Featuring the topology with the unsupervised machine learning

Images of line drawings are generally composed of primitive elements. One of the most fundamental elements to characterize images is the topology; line segments belong to a category different from closed circles, and closed circles with different winding degrees are nonequivalent. We investigate images with nontrivial winding using the unsupervised machine learning. We build an autoencoder model with a combination of convolutional and fully connected neural networks. We confirm that compressed data filtered from the trained model retain more than 90% of correct information on the topology, evidencing that image clustering from the unsupervised learning features the topology.Comment: 14 pages, 7 figure

Topological quantum phase transitions retrieved through unsupervised machine learning

The discovery of topological features of quantum states plays an important role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological quantum phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work, we show that the unsupervised manifold learning can successfully retrieve topological quantum phase transitions in momentum and real space. Our results show that the Chebyshev distance between two data points sharpens the characteristic features of topological quantum phase transitions in momentum space, while the widely used Euclidean distance is in general suboptimal. Then a diffusion map or isometric map can be applied to implement the dimensionality reduction, and to learn about topological quantum phase transitions in an unsupervised manner. We demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model, the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and further provide implications and demonstrations for learning in real space, where the topological invariants could be unknown or hard to compute. The interpretable good performance of our approach shows the capability of manifold learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.Comment: Accepted by Physical Review B (2020); Updated version including implications and demonstrations for learning in real space, and with appendices; Typos were corrected; 14 pages, 12 figure