Projects
Classifying monoaminergic and peptidergic neurons of fruit flies.
I am assisting bioscientists, who are studying the affects of water and food deprivation on fruit flies (drosophila melanogaster). As part of their study, they obtained several gene expression (scRNA-seq) data sets from their neurons, and are clustering them based on cell sub-types in their subsequent analysis (among other things). This data is challenging to work with due to inherent variability and noise in the data, lack of ground truth (often even no clear key for validation of results), data set size and dimension. I am trying to address several of these issues by standard and topological methods for data analysis. I have been using kcluster, a clustering method developed by O. Bobrowski and P. Škraba, especially frequently. I recently presented my work on this project in AATRN Applied topology seminar (a recording can be found here).
Measuring distances between directed topological spaces.
In collaboration with Lisbeth Fajstrup, Brittany Terese Fasy, Wenwen Li, Lydia Mezrag, Tatum Rask, and Francesca Tombari. (Ongoing)
This collaboration began at the third workshop for Women in Computational Topology, Bernoulli Center, Lausanne, 17.-21. July 2023.Directed topological spaces are topological spaces with a notion of direction defined via a family of directed paths on it. They can be used to model time-dependent processes or directed networks such as transportation and social networks. We are developing a new similarity measure based on the standard distance between metric spaces, the Gromov--Hausdorff distance. Depending on which of its (equivalent) definitions we generalize to directed spaces, we obtain two different measures. One captures the metric structure induced by the directed paths, while the other better compares the directed structure on spaces in question. We are preparing many interesting examples as well. Hopefully, we will publish our first results soon, so stay tuned!
Study of linear regions of deep neural networks with ReLU (or Maxout) activation.
Linear regions of a neural network are convex subsets of the input space on which each unit of each layer "makes the same decision" (this is meant in terms of the chosen activation function and can be interpreted differently depending on the choice). My collaborators and I have been studying how these regions change during the training process in hopes of gaining insight into which initialization strategies perform best, and whether or not the linear regions of a network can hint at the task the network was trained for.
Ladder decomposition for morphisms of persistence modules.
The output of persistent homology is an algebraic object called a persistence module. It admits a decomposition into a direct sum of interval persistence modules described entirely by the barcode invariant. We investigated when a morphism of persistence modules admits an analogous direct sum decomposition. Jacquard et al. showed that a ladder decomposition can be obtained whenever the barcodes of the persistence modules in question do not have any strictly nested bars. We refine this result and show that even in the presence of nested bars, a ladder decomposition exists when the morphism is sufficiently close to being invertible relative to the scale of the nested bars.
Hands-on tropical geometry.
In this project, we gave a comprehensive guide on how to create 3D-printable models of tropical surfaces, tropical curves, and combinations thereof. Code templates for creating models in polymake, exporting them to OpenSCAD and then to an appropriate file format have been made available in the official Polymake distribution. A user guide can be found here.