  In our approach, we consider the data as instances of a random field within a
relevant Bochner space. Our key observation is that the classes can
predominantly reside in two distinct subspaces. To uncover the separation
between these classes, we employ the Karhunen-Loeve expansion and construct the
appropriate subspaces. This allows us to effectively reveal the distinction
between the classes. The novel features forming the above bases 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. 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 for 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 large unbalanced datasets. We demonstrate it on a number of
high-dimensional datasets. This approach yields significant increases in
accuracy of ML methods compared to using the same ML algorithm with the
original feature data. Our tests on the Alzheimer's Disease ADNI dataset shows
a dramatic increase in accuracy (from 48% to 89% accuracy). Furthermore, tests
using unbalanced semi-synthetic datasets created from the benchmark GCM dataset
confirm increased accuracy as the dataset becomes more unbalanced.
