Deep multi-task mining Calabi–Yau four-folds

A complete intersection Calabi–Yau manifold is fully defined by its configuration matrix. This matrix encodes the polynomial degrees and ambient space factors in the following way:

$$\begin{align} \mathcal{M} = \left[ \begin{array}{c||ccc} n_0 & p^0_1 & \cdots & p^0_{K} \\ \vdots & \vdots & \ddots & \vdots \\ n_r & p^{r}_1 & \cdots & p^{r}_K \\ \end{array} \right]_{\chi}. \end{align} \tag{ 1 }$$

Each $p^i_j \in \mathbb{N}$ is the degree of the jth polynomial in the homogeneous coordinates of the ith complex projective space with dimension n_i . The Calabi–Yau condition is translated in the configuration matrix by requiring that

$$\begin{align} n_i + 1 = \sum_{j = 1}^K p^i_j. \end{align} \tag{ 2 }$$

The Euler number χ is given in the subscript and can be directly computed by integrating the fourth Chern class or from the four non-trivial Hodge numbers as

$$\begin{align} \chi = 4 + 2 h^{(1,1)} - 4h^{(2,1)} + 2 h^{(3,1)} + h^{(2,2)}. \end{align} \tag{ 3 }$$

A second linear relationship between the Hodge numbers can be derived by combining the indices $\chi_q = \chi(\mathcal{M}, \wedge^q T\mathcal{M}^*)$ [13] leading to

$$\begin{align} 44 = - 4 h^{(1,1)} + 2 h^{(2,1)} - 4 h^{(3,1)} + h^{(2,2)}. \end{align} \tag{ 4 }$$

The configuration matrices have been generated from an initial set of matrices and subsequently applying the splitting procedure [8, 12], finding new manifolds and discarding equivalent descriptions. In this way, a total of 921 497 topological distinct types of CICY manifolds were found, with 905 684 of them not being direct products of lower dimensional manifolds.

The Hodge number distributions are presented in figure 1. The mean, maximum and minimum values are

$$\begin{align} & \langle h^{(1,1)} \rangle = 10.1^{24}_{1}, \quad \langle h^{(2,1)} \rangle = 0.817^{33}_{0}, \nonumber \\ &\langle h^{(3,1)} \rangle = 39.6^{426}_{20}, \quad \langle h^{(2,2)} \rangle = 241^{1752}_{204}. \end{align} \tag{ 5 }$$

Notice that the distributions of the Hodge numbers are, in general, imbalanced: for instance, $h^{(2,1)}$ vanishes for 70% of the configuration matrices in the dataset. We find that 54.5% are favourable (i.e. $h^{(1,1)}$ is equal to the number of projective spaces), less than the 61.9% for CICY three-folds ⁶ . Hence, for slightly more than half of the cases we have $h^{(1,1)} = r$, the number of projective ambient space factors. This number is important as it should be the baseline to compare any algorithm against.

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Problems in image recognition are usually formulated as classification tasks. Take the ImageNet dataset which consists of $14 \times 10^{6}$ data points with over 21 000 classes. That is about one order of magnitude larger, both in samples and classes, than predicting $h^{(2,2)}$. In this section, we will train one neural network to classify each of the four non-trivial Hodge numbers independently. We will use an architecture based on Inception modules [16–18] as was done for the best performing predictors of the CICY three-fold Hodge numbers [21, 22]. This specific architecture has been shown to lead to the best performance on the configuration matrices when using 1d kernels of maximal size. This partially reflects the fact that scanning coordinates in each projective space and a single variable over all projective spaces helps in better learning the connections between the different hypersurfaces of the CICYs (see figure 2). The choice of maximal 1d kernel is, in fact, motivated by the mathematical machinery required to compute Hodge numbers. There, one has to compute the dimension of ambient space cohomology group representations, which are stacked for each projective space. These ambient space representations arise after splitting up the Koszul resolution

$$\begin{align} 0~\to \wedge^K \mathcal{N}^* \to \cdots \to \mathcal{N}^* \to \mathcal{O}_\mathcal{A} \to \mathcal{O}_\mathcal{A}|_\mathcal{M} \to 0\; . \end{align} \tag{ 6 }$$

which contains the antisymmetric products $\wedge^s \mathcal{N}^*$ of the defining hypersurfaces ($\mathcal{N}$ denotes the normal bundle, which contains the information about the polynomial degrees $p^i_j$). Moreover, using an Inception based architecture lead to a performance increase of misclassification rate on the ImageNet dataset from 15.3% for AlexNet [15] using a standard convolutional architecture to 6.7% for the first version of GoogLeNet [16].

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We proceed as in earlier studies [21, 22] by considering different train:val:test splits with respectively 10%, 30%, 50% and 80% training and 10% validation data. The architecture hyperparameters have been optimized using Bayesian Optimization Hyperband [44, 45] on the problem of predicting $h^{(3,1)}$. The same hyperparameters have then been used to also classify the other three Hodge numbers.

We opted to present the results of neural networks with a comparable number parameters $840\,000 \pm 10\,000$ to the number of configuration matrices. This architecture comprises four Inception modules, with respectively $3 \times 64$ and 16 filters, utilizing batch normalization for better gradient propagation into the earlier layers [16–18, 46]. Figure 2 decomposes an Inception module into its different ingredients. The convolutional kernels scan over the configuration matrix dimensions, i.e. the maximal number of possible projective ambient spaces (16) and the maximal number of polynomial constraints (20). The Inception modules are followed by three dense layers with 16 units, ReLU activation function and dropout layers with a 0.2 rate to contrast overfitting. Furthermore, we employ $\ell_1$ ($1 \times 10^{-5}$) and $\ell_2$ ($1 \times 10^{-6}$) regularization for all weights in the network. The last layer contains a softmax activation function with $\{h^{(i,j)}_{\textrm{min}}, \dots, h^{(i,j)}_{\textrm{max}} \}$ classes. The network is trained with Adam optimizer and an initial learning rate of $4 \times 10^{-4}$ on a 32 mini-batch size. This architecture is still trainable in a reasonable amount of time on a desktop computer with access to a GPU. In comparison to earlier studies [21, 22], we found that leaving the outliers inside the training data does not negatively impact the results.

Figure 3 shows in the top row the training loss and accuracy, and in the bottom row validation accuracy tracked over the training process and test accuracy for the best-performing model. The best-performing model is the one with the highest validation accuracy, which one would get when employing early stopping on that metric. It is important to track the best performing models as sometimes the loss starts increasing again as visible from the $h^{(2,1)}$ curve. The error bars are computed from the different training ratios and the budget on the x-axis is given by

$$\begin{align} \textrm{budget} & = \textrm{number of epochs} \times \frac{\textrm{percentage of training data}}{80}. \end{align} \tag{ 7 }$$

We observe that $h^{(1,1)}$ is predicted with almost perfect accuracy for any training ratio, while the accuracies of the other three Hodge numbers improve with more training data. However, when the training data contains more than 30% of the samples one has diminishing returns for the accuracy. This is in line with previous observations for the CICY three-folds [21, 22]. Even though the hyperparameters have been optimized to learn $h^{(3,1)}$, it is the worst performing value. This is interesting as figure 1 shows that the distribution of $h^{(2,2)}$ spans a longer range, contains more outliers and has a thicker tail. The plots show that we avoid overfitting to the training data.

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Table 1 collects the accuracy at different training ratios and the mean-value for the four different training ratios of the best performing model ⁷ . In the training process, we employed learning rate decay with a factor of 0.4, when the validation accuracy did not improve for epochs equivalent to 0.15× budget. This is clearly visible from the loss and accuracy plots in the top row and accounts for the down- and up-stairs steps. Summarizing the results, we find that the hyperparameters found for predicting the worst performing Hodge number $h^{(3,1)}$ also generalize well to the other three Hodge numbers. This is a first indication that the prediction of Hodge numbers could benefit from multi-task learning.

Table 1. Comparison of the test accuracy for different training ratios.

	$h^{(1,1)}$	$h^{(2,1)}$	$h^{(3,1)}$	$h^{(2,2)}$
10%	0.99	0.87	0.59	0.62
30%	1.00	0.91	0.67	0.73
50%	1.00	0.94	0.68	0.75
80%	1.00	0.95	0.70	0.75
Mean	1.00	0.92	0.66	0.71

We use the same dataset presented for the classification objective in the previous section. Given the strong class imbalance, we select the training set by using a stratified approach on $h^{(2,1)}$ in order to preserve the distribution of the samples. The validation set is then chosen totally at random, using 10% of the samples. The remaining samples form the test set. We preprocess the input data by simply rescaling the entries of the configuration matrices in the training set to the interval $[ 0,\, 1 ]$. Matrices in the validation and test sets are rescaled accordingly, using the statistics obtained from the training set.

The outputs of CICYMiner are, in fact, floating point numbers $\tilde{h}^{(i,\,j)} \in \mathbb{R}^+$, as it is typical in regression tasks. They ultimately need to be rounded to integers to be directly compared with the true values and to compute the accuracy. The distributions of the Hodge numbers have not been rescaled as training led to lower accuracy when this strategy was adopted. The specialised branches of the network are, in fact, deep enough to apply the proper scaling starting from a shared representation and correctly learn the output distribution of the Hodge numbers.

In order to test the robustness and versatility of the network, we choose to keep the outliers in the training set. In multi-task learning architectures, they may strongly affect the behavior of the network and may need robust loss functions during training [49]: this problem is directly addressed in what follows. On the other hand, what represents an outlier for a certain task, can be valuable information for another [50], hence the choice of keeping the outliers in the training set. Empirically, we also experienced a decrease in accuracy when trying to find a good outlier exclusion strategy.

In this case, training occurred over a fixed amount of 300 epochs, due to time restrictions on the cluster computing infrastructure. Training takes approximately 5 d on a single NVIDIA V100 GPU. We use the Adam [51] stochastic gradient descent with an initial learning rate of $1 \times 10^{-3}$ and a mini-batch size of 64 configuration matrices. Due to the long training time, the optimization was done using a grid search over a reasonable amount of choices of hyperparameters. The network is ultimately made of $1 \times 10^{7}$ trainable parameters, accounting for both the shared representation and the eight sub-networks learning Hodge numbers and their auxiliary outputs. In terms of typical computer vision multi-task learning, we still deal with a small network: for instance, the original Inception network by Google has $0.7 \times 10^{7}$ parameters for a single classification task [16–18].

We already motivated the choice of keeping the outliers in the training set. We address the arising issues by employing a Huber loss function [52]:

$$\begin{equation} \mathcal{H}^{\{k \}}_{\delta}( x ) = \begin{cases} \displaystyle \frac{1}{2} \sum\limits_{n = 1}^k \sum\limits_{i = 1}^{N_k}\, \omega_n \big( x^{(i)} \big)^2, & \left| x^{(i)} \right| \leqslant \delta \\ \displaystyle \delta \sum\limits_{n = 1}^k \sum\limits_{i = 1}^{N_k}\, \omega_n \left( \big| x^{(i)} \big| - \frac{\delta}{2} \right), & \left| x^{(i)} \right| \gt \delta \end{cases} \end{equation} \tag{ 8 }$$

where ω_n for $n = 1, 2, \dots, k$ are the loss weights of the different branches of the CICYMiner, δ is a hyperparameter of the model and $x^{(i)}$ is the residual error of the ith sample. The choice of the loss turns out to be extremely useful in this regression task, as it behaves as a $\ell_2$ loss for small residuals, and it is linear for larger errors. Robustness is thus implemented as a continuous interpolation between the quadratic and linear behaviour of the loss function. This is a solution usually adopted for classification [50] where combinations of $\ell_1$, $\ell_2$ and Frobenius norm are used for robustness.

In our best implementation, we used δ = 1.5, and loss weights 0.05, 0.3, 0.25 and 0.35 for $h^{(1,1)}$, $h^{(2,1)}$, $h^{(3,1)}$ and $h^{(2,2)}$, respectively (the auxiliary branches use the same values as the principal ones). The learning rate is set to reduce by a factor of 0.3 after 75 epochs without improvements in the total loss of the validation set (as a reference, at 80% training ratio, this hard reduction mechanism triggered only once between epochs 270 and 300).

The final results are presented in figure 5 and in the last row of table 2. As shown in the learning curve, $h^{(1,1)}$ reaches perfect accuracy with just 10% of the training data, in alignment with previous attempts [35] and the classification results of the previous section. $h^{(2,2)}$ is in general the most difficult label to train and it is strongly dependent on the training ratio. The network appears to be underfitting the distributions of the Hodge numbers, and validation loss is still decaying after 300 epochs: it would be interesting to run training for longer time, in order to study the behaviour of the network. At a training ratio of 30% the network reaches perfect accuracy on $h^{(1,1)}$, while $h^{(2,1)}$ gets to 97%. $h^{(3,1)}$ remains at 81%, while $h^{(2,2)}$ reaches barely 49%. Increasing the number of training samples is, in general, beneficial for all Hodge numbers: $h^{(1,1)}$ and $h^{(2,1)}$ reach 100%, while the accuracy of $h^{(3,1)}$ and $h^{(2,2)}$ rises to 96% and 83%, respectively, when the training ratio reaches 80%. For the first three outputs in table 2, the regression metrics, mean squared error (MSE) and mean absolute error (MAE), show the ability to effectively learn the discreteness of the Hodge numbers: both metrics show, in fact, values which can be confidently rounded to well defined integer results (i.e. $\textrm{MAE} \ll 0.50$ and $\textrm{MSE} \ll 0.25$).

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Table 2. Comparison of the accuracy obtained by similar models at 80% training ratio. Regression metrics are also specified for CICYMiner at the same ratio.

	$h^{(1,1)}$	$h^{(2,1)}$	$h^{(3,1)}$	$h^{(2,2)}$
+att	1.00	0.99	0.96	0.81
MSE loss	1.00	0.97	0.92	0.50
No aux	1.00	0.84	0.92	0.72
bs-256	1.00	0.99	0.94	0.65
Layer norm	1.00	0.99	0.92	0.66
CICYMiner	1.00	1.00	0.96	0.83
MSE ($1 \times 10^{-4}$)	1.3	98	560	6800
MAE ($1 \times 10^{e-3}$)	7.8	19	130	360

The good performance of the first three Hodge numbers suggests the possibility to use relations such as the Euler characteristic (3), which can be computed from combinatorics, and the linear constraint (4). Using the latter to compute $h^{(2,2)}$ leads to an accuracy of 96% on the test set, using the best results at 80% training ratio. Using (3) and (4) together, $h^{(3,1)}$ and $h^{(2,2)}$ can reach perfect accuracy at 80% training ratio. Using CICYMiner it is therefore possible to compute all four Hodge numbers with 100% of accuracy.

CICYMiner introduces new elements, with respect to previous attempts at predicting Hodge numbers of CICYs [21, 35], namely:

• (a)  
  Huber loss for robustness;
• (b)  
  auxiliary branches.

In this section, we separately analyse each new aspect, together with other variations of the architecture. Specifically, we analyse the impact of the batch normalization used in the Inception modules. We also address the use of attention mechanisms [53], used in the DeepMiner model, which in our case did not lead to an improvement in accuracy, but rather to a faster training.

We proceed by modifying the backbone structure of CICYMiner. We first introduce the attention mechanism used in [48] for comparison. The Spatial Attention Module ($\mathrm{SAM}$) and CHannel Attention Module ($\mathrm{CHAM}$) are presented in figure 6: the full attention mechanism is the composition $\mathrm{CHAM} \circ \mathrm{SAM}$ used between each Inception module in the main branch of the task-specific architecture in figure 4. We also analyse the performance of the model by simply removing the auxiliary branches in the top layers of the network. Then, as opposed to the Huber loss, we test the predictions using the usual MSE used in most regression tasks. We finally change the size and type of the normalization strategy used in the architecture: we first train a network with a mini-batch size of 256 samples, and we then compare the results with a Layer Normalization [54] strategy. Results are summarised in figure 7 and numerically reported in table 2. CICYMiner leads to the best overall performance for all four Hodge numbers. The distributions of the residuals, $x^{(i)}$ appearing in the Huber Loss (8), show in figure 8 a homoscedastic behaviour (no correlations between predictions and absolute value of the residuals), which ultimately supports the completeness of the model and its ability to properly predict the four Hodge numbers correctly.

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The use of a different loss function, which is not robust against outliers, led to largest drop in accuracy, overall: the difference starts to be consistent even for $h^{(2,1)}$ and $h^{(3,1)}$, which do not present many outliers in figure 1. The accuracy plummets when considering $h^{(2,2)}$, as expected. The presence of outliers is also evident when increasing the mini-batch size: $h^{(2,2)}$ suffers the largest decrease in accuracy due to such normalisation strategy. At the same time, the introduction of a batch-size independent Layer Normalization strategy, which normalizes each sample over the channel direction rather than the batch dimension, leads to a similar decrease. The presence of outliers seems, therefore, a delicate issue for which the size of the mini-batches plays a relevant role.

A related aspect is represented by the ablation study on the auxiliary branches. As their role is to mine a richer variety of features to stabilise the shared representation, and learn better approximations of the output, the accuracy drops significantly in the case of highly imbalanced distributions. The largest drop impacts $h^{(2,1)}$ which suffers from predictions shifting towards zero. This shows that we indeed need a mechanism to get as much training information as possible through the addition of transformations and auxiliary branches, as in the CICYMiner.

Finally, we analyse the impact of the attention modules: we insert such additional layers to improve the predictions of $h^{(3,1)}$ and $h^{(2,2)}$ only, as other Hodge numbers do not need additional transformations. The results do not strongly differ from the case without the attention modules, though $h^{(2,2)}$ drops by 2% in accuracy. It therefore seems that the attention modules do not help the predictions in this case, supported by the naive intuition that the configuration matrices do not suggest the development of a sequence model, such as in natural language processing (NLP) or deep learning for video sequences. However, the accuracy reached by the model occurs at around 100 training epochs, rather than 300 as in other cases. The loss function then presents a slight increase after that. The use of attention modules, together with an early stopping strategy, may therefore significantly cut the training time in this context.