Topological Representation of Rare States Using Combination of Persistent Homology and Complexity Measures

Identification of rare states and training models with limited data is fundamentally challenging for mainstream machine learning. Alternative approaches include one-shot learning using similarities to reference classes, meta-learning training on many related tasks and transfer learning using relevant pre-trained model. However, their performance quickly deteriorates with decreasing number of available reference classes and related tasks or lack of relevant problem for transfer learning. Previously, we proposed ensemble decomposition learning (EDL) where boosting-ensemble components trained on just two broad classes provide large number of implicit reference classes. Domain-expert knowledge such as complexity measures can be directly incorporated within EDL to reduce dependence on training data. However, success of EDL and similar approaches requires variety of complexity measures sufficiently flexible for further tuning given enough data which is not always available. Therefore, addition of complementary measures not requiring fine-tuning is important. Persistent homology (PH), one of computational topology tools, offers noise-tolerant topological summary of data set. Direct application of PH to high-dimensional data is often prohibitive and requires domain-specific dimensionality reduction. Here we suggest that PH computed on complexity measures rather than raw data could provide robust complementary metrics for enhancement of rare state representation as illustrated in the context of personalized medicine application using data from www.physionet.org.