bibkey: singerwu2011vectordiffusion authors: Amit Singer; Hau-tieng Wu year: 2011 title: Vector Diffusion Maps and the Connection Laplacian doi: 10.48550/arXiv.1102.0075 url: https://arxiv.org/abs/1102.0075v1 claim: Section 3 constructs weighted orthogonal transport blocks and their degree matrix, with reverse-edge transpose symmetry, as a graph approximation associated with the connection Laplacian. strata_touched: [] license: citation-only triage: anchor

Orthogonal edge transport and connection operators

The arXiv preprint §3, equations (3.1)–(3.3), defines the block matrix S(i,j)=w_ij O_ij and scalar degree blocks D(i,i)=deg(i)I, and explains symmetry from O_ijᵀ=O_ji and equal reverse weights. It then uses the normalized averaging operator. Its manifold approximation involves local PCA and aligned tangent frames with their own sampling hypotheses.

The FIB boundary geometry volume §19 uses the corresponding unnormalized quadratic operator with explicitly supplied SO(3) transports. Labeled parallel edges are summed separately. Its common-holonomy fixed-vector calculation, repeated-walk frame-energy bound and edge-addition examples are finite-network deductions, not a claim that FIB syntax supplies aligned tangent frames or satisfies the paper’s manifold limit assumptions. The DOI above identifies the preprint rather than a journal version.