Data or patterns (e.g., signals and images) emanating from physical sensors often exhibit complicated nonlinear structures in high dimensional spaces, which post challenges in constructing effective models and interpretable machine learning algorithms.  When data is generated through deformations of certain templates, transport transforms often linearize data clusters which are non-linear in the original domain. We will describe several transport transforms and their mathematical properties related to convexification, enabling efficient modeling of data classes as subspaces in the transform domain. We will show how to leverage such representations to solve various classification and estimation problems with high accuracy and low computational cost. This talk is based on joint work with Akram Aldroubi, Gustavo Rohde, Hasnat Rubaiyat, Xuwang Yin, M Shifat Rabbi, Yan Zhuang, Soheil Kolouri and Jonathan Nichols.