Multidimensional scaling in dually flat spaces

摘要

Formulations of multidimensional scaling (MDS) in dually flat spaces are proposed. First the space supposed in the classical MDS is extended to a tangent space around a specific point in a dually flat space. We see that Riemannian metric of the tangent point plays a key role in the extension. Next, in order to remove the restriction of symmetry in dissimilarities, the affine connection is incorporated. We pay attention to the fact that it is an affine connection term that causes an asymmetry in dissimilarities in infinitesimal space. To mitigate the difficulty in treating the affine connection term, an approximation is shown and we can see the effect of the affine connection term to modify the effective Riemannian metric. Finally a numerical example is shown.