Data points in many scientific experiments originate from an ordered structure, yet this ordering is often unavailable.We consider noisy data points with the correct ordering to be recovered. The underlying structure naturally places the data on a 1-dimensional manifold. Because eigenfunctions of 1-dimensional manifold Laplacian are trigonometric functions, and the manifold Laplacian can be approximated by the graph data Laplacian, the data ordering can be recovered by inverting the data Laplacian eigenvectors.We propose two spectral algorithms, one for the periodic structure (closed loop) and one for the non-periodic structure (open curve). We have derived the uniform error bound for the algorithms, which is composed of two parts: the discretization error between the manifold eigenfunctions and the noiseless graph Laplacian eigenvectors, and the eigenvectors error caused by data noise. In numerical studies, our spectral seriation algorithms outperform other manifold learning methods. The superior performance of our algorithms is demonstrated further on a biomolecule data example.

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