22 lines
2.9 KiB
TeX
22 lines
2.9 KiB
TeX
We have proved that a proper data analysis can lead to improvements in predictions of Hodge numbers \hodge{1}{1} and \hodge{2}{1} for \cicy $3$-folds.
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Moreover more complex neural networks inspired by 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.
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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}.
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Hence it would be interesting to push further our study to improve the accuracy.
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Possible solutions would be to use a deeper Inception network, find a better architecture including engineered features, and refine the ensembling.
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Another interesting question to probe is related to representation learning, i.e.\ finding a better description of the \cy.
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Indeed one of the main difficulty in making predictions is the redundancy of the possible descriptions of a single manifold.
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For instance we could try to set up a map from any matrix to its favourable representation (if it exists).
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This could be the basis for the use of adversarial networks~\cite{Goodfellow:2014:GenerativeAdversarialNets} capable of generating the favourable embedding from the first.
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Or on the contrary one could generate more matrices for the same manifold in order to increase the size of the training set.
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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).
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Techniques such as (variational) autoencoders~\cite{Kingma:2014:AutoEncodingVariationalBayes, Rezende:2014:StochasticBackpropagationApproximate}, cycle GAN~\cite{Zhu:2017:UnpairedImagetoimageTranslation}, invertible neural networks~\cite{Ardizzone:2019:AnalyzingInverseProblems}, graph neural networks~\cite{Gori:2005:NewModelLearning, Scarselli:2004:GraphicalbasedLearningEnvironments} or techniques from geometric deep learning~\cite{Monti:2017:GeometricDeepLearning} could be helpful.
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Finally our techniques apply directly to \cicy $4$-folds~\cite{Gray:2013:AllCompleteIntersection, Gray:2014:TopologicalInvariantsFibration}.
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However there are many more manifolds in this case (around \num{e6}) and more Hodge numbers, 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).
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Another interesting class of manifolds to explore with our techniques are generalized \cicy $3$-folds~\cite{Anderson:2016:NewConstructionCalabiYau}.
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These and others will indeed be ground for future investigations.
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