥E∥_F and r=rank(E)/2. The nearest exact orthogonal projector P^* satisfies ∥P^*-P ̃∥_F=∥P ̃^2-P ̃∥_F+O(ϵ^6). Non-tangent perturbations are shown to yield linear O(ϵ) defect, proving quadratic cancellation is unique to tangent structure. Propagation through retractions remains first-order. Rank-dependent (ϵ&lt;√(u/p)) and rank-independent (ϵ≲2^(1/4) √u) stability criteria are provided. Numerical experiments on random projectors and on a real-data CIFAR-10 projector confirm all theoretical predictions to machine precision.</p>