aainstitutetext:  Centre for Theoretical Physics, School of Physics and
Astronomy, Queen Mary University of London, 327 Mile End Road, London E1 4NS,
UKbbinstitutetext:  London Institute for Mathematical Sciences, Royal
Institution, London W1S 4BS, UKccinstitutetext:  Department of Mathematics,
City, University of London, EC1V 0HB, UKddinstitutetext:  Merton College,
University of Oxford, OX1 4JD, UKeeinstitutetext:  School of Physics, NanKai
University, Tianjin, 300071, P.R. China

# Machine Learning Calabi-Yau Hypersurfaces

David S. Berman b,c,d,e Yang-Hui He c,b Edward Hirst<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We revisit the classic database of weighted-$\mathbb{P}^{4}$s which admit
Calabi-Yau 3-fold hypersurfaces equipped with a diverse set of tools from the
machine-learning toolbox. Unsupervised techniques identify an unanticipated
almost linear dependence of the topological data on the weights. This then
allows us to identify a previously unnoticed clustering in the Calabi-Yau
data. Supervised techniques are successful in predicting the topological
parameters of the hypersurface from its weights with an accuracy of
$R^{2}>95\%$. Supervised learning also allows us to identify
weighted-$\mathbb{P}^{4}$s which admit Calabi-Yau hypersurfaces to $100\%$
accuracy by making use of partitioning supported by the clustering behaviour.

††preprint:  QMUL-PH-21-55 LIMS-2021-017

##  1 Introduction

Artificial intelligence has now permeated through all disciplines of human
enterprise. Machine-learning (ML) has become, in this age driven by big data,
as indispensable a tool as calculus was to the Age of Enlightenment
lecun2015deep . Perhaps surprisingly, noise-less, pure mathematical data can
often be learned at high precision, indicating underlying formulae which have
not yet been uncovered from traditional analyses. Examples of this data driven
approach to mathematics may be seen in applications of ML to: the string
theory landscape He:2017aed ; He:2017set ; Carifio:2017bov ; Krefl:2017yox ;
Ruehle:2017mzq ; abstract algebra He:2019nzx ; modern number theory
Alessandretti:2019jbs ; He:2020eva ; and graph theory He:2020fdg . It is hoped
that ML might reveal structures in the very nature of mathematics He:2021oav
and mathematical intuition davies2021advancing , deeply embedded in the
mathematical data. Apart from ML itself, the tools that have been developed to
enable ML have provided significant new capabilities for old problems. This is
exemplified by the use of the autodifferentiation capabilities of Tensorflow
to explore the possible vacua of various gauged supergravities, see for
example Comsa:2019rcz ; Bobev:2019dik ; Krishnan:2020sfg ; Bobev:2020ttg ;
Berman:2021ynm ; Berman:2022jqn .

Amongst its various virtues, string theory pioneered the data-mining of such
mathematical data. One should be mindful that this was done shortly after the
beginnings of string phenomenology in the late 1980s, long before the dawn of
the modern era of “Big Data” and modern readily available ultra-fast computing
power. Indeed, when Calabi-Yau manifolds were realized to be Candelas:1985en
the standard solution to vacuum configurations (see Bao:2020sqg  for a brief,
and He:2018jtw  a longer, pedagogical review), and hence low-energy particle
physics, a programme was introduced by the physics community to compile one of
the first databases in algebraic geometry.

These were some of the earliest appearances of “big data” in mathematics,
beyond compiling digits of $\pi$ or large primes. The first dataset was the
so-called CICYs, which stands for “complete intersection Calabi-Yau manifolds”
in products of complex projective spaces Candelas:1987kf ; Green:1986ck ;
which can be thought of as a generalization of the famous quintic 3-fold in
$\mathbb{P}^{4}$. Appropriately, one of the first ML experiments in geometry
was performed on this dataset He:2017aed . Over the last few years, the
initial success has been vastly improved by using more and more sophisticated
neural network (NN) architectures and machine learning techniques Bull:2018uow
; Bull:2019cij ; Krippendorf:2020gny ; He:2020lbz ; Douglas:2020hpv ;
ashmore2021machine ; Anderson:2020hux ; Erbin:2020tks ; Erbin:2020srm ;
Erbin:2021hmx ; Larfors:2021pbb ; Bao:2020nbi ; Bao:2021auj ; Bao:2021olg ;
Jejjala:2020wcc ; Brodie:2021nit ; Cole:2021nnt ; Halverson:2021aot ;
gao2021machine ; Cole:2019enn ; Krippendorf:2021uxu .

Yet, the CICY dataset has a peculiarity: it is skewed toward negative Euler
number. This would have occurred to Candelas et al. since they knew about
mirror symmetry. Since the exchange of the two Hodge numbers
$(h^{1,1},h^{2,1})$ would reverse the sign of the Euler number $\chi$; the
conjecture that to every Calabi-Yau 3-fold with $(h^{1,1},h^{2,1})$ there is a
mirror with these exchanged would imply the negation of $\chi$.

Therefore, as the second database of geometries in string theory, another
generalization of the quintic was undertaken by placing weights on the ambient
$\mathbb{P}^{4}$ and considering a single, generic Calabi-Yau hypersurface
therein CANDELAS1990383 . This produced a much more balanced set of Calabi-Yau
3-folds with $\pm$ Euler numbers, and the rough outline of the famous “mirror
plot” of the distributions of $2(h^{1,1}-h^{2,1})$ vs $(h^{1,1}+h^{2,1})$
could already be seen to emerge.

All these datasets were subsequently subsumed into the dataset created through
the extraordinary work of Kreuzer and Skarke Kreuzer:2000xy ;
Skarke1996WEIGHTSF . This set contains the Calabi-Yau hypersurfaces in toric
varieties. (Since weighted projective spaces are special types of toric
varieties, the set described in CANDELAS1990383  is a subset of the Kreuzer-
Sharke set.)

However, the Kreuzer-Sharke dataset is of astronomical size, containing some
half-billion members. While ML of this set is in progress Bao:2021ofk , the
much more manageable list of hypersurfaces in weighted $\mathbb{P}^{4}$,
numbering around 8000 (comparable to the CICYs) is natural choice of
geometries to study and apply the latest methods from data science.

Thus our motivation is clear. We shall re-visit the classic database of
CANDELAS1990383  with a modern perspective, continuing the paradigm of
machine-learning the string theory landscape and the resulting emergent
mathematical structures, using tools from the sci-kit learn library scikit-
learn  implemented in python. The paper is organized as follows. In §2 we
begin with a rapid review of the mathematics of our Calabi-Yau hypersurfaces,
emphasizing on the data structure. In §3, we proceed with analyses of the
data, using methods which were unavailable at the time of their creation, such
as principle component analysis and topological data analysis. We then use
neural-networks to machine learn the dataset in §4. We conclude with a summary
and outlook in §5.

All of the data and code are freely available on GitHub at:
https://github.com/edhirst/P4CY3ML.git

##  2 Characterising the Calabi-Yau Hypersurfaces

The dataset of focus in this study is that of weighted projective spaces
$\mathbb{P}^{4}_{\mathbb{C}}(w_{i})$, which admit Calabi-Yau (CY) three-fold
hypersurfaces within them.

This dataset was constructed in the early 90s alongside other efforts to
expand the CY landscape for use in Landau-Ginzburg models and superstring
compactification KS1992 ; KS1994 ; CANDELAS1990383 ; Kreuzer:2000xy ; COK1995
. The dataset is readily available at:
http://hep.itp.tuwien.ac.at/~kreuzer/CY/, whilst another copy is given with
this study’s scripts on the corresponding GitHub repository.

A generic weighted projective space generalises the concept of a projective
space, defined by taking some $\mathbb{C}^{n+1}$ with coordinates
$\\{z_{1},z_{2},...,z_{n+1}\\}$ and performing an identification with weights
$w_{i}$ such that

$(z_{1},z_{2},...,z_{n+1})\sim(\lambda^{w_{1}}z_{1},\lambda^{w_{2}}z_{2},...,\lambda^{w_{n+1}}z_{n+1})\,,$
(2.1)

$\forall\lambda\in\mathbb{C}$, hence defining the projective space
$\mathbb{P}^{n}$ with these $n+1$ homogeneous coordinates. For the projective
spaces in consideration $n$ takes the value of 4, and the space is hence
defined with 5 weights. These weights are coprime as a set, such that the
projective space definition is free from redundancy from different weight
combinations.

Within these weighted projective spaces one can embed hypersurfaces defined by
polynomials in these homogeneous coordinates. Of particular interest to
physicists are those hypersurfaces which are CY in nature. A defining property
of CY manifolds is the vanishing of the first Chern class, and for this to
hold within the projective space the hypersurface’s polynomial has to have
degree $d=\sum_{i}(w_{i})$.

It should be noted that the identifications that are used in constructing the
projective space lead to singular sets, which the hypersurfaces can intersect
with suitable resolution. To be consistently defined over these singular sets
another property of the polynomial is required: transversity. The transversity
property implies that the polynomial equation and its derivative share no
common solutions, and this condition translates into a condition on the
projective space weights:

$\forall w_{i}\;\exists
w_{j}\;s.t.\;\frac{\sum_{k}(w_{k})-w_{j}}{w_{i}}\in\mathbb{Z}^{+}.$ (2.2)

However as exemplified in CANDELAS1990383 , this condition is necessary but
not sufficient for the surface to be CY. It is therefore of interest to
consider the extent to which each of these 5-vector weights properties
contribute to determine the very special CY property; and it is this question
we look to probe with new tools from data analysis, and machine-learning.

It has been shown that only a finite number of possible weights permit these
CY hypersurfaces. In fact, the dataset of weights consists of just 7555
5-vectors of transverse coprime integers.

Beyond learning the CY property explicitly, we are also interested in
exploring if the topological features of the Calabi-Yau can be learnt from the
weights. Of specific importance are the non-trivial Hodge numbers, $h^{1,1}$
and $h^{2,1}$, which describe the manifolds cohomology, and the Euler number,
$\chi$. These all have a variety of uses in determining physical phenomena.
The formula for Hodge numbers comes from expansion of the Poincaré polynomial
$Q(u,v):=\sum_{p,q}h^{p,q}u^{p}v^{q}$, the generating function of the Hodge
numbers $h^{p,q}$; whilst the formula for Euler number has a direct form
Vafa:1989xc ; KS1992 ; KLRY1998 ; Batyrev:2020ych . Specifically these are

$\begin{split}Q(u,v)&=\frac{1}{uv}\sum_{l=0}^{\sum_{i}(w_{i})}\bigg{[}\prod_{\tilde{\theta}_{i}(l)\in\mathbb{Z}}\frac{(uv)^{q_{i}}-uv}{1-(uv)^{q_{i}}}\bigg{]}_{int}\bigg{(}v^{size(l)}\bigg{(}\frac{u}{v}\bigg{)}^{age(l)}\bigg{)}\;,\\\
\chi&=\frac{1}{\sum_{i}(w_{i})}\sum_{l,r=0}^{\sum_{i}(w_{i})-1}\bigg{[}\prod_{i|lq_{i}\&rq_{i}\in\mathbb{Z}}\bigg{(}1-\frac{1}{q_{i}}\bigg{)}\bigg{]}\;,\end{split}$
(2.3)

for weights $w_{i}$, normalised weights $q_{i}=w_{i}/\sum_{i}(w_{i})$, and
$u,v$ are the dummy variables of the Poincaré polynomial. For $Q(u,v)$,
$\tilde{\theta}_{i}(l)$ is the canonical representative of $lq_{i}$ in
$(\mathbb{R}/\mathbb{Z})^{5}$, $age(l)=\sum_{i=0}^{4}\tilde{\theta}_{i}(l)$
and $size(l)=age(l)+age(\sum_{i}(w_{i})-l)$. Note also for $\chi$, where
$\forall\,i\ lq_{i}\ or\ rq_{i}\notin\mathbb{Z}$ then the product takes value
1 Batyrev:2020ych .

Both formulas require significant computation, involving many non-trivial
steps. Even if we realize this dataset in the language of the toric geometry
of Kreuzer:2000xy ; batyrev2011calabi , the formulae involve non-trivial sums
over faces of various dimension. It is consequently interesting to examine the
performance of machine-learning methods in learning the Euler number/Hodge
numbers from the weights, and perhaps predicting the results through the use
of possible hidden structures which we hope to uncover in the weights and
weight distributions.

##  3 Data Analysis

Before we apply the supervised machine-learning methods described in §4, let
us provide some analysis of the fundamentals of the dataset through the use of
principal component analysis (PCA), topological data analysis (TDA), and other
unsupervised machine-learning methods.

###  3.1 Datasets

In addition to the CY dataset which forms the central focus of this study, we
will construct some auxiliary datasets that will help in assessing the
learning of the Calabi-Yau property. These are equivalent datasets of
5-vectors that possess fewer of the necessary properties required to meet the
Calabi-Yau property.

The 4 datasets (including the original CY dataset) are composed of:

(a)

7555 5-vectors of positive random integers,

(b)

7555 5-vectors of positive random coprime integers,

(c)

7555 transverse 5-vectors of positive random coprime integers,

(d)

7555 Calabi-Yau 5-vectors.

These datasets were specifically constructed so as not to form a filtration,
therefore at each stage the dataset generated was ensured to not include data
which satisfies the additional conditions at the next level. To clarify, each
5-vector in set (a) had weights which shared a common factor, in set (b) all
5-vectors did not satisfy condition 2.2, and those in set (c) where not in the
CY list of (d).

To introduce a consistency across the datasets, all the 5-vectors entries are
sorted in increasing order. Initially the weights for each of the datasets
(a-c) were sampled using a discretised uniform distribution, $U(1,2000)$,
bound above by 2000 to mimic the highest value in the CY dataset of 1743.
However as shown in figure 1(a) the weights follow a distribution far from
that of a uniform distribution. Therefore to make the generated data more
representative, an exponential distribution was fitted to the histogram of all
weights in the CY dataset, as shown in figure 3.2. Fitting was performed using
the scipy library.

This exponential distribution was instead then used to sample weights, and as
can be seen in figures 3.3, the frequency distributions of the weights for
each of the datasets align much closer to that of the CY data. For reference
the weight histograms are shown for the uniform distribution sampling in
appendix A.1.

##### Aside: Coprimality

It is interesting to note that the probability of $k$ randomly chosen integers
being coprime is: $1/\zeta(k)$; via the Riemann zeta function. Hence the
probability of a random 5-vector of integers being coprime is $\sim 0.964$,
and therefore the dataset (b) is relatively more common than the dataset (a).
Effectively it is easy to randomly produce weighted projective spaces.

(a)

(b)

Figure 3.1: Frequency distribution of each of the CY 5-vector weights, $w_{i}$
(labelled by $i:1-5$). Figure (b) shows the same data as (a), but restricted
to lower entries so as to highlight the low value behaviour, due to the entry
sorting. Figure 3.2: Frequency distribution for all weights occurring across
all 5-vectors in the CY dataset. Plot also shows the fitted exponential
distribution, with scale parameter 49.536 (to 3 decimal places).

(a) Random Integers

(b) Random Coprime Integers

(c) Random Transverse Coprime Integers

Figure 3.3: Frequency distributions for 5-vector weights, $w_{i}$ (labelled by
$i:1-5$), for each of the generated datasets. The weights were generated using
an exponential distribution fitted to the CY weight data; and hence
distributions show similar behaviour across all datasets, and to the CY
dataset.

###  3.2 Principal Component Analysis

Datasets of vectors can be analysed for hidden structures through the use of
principal component analysis (PCA). This method, generally considered to be
the simplest form of unsupervised machine-learning, diagonalises the data’s
covariance matrix and sorts the respective eigenvalues and eigenvectors.

The covariance matrix of a dataset, computes the covariance between each
pairing of the constituent vector elements, defined as:

$K_{ij}=E(w_{i}-E(w_{i}))\cdot E(w_{j}-E(w_{j}))\;.$ (3.1)

Since our weight entries are within the field of integers the covariance
matrix is symmetric. Diagonalising this matrix finds the orthogonal linear
combinations of the vector entries which dictate the directions of largest
variance. Therefore the result of this diagonalisation is to identify the
data’s principal components, which are then sorted in decreasing order
according to their respective eigenvalues. The first component then gives the
direction where the data has the most alignment and hence the highest
variance, with successive decreasing variance until the final entry gives the
direction along which the data has the lowest variance.

The structure of the dataset can then be most easily observed through
consideration of these principal components. In this study PCA was applied to
each of the datasets under consideration independently.

In each case the variance eigenvalues were at least 5 times larger for the
first principal component compared to the others. In particular for the CY
dataset the first principal component was 2 orders of magnitude larger than
the others. This indicates that much of the variation, and hence data
structure, is dominated by a single dimension.

Usually a scaling is applied to the data prior to the PCA. The ’scaling’
process both centres and scales the data such that each entry (i.e. weight)
has mean value 0 and standard deviation 1 across the dataset; hence replacing
each weight by its respective standardised score. However for this analysis
scaling was not used since the data’s general structure is based on the
relative sizes between the weights (which are sorted according to their size).
These relative sizes between weights across each vector are lost through the
scaling process, which scales each weight independent of the rest of the
vector.

As the data is not scaled one may think that the latter weights of each vector
would dominate the behaviour (since the weights are ordered). This would lead
the covariance matrix to be near-diagonal, and the principal components would
align closely to the original vector entries. However, as shown by the
covariance matrix for the CY dataset in equation 3.2, the matrix is not
diagonal and the eigenvectors have significant contribution from multiple
components.

$\scriptsize{K_{CY}=\begin{pmatrix}41&43&109&250&404\\\ 43&119&278&642&1017\\\
109&278&1795&3626&5562\\\ 250&642&3626&8588&12941\\\
404&1017&5562&12941&20018\end{pmatrix},\;\varepsilon_{CY}=\begin{pmatrix}0.016&0.041&0.229&0.531&0.815\\\
0.021&0.036&-0.973&0.100&0.205\\\ 0.120&0.206&0.034&-0.823&0.514\\\
0.417&0.875&0.023&0.173&-0.172\\\
0.900&-0.435&0.003&0.018&-0.008\end{pmatrix},\;\lambda_{CY}=\begin{pmatrix}30071\\\
233\\\ 161\\\ 74\\\ 21\end{pmatrix},}$ (3.2)

for eigenvectors as rows of $\varepsilon_{CY}$ with respective eigenvalues in
$\lambda_{CY}$; where covariance and eigenvalue entries are given to the
nearest integer, and eigenvector entries to 3 decimal places. This implies
that the PCA structure is more subtle than a trivial projection. The
covariance matrices, eigenvectors and eigenvalues for the other datasets are
provided for comparison in appendix A.2.

To relatively compare the datasets’ PCAs, the normalised vectors of
eigenvalues are given in equation 3.3, for the random ’R’, coprime ’C’,
transverse ’T’, and Calabi-Yau ’CY’ datasets respectively. They show that the
first component significantly dominates, and hence lower dimensional
representation of the data through PCA will usefully depict the data’s
underlying linear structure.

$\footnotesize{\lambda_{R}=\begin{pmatrix}0.75534\\\ 0.16297\\\ 0.05274\\\
0.02059\\\ 0.00837\end{pmatrix},\quad\lambda_{C}=\begin{pmatrix}0.74845\\\
0.16856\\\ 0.05417\\\ 0.01997\\\
0.00885\end{pmatrix},\quad\lambda_{T}=\begin{pmatrix}0.91388\\\ 0.04211\\\
0.02578\\\ 0.01334\\\
0.00489\end{pmatrix},\quad\lambda_{CY}=\begin{pmatrix}0.98399\\\ 0.00764\\\
0.00525\\\ 0.00242\\\ 0.00070\end{pmatrix}.}$ (3.3)

Hence for the sake of visualisation, the first 2 components of each
datapoint’s principal component projection are plotted as a 2-dimensional
scatter diagram for each dataset. These components show the directions with
the most variation, and hence display the underlying structure most clearly.
The 2d PCA plots are given in figure 3.4, for each of the 4 datasets
considered.

(a) Random Integers

(b) Random Coprime Integers

(c) Random Transverse Coprime Integers

(d) CY Weights

Figure 3.4: 2d PCA plots for the 4 considered datasets. As more of the
conditions are added, more structure appears, in particular there is some form
of distinct class separation for the CY weights.

The cone-like bounding structure of all plots shows the effects of the weight
ordering. This is simply that as the first component’s value increases (most
correlated to the largest, and hence last, weight in the 5-vector) the range
of values the second component (roughly correlated to the second-largest /
second-last weight) can take increases. Or put more simply, the second-last
weight takes values up to the size of the last weight and so this places cone-
like bounds on the plots. All plots also show higher densities at lower values
of the principal components which is also related to this effect.

The PCA plots show that as more of the necessary conditions are added to the
datasets, more structure is apparent in the projected outputs. First note, the
coprime condition causes a negligible change to the distribution of weights.
The transverse condition however has a significant effect. The second
components become much more limited and the data begins to separate into
approximately two forks. Most exciting, is the jump to the full Calabi-Yau
data. Now the PCA shows a clear clustering of the 5-vectors at higher values
of the first principal component. This distinct separation into clear lines of
datapoints shows a rich structure to the weights of Calabi-Yau projective
spaces, which is not present for spaces with just the transverse condition.
The reasons for this separation are unclear, however we make conjectural
statements about a potential relation to the spaces’ Hodge numbers due to a
similar structural separation in section §3.4.

A final note is that the PCA used here was explicitly linear, and hence probes
the simplest kind of implicit structure. More technically involved methods of
principal component analysis involve kernel methods, called ’kernel-PCA’.
Kernel methods were also used to analyse these datasets, for a variety of
traditional kernels (including Gaussian, sigmoid, and an array of polynomial
kernels), and functionality for this is provided in the respective code
scripts. However, none of these methods produced as distinct a clustering
separation as that for the linear kernel. Indicating, that surprisingly, the
most prominent implicit structure of the Calabi-Yau weights takes a linear
form.

###  3.3 Topological Data Analysis

Principal Component Analysis allows for a 2d visualisation of the 5d CY data.
Through the PCA with linear kernel, a linear clustering structure was
uncovered in the data. To visualise the extension of this behaviour to the
full 5d space we turn to tools from topological data analysis; specifically
persistent homology.

The persistent homology of a dataset is constructed through a filtration of
Vietoris-Rips complexes. The full CY dataset is first plotted in
$\mathbb{R}^{5}$ with each weight a coordinate, such that each
weighted-$\mathbb{P}^{4}$ is now represented by a point (0-simplex) in the
$\mathbb{R}^{5}$ space due to its respective 5-vector.

5-sphere’s of radius $d$ are then drawn around each point, and the range of
$d$ values are taken from $0\longmapsto\infty$. Initially all the spheres will
be independent with no overlap, but as $d$ increases the spheres will begin to
overlap more frequently. The complex is then constructed by drawing an
$n$-simplex between $n$ points where all their spheres overlap.

Therefore as $d$ increases more simplices are added to the complex, and at
each $d$-value where there is a change to the complex we have a stage in the
complex’s filtration. The complex hence grows up until a point where all
possible simplices lie in the complex. This is where the filtration terminates
(no further changes as $d\longmapsto\infty$).

The role of persistent homology in the analysis of this filtration is to
examine how long cycles of $n$-simplices last throughout the filtration before
they become filled by the $(n+1)$-simplices they bound. Specifically $H_{n}$
examines how long cycles of $n$-simplices exist until becoming filled by
$(n+1)$-simplices.

This persistent homology for the CY data was computed for $H_{0}$ and $H_{1}$
(higher $H_{n}$ up to $n=4$ can be computed in 5d space but are incredibly
computationally expensive in terms of memory for $n\geq 2$). The persistence
diagram for this analysis is shown in figure 3.5, where the diagram plots all
members of $H_{0}$ and $H_{1}$ as points with their respective $d$ values of
birth (cycle creation) and death (cycle filling). For specific computation of
the persistent homology the python package ’ripser’ was used ctralie2018ripser
; whilst to review previous application of these techniques to the string
landscape please see Cirafici:2015pky ; Cole:2018emh .

As can be seen from the diagram all the members of $H_{0}$ are blue points
born at $d=0$, these are each of the 0-cycles (i.e. 0-simplices / datapoints)
that exist until they are connected by an edge (1-simplex) to any other
datapoint. The behaviour shows that there are some datapoints that are
significantly far away from the rest of the data and hence join/die much later
in the filtration. These points are those with large weight values such that
they are far from the origin in the $\mathbb{R}^{5}$ embedding.

Conversely all members of $H_{1}$ are points in orange, and as expected all
these 1-cycles (i.e. cycles of 1-simplices/edges which are not boundaries of
2-simplices/triangles) lie close to the diagonal line in the persistence
diagram. This behaviour indicates a short life of each cycle, a behaviour
typical of noise in the dataset. Since traditionally it is only points far
from the diagonal that indicate significant persistent structure, there is
hence not higher dimensional structure formation or non-trivial topology in
the data which would deter from the linear clustering behaviour seen through
the PCA.

Figure 3.5: Persistent diagram for the $H_{0}$ and $H_{1}$ homology groups of
the CY data’s Vietoris-Rips complex filtration.

###  3.4 Analysis of CY Topological Properties

In addition to the weights used to represent these Calabi-Yau weighted
projective spaces, the non-trivial Hodge numbers, $\\{h^{1,1},h^{2,1}\\}$, for
the surfaces are also provided with the KS databases Kreuzer:2000xy , and
repeated with this study’s GitHub.

This provides more information for analysis the spectrum of CY weights. Simple
plotting of these weights produces an astonishingly familiar structure, one
which is exemplified best when the CY’s Hodge numbers are plotted against the
final (and hence largest) weight, as shown in figure 3.6.

The behaviour in figure 6(a) shows a similar form of fork-like splitting of
the datapoints as in the PCA of figure 4(d), even with a central fork
particularly more dominant than the others. This seemingly linear behaviour
between final weight and $h^{1,1}$ is quite surprising, and here again the CY
hypersurfaces appear to be separating themselves into classes, according to
the ratio of $h^{1,1}$ to the final weight, $w_{5}$. On the contrary, the
behaviour in figure 6(b), follows the familiar mirror symmetry plot
CANDELAS1990383 , complimenting the linear behaviour with $h^{1,1}$ such that
their combination will preserve this structure.

Similar behaviour also occurs for the other weights in the 5-vectors, despite
less obvious clustering. Plots of these relations are given in appendix A.3.

To further examine this clustering phenomena we plot a histogram of the ratio
$h^{1,1}/w_{5}$ in figure 3.7. Note for this plot only datapoints with
$w_{5}>250$ were used since this was where the class separation was more
prominent such that the cluster identification would be improved. As can be
seen from the peaks in the figure, there is a clear clustering behaviour.
Therefore we reexamine this data of ratios with the use of K-Means clustering.

(a)

(b)

Figure 3.6: Distribution of Calabi-Yau weighted projective spaces, according
to their final (and largest) weight and (a) $h^{1,1}$ or (b) $h^{2,1}$
respectively. Figure 3.7: Frequency of the ratio between $h^{1,1}$ and the
largest weight, $w_{5}$, for the CY data with $w_{5}>250$ (where structure
more prominent). Peaks indicate a natural clustering.

####  3.4.1 Clustering for $h^{1,1}$ Classes

As motivated by the formation of a set of linear relationships between $w_{5}$
and $h^{1,1}$ shown in figure 6(a), and the peak occurrence in the histogram
of ratios in figure 3.7, unsupervised clustering methods were used to examine
this behaviour.

The ’outer’ ratio data used to produce the histogram plot, where clustering
was more prominent, provides a very suitable database for 1-dimensional
clustering. The method used was K-Means clustering, which takes an input
predefined number of clusters, initialises mean values for each cluster, and
iteratively updates these means such that the final sum of squared distances
from each datapoint to its nearest cluster’s mean is minimised. This measure
is known as the K-Means inertia,

$\mathscr{I}=\sum_{\mathscr{C}}\sum_{i\in\mathscr{C}}(\mu_{\mathscr{C}}-r_{i})^{2}\;$
(3.4)

for clusters, $\mathscr{C}$, with respective means, $\mu_{\mathscr{C}}$, and
all datapoints, $i$, exclusively in their nearest cluster with ratios,
$r_{i}$.

Determining the optimal number of clusters to use is a standard problem in
K-Means clustering, to motivate this choice we use a novel measure we call
’scaled-max-inertia’. This measure identifies the maximum squared-distance any
point is from its closest cluster centre, normalises it according to that
maximum squared-distance from using only one cluster, and adds a weight factor
to penalise using an excessive number of clusters. We define this to be:

$\mathscr{I}_{max}=\frac{\text{Max}_{i}(\mu_{\mathscr{C}}-r_{i})^{2}}{\text{Max}_{i}(\mu_{1}-r_{i})^{2}}+\frac{(k-1)}{100}\;,$
(3.5)

where $\text{Max}_{i}$ determines the maximum over all ratios, $r_{i}$,
examining the squared distance to either the closest cluster’s mean,
$\mu_{\mathscr{C}}$ or the single cluster’s mean, $\mu_{1}$; then weighting by
the number of clusters, $k$. A plot of scaled-max-inertia against number of
clusters identifies an optimum of 10 clusters, as shown in figure 3.8.

Figure 3.8: Plot of Scaled Max-Inertia as the number of clusters used for
K-Means clustering varies. The minimum identifies an optimum number of
clusters: 10.

Using the optimal number of 10 clusters, the separation matches up
exceptionally for the outer data, as shown by plots of the cluster bounds in
figure 3.9. The clusters sizes for the clusters moving anticlockwise about the
plot, for increasing ratio, are:
$[103,354,454,734,626,623,643,895,1419,1704]$, highlighting that there is a
greater density of points at low $w_{5}$ as expected, since this was why
’outer’ data was focused on for clustering.

To measure the clustering performance we use the standard Inertia measure over
the full dataset, however normalised by the number of ratios across the
dataset, $\hat{\mathscr{I}}$, and an equivalent measure also normalised by the
range of the ratios:

$\hat{\mathscr{I}}=0.0266\,,\quad\frac{\hat{\mathscr{I}}}{max(r_{i})-min(r_{i})}=0.00084\,,$
(3.6)

These values show that clustering performed exceptionally well, as each ratio
in the full CY dataset was less than 0.1% of the ratio-range away from its
nearest cluster. Therefore confirming the distinct linear behaviour observed,
as well as the class separation. The distinct classes of CY 5-vectors are also
provided in the GitHub.

Figure 3.9: Plot of the bounds of the 10 clusters produced on the outer data
($w_{5}>250)$ via K-Means clustering.

##  4 Machine Learning

After use of unsupervised ML methods in section §3, we now turn to use of
supervised ML methods for learning of the topological properties, as well as
the CY property.

###  4.1 Architectures

The problems addressed by supervised ML in this study fit into both of the
field’s typical styles: regression, and classification.

The first set of problems learnt in section §4.2 learn the topological Hodge
numbers (and related Euler number) from the CY 5-vectors of weights. Since the
output Hodge numbers can take a large range of integer values the problem was
formulated as a regression problem. For this a Multi-Layer Perceptron
Regressor was used to learn each output from the input weights. This regressor
is a type of Neural Network, and the one used specifically had layer sizes of
[32,64,32], with ReLU activation, and used the Adam kingma2017adam  optimiser
method to minimise a mean-squared error loss. The fitting used a batchsize of
200, and ran up to 200 epochs until the tolerance of 0.0001 was reached for
loss updating.

The second set of problems considered in section §4.3 sort to determine which
dataset a 5-vector belonged to, either by binary classification between each
dataset and the CY dataset, or a multiclassification among all 4 datasets.
Since these were classification problems an array of different classifiers
were used to perform the learning.

The first classifier was a Logistic Regressor, as perhaps the simplest form of
classifier. This Logistic Regressor had a tolerance of 1 for learning the
weight behaviour, a C-value of 100 such that there was a low amount of
regularisation, and used Newtons method for solving, such that
multiclassification could also be performed. The second classifier was a
Support Vector Machine with a simple linear kernel, and here a higher
regularisation due to a C-value 1. The third and final classifier used was a
Neural Network Classifier (Multi-Layer Perceptron also), this time with the
same hyperparameters as the Regressor except now with a cross-entropy loss
function.

####  4.1.1 Measures

To assess learning performance consistent measures are required. For this,
dependent on the problem being a regression or classification, different
measures were selected as follows.

##### Regressors:

The most standard regressor measure is Mean-Squared Error, MSE, which was used
for the regressor loss function. However MSE should be considered in relation
to the square of the range of output values to be useful, hence a preferable
measure also used was Mean-Absolute-Percentage Error, MAPE. Although it should
be noted MAPE has its own drawbacks where it is incalculable when $y_{true}=0$
for any of the date inputs. Both these measures are unbounded above and take
optimal value of 0 which indicates perfect prediction.

The final regressor measure used was $R^{2}$, this evaluates how well a
regressor is performing by comparing the proximity of the predicted output to
the proximity of the mean (which would be the prediction for a null model
regressor). For this measure 1 is optimal, 0 means that prediction is not
better than just predicting the true mean each time, and $<0$ means worse than
just predicting the mean. The equations and output bounds for these measures
are given in equation 4.1.

$\begin{split}MSE&=\frac{1}{n}\sum(y_{pred}-y_{true})^{2}\qquad\
\in[0,\infty)\;,\\\
MAPE&=\frac{1}{n}\sum\bigg{|}\frac{y_{pred}-y_{true}}{y_{true}}\bigg{|}\qquad\
\in[0,\infty)\;,\\\
R^{2}&=1-\frac{\sum(y_{true}-y_{pred})^{2}}{\sum(y_{true}-y_{truemean})^{2}}\in(-\infty,1]\;,\end{split}$
(4.1)

summing over all predicted, $y_{pred}$, and true, $y_{true}$, outputs in the
test data. In addition for $R^{2}$ the mean of the true values over the test
data outputs, $y_{truemean}$, was also used.

##### Classifiers:

Trained classifiers predict on input test data by assigning them to classes,
this leads to a natural sorting of true (row) vs predicted (column) class
frequencies over all the test data, arranged into a confusion matrix, CM. From
the confusion matrix the normalised sum over the diagonal gives the Accuracy
which is the proportion of test data correctly classified. However simple
accuracy has problems associated to bias data, therefore a better measure of
learning is Matthew’s Correlation Coefficient, MCC. Both these measures have
optimum learning with values of 1, where all test data inputs are allocated to
their true class. Equations for these two measures used are given in equation
4.2.

$\begin{split}CM&=\begin{pmatrix}TP&FN\\\ FP&TN\end{pmatrix}\,,\\\
Accuracy&=\frac{TP+TN}{TP+TN+FP+FN}\in[0,1]\,,\\\ MCC&=\frac{TP\cdot TN-
FP\cdot
FN}{\sqrt{(TP+FP)\cdot(TP+FN)\cdot(TN+FP)\cdot(TN+FN)}}\in[-1,1]\,,\end{split}$
(4.2)

for the binary classification case, where generalisations exist for the
multiclassification case.

For all problems 5-fold cross-validation was used, whereby 5 independent
versions of each architecture were trained and tested on 5 different
train:test partitions of the data, and the learning measures then averaged and
standard error computed.

###  4.2 ML Topological Parameters

Topological parameters provide key information about a Calabi-Yau manifold
which are essential in the computation of physical phenomena when these
manifolds are used for superstring compactifications.

This CY subset from weighted $\mathbb{P}^{4}$s provides a simple scenario
whereby the Hodge numbers (and hence Euler number) can be computed directly
from the weights of the toric space that the Calabi-Yau is a hypersurface of.
Although it should be noted these formulas are quite non-trivial, as discussed
in section §2.

Both of these formulas, given in equation 2.3, require greatest common divisor
computations throughout their evaluation. Machine-learning methods famously
perform badly when approximating these styles of equations and so one would
expect the simple Neural Network architecture used here to not be particularly
successful.

The results for the machine-learning of the non-trivial Hodge numbers, and the
Euler number are given in table 4.1. The Hodge number data, provided by
Kreuzer:2000xy , is also made available on the GitHub with the Calabi-Yau
weight data, and from here the Euler numbers can be calculated using
$\chi=2(h^{1,1}-h^{2,1})$.

Measure | Property
---|---
$h^{1,1}$ | $h^{2,1}$ | $[h^{1,1},h^{2,1}]$ | $\chi$
$R^{2}$ |  | 0.9630
---
$\pm$ 0.0015
| 0.9450
---
$\pm$ 0.0133
| 0.9470
---
$\pm$ 0.0041
| 0.9510
---
$\pm$ 0.0023
MAPE |  | 0.1493
---
$\pm$ 0.0027
| 0.2519
---
$\pm$ 0.0152
| 0.2375
---
$\pm$ 0.018
-
MSE |  | 166.9
---
$\pm$ 10.0
| 147.0
---
$\pm$ 35.6
| 186.9
---
$\pm$13.9
| 1746.1
---
$\pm$ 82.4
Table 4.1: Learning each of the topological parameters from the Calabi-Yau
5-vectors of weights. Note the final column is Euler number
$\chi=2(h^{1,1}-h^{2,1})$, and since it can evaluate to 0 its MAPE value is
not defined. Measurement of learning performance uses 5-fold cross-validation
to provide an average and standard error on each measure’s value.

The results show a surprisingly successful predictive ability for the Hodge
numbers and Euler number, particularly with $R^{2}$ values exceeding 0.9. The
MAPE values show the Hodge numbers are consistently predicted to be only
around 20% off from their true values, whilst the MSE values provide a less
physical measure of learning but are included for reference since they were
used as the regressor loss.

Considering the complexity of the equation forms in equation 2.3, it is
impressive the Neural Network can learn any correlating behaviour for
computation of Hodge numbers or Euler number from the weights alone. In
addition, the relatively better performance in learning $h^{1,1}$ may be due
to the apparent linear relationship to the weights as exemplified in section
§3.4.

###  4.3 ML CY Property

The conditions for a 5-vector of weights to represent a weighted projective
space which can admit a Calabi-Yau hypersurface are highly non-trivial. As
discussed in section §3.1, the necessary conditions of coprimality and
transversity are probed through generation of equivalent datasets, with which
the CY dataset can be compared.

Due to the exponential generation techniques making these weights more
representative, differentiating which dataset a 5-vector belongs to is not
possible by eye. Therefore it is natural to wish to consider the effectiveness
of machine-learning to this classification problem: learning the Calabi-Yau
nature.

Introduced in section §4.1, three architectures were used to learn to
differentiate the Calabi-Yau weights from each of the other datasets: random
integers, coprime random integers, and transverse coprime random integers in
binary classification problems. Furthermore they were also used to
differentiate all 4 datasets in a multiclassification problem.

Results for this learning are given in table 4.2. Measures show that Neural
Networks can well differentiate the Calabi-Yau weights from each of the other
datasets. As expected there is minimal difference due to introduction of
coprimality, since this is a common behaviour for 5-vectors as mentioned in
section §3.1. Once transversity was included into the dataset, the binary
classification performance dropped. However performance was still surprisingly
good.

A further surprise was the equally good performance of the Logistic Regressor
and Support Vector Machine. These simple architectures could accurately
classify approximately three-quarters of the data even without using
transversity (where this condition was in both CY and compared dataset).

Architecture | Measure | Dataset
---|---|---
Random | Coprime | Transverse | All
Logistic Regressor | Accuracy |  | 0.7152
---
$\pm$ 0.0035
| 0.7199
---
$\pm$ 0.0037
| 0.7430
---
$\pm$ 0.0065
| 0.4825
---
$\pm$ 0.0035
MCC |  | 0.4352
---
$\pm$ 0.0065
| 0.4467
---
$\pm$ 0.0073
| 0.5003
---
$\pm$ 0.0121
| 0.3141
---
$\pm$ 0.0043
Support Vector Machine | Accuracy |  | 0.7253
---
$\pm$ 0.0029
| 0.7116
---
$\pm$ 0.0029
| 0.7464
---
$\pm$ 0.0014
| 0.4732
---
$\pm$ 0.0070
MCC |  | 0.4605
---
$\pm$ 0.0054
| 0.4374
---
$\pm$ 0.0054
| 0.5174
---
$\pm$ 0.0029
| 0.3060
---
$\pm$ 0.0078
Neural Network | Accuracy |  | 0.9189
---
$\pm$ 0.0037
| 0.9178
---
$\pm$ 0.0030
| 0.7575
---
$\pm$ 0.0024
| 0.5881
---
$\pm$ 0.0048
MCC |  | 0.8380
---
$\pm$ 0.0073
| 0.8377
---
$\pm$ 0.0056
| 0.5306
---
$\pm$ 0.0059
| 0.4615
---
$\pm$ 0.0072
Table 4.2: Machine-learning results for three different architectures
performing binary classification between the CY data and each specified
dataset; and in addition multiclassification across all 4 datasets (labelled
’All’). Learning is measured using Accuracy and MCC with 5-fold cross-
validation to provide an average and standard error on each measure’s value.

Multiclassification of all datasets was not as strong. However within these
measures the identification of the Calabi-Yau data was considerably better,
with most of the performance reduction due to misclassifying between random,
coprimality, and transversity. To exemplify this we give a sample confusion
matrix for the multiclassification with the Logistic Regressor:

$\footnotesize{CM_{LR}=\begin{pmatrix}0.116&0.013&0.029&0.091\\\
0.076&0.083&0.074&0.020\\\ 0.074&0.078&0.062&0.019\\\
0.026&0.004&0.008&0.228\end{pmatrix}}\;,$ (4.3)

where row indicates true class and column predicted class for each of: random,
coprime, transverse, CY respectively. The final entry shows nearly all the
Calabi-Yau data is correctly classified (0.25 indicates the full quarter of
the accumulated datasets). Therefore measures will indicate lower performance
where the other conditions cannot be differentiated, and it is likely that
these conditions are not the most prominent conditions to indicate the Calabi-
Yau property.

To further examine the learning performance we next look explicitly at the
misclassifications of the Calabi-Yau data, using again links to the Hodge
numbers to identify areas of difficulty.

####  4.3.1 Misclassification Analysis with Hodge Numbers

Since the Logistic Regressor performed comparably to the other architectures,
and is a significantly simpler architecture than the neural network, its use
for misclassification analysis seemed the most appropriate.

Due to the simple structure, only 50 5-vectors in each non-CY dataset were
used to train the regressor with another 50 CY 5-vectors. The regressor was
then used to predict the class of all the CY data, producing accuracies of:
78%, 81%, 61% when trained with each of the random, coprime and transverse
datasets respectively.

Perhaps more curious is the distribution of these CY misclassifications with
respect to their Hodge numbers, plotted in figure 4.1. Training Random and
Coprime datasets in both cases leads to perfect classification of CY spaces
with high $h^{2,1}$, whereas training with Transverse data leads to perfect
classification with high $h^{1,1}$.

For reference both other architectures had similar performance with respect to
Hodge numbers, as documented in appendix A.4.

(a) Random Integers
(1899 misclassified)

(b) Random Coprime Integers
(1847 misclassified)

(c) Random Transverse Coprime Integers
(2739 misclassified)

Figure 4.1: A Logistic Regressor, trained on $50$ CY 5-vectors and $50$ non-CY
5-vectors, predicts whether all of the CY 5-vectors are CY or not. The plot
shows the distribution of the CY surfaces according to their Hodge numbers.
Those in blue are misclassified as non-CY, those in orange are correctly
classified to be CY. The non-CY vectors come from datasets of Random, Coprime,
or Transverse 5-vectors respectively.

To examine further this relationship, we bin the CY data according to each of
the Hodge numbers and only train and test on 5-vectors in each bin’s range.
This is detailed in section §4.3.2.

####  4.3.2 Hodge Partitioning

To investigate the dependence of the learning performance on the Hodge
numbers, the CY dataset was binned in two independent ways. The first was
according to $h^{2,1}$, and the second according to $h^{1,1}$. The bin bounds
were optimised such that an approximately consistent number of CYs had Hodge
numbers within each bin’s bounds, with a preset number of 50 bins used
(selected to have a suitable bin size $>100$). Plots of these bin frequencies
are given in figures 2(a) and 2(b).

This produced a CY dataset associated to each bin, with which a non-CY
5-vector dataset was randomly sampled. For the $h^{2,1}$ partition the Random
dataset was used to sample as many non-CY 5-vectors for each bin, such that
the datasets were balanced. As training-behaviour for the Random and Coprime
datasets was so similar, only the Random dataset was used in this
investigation. Conversely, for the $h^{1,1}$ partition the Transverse dataset
was used. These choices of non-CY datasets used for training were selected
such that they aligned with the predicted behaviour of section §4.3.1, where
Random-training improves high-$h^{2,1}$ performance, and Transverse-training
improves high-$h^{1,1}$ performance.

For each bin’s now balanced dataset an independent Logistic Regressor (with
architecture as before) was initialised, trained and tested. A random 80%
sample of the data was used for training, with testing on the remaining 20%
complement. For each bin, the initialisation, training, and testing was
repeated 20 times, such that variances on the measures could be calculated.
Accuracies were recorded for each bin regressor, as well as the final 5
weights used to define the LR.

Accurracies across the bins for both partitions are given in figures 3(a) and
3(b), with their respective accuracy variances in 3(c) and 3(d). There are
near perfect predictions at the upper ends of these partitions, with
relatively very small variances. Determination of the CY property is hence
considerably easier for surfaces whose Hodge numbers take extreme values, and
pre-training against data with or without the transverse condition can
significantly aid learning depending on what values the Hodge numbers take.

Finally, the 5 averaged LR weights are plotted for each bin (with respective
variances surrounding them) in figures 3(e) and 3(f). As can be seen by
comparing the relative weight sizes, in both cases at the higher ends of the
partitions the first two weights particularly dominate the regression. Since
each LR weight aligns with the projective space weight, this indicates at
these extremes where learning is particular strong, only the first two (i.e.
lowest) weights are needed to identify whether the weighted projective space
admits a Calabi-Yau hypersurface. Where only the CY dataset has the
transversity property (i.e. training against Random) the first weight is the
most significant, whilst where transversity is in both datasets (i.e. training
against Transverse) the second weight is the most significant.

(a) Bin frequencies for $h^{2,1}$ partition

(b) Bin frequencies for $h^{1,1}$ partition

(a) LR (Random-trained) Accuracies for $h^{2,1}$ partition

(b) LR (Transverse-trained) Accuracies for $h^{1,1}$ partition

(c) Variances of the LR (Random-trained) accuracies for $h^{2,1}$ partition

(d) Variances of the LR (Transverse-trained) accuracies for $h^{1,1}$
partition

(e) LR (Random-trained) weights for $h^{2,1}$ partition, plotted with variance
bars

(f) LR (Transverse-trained) weights for $h^{1,1}$ partition, plotted with
variance bars

Figure 4.3: Relevant plots for Logistic Regressor learning of the 5-vectors
being CY or non-CY. Where the non-CY data was the Random data then binning was
according to $h^{2,1}$, where it was Transverse data then according to
$h^{1,1}$. The CY data was binned according to either $h^{1,1}$ or $h^{2,1}$
Figures (a) & (b) show the number of CYs in each Hodge partition bin (half the
dataset used in each case as non-CYs cannot be plotted without known Hodge
numbers). Figures (c) & (d) show the average accuracies for the LR learning in
each case, with (e) & (f) the respective variances (very small comparatively).
Finally, figures (g) & (h) show the averaged trained LR weights, plotted with
their variances as bands about the average values.

##  5 Summary & Outlook

Through the use of unsupervised machine-learning methods we were able to
identify a linear clustering structure of the weighted projective spaces that
admit Calabi-Yau hypersurfaces. This structure was first observed through PCA,
corroborated with TDA, and then observed again due to relations with the
hypersurface’s Hodge numbers.

Supervised machine-learning methods then learnt to predict Hodge numbers from
the weights directly to a surprisingly exceptional accuracy, perhaps making
use of this simple structure. In addition, simple classifier architecture
could detect whether a generic weighted-$\mathbb{P}^{4}$ admitted a Calabi-Yau
hypersurface from the weights alone, and with specific pre-training could
reach perfect performance at certain extremes of Hodge numbers.

Further analysis into this Calabi-Yau clustering behaviour for
weighted-$\mathbb{P}^{4}$s would hope to uncover its source, simultaneously
explaining the success of machine-learning techniques on this dataset.

## Acknowledgement

The authors would like to thank Prof. V. Batyrev for clarifying discussion.
DSB is partially supported by Pierre Andurand. YHH would like to thank STFC
for grant ST/J00037X/1. EH would like to thank STFC for a PhD studentship.

##  Appendix A Appendix

###  A.1 Uniformly Sampled Weight Distributions

For reference, the weight frequency distributions for two of the three
generated datasets: (a) Random integers, (b) Random coprime integers; as
discussed in section §3.1, are shown below A.1, where the weights were sampled
uniformly using a discretisation of $U(1,2000)$.

Note the dataset of transverse random coprime integers could not be generated
using a uniform distribution. Since the probability of five random integers in
this range each dividing another weight negated from the sum is so improbable,
no examples were generated running the code for a multiple days on a
supercomputing cluster. However generation with an exponential distribution
took the order of minutes. Hence the transverse property most likely is has a
significant contribution to the exponential weight distribution behaviour of
the CY data.

(a) Random Integers

(b) Random Coprime Integers

Figure A.1: Frequency distributions for 5-vector weights, $w_{i}$ (labelled by
$i:1-5$), for the generated datasets of random integers and random coprime
integers. Weights were generated using a discretised uniform distribution,
$U(1,2000)$. Distributions show a spread across the range (accounting for the
sorting), and hence do not well mimic the CY dataset.

###  A.2 Additional PCA Information

Further to the PCA information provided for the CY dataset in section §3.2,
the covariance matrices, eigenvectors, and eigenvalues are given for the other
three datasets here. They are respectively labelled ’R’ for the random
dataset, ’C’ for coprime dataset, and ’T’ for transverse dataset. The
covariance matrices, $K$, and eigenvalues, $\lambda$, are given to the nearest
integer, whilst eigenvectors, rows of $\varepsilon$, are given to 3 decimal
places.

$\scriptsize{K_{R}=\begin{pmatrix}97&98&98&96&107\\\ 98&251&250&245&255\\\
98&250&530&514&542\\\ 96&245&514&1122&1157\\\
107&255&542&1157&3614\end{pmatrix},\;\varepsilon_{R}=\begin{pmatrix}0.039&0.094&0.191&0.375&0.902\\\
-0.121&-0.298&-0.519&-0.669&0.424\\\ -0.253&-0.517&-0.520&0.626&-0.085\\\
-0.469&-0.591&0.640&-0.145&0.006\\\
-0.837&0.535&-0.117&0.006&0.003\end{pmatrix},\;\lambda_{R}=\begin{pmatrix}4241\\\
915\\\ 296\\\ 116\\\ 47\end{pmatrix},}$ (A.1)

$\scriptsize{K_{C}=\begin{pmatrix}100&100&101&91&89\\\ 100&254&255&254&249\\\
101&255&527&534&527\\\ 91&254&534&1166&1163\\\
89&249&527&1163&3418\end{pmatrix},\;\varepsilon_{C}=\begin{pmatrix}0.036&0.098&0.199&0.400&0.889\\\
-0.124&-0.297&-0.514&-0.657&0.448\\\ -0.284&-0.532&-0.497&0.617&-0.095\\\
-0.457&-0.570&0.662&-0.168&0.009\\\
-0.833&0.543&-0.109&-0.003&0.000\end{pmatrix},\;\lambda_{C}=\begin{pmatrix}4091\\\
921\\\ 296\\\ 109\\\ 48\end{pmatrix},}$ (A.2)

$\scriptsize{K_{T}=\begin{pmatrix}6&7&8&12&19\\\ 7&20&25&35&55\\\
8&25&62&85&125\\\ 12&35&85&173&246\\\
19&55&125&246&417\end{pmatrix},\;\varepsilon_{T}=\begin{pmatrix}0.040&0.114&0.264&0.507&0.812\\\
0.102&0.332&0.712&0.349&-0.501\\\ 0.198&0.467&0.321&-0.746&0.286\\\
-0.428&-0.660&0.556&-0.253&0.091\\\
-0.875&0.473&-0.105&0.018&-0.001\end{pmatrix},\;\lambda_{T}=\begin{pmatrix}620\\\
29\\\ 17\\\ 9\\\ 3\end{pmatrix}.}$ (A.3)

###  A.3 Additional Hodge Plots

Further to the plots of the two non-trivial Hodge numbers of the CY surfaces,
$\\{h^{1,1},h^{2,1}\\}$, against the final 5-vector weights in section §3.4,
additional plots of these Hodge numbers against the other weights are given
here in figure A.3 for reference.

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

Figure A.3: Plots of the non-trivial Hodge numbers $\\{h^{1,1},h^{2,1}\\}$
against each of the first 4 weights in the CY 5-vectors. Behaviour is similar
to that with the final weight, showing a linear relationship to $h^{1,1}$ and
a relationship preserving the mirror symmetry structure for $h^{2,1}$.

###  A.4 Additional Misclassification Analysis

Distributions of correctly and incorrectly classified CY 5-vectors for each of
the other architectures (Support Vector Machine and Neural Network), trained
on 50 CY and 50 non-CY 5-vectors, are given in figure A.5. Note the
architectures had the same hyperparameters as in previous investigation of
section §4.3.

The behaviour is similar to that for the Logistic Regressor, where training
with Random 5-vectors improves determination for high $h^{2,1}$, whilst
training with Transverse 5-vectors improves determination for high $h^{1,1}$.

(a) SVM trained with Random

(b) NN trained with Random

(a) SVM trained with Coprime

(b) NN trained with Coprime

(c) SVM trained with Transverse

(d) NN trained with Transverse

Figure A.5: Classified and misclassified CY 5-vectors plotted with respect to
Hodge numbers, where prediction was performed by either of the architectures:
Support Vector Machine (SVM), or Neural Network (NN); trained with each of the
non-CY datasets respectively.

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