Trajectory Seriation via Spectral Tangent Alignment and Global Embedding

This paper addresses the problem of linear seriation: recovering the intrinsic order of noisy samples drawn from an unknown one-dimensional manifold embedded in a higher-dimensional space. We propose a multi-stage approach that first robustly estimates local tangent directions using Principal Component Analysis (PCA) on neighborhoods, establishing theoretical consistency for these local estimates. Global orientation consistency of these tangents is then achieved through a spectral relaxation of a pairwise alignment objective. Finally, a globally consistent 1D embedding is computed by solving a carefully formulated linear system (or equivalently, a spectral problem on a derived Laplacian) that aligns the embedding with the oriented local projections. This method effectively leverages local geometric information while ensuring global coherence, producing an ordering robust to noise, curvature, and initial data rotation. We demonstrate its performance on simulated manifold data and discuss the theoretical underpinnings of its core components.