  In this paper, we propose a new method for determining shared features of and
measuring the distance between data sets or point clouds. Our approach uses the
joint factorization of two data matrices $X_1,X_2$ into non-negative matrices
$X_1 = AS_1, X_2 = AS_2$ to derive a similarity measure that determines how
well the shared basis $A$ approximates $X_1, X_2$. We also propose a point
cloud distance measure built upon this method and the learned factorization.
Our method reveals structural differences in both image and text data.
Potential applications include classification, detecting plagiarism or other
manipulation, data denoising, and transfer learning.
