Stochastic coordinate transformations with applications to robust machine learning

In this paper we introduce a set of novel features for identifying underlying stochastic behavior of input data using the Karhunen-Loeve expansion. These novel features are constructed by applying a coordinate transformation based on the recent Functional Data Analysis theory for anomaly detection. The associated signal decomposition is an exact hierarchical tensor product expansion with known optimality properties for approximating stochastic processes (random fields) with finite dimensional function spaces. In principle these low dimensional spaces can capture most of the stochastic behavior of `underlying signals' in a given nominal class, and can reject signals in alternative classes as stochastic anomalies. Using a hierarchical finite dimensional expansion of the nominal class, a series of orthogonal nested subspaces is constructed for detecting anomalous signal components. Projection coefficients of input data in these subspaces are then used to train a Machine Learning (ML) classifier. However, due to the split of the signal into nominal and anomalous projection components, clearer separation surfaces of the classes arise. In fact we show that with a sufficiently accurate estimation of the covariance structure of the nominal class, a sharp classification can be obtained. This is particularly advantageous for situations with large unbalanced datasets. We formulate this concept and demonstrate it on a number of high-dimensional datasets in cancer diagnostics. This approach yields significant increases in accuracy over ML methods that use the original feature data. This method leads to a significant increase in precision and accuracy over the current top benchmarks for the Global Cancer Map (GCM) gene expression network dataset. Furthermore, tests from unbalanced semi-synthetic datasets created from the GCM data confirmed increased accuracy as the dataset becomes more unbalanced.