The Cumulative Distribution Transform and Linear Pattern Classification
This addresses pattern identification problems in sensor data analysis, offering a new tool for simplifying classification tasks, though it appears incremental as it builds on existing transform-based methods.
The paper introduces the Cumulative Distribution Transform (CDT), which interprets patterns as probability density functions to simplify classification by converting Lagrangian variations into Eulerian ones in transform space, enabling linear classification and demonstrating effectiveness with computational experiments on real and simulated data.
Discriminating data classes emanating from sensors is an important problem with many applications in science and technology. We describe a new transform for pattern identification that interprets patterns as probability density functions, and has special properties with regards to classification. The transform, which we denote as the Cumulative Distribution Transform (CDT) is invertible, with well defined forward and inverse operations. We show that it can be useful in `parsing out' variations (confounds) that are `Lagrangian' (displacement and intensity variations) by converting these to `Eulerian' (intensity variations) in transform space. This conversion is the basis for our main result that describes when the CDT can allow for linear classification to be possible in transform space. We also describe several properties of the transform and show, with computational experiments that used both real and simulated data, that the CDT can help render a variety of real world problems simpler to solve.