Multiple kernel interpolation for inverting non-linear dimensionality reduction and dimension estimation

The problem of stably inverting a non-linear dimensionality reduction map has applications in data visualization and machine learning, besides being of theoretical interest. In this paper, we propose a meshfree interpolation method for obtaining such inverse maps using a non-negative linear combination of multiple interpolants. We show that the proposed scheme can improve upon the approximation power of its individual constituent kernels, and discuss the conditions under which its parameters can be uniquely estimated. We also provide an approach for estimating the intrinsic dimensionality (ID) of manifolds using the proposed inverse map. Experiments using multiple kernel interpolation for reconstruction of novel test data and ID estimation show an improved or similar performance compared to existing techniques.