Unsupervised mapping of phase diagrams of 2D systems from infinite projected entangled-pair states via deep anomaly detection

Both referees pointed out that because the training can be done with just one example, the autoencoder might be superfluous. Referee 1 suggested using a similarity measure equivalent to the loss function for the autoencoder and referee 2 suggested using inner products. So, if we understand correctly, the idea is to perform a kind of data-driven geometric analysis in the spirit of machine learning but w/o machine learning (i.e. neural networks). There has been work done in similar directions, e.g., Ref. [38] showed that phase boundaries can also be determined via inner products of quantum states. One point that we made in the paper is that inner products between quantum states are in fact expensive for 2D tensor networks and can be avoided with our proposed method.

If we understood correctly, the referees raise the very interesting point that overlaps (or similarities) could also be computed from the reduced data that is used for the machine learning protocol. Indeed we find that this is possible and leads to comparable results in quality. For the inner product it is a bit hard to argue as we see that the contrast is arguably small on a range of 0.01 (between 1.00 and 0.99). But in both cases, undeniably, the results are qualitatively comparable. We find this to be very curious and allowed ourselves to add it to the manuscript (Fig. 4).

The beauty of ML methods, like the anomaly detection scheme in discussion here, is that it is a very general framework that is capable of adapting to a data-specific problem by having an over-parametrized objective function that is optimized for the given data. This is in general very powerful as it is very flexible in the problems it can be applied to. Yet, of course, there is never the method to detect phase transitions (and don’t claim that ours is). We saw that for our problem, i.e. for the model and data at hand, this might not be necessary and simple geometric analysis can be sufficient. One very interesting question is whether this is true for just this example, for some cases or maybe even a majority / all cases? The answer to this question is beyond the scope of our paper but an interesting one worth pointing out and investigating next. Having a situation in which our ML approach detects a phase transition and “no other existing approach works” would of course be interesting but probably very hard to find, if even possible. In our opinion, the main virtue of our approach is its generality and flexibility to adapt to the problem of interest by adjusting the network parameters. We benchmarked on a model that is well understood and demonstrated that it could be combined with state-of-the-art iPEPS simulations.