Classifying experimental image data often requires manual identification of qualitative features, which is difficult to automate. Existing automated approaches based on deep convolutional neural networks can achieve accuracy comparable to human class
ifiers, but require extensive training data and computational resources. Here we show that the emerging framework of topological data analysis can be used to rapidly and reliably identify qualitative features in image data, enabling their classification using easily-interpretable linear models. Specifically, we consider the task of identifying dark solitons using a freely-available dataset of 6257 labelled Bose-Einstein condensate (BEC) density images. We use point summaries of the images topological features -- their persistent entropy and lifetime $p$-norms -- to train logistic regression models. The models attain performance comparable to neural networks using a fraction of the training data, classifying images 30 times faster.
The machine learning technique of persistent homology classifies complex systems or datasets by computing their topological features over a range of characteristic scales. There is growing interest in applying persistent homology to characterize phys
ical systems such as spin models and multiqubit entangled states. Here we propose persistent homology as a tool for characterizing and optimizing band structures of periodic photonic media. Using the honeycomb photonic lattice Haldane model as an example, we show how persistent homology is able to reliably classify a variety of band structures falling outside the usual paradigms of topological band theory, including moat band and multi-valley dispersion relations, and thereby control the properties of quantum emitters embedded in the lattice. The method is promising for the automated design of more complex systems such as photonic crystals and Moire superlattices.
