Manifold Learning with Intrinsic Distance Estimation Using Kernelized Linear Model for Metric Tensors
Katsuhiko Kojima, Yoshifumi Kusunoki, Keiji Tatsumi · 2020
We often observe manifolds embedded in a high-dimensional space, which are generated by a latent low-dimensional system via non-linear measurements. If we obtain push-forward metric tensors of the manifold, we can reconstruct a data representation which is intrinsic and isometric with respect to the latent geometry. For that purpose, assuming samples can be drawn from Gaussian distributions in the latent low-dimensional system, we study methods to estimate metric tensors. If we don't have enough amount of samples, the estimation will be failed. One solution to that problem is artificial neural networks, however it can be time-consuming. So, we propose a kernelized linear model which is easy to be implemented and has capability of estimating the metric tensor under the insufficient circumstance. Estimation capability of the proposed method is evaluated in the numerical experiment.