The pre-image problem for Laplacian Eigenmaps utilizing L ₁ regularization with applications to data fusion

As the popularity of non-linear manifold learning techniques such as kernel PCA and Laplacian Eigenmaps grows, vast improvements have been seen in many areas of data processing, including heterogeneous data fusion and integration. One problem with the non-linear techniques, however, is the lack of an easily calculable pre-image. Existence of such pre-image would allow visualization of the fused data not only in the embedded space, but also in the original data space. The ability to make such comparisons can be crucial for data analysts and other subject matter experts who are the end users of novel mathematical algorithms. In this paper, we propose a pre-image algorithm for Laplacian Eigenmaps. Our method offers major improvements over existing techniques, which allow us to address the problem of noisy inputs and the issue of how to calculate the pre-image of a point outside the convex hull of training samples; both of which have been overlooked in previous studies in this field. We conclude by showing that our pre-image algorithm, combined with feature space rotations, allows us to recover occluded pixels of an imaging modality based off knowledge of that image measured by heterogeneous modalities. We demonstrate this data recovery on heterogeneous hyperspectral (HS) cameras, as well as by recovering LIDAR measurements from HS data. (paper)