From contours of 2D shapes to knotted proteins, many areas of statistics and physical sciences are concerned with data in the form of geometric loops - embeddings of the circle into a Euclidean space. It is desirable to find a representation of the geometry of these loops that is invariant under Euclidean symmetries, and indifferent to how the loops are parametrised. In this joint work with Ka Man Yim (Oxford) and Otto Sumray (MPI Dresden), we propose a novel representation of a loop’s geometry by rep resenting the distance matrix of the loop as a Morse function on a Möbius band - the second finite subset space of the circle. By considering the persistent homology of this function, we obtain a representation of the loop’s geometry that has the desired symmetry invariance. We show that this feature map is stable with respect to perturbations, give a geometric interpretation of the features obtained and explore analogues involving higher order finite subset spaces.

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