Rectifying Non-euclidean Similarity Data through Tangent Space Reprojection

Abstract

This paper concerns the analysis of shapes characterised in terms of dissimilarities rather than vectors of ordinal shape-attributes. Such characterisations are rarely metric, and as a result shape or pattern spaces can not be constructed via embeddings into a Euclidean space. The problem arises when the similarity matrix has negative eigenvalues. One way to characterise the departures from metricty is to use the relative mass of negative eigenvalues, or negative eigenfraction. In this paper, we commence by developing a new measure which gauges the extent to which individual data give rise to departures from metricity in a set of similarity data. This allows us to assess whether the non-Euclidean artifacts in a data-set can be attributed to individual objects or are distributed uniformly. Our second contribution is to develop a new means of rectifying non-Euclidean similarity data. To do this we represent the data using a graph on a curved manifold of constant curvature (i.e. hypersphere). Xu et. al. have shown how the rectification process can be effected by evolving the hyperspheres under the Ricci flow. However, this can have effect of violating the proximity constraints applying to the data. To overcome problem, here we show how to preserve the constraints using a tangent space representation that captures local structures. We demonstrate the utility of our method on the standard “chicken pieces” dataset.