Matthew W. Wheeler

Matthew Wheeler, Jose Bouza, Peter Bubenik

We use topological data analysis (TDA) to study how data transforms as it passes through successive layers of a deep neural network (DNN). We compute the persistent homology of the activation data for each layer of the network and summarize this information using persistence landscapes. The resulting feature map provides both an informative visual- ization of the network and a kernel for statistical analysis and machine learning. We observe that the topological complexity often increases with training and that the topological complexity does not decrease with each layer.

Matthew W. Wheeler

Many modern data sets are sampled with error from complex high-dimensional surfaces. Methods such as tensor product splines or Gaussian processes are effective/well suited for characterizing a surface in two or three dimensions but may suffer from difficulties when representing higher dimensional surfaces. Motivated by high throughput toxicity testing where observed dose-response curves are cross sections of a surface defined by a chemical's structural properties, a model is developed to characterize this surface to predict untested chemicals' dose-responses. This manuscript proposes a novel approach that models the multidimensional surface as a sum of learned basis functions formed as the tensor product of lower dimensional functions, which are themselves representable by a basis expansion learned from the data. The model is described, a Gibbs sampling algorithm proposed, and is investigated in a simulation study as well as data taken from the US EPA's ToxCast high throughput toxicity testing platform.