aainstitutetext: Department of Mathematics, City, University of London, EC1V
0HB, UKbbinstitutetext: London Institute for Mathematical Sciences, Royal
Institution, London W1S 4BS, UK

# Machine Learning for Hilbert Series

Edward Hirst<EMAIL_ADDRESS>
###### Abstract

Hilbert series are a standard tool in algebraic geometry, and more recently
are finding many uses in theoretical physics. This summary reviews work
applying machine learning to databases of them; and was prepared for the
proceedings of the Nankai Symposium on Mathematical Dialogues, 2021.

##  1 Introduction

Mathematically, Hilbert series (HS) provide a means of counting regular
functions on varieties. Physically, they are used in a range of scenarios,
including to count BPS operators Benvenuti:2006qr ; Feng:2007ur ; Gray:2008yu
; Hanany:2008sb ; Chen:2011wn ; Jokela:2011vg , determine moduli spaces
Benvenuti:2010pq ; Hanany:2012dm ; Buchbinder:2019eal ; Forcella:2008ng , find
standard model invariants Hanany:2010vu ; Lehman:2015coa ; Xiao:2019uhh , deal
with string compactifications Braun:2012qc , construct effective Lagrangians
Lehman:2015via ; Henning:2015daa ; Kobach:2017xkw ; Anisha:2019nzx ;
Marinissen:2020jmb ; Graf:2020yxt , as well as for a range of other quiver-
related computations Cremonesi:2013lqa ; Bourget:2020xdz ; Bourget:2021siw ;
Bourget:2019rtl ; Bourget:2021xex ; Closset:2020scj ; Franco:2005sm ;
Argurio:2020dko ; Franco:2022ziy ; Dey:2013fea ; Flo2021 .

In parallel, machine learning (ML) is fast becoming a definitive methodology
for data analysis and pattern recognition. The most common tool being neural
networks (NNs), which are many-parameter non-linear functions used to learn
properties of a dataset. Beyond supervised methods, principal component
analysis (PCA) is an established unsupervised method, used for examining
clustering in datasets. PCA puts focus on a data representation’s largest
principal components, which are found through diagonalisation of the data’s
covariance matrix. ML has also been successfully applied to a variety of
physically motivated scenarios, including: Calabi-Yau manifolds He:2017aed ;
He:2017set ; Carifio:2017bov ; Krefl:2017yox ; Ruehle:2017mzq ;
Candelas:2016fdy ; Ashmore:2019wzb ; Berman:2021mcw ; Bao:2020sqg , polytopes
Bao:2021ofk ; Berglund:2021ztg , graph theory He:2020fdg , knot theory
Craven:2021ckk ; Gukov:2020qaj , amoebae Bao:2021olg , brane webs Arias-
Tamargo:2022qgb , integrability Krippendorf:2021lee , Seiberg duality among
quivers Bao:2020nbi , and the related dessin d’enfant Galois orbits He:2020eva
.

In this summary, work from Bao:2021auj  using neural networks to learn
geometric properties of a variety from coefficients in its respective HS is
described. The work was carried out in python with the use of Tensorflow
tensorflow2015-whitepaper  for machine learning implementation.

##  2 Defining Hilbert Series

Within algebraic geometry HS encode information about a complex variety $X$,
embedded in complex weighted projective space
$\mathbb{P}_{\mathbb{C}}(p_{0}^{q_{0}},\ldots,p_{s}^{q_{s}})$, such that
weight $p_{i}$ occurs $q_{i}$ times. The variety’s coordinate ring is denoted
$R=\mathbb{C}[x_{0},\ldots,x_{k}]/I$, for complex variables $x_{i}$, and
homogeneous ideal $I$ generated by the polynomials defining the variety. Any
specific embedding of the variety then induces a grading on this coordinate
ring Dolga , which can be written $R=\bigoplus_{i\geq 0}R_{i}$. The HS is the
generating function for the dimensions of these graded pieces of $R$,

$H(t;X)=\sum\limits_{i=0}^{\infty}(\dim_{\mathbb{C}}R_{i})t^{i}\;;$ (2.1)

such that $\dim_{\mathbb{C}}R_{i}$ defines the number of independent degree
$i$ polynomials on the variety, and $t$ is a dummy variable used to keep track
of the grading order (known as the fugacity in the context of quivers).

This concept can be generalised to any $n$-multiple grading of the ring, such
that
$R=\bigoplus_{i_{1},i_{2},...,i_{n}\geq 0}R_{i_{1},i_{2},...,i_{n}}$, giving
HS

$H(t_{1},t_{2},...,t_{n};X)=\sum\limits_{i_{1},i_{2},...,i_{n}=0}^{\infty}(\dim_{\mathbb{C}}R_{i_{1},i_{2},...,i_{n}})t_{1}^{i_{1}}t_{2}^{i_{2}}...t_{n}^{i_{n}}\;;$
(2.2)

which in the physics literature is called the ‘refined’ HS and each fugacity
is a gauge invariant operator. Note it is simple to unrefine by setting
$t_{k}=t\ \forall\ k$.

Figure 2.1: The quiver for a $\mathcal{N}=1$ $U(N)$ gauge theory with three
adjoint chiral multiplets.

In the context of supersymmetric gauge theory in string theory, when one knows
the matter content and superpotential defining the theory they often wish to
compute the geometric information about the moduli space of vacua. Whilst in
the case of the moduli space being toric one can determine it from the
combinatorics of the toric diagram Hanany:2005ss , where the moduli space is
not toric the process is more difficult. Hence in general, single-trace gauge
invariant operators must be counted order by order to construct the HS to then
learn about this moduli space and the underlying variety it is a cone over.

The process of computing a HS is nicely exemplified with the prototypical
example of the clover quiver in $\mathcal{N}=1$ supersymmetry, see figure 2.1,
with $U(N)$ gauge group and superpotential given by tracing over a certain
choice of cycle combinations: $W=\Tr(X_{1}X_{2}X_{3}-X_{1}X_{3}X_{2})$. The
$X_{k}$ are the chiral supermultiplets in a representation (here adjoint) of
the gauge group, and can be related to the fugacities $t_{k}=\Tr(X_{k})$ in
this case as $\Tr(X_{k})$ for $k\in\\{1,2,3\\}$ represent the 3 order 1
generators.

Higher order generators are found using the information of the superpotential
choice, computing all the F-terms as
$0=\partial_{X_{1}}W=\partial_{X_{2}}W=\partial_{X_{3}}W$ shows all the 3
generators are freely commuting such that single trace operators can be
constructed at each order from combining the operators in all ways.

Therefore in the refined case we have $\begin{pmatrix}n+2\\\
2\end{pmatrix}=\frac{(n+1)(n+2)}{2}$ single-trace operators at each order $n$,
of the form $\Tr(X_{1}^{a}X_{2}^{b}X_{3}^{c})$ such that $(a,b,c)$ are all
possible combinations that sum to $n$. Hence the HS starts

Order $\displaystyle\ |\ \\#\ $ $\displaystyle|\ \text{Operators}$
$\displaystyle 0$ $\displaystyle\ |\ \,1\ $ $\displaystyle|\ 1$ $\displaystyle
1$ $\displaystyle\ |\ \,3\ $ $\displaystyle|\ \text{Tr}(X_{1}),\
\text{Tr}(X_{2}),\ \text{Tr}(X_{3})$ $\displaystyle 2$ $\displaystyle\ |\ \,6\
$ $\displaystyle|\ \text{Tr}(X_{1}^{2}),\ \text{Tr}(X_{2}^{2}),\
\text{Tr}(X_{3}^{2}),\ \text{Tr}(X_{1}X_{2}),\ \text{Tr}(X_{1}X_{3}),\
\text{Tr}(X_{2}X_{3})$

for the first 3 orders. Consequently, if we consider the simplest case of
$N=1$ where the multiplet representations, $X_{k}$, are just complex numbers,
the HS in the refined and unrefined cases are

$\displaystyle H(t_{1},t_{2},t_{3};X)$
$\displaystyle=\frac{1}{(1-t_{1})(1-t_{2})(1-t_{3})}\;,$ (2.3) $\displaystyle
H(t;X)$ $\displaystyle=\frac{1}{(1-t)^{3}}\;,$ (2.4)

respectively. Since there are 3 independent generators with no relations (seen
via the trivial numerator), the moduli space is therefore simply
$\mathbb{C}^{3}$. This can be shown by taking the plethystic logarithm of the
HS where the first positive terms in the expansion are the generators, and the
next are the relations (of which here there are none).

Note one may consider higher $N$ where the moduli space is then the $N$-th
symmetric product of $\mathbb{C}^{3}$, deriving from the permutations of the
$X_{k}$ matrices being invariant under the trace. Additionally, using the
plethystic exponential creates multi-trace operators of the form
$\Tr(...)\Tr(...)...\Tr(...)$, which are also valid gauge-invariant operators
but are instead from analysis of these symmetric products of the moduli space
Benvenuti:2006qr .

In the work summarised here the focus is a single grading, and hence the
unrefined HS, as in equation (2.1), which via Hilbert-Serre theorem
AtiyahMacdonald  can be written in two equivalent forms:

$\displaystyle H(t;X)$
$\displaystyle=\frac{P(t)}{\prod\limits_{i=0}^{s}(1-t^{p_{i}})^{q_{i}}}\;,$
(2.5) $\displaystyle H(t;X)$
$\displaystyle=\frac{\tilde{P}(t)}{(1-t^{j})^{dim+1}}\;,$ (2.6)

for polynomials $P$ and $\tilde{P}$ in the fugacity $t$, and where the
variety, $X$, has Gorenstein index $j$ and dimension $dim$. From (2.6) one can
tell if the variety is Gorenstein if and only if the numerator polynomial
$\tilde{P}$ is palindromic Stanley1978 . Likewise one can also use the HS to
distinguish whether a variety is a complete intersection, which happens when
the expansion of the HS plethystic logarithm terminates at finite order
Benvenuti:2006qr ; Feng:2007ur .

We can therefore see that our example HS in (2.4) is trivially Gorenstein, and
also trivially a complete intersection as their are no relations (and hence
also no syzygies) among the generators to occur in the plethystic logarithm
expansion. One can also extract the dimension, however since the moduli space
is formed from taking a specific cone over the variety its dimension is one
larger than that given by equating (2.4) to (2.6). For our example it hence
has a 3-dimensional $\mathbb{C}^{3}$ moduli space, which may be considered to
be a cone over the 2-dimensional Fano $\mathbb{P}_{\mathbb{C}}^{2}$ variety.

##  3 Machine Learning

With data about the low order terms in the HS, one wishes to determine
geometric information about the underlying algebraic variety. In the work
summarised here, NN regressors were used to predict the defining weights
$(p_{i})$ of the weighted projective space each variety is embedded in, and NN
classifiers learnt the Gorenstein index $j$ and dimension $dim$ of the variety
itself. Further to these, three binary classification investigations were
performed with NN classifiers to determine whether input HS expansion
coefficients came from varieties which were Gorenstein, were complete
intersections, or were ‘fake’ such that they were generated from sampling as
described below.

### Data Information

HS data used for this work was retrieved from the GRDB grdb ; fanodata ;
kasprzyk2010canonical , consisting of HS associated to three-dimensional
$\mathbb{Q}$-Fano varieties with Fano index one ABR:Fano ; BK22 .
Investigations also used generated ‘fake’ data, created through sampling the
hyperparameters of the HS equations (2.5) and (2.6) from distributions fitted
to the ‘real’ GRDB data. The sampling discarded hyperparameter combinations
which did not satisfy certain physical conditions, encouraging the data to be
more representative; and also discarded any repeats of GRDB data.

For both HS styles of input data mentioned, which we call ‘real’ and ‘fake’
respectively, the HS closed forms were Taylor expanded and expansion
coefficients saved. These lists of coefficients form vectors of integers, and
make up the ML input for the investigations carried out. Two types of
coefficient vector were used as inputs, one from the start of the series
(coefficients 0-100) and one from deeper into it (coefficients 1000-1009). The
reason to also learn with coefficients sampled from deeper in the series was
to provide intuition on the importance of the variety’s orbifold points, which
have a greater relative significance than the ‘initial part’ at higher orders.

Output data depended on the investigation. For the regression of embedding
weights, these were sorted 3-vectors of integers (sampled in the range
[1,10]). For the multi-classification the outputs were single integers in the
range [1,5], which were the values of either the variety’s Gorenstein index or
its dimension respectively. The final binary classifications just outputted 0
or 1 dependent on whether the Gorenstein or complete intersection properties
were respectively satisfied, or whether the data came from the GRDB or was
generated by us.

### NN Architectures

For all investigations the NNs used were built of 4 dense layers of 1024
neurons, all with ReLU activation and 0.05 dropout factor. Training was in
batches of 32 for 20 epochs using the Adam Adam  optimiser to minimise either
the regression log(cosh) loss function or the classification cross-entropy
loss function. 5-fold cross-validation provided confidence on the metrics used
to evaluate the learning via averaging and standard error.

In the regression investigation, mean absolute error (MAE) evaluated learning
by computing the average absolute difference between each true embedding
weight and the NN predicted embedding weight. This measure evaluates in the
range $[0,\infty)$ with 0 indicating perfect learning, where the embedding
weights are always correctly predicted. In the classification investigations,
Matthew’s correlation coefficient (MCC), which is an application of Pearson’s
correlation coefficient to binary variables, evaluated the learning. This
measure evaluates in the range $[-1,1]$ with 1 indicating perfect learning.

### Results

ML results are provided in table 3.1 for each of these 5 investigations.
Learning of the varieties’ ambient space embedding weights predicted each
weight within an average range of 1 from the correct values when using higher
order coefficients. Additionally the geometric parameters of equation (2.6)
were learnt exceptionally well, but now better from the lower order
coefficients.

Each of the binary classifications also evaluated with MCC values exceeding
0.9, demonstrating strong learning. Both the Gorenstein and ‘fake’ (non-GRDB)
properties were better identified with higher order coefficients, whilst the
complete intersection property was only learnt from lower orders but performed
equally well.

All these results demonstrate the success of ML in learning geometric
properties of varieties from HS, and hopefully further encourages the use of
ML techniques in the context of Hilbert series and beyond.

| Investigation
---
| Embedding
---
weights
| Gorenstein
---
index
Dimension |  | Gorenstein
---
property
| Complete
---
Intersection
GRDB
Output Ranges | 3 x [1,10] | [1, 5] | [1, 5] | Binary | Binary | Binary
Measure | MAE | MCC | MCC | MCC | MCC | MCC
Orders | 0 - 100 |  | 1.94
---
$\pm$ 0.11
| 0.916
---
$\pm$ 0.010
| 0.993
---
$\pm$ 0.006
| 0.717
---
$\pm$ 0.155
| 0.910
---
$\pm$ 0.022
| 0.717
---
$\pm$ 0.155
1000 - 1009 |  | 1.04
---
$\pm$ 0.12
| 0.727
---
$\pm$ 0.022
| 0.822
---
$\pm$ 0.031
| 0.919
---
$\pm$ 0.073
- |  | 0.919
---
$\pm$ 0.073
Table 3.1: Machine learning results for each of the investigations. Neural
networks learnt geometric properties associated to an algebraic variety from
coefficients in their Hilbert series expansions. Regression was assessed with
mean absolute error (MAE) and classification with Matthew’s correlation
coefficient (MCC), training using 5-fold cross-validation to provide a
standard error on the averaged measures.

### PCA

Whilst all the learning so far uses NNs from supervised ML, PCA provides
another avenue for analysis of this data from the conjugate field of
unsupervised ML.

In PCA, the covariance matrix is computed between all vector entries, which is
then subsequently diagonalised and eigenvectors sorted in decreasing order of
variance eigenvalue. Each data vector is then re-expressed in this eigenvector
basis, which in this case of considering the order 0-100 coefficients gives
vectors with 101 entries. Since this is too high-dimensional to visualise, the
first 2 principal components of each datapoint are plotted to provide a means
for examining the data’s clustering, these plots can be seen for the 3 binary
classification investigations in figure 3.1.

Here, 2d PCA was performed for each of the binary classification
investigations, and the datapoints in each class plotted separately in
different colours. Whilst there is no clear separation for the Gorenstein
data, the complete intersection and ‘real’/‘fake’ data do show some
separation, indicating there is some linear structure that the NNs can take
advantage of for classification. In particular the occurrence of this for the
‘fake’ data demonstrates the difficulty of generating representative HS.

(a) Gorenstein

(b) Complete Intersection

(c) ‘real’ / ‘fake’

Figure 3.1: 2d principal component analysis plots of the Hilbert series 0-100
coefficient data, labelled by each of the binary classification investigations
respectively.

## Acknowledgements

The author wishes to thank Jiakang Bao, Yang-Hui He, Johannes Hofscheier,
Alexander Kasprzyk, and Suvajit Majumder for collaboration on this work;
Jiakang Bao again for clarifying discussion; and also STFC for the PhD
studentship.

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