Unsupervised learning using topological data augmentation

Unsupervised machine learning is a cornerstone of artificial intelligence as it provides algorithms capable of learning tasks, such as classification of data, without explicit human assistance. We present an unsupervised deep learning protocol for finding topological indices of quantum systems. The core of the proposed scheme is a “topological data augmentation” procedure that uses seed objects to generate ensembles of topologically equivalent data. Such data, assigned with dummy labels, can then be used to train a neural network classifier for sorting arbitrary objects into topological equivalence classes. Importantly, we also show how to retrieve the local quantities corresponding to the learned topological indices from the intermediate outputs of the trained network. Our protocol is explicitly illustrated on two-band insulators in one and two dimensions, characterized by a winding number and a Chern number respectively. Using the augmentation technique also in the classification step, to classify a family of topologically equivalent objects instead of a single object, we can achieve accuracy arbitrarily close to $100\%$ even for indices outside the training regime. Apart from the method's applicability to topological classification, it also provides a new perspective on data augmentation in supervised machine learning, where given sufficient mathematical structure the set of category-preserving deformations can be rigorously defined.

Figure 1

A schematic illustration of the topological data augmentation protocol for spotting topological equivalences, applied to three geometric objects of different genus in two dimensions (2d). By augmenting a parent object, we create topologically equivalent ensembles and use them as input data to the learning by confusion method by Van Nieuwenburg et al. [56]. A network trained on distinct ensembles can then be used for further classification of unknown objects.

Figure 2

Examples of 1d Bloch Hamiltonians $H\left(k\right)$ from symmetry class AIII: (a) $h_x\left(k\right)=cos\left(k\right)$ and $h_y\left(k\right)=sin\left(k\right)$ for $k\in [0,\pi )$ and $h_x\left(k\right)=cos(-k)$ and $h_y\left(k\right)=sin(-k)$ for $k\in [\pi ,2\pi )$ with $\omega =0$; (b) $h_x\left(k\right)=cos\left(k\right)$ and $h_y\left(k\right)=sin\left(k\right)$ for $k\in [0,2\pi )$ with $\omega =1$.

Figure 3

The neural network employed by us for classifying 1d band insulators. It takes in data of size $201\times 2$ and consists of $1\times$ Conv1D layer of 128 feature maps with receptive field of size 2, $3\times$ Conv1D layers of 64, 32, and 1 feature maps with receptive fields of size 1, and a summation layer.

Figure 4

A histogram plot of the network's output $\left(y\right)$ evaluated on a test dataset. The network is trained using a trivial reference ensemble labeled “0” and an ensemble labeled “1” derived by data augmentation from the parent Hamiltonian (a) with $\omega =0$ or (b) with $\omega =1$ from Fig. 2. In panel (b), the data corresponding to the trivial (nontrivial) ensemble is illustrated in blue (red).

Figure 5

The number of input child systems $N$ vs network's output $y$ corresponding to (a) a dataset of ${10}^4$ random Bloch Hamiltonians and [(b)–(e)] datasets of ${10}^3$ samples generated from four randomly chosen parent systems.

Figure 6

The outcome of our protocol performed on a randomly selected 1d parent system: The number of input systems $N$ vs the network's output $y$ evaluated on (a) the test set with the data corresponding to the trivial reference (random parent system) depicted in blue (red), and (b) a dataset of ${10}^4$ random 1d Bloch Hamiltonians.

Figure 7

Samples of feature maps from the last convolutional layer of the network in Fig. 3 (right), compared to the explicitly calculated difference in neighboring angles $\mathrm{\Delta }\theta$ (left). These are summed up to give the network output “index” and the exact winding number respectively. The close resemblance shows that the network has learned to calculate the relevant local quantity with high precision. (For visual clarity, only every fourth of $200k$ values are shown.)

Figure 8

Examples of 2d band insulators: (a) $h_x=sin\left(k_x\right),h_y=sin\left(k_y\right),h_z=3+cos\left(k_x\right)+cos\left(k_y\right)$ for $k\in [0,2\pi )$ with $C=0$; (b) $h_x=sin\left(k_x\right),h_y=sin\left(k_y\right),h_z=1.5+cos\left(k_x\right)+cos\left(k_y\right)$ for $k\in [0,2\pi )$ with $C=1$.

Figure 9

Correspondence between a grid triangle of the discretized 2d momentum space $\left(\vec{k}\right)$ and a triangle on a 3d sphere $\left(\vec{h}\right)$ used for interpolating from discretized to continuous Bloch Hamiltonians $H\left(\vec{k}\right)=\vec{h}\left(\vec{k}\right)·\vec{\sigma }$.

Figure 10

The neural network used for classifying topological insulators in 2d: $1\times$ Conv2D of 512 feature maps with $2\times 2$ receptive field, $5\times$ Conv2D of 256, 128, 64, 32, and 1 feature maps with $1\times 1$ receptive fields, and a summation layer. The input is of size $21\times 21\times 3$.

Figure 11

The number of input child systems $N$ vs the net's output $y$ evaluated on the test dataset. Panels (a) and (b) correspond to the parent band insulators (a) and (b) from Fig. 8. In panel (b), we illustrate the data corresponding to the trivial (nontrivial) ensemble in blue (red).

Figure 12

The number of input child systems $N$ vs network's output $y$ evaluated on (a) ${10}^4$ randomly selected Bloch Hamiltonians and [(b)–(e)] ensembles of ${10}^3$ topologically equivalent Bloch Hamiltonians generated from four distinct random parent systems. Red marks denote the outputs evaluated on the parent states. In panels (b) and (e), the parent states are outliers of the respective ensembles which are peaked at the correct indices.

Figure 13

Samples of feature maps from the last convolutional layer of the network in Fig. 10 (right), compared to the explicitly calculated Berry curvature in discretized momentum space [21] (left). These maps are summed up to give the network output and the exact Chern number respectively. This demonstrates that network has learnt to calculate the relevant local quantity.

Figure 14

The outcome of our protocol performed on a randomly selected 2d parent system: The number of input systems $N$ vs the network's output $y$ corresponding to (a) the test set of ${10}^3$ Bloch Hamiltonians and (b) a set of ${10}^4$ randomly chosen 2d band insulators. In panel (b), the data corresponding to the trivial reference (randomly selected parent 2d band insulator) is illustrated in blue (red).

Figure 15

(a) Six triangles on the momentum grid that get affected by a rotation of the vector at site $(i,j)$. (b) Three vectors $v_1,v_2,v_3$ defining the corresponding spherical triangle, with $v_3$ to be rotated around $z$ axis by an angle $\mathrm{\Delta }\varphi$ to $v_4$. (c) Vectors $v_{x1}$ and $v_{x2}$ being at the intersection of the equatorial planes $(v_1,v_2)$ and $(v_3,v_4)$.