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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-10 19:11:29 +02:00
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We have proved how a proper data analysis can lead to improvements in predictions of Hodge numbers \hodge{1}{1} and \hodge{2}{1} for \cicy $3$-folds.
Moreover considering more complex neural networks inspired by the computer vision applications~\cite{Szegedy:2015:GoingDeeperConvolutions, Szegedy:2016:RethinkingInceptionArchitecture, Szegedy:2016:Inceptionv4InceptionresnetImpact} allowed us to reach close to \SI{100}{\percent} accuracy for \hodge{1}{1} with much less data and less parameters than in previous works.
While our analysis improved the accuracy for \hodge{2}{1} over what can be expected from a simple sequential neural network, we barely reached \SI{50}{\percent}.
Hence it would be interesting to push further our study to improve the accuracy.
Possible solutions would be to use a deeper Inception network, find a better architecture including engineered features, and refine the ensembling.
Another interesting question to probe is related to representation learning, i.e.\ finding a better description of the \cy.
Indeed one of the main difficulty in making predictions is the redundancy of the possible descriptions of a single manifold.
For instance we could try to set up a map from any matrix to its favourable representation (if it exists).
This could be the basis for the use of adversarial networks~\cite{Goodfellow:2014:GenerativeAdversarialNets} capable of generating the favourable embedding from the first.
Or on the contrary one could generate more matrices for the same manifold in order to increase the size of the training set.
Another possibility is to use the graph representation of the configuration matrix to which is automatically invariant under permutations~\cite{Hubsch:1992:CalabiyauManifoldsBestiary} (another graph representation has been decisive in~\cite{Krippendorf:2020:DetectingSymmetriesNeural} to get a good accuracy).
Techniques such as (variational) autoencoders~\cite{Kingma:2014:AutoEncodingVariationalBayes, Rezende:2014:StochasticBackpropagationApproximate, Salimans:2015:MarkovChainMonte}, cycle GAN~\cite{Zhu:2017:UnpairedImagetoimageTranslation}, invertible neural networks~\cite{Ardizzone:2019:AnalyzingInverseProblems}, graph neural networks~\cite{Gori:2005:NewModelLearning, Scarselli:2004:GraphicalbasedLearningEnvironments} or more generally techniques from geometric deep learning~\cite{Monti:2017:GeometricDeepLearning} could be helpful.
Finally, our techniques apply directly to \cicy $4$-folds~\cite{Gray:2013:AllCompleteIntersection, Gray:2014:TopologicalInvariantsFibration}.
However there are much more manifolds in this case, such that one can expect to reach a better accuracy for the different Hodge numbers (the different learning curves for the $3$-folds indicate that the model training would benefit from more data).
Another interesting class of manifolds to explore with our techniques are generalized \cicy $3$-folds~\cite{Anderson:2016:NewConstructionCalabiYau}.
These and others will indeed be ground for future investigations.
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