Topological Features In Cancer Gene Expression Data

We present a new method for exploring cancer gene expression data based on tools from algebraic topology. Our method selects a small relevant subset from tens of thousands of genes while simultaneously identifying nontrivial higher order topological features, i.e., holes, in the data. We first circumvent the problem of high dimensionality by dualizing the data, i.e., by studying genes as points in the sample space. Then we select a small subset of the genes as landmarks to construct topological structures that capture persistent, i.e., topologically significant, features of the data set in its first homology group. Furthermore, we demonstrate that many members of these loops have been implicated for cancer biogenesis in scientific literature. We illustrate our method on five different data sets belonging to brain, breast, leukemia, and ovarian cancers.Comment: 12 pages, 9 figures, appears in proceedings of Pacific Symposium on Biocomputing 201

Topological classifier for detecting the emergence of epileptic seizures

Objective An innovative method based on topological data analysis is introduced for classifying EEG recordings of patients affected by epilepsy. We construct a topological space from a collection of EEGs signals using Persistent Homology; then, we analyse the space by Persistent entropy, a global topological feature, in order to classify healthy and epileptic signals. Results The performance of the resulting one-feature-based linear topological classifier is tested by analysing the Physionet dataset. The quality of classification is evaluated in terms of the Area Under Curve (AUC) of the receiver operating characteristic curve. It is shown that the linear topological classifier has an AUC equal to 97.2% while the performance of a classifier based on Sample Entropy has an AUC equal to 62.0%

Harmonic Persistent Homology

We introduce harmonic persistent homology spaces for filtrations of finite simplicial complexes. As a result we can associate concrete subspaces of cycles to each bar of the barcode of the filtration. We prove stability of the harmonic persistent homology subspaces under small perturbations of functions defining them. We relate the notion of "essential simplices" introduced in an earlier work to identify simplices which play a significant role in the birth of a bar, with that of harmonic persistent homology. We prove that the harmonic representatives of simple bars maximizes the "relative essential content" amongst all representatives of the bar, where the relative essential content is the weight a particular cycle puts on the set of essential simplices.Comment: 35 pages, 5 figures. Comments welcom