Tangent-based manifold approximation with locally linear models

• We relate manifold approximation with affine spaces to the structure of the manifold.
• We employ the variation of local tangent spaces as a measure of manifold linearity.
• We propose a new algorithm for manifold approximation based on our linearity measure.
• We provide an effective upper bound for the dissimilarity measure used in our scheme.
• We prove the effectiveness of our scheme in both synthetic and real signals.

In this paper, we consider the problem of manifold approximation with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that represent manifold data accurately while preserving the manifold׳s structure. For this purpose, we employ a greedy technique that partitions manifold samples into groups, which are approximated by low dimensional subspaces. We start by considering each manifold sample as a different group and we use the difference of local tangents to determine appropriate group mergings. We repeat this procedure until we reach the desired number of sample groups. The best low dimensional affine subspaces corresponding to the final groups constitute our approximate manifold representation. Our experiments verify the effectiveness of the proposed scheme and show its superior performance compared to state-of-the-art methods for manifold approximation.

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