Machine learning detects terminal singularities
Authors: Tom Coates
Alexander M. Kasprzyk
Sara Veneziale
Venue: NeurIPS 2023 poster (2023)
Keywords: mathematics
geometry
Fano varieties
terminal singularities
theorem discovery
neural network classifier
supervised learning
TL;DR: We show that machine learning methods can solve a highly challenging problem in mathematics for which there is no theoretical understanding, and use this to give the first sketch of the classification of an important class of 8-dimensional shapes.

Abstract: Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are `atomic pieces’ of more complex shapes – the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Programme.

Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification. We focus on eight-dimensional positively-curved algebraic varieties that have toric symmetry and Picard rank two, and develop a neural network classifier that predicts with 95% accuracy whether or not such an algebraic variety is Q-Fano. We use this to give a first sketch of the landscape of Q-Fano varieties in dimension eight.

How the neural network is able to detect Q-Fano varieties with such accuracy remains mysterious, and hints at some deep mathematical theory waiting to be uncovered. Furthermore, when visualised using the quantum period, an invariant that has played an important role in recent theoretical developments, we observe that the classification as revealed by ML appears to fall within a bounded region, and is stratified by the Fano index. This suggests that it may be possible to state and prove conjectures on completeness in the future.

Inspired by the ML analysis, we formulate and prove a new global combinatorial criterion for a positively curved toric variety of Picard rank two to have terminal singularities. Together with the first sketch of the landscape of Q-Fano varieties in higher dimensions, this gives strong new evidence that machine learning can be an essential tool in developing mathematical conjectures and accelerating theoretical discovery.

Machine learning detects terminal singularities
Tom Coates
Alexander M. Kasprzyk
Department of Mathematics
School of Mathematical Sciences University of Nottingham Nottingham, NG7 2RD UK a. m. kasppreykkhottinggamm..mm ac
Imperial College London
180 Queen's 's Gate
London, SW7 2AZ UK wwwtest@imeriilllcoooo uk
Sara Veneziale*
Department of Mathematics
Imperial College London
180 Queen's 's Gate London, SW7 2AZ UK
8. vevnezzialee11iieriilll....
Abstract
Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are 'atomic pieces' of more complex shapes - the process breaking shape into simpler pieces in this sense is called the Minimal Model Programme.
Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification. We focus eight-dimensional positively-curved algebraic varieties that have toric symmetry and Picard rank two, and develop a neural network classifier that predicts with 95% accuracy whether not such algebraic variety is Q-Fano. We use this to give first sketch of the landscape of Qa-F varieties dimension eight.
How the neural network is able detect Q-Fano varieties with such accuracy remains mysterious, and hints at some deep mathematical theory waiting to be uncovered. Furthermore, when visualised using the quantum period, an invariant that has played an important role in recent theoretical developments, we observe that the classification as revealed by ML appears to fall within bounded region, and is stratified by the Fano index. This suggests that it may be possible to state and prove conjectures on completeness in the future.
Inspired by the ML analysis, we formulate and prove new global combinatorial criterion for positively curved toric variety of Picard rank two to have terminal singularities. Together with the first sketch of the landscape of Q-Fano varieties in higher dimensions, this gives strong new evidence that machine learning can an essential tool in developing mathematical conjectures and accelerating theoretical discovery.
*Corresponding author.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Introduction
Systems of polynomial equations occur throughout mathematics and science; see e.g. 17 18 35
Solutions of these systems define shapes called algebraic varieties. Depending on the equations involved, algebraic varieties can be smooth (as in Figure Tab or have singularities (as in Figures [b and Tep. In this paper we show that machine learning methods can detect a class of singularities called terminal singularities.
(a) x² + x² = x² +1
(b) x² +
(c) x² +2 =
Figure 1: Algebraic varieties in R with different defining equations.
key class of algebraic varieties are Fano varieties: positively curved shapes that are basic building blocks in algebraic geometry. Fano varieties are 'atomic pieces" of more complex shapes, in the sense of the Minimal Model Programme 7 25 26]. Running the Minimal Model Programme that is, breaking an algebraic variety X into atomic pieces involves making birational transformations of X. These are modifications on subsets with zero volume (and codimension at least one), and can either introduce remove singularities. The building blocks that emerge from this process not necessarily smooth: they satisfy a weaker condition called Q-factorialityy and can have mild singularities called terminal singularities (38). Fano varieties that are Q--fcitoiall and have terminal singularities are called Fano varieties.
The classification of Q-Fano varieties is therefore a long-standing problem of great importance 4 15 27 33
34 one can think of this as building Periodic Table for geometry. But, despite more than a century of study, very little is known. In what follows we exploit the fact that machine learning can detect terminal singularities to give the first sketch of part of the classification of higher-dimensional Q-Fano varieties.
We probe the classification of Q-Fano varieties using a class of syghlyssymmetrical shapes called toric varieties. (For example, the algebraic varieties pictured in Figure arre toric varieties.) Toric varieties are particularly suitable for computation and machine learning, because their geometrie properties are encoded by simple combinatorial objects. We consider Fano toric varieties of Picard rank two. These can be encoded using a x N matrix of non-negative integers called the weight matrix; here the dimension of the toric variety is N - 2.
To determine whether such a toric variety X is Q-Fano variety we need to check whether X is Q-factorial, and whether the singularities X are terminal. Checking Q-factoriality from the weight matrix of X turns This is because there is no satisfactory theoretical understanding of the problem. We lack global criterion for detecting terminality in terms of weight data (such as 24 in simpler setting) and so have to fall back on first enumerating all the singularities to analyse, and then checking terminality for each singularity. Each step a challenging problem in discrete geometry: the first step involves building different combinatorial object associated to the n nz-dimensional toric variety X, which is a collection of cones in R called the fan EXX); the second step involves checking for various cones in the fan whether or not they contain lattice points on or below a certain hyperplane. To give sense of the difficulty of the computations involved, generating and post-processing our dataset of 10 million toric varieties in dimension eight took around 30 CPU years.
be straightforward (see } but checking terminality is extremely challenging.
Q-factorial if is normal and, addition, each rank- one reflexive sheaf on X, some tensor power of E' is a line bundle. This implies that the dimension of the singular locus in is at most dim X - 2. and that some tensor power of the canonical sheaf (of top-degree differential forms) is line bundle.
2 algebraic variety

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To overcome this difficulty, and hence to begin to investigate the classification of Q-Fano varieties in dimension eight, we used supervised machine learning. We trained feed-forward neural network classifier on balanced dataset of 5 million examples; these are eight-dimensional Q-factorial Fano torie varieties of Picard rank two, of which 2.5 million are terminal and 2.5 million non-terminal. Testing on a further balanced dataset of 5 million examples showed that the neural network classifies such toric varieties as terminal or non-terminal with an accuracy of 95%. This high accuracy allowed us to rapidly generate many additional examples that are with high probability Q- Q--Fano varieties that is, examples that the classifier predicts have terminal singularities. This ML-assisted generation step is much more efficient: generating 100 million examples in dimension eight took less than 120 CPU hours.
The fact that the ML classifier can detect terminal singularities with such high accuracy suggests that there new mathematics waiting to be discovered here there should be a simple criterion in terms of the weight matrix to determine whether or not toric variety X has terminal singularities. In we take the first steps in this direction, giving in Algorithm a new method to check terminality directly from the weight matrix, for toric varieties of Picard rank two. A proof of correctness is given in the Supplementary Material. This new algorithm is fifteen times faster than the naive approach that we used to generate our labelled dataset, but still several orders of magnitude slower than the neural network classifier. We believe that this is not the end of the story, and that the ML results suggest that simpler criterion exists. Note that the neural network classifier cannot be doing anything analogous to Algorithm|] [I] the algorithm relies on divisibility relations between entries of the weight matrix (GCDs etc.) that are not visible to the neural network, they are destroyed by the rescaling and standardisation that is applied to the weights before they are fed to the classifier.
In $ we use the ML-assisted dataset of 100 million examples to begin to explore the classification of Q-Fano varieties in dimension eight. We visualise the dataset using the regularized quantum period, an invariant that has played an important role in recent theoretical work on Q-Fano classification, discovering that an appropriate projection of the data appears to fill out a wedge-shaped region bounded by two straight lines. This visualisation suggests some simple patterns in the classification: for example, the distance from one edge of the wedge appears to be determined by the Fano index of the variety.
Our work is further evidence that machine learning can be an indispensable tool for generating and guiding mathematical understanding. The neural network classifier led directly to Algorithm [] new theoretical result, by revealing that the classification problem was tractable and thus there was probably new mathematics waiting to be found. This is part of a new wave of application of artificial where machine learning methods drive
intelligence to pure mathematics 10 16. 20 39.41, theorem discovery.
genuinely novel contribution here, though, is the use of machine learning for data generation and data exploration in pure mathematics. Sketching the landscape of siggher-iimesiioall Q-Fano varieties using traditional methods would be impossible with the current theoretical understanding. and prohibitively expensive using the current exact algorithms. Training a neural network classifier however, allows us to explore this landscape easily landscape that is unreachable with current mathematical tools.
Why dimension eight? We chose to work with digteniimensional varieties for several reasons. It important to distance ourselves from the surface case (dimension two), where terminality is trivial condition. A two-dimensional algebraic variety has terminal singularities if and only if it is smooth. On the other hand, we should consider dimension where we can generate sufficient amount of data for machine learning (the analogue of our dataset in dimension three, for example, contains only 34 examples 23) and where we can generate enough data to meaningfully probe the classification. Moreover, we work in Picard rank two because there already exists a fast combinatorial formula to check terminality in rank one 24 Picard rank two is the next natural case consider.
Data and code availability The datasets underlying this work and the code used to generate them are available from Zenodo under a CCO license [T]. Data generation and post-processing was carried out using the computational algebra system Magma V2.27-3
5 The machine learning model was built using PyTorch 13.1 (66] and scikit-learn 11. [37]. All code used and trained models are
available from BitBucket under an MIT licence 12]
3

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The action is 2 Mathematical background
The prototypical example of a Fano variety is projective space p--11, which can be thought of as the quotient of CN \\ (0) by CX acting as follows:
X (Z1....,,,0) EN) = (AzI, AZN)
Fano toric varieties of Picard rank two arise similarly. They can be constructed as the quotient of CN SS where S is union of subspaces, by an action of (CX)². This action, and the union of subspaces S, is encoded by a weight matrix:
a1
GN
(2.1)
bi
bn
Here we assume that all (a,, b.) E 2² {O} lie in strictly convex cone C' C IR,
(A, (Z1, ,N) (A 1 21, 21...., nay AaN nN zN)
and S = S+ US is the union of subspaces S+ and S_, where
S = ((21,...,,, EN) ) S_ -(2 2N) 2; Of b/a}
= 0f bi/0: > b/a}
(2.2)
and Ei u; E of dimension N 2. We assume in addition that both 5 and S_ have dimension least two; this implies that the second Betti number of X is two, that is, X has Picard rank two.
=== b:: see [6]. The quotient X - (CN \ () is an algebraic variety
Since we have insisted that all columns (a,, b,) lie in a strictly convex cone C, we can always permute columns and apply an (L)(Z) transformation to the weight matrix to obtain matrix in standard form:
02
(2.3)
b2
bn
where all entries are non-negative, the columns are cyclically ordered anticlockwise, and a,y < by- This transformation corresponds renumbering the co-ordinates of CNV and reparametrising the torus (Cx)² that acts, and consequently leaves the quotient variety X that we construct unchanged.
We will consider weight matrices (2.1) that satisfy an additional condition called being well-formed. An r N weight matrix is called standard if the greatest common divisor of its r r minors one, and is well-formed if every submatrix formed by deleting column is standard (2] Considering only well-formed weight matrices guarantees that toric variety determines and is determined by its weight matrix, uniquely up to SL, (2)rttassoooooomaaa
Testing terminality As mentioned in the introduction, an n--dimensional toric variety determines collection Σ(X) of cones in RR called the fan of X. A toric variety is completely determined by its fan. The process of determining the fan Σ(X) from the weight matrix (2.I) is explained in the Supplementary Material; this is challenging combinatorial calculation. In the fan Σ(X). the onc-dimensional cones are called rays. For Fano toric variety X, taking the convex hull of the first lattice point each ray defines convex polytope P, and X has terminal singularities if and only if the only lattice points in P are the origin and the vertices. Verifying this a conceptually straightforward but computationally challenging calculation in integer linear programming.
3 Data generation
We generated a balanced, labelled dataset of ten million Q-factorial Fano toric varieties of Picard rank two and dimension eight. These varieties are encoded, as described above, by weight matrices. We generated 2 x 10 integer-valued matrices in standard form, in (2.3). with entries chosen uniformly random from the set (o...., , 7). were both chosen uniformly at random from the set (1,...: 7). and the value for an, which was
Minor exceptions to this were the values for on and bx, which
chosen uniformly at random from the set {O....,bx 1}. Once random weight matrix was generated, retained it only if it satisfied:

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Table 1: Final network architecture and configuration.
Hyyerrparameter
Value
Hyperparameter
Value
Layers (512, 768, 512)
ummentum
0.99
Batch size 128
LeakyRelu slope
0.01
Initial learning rate 0.01
1. None of the columns are the zero vector.
2. The sum of the columns is not multiple of any of them.
3. The subspaces S4 and S_ in 2.2 are both of dimension at least two.
4. The matrix is well-formed.
The first condition here was part of our definition of weight matrix; the second condition equivalent to X being Q-factorial; the third condition guarantees that X has Picard rank two; and the fourth condition was discussed above.
We used rejection sampling to ensure that the dataset contains an equal number of terminal and non-terminal examples. Before generating any weight matrix, boolean value was set to True (terminal) or False (non-terminal). Once random weight matrix that satisfied conditions ( above generated, checked if the corresponding toric variety terminal using the method discussed in $2] If the terminality check agreed with the chosen boolean, the weight matrix was added to dataset; otherwise the generation step was repeated until match was found.
As discussed, different weight matrices can give rise to the same toric variety. Up to isomorphism, however, toric variety X is determined by the isomorphism class of its fan. We deduplicated our dataset by placing the corresponding fan [X),, which we had already computed order to test for terminality, in normal form 19 29) In practice, very few duplicates occurred.
4 Building the machine learning model
We built a neural network classifier to determine whether a Q-factorial Fano variety of Picard rank two and dimension eight is terminal. The network was trained on the features given by concatenating the two rows of a weight matrix, [a1,.... a10, bi...., b10]. The features were standardised by translating their mean to zero and scaling to variance one. The network, multilayer perceptron, is fully connected feedforward neural network with three hidden layers and leaky ReLu activation function. It was trained the dataset described in } using binary cross-entropy as loss function, stochastic mini-bated gradient descent optimiser and using early-stopping, for a maximum of 150 epochs and with learning rate reduction on plateaux. We tested the model on balanced subset of 50% of the data (5M); the remainder was used for training (40%; 4M balanced) and validation (10%; 1M).
Hyperparameter tuning was partly carried out using RayTune (31) small portion of the training data, via random grid search with Asyne Successive Halving Algorithm (ASHA) scheduler (30], for 100 experiments. Given the best configuration resulting from the random grid search, we then manually explored nearby configurations and took the best performing one. The final best network configuration is summarised in Table T
By trying different train-test splits, and using 20% of the training data for validation throughout, we obtained the learning curve in Figure 2a This shows that a siliiivvllatteeeee split of 4M-1M-5M produced an accurate model that did not overfit. Training this model gave the loss learning curve in Figure 2b and final accuracy (on the test split of size 5M) of 95%.
5 Theoretical result
The high accuracy of the model in $ was very surprising. explained the introduction, Q-Fano varieties are of fundamental importance in algebraic geometry. However, asking whether a Fano variety has terminal singularities is, in general, an extremely challenging geometric question. In the case of Fano toric variety would typically proceed by constructing the fan, and then performing
5

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Train Test
0.6
036
Validation
Accuracy
0.50
0.80
0.86
Training samples
106
60
Epochs
(b)
Figure 2: (a) Accuracy for different train-test splits; (b) epochs against loss for the network trained on 5M samples.
cone-by-cone analysis of the combinatorics. This is computationally expensive and unsatisfying from a theoretical viewpoint. The success of the model suggested that a more direct characterisation possible from the weight matrix alone. An analogous characterisation exists in the simpler case of weighted projective spaces [24), which have Picard rank one, however no such result higher Picard rank known prior training this model.
Inspired by this we prove theoretical result, Proposition 5.3 which leads to a new algorithm for checking terminality directly from the weight matrix, for @-factorial Fano toric varieties of Picard rank two. Consider a weight matrix as in (2.l that satisfies conditions toric variety X that it determines. As discussed in $2) and explained in detail in the Supplementary Material, X determines convex polytope in R- with N vertices given by the first lattice points on the N rays of the fan. Each of the vertices of P is lattice point (i.e., lies in N2N C IR and X has terminal singularities and only if the only lattice points in are the vertices cI,...,CN and the origin.
from $3 and the
Definition 5.1. Let A; denote the simplex in R-2 with vertices e1...., ......, where is omitted. We say that A; mostly empty if each lattice point in A, either vertex or the origin. Notation 5.2. Let {Ø} denote the fractional part [x] of rational number x.
Proposition 5.3. Consider weight matrix
a
aN bN
that satisfies conditions 7 from B,b, 9t. Set
Let gr = gedfar, bi and let Ai, B, be integers such that Aqa;
a b byo
9i
j=1
8 - A,a, B,b,
8,
g
ag ft
godlgi 8.}
j=1
noting that all these quantities are integers. Then As mostly empty if and only if for all l (O..... fs - 1} and E {O..... - 1} such that
N
gi
have that
31
k
J
for all j.

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Let = {i a, - 6,a 0}. 8- a,b 1000-yp b < 0}, and let be either s.. Then A.,
i forms triangulation of P. Thus X has terminal singularities if and only if A, mostly empty for each E T. This leads to Algorithm
Algorithm 1 Test terminality for weight matrix W [[01.... aNl. [b....., bN]].
1: Set E 0g Σ ii= b;
2: Set 84 = { a, b b,a 0) and s = {i a. bia < 0}.
3: Set I to be the smaller of 8+ and s..
4:
for E do
5:
Test if A, is mostly empty, using Proposition 5.3 6: if A; is not mostly empty then
7:
return False.
8: end if
9:
end for
10: return True.
Comparisons Testing on 100 randomly-chosen examples indicates that Algorithm is ap- proximately 15 times faster than the fan-hased approach to checking terminality that we used when labelling our dataset (0.020s per weight matrix for Algorithm | versus 0.305s for the standard ap- proach implemented in Magma). On single examples, the neural network classifier is approximately 30 times faster than Algorithm|] [T] The neural network also benefits greatly from batching, whereas the other two algorithms do not: for batches of size 10000, the neural network is roughly 2000 times faster than Algorithm
6 The terminal toric Fano landscape
Having trained the terminality classifier, we used it to explore the landscape of Q-Fano toric varieties with Picard rank two. To do so, we built a large dataset of examples and analysed their regularized quantum period, numerical invariant of Q-Fano varieties [8]. For smooth low-dimensional Fano varieties, is known that the regularized quantum period a complete invariant [9] This is believed to be true in higher dimension, but is still conjectural. Given Q--Fno variety X, its regularized quantum period a power series
(x
Σ cata
where = 1, = 0 dr and is the number of degree-d rational curves in X that satisfy certain geometric conditions. Formally speaking. ra degree-d, genus-zero Gromov-Wint invariant (28). The period sequence of X is the sequence (ca) of coefficients of the regularized quantum period. This sequence grows rapidly. In the case where is a Q-Fano toric variety of Picard rank two, rigorous asymptotics for this growth are known.
Theorem 6.1 (Theorem 5.2, [10). Consider weight matrix
[ai bi
aN
bn
for Q-factooial Fano toric variety X of Picard rank two. Let EN ag and b E by and let [w:v] E Ip1 be the unique real root of the homogeneous polynomial
N
[a. + b,v)aabb II (a.p + b.v)
(6.1)
such that asjt + bay 2 O for all E (1,2,.... NY. Lei (ca) be the corresponding period sequence. Then non-zero coefficients ca satisfy
log Ca Ad dim X
log

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10
10
Fano Index
Prequency
100
10
13
2/2
10"
A
(a)
(b)
Figure 3: A dataset of 100M probably-Q- Q-Fano toric varieties of Picard rank and dimension eight, projected to I using the growth coefficients A and B from (6.2). In (a) we colour by Fano index, while (b) we colour heatmap according the frequency.
as d xc, where
N Σ
A =
Pe log Ps
1=1
(6.2)
dim X
N
(a.b bya)²
B
log(2
log P:
log
2 t=1
2
C:ps
Here pag + vb; pa
so that E,P; = 1, and L = ged (a, b} is the Fano index. whb
In Figure we picture our dataset of Q-Fano varieties by using the coefficients A and B to project it R²;; for the corresponding images for terminal Fano weighted projective spaces, see [10] Figure 7a]. Note the stratification by Fano index. Although many weight matrices can give rise to the same toric variety, our context we are using well-formed weight matrices standard form (2.3) and so most two weight matrices can give rise to the same toric variety. We removed any such duplicates from our dataset, so the heatmap in Figure 3b reflects genuine variation in the distribution of Q-Fano varieties, rather than simply the many-to-one correspondence between weight matrices and toric varieties.
Data generation tion workflow that combines algorithmic checks and our machine learning model, as follows.
The dataset pictured in Figure 3 was generated using Al-assisted data genera-
Generate random 2 x 10 matrix with entries chosen uniformly from {0, 2, 4,
Cyclically order the columns and only keep the matrix if is in standard form, as in (2.3)
Check conditions [1 from
Predict terminality using the neural network classifier from $4 only keeping examples that are classified as terminal and storing their probabilities.
Set μ 1 in 6.1 and solve the univariate real polynomial in the correct domain to obtain the solution v).
Calculate the coefficients A and B using the formulae in (6.
The final dataset is composed 100M samples. Each of these represents Q-fctoriil toric Fano variety of dimension eight and Picard rank two that the classifier predicts is a Q-Fano variety.
Data analysis We note that the vertical boundary in Figure 3 is not surprise. In fact, we can apply the log-sum inequality to the formula for A to obtain
N E Pi N
Σ ps log(p)) S
Σ
Pi
log
log(N)

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Upper Bound =
Uppar bound 6
Upper Bound
Upeer Bound =
Uppar Bourd =
Upser Bound
142
4719
6741
151
2471
A483
Prodkted
Preditedd
Wredkted
Predited
Predited
Predited
Figure 4: Confusion matrices for the neural network classifier on in-sample and out-of-sample data. In each case balanced set of 10000 random examples was tested.
In our N 10, and the vertical boundary that see in Figure Ga is the line log ~ 2.3. We also see what looks like linear lower bound for the cluster; similar bound was observed, and established rigorously, for weighted projective spaces in [10].
Closer analysis (see the Supplementary Material) reveals large overlapping clusters that correspond to Fano varieties of different Fano index. Furthermore the simplest toric varieties of Picard rank two products of projective spaces, and products of weighted projective spaces appear to lie in specific regions of the diagram.
Limitations and future directions
The main message of this work new proposed Al-assisted workflow for data generation in pure mathematics. This allowed us to construct, for the first time, an approximate landscape of objects of mathematical interest (Q-Fano varieties) which is inaccessible by traditional methods. We hope that this methodology will have broad application, especially to other large-scale classification questions in mathematics, of which there are many [ 21)
Our approach has some limitations, however, which we enumerate here. Some of these limitations suggest directions for future research. A key drawback, common most ML models, is that our classifier performs poorly on out-of-sample data. Recall from $ that the dataset we generated bounded the entries of the matrices by seven. For weight matrices within this range the model is extremely accurate (95%), however this accuracy drops off rapidly for weight matrices that fall outside of this range: 62% for entries bounded by eight; 52% for entries bounded by nine; and 50% for entries bounded by ten. See Figure for details. Note that the network quickly degenerates to always predicting non-terminal singularities.
Furthermore the training process seems to require more data than we would like, given how computa- tionally expensive the training data is generate. It possible that more sophisticated network architecture, that is better adapted to this specific problem, might require less data to train.
Mathematicallyy, our work here limited to toric varieties, and furthermore only to toric varieties of Picard rank two. Finding a meaningful vectorisation an arbitrary algebraic variety looks like an impossible task. But if one is interested in the classification of algebraic varieties up deformation, this might be less of a problem than it first appears. Any smooth Fano variety low dimensions is, up deformation, either toric variety, toric complete intersection, locus [9] [22]; one might hope that this also covers a substantial fraction of the Q- landscape. Each of these classes of geometry is controlled by combinatorial structures, and it is possible to imagine a generalisation of our vectorisation by weight matrices to this broader context.
quiver flag zero
Generalising Q-factorial Fano toric varieties in higher Picard rank will require more sophisticated approach to equivariant machine learning. In this paper, we could rely on the fact that there is a normal form 2..3 for rank-two weight matrices that gives almost unique representative of each SL2(2) x Sororbit weight matrices. For higher Picard rank r we need to consider weight matrices Sy. Here normal form is known, so to work G-equivariantly we will need to augment our dataset, to fill out the different G-orbits, or to use invariant functions of the weights features. The latter option, geometrically speaking, is working directly with the quotient space.
up to the action of G SL, (Z) )
The best possible path forward would be to train explainable model that predicted terminality from the weight data. This would allow us to extract from the machine learning not only that the problem is tractable, but also precise mathematical conjecture for the solution. At the moment,

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however, we are very far from this. The multilayer perceptron that we trained is a black-box model, and post-hoe explanatory methods such as SHAP analysis [32] yielded little insight: all features were used uniformly, as might be expected. We hope to return to this point elsewhere.
Acknowledgments and Disclosure of Funding
TC was partially supported by ERC Consolidator Grant 682603 and EPSRC Programme Grant EP/N03189X/1 AK supported by EPSRC Fellowship EP/N02251// 3/1. SV is supported by the Engineering and Physical Sciences Research Council [EP/S021550/1] the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. The authors would like to thank Hamid Abban, Alessio Corti, and Challenger Mishra for many useful conversations, and the anonymous referees for their insightful feedback and suggestions. The authors have no competing interests.
References
[1] Jeffrey Adams, Annegret Paul, Ran Cui, Susana Salamanca Riba, Peter Trapa. Mare van Leeuwen, and David Vogan. Atlas of Lie groups and representations. Online, 2016.
org
[2] Hamid Ahmadinezhad. On pliability of del Pezzo fibrations and Cox rings. J. Reine Angew. Math., 723: 101-125, 2017.
2014-0009
[3] Michael F. Atiyah, Nigel J. Hitchin, Vladimir G. Drinfeld, and Yuri I. Manin. Construction of instantons. Phys. Lett. A, 6858)1185-187, 1978. doi 046000066775596001899144444
[4] Caucher Birkar. Singularities linear systems boundedness of Fano varieties. Ann, of Math. (2), (2):347-405, 2021. 400000004007/mmml..2222222
[5] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language.
J. Symbolic Comput.. 24(3-4):235-265 1997. doi 10. 10.1006/jsco. wtt.ss.o..99...11. 1996.0125 Computational algebra and number theory (London, 1993).
Gavin Brown, Alessio Corti, and Francesco Zucconi. Birational geometry of 3-fold Mori fibre spaces. In The Fano Conference, pages 235-275. Univ. Torino, Turin, 2004.
[7] Paolo Cascini. New directioons in the minimal model program. Boll. Unione Mat. Ital., 14(1):179-190, 2021. TO. 1007/540577400002222550
[8] Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, and Alexander M. Kasprzyk. Mirror symmetry and Fano manifolds. In European Congress of Mathematics, pages 285-300. Eur. Math. Soc., Zürich, 2013. 1o:110.4770/100
[9] Tom Coates, Alessio Corti, Sergey Galkin, and Alexander M. Kasprzyk. Quantum periods for 3-dimensional Fano manifolds. Geom. Topol., 20(1):103-256) 2016. doi 10.2 40/g
[10] Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale. Machine learning the dimension of a Fano variety. Nat. Commun,, 14:5526, 2023. 1MM,T0..00884444447222-111113313
[11] Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale. A dataset of 8- s-dimensional Q-factorial Fano varieties of Picard rank 2. Zenodo, 2023. 120.5800
[12] Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale. Supporting code. [tttps://bitbuckeer org fanosearch/ ml1 honbeaarcch/m__ttemmmaaaaii/// 2023.
[13] John Cremona. The I-ffunctions and modular forms database project. Found. Comput. Math., 16(6):
1541-1553, 2016. doi TO. 10.10007/1000880116669000
[14] Alex Davies, Petar Veličković, Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Tomašev, Richard Tanbur, Peter Battaglia, Charles Blundell, András Juhász, Mare Lackenby, Geordie Williamson, Demis Hassabis, and Pushmeet Kohli. Advancing mathematics by guiding human intuition with Al. Nature, 600;
70-74, 2021. doi
[15] Pasquale Del Pezzo. Sulle superficie dell'n" nM ordine immerse nello spazio ad dimensioni. Rend, del Circolo Mat. di Palermo. 1224-225. 1887.
10

---

[16] Harold Erbin and Riccardo Finotello. manifolds: methodological study. doi:10.1 100p10001000/pyysee.11......11.0
Machine learning for complete intersection Calabi-Yau Phys. Rev. D, 103(12):Paper No. 126014, 40, 2021.
[17] Nicholas Eriksson, Kristian Ranestad, Berdd Surrffels, and Seth Sullivant. Phylogenetic algebraic geometry. In Projective varieties with unexpected properties, pages 237-255. Walter de Gruyter, Berlin, 2005.
[18] Brian R. Greene. String thcory on Calabi Yau manifolds. In Fields, strings and duality (Boulder, CO, 1996), pages 543-726. World Sci. Publ., River Edge, NJ, 1997.
[19] Roland Grinis and Alexander Kasprzyk. Normal forms of convex lattice polytopes. #xiv:1301.6660 [O] 2013.
[20] Yang-Hui He. Machine-leaming mathematical structures. International Journal of Data Science in the Mathematicul Sciences. 1:23 47, 2023.
[21] Yang- Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov. Murmurations of elliptic curves. arXiv:2204. 10140 [math.NT]] 2022.
[22] Elana Kalashnikov. Four-dimensional Fano quiver flag zero loci. Proc. Royal Society A., 475(2225): 20180791, 23, 2019. doi 10.1099//spa.2118..777
[23] Alexander M. Kasprzyk. Toric Fano three-folds with terminal singularitics. Tohoku Math. J. (2), 58(1): 101-121, 2006. 46i:000.744//mm1145555550
[24] Alexander M. Kasprzyk. Classifying terminal weighted projective space. 2013.
[math.AG]
[25] János Kollár. The structure algebraic threefolds: an introduction Mori's program. Bull. Amer. Math. Soc. (N.S.). 17(2):11-273,, 1987. 00010000000000222200099999881819
[26] János Kollár and Shigefumi Mori. Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. do1100.0017/BB0778555 TeA2009988055666666660 1662560
[27] János Kollár, Yoichi Miyaoka, Shigefumi Mori, and Hiromichi Takagi. Boundedness of canonical Q- Q-Fano 3-folds. Proc. Japan Acad. Ser. Math. Sci., 76(5):73-77, 2000. doi 10.3792/ppaa..6.73
[28] Maxim Kontsevich and Yuri Manin. Gromov- Witten classes, quantum cohomology. and enumerative geometry. In Mirror symmetry, II, volume of AMS/IP Stud. Adv. Math., pages 607-653. Amer. Math. Soc., Providence, RI, 1997. doi 10. 00000////mip/0001/211
[29] Maximilian Kreuzer and Harald Skarke. PALP: package for analysing lattice polytopes with applications to toric geometry. Comput. Phys. Comm., (57(1)887-006 2004. doi 10.1016/500100666553333000
[30] Liam Li, Kevin Jamieson, Afshin Rostamizadeh, Ekaterina Gonina, Jonathan Ben-Tzur, Moritz, Hardt, Ben- jamin Recht, and Amect Talwalkar, A system for massively parallel hyperparameter tuning. Proceedings of Machine Learning and Systems, 2:230-246, 2020.
[31] Richard Liaw, Eric Liang, Robert Nishihara, Philipp Moritz, Joseph E Gonzalez, and Ion Stoica. Tune: A research platform for distributed model selection and training. www.11807.05113. [cs.LG] 2018.
[32] Scott M Lundberg and Su-In Lee. A unified approach interpreting model predictions. Advances Neural Information Processing Systems, 30, 2017.
[33] Shigefumi Mori and Shigeru Mukai, Classification of Fano 3-folds with Bg 2 2. Manuscripta Math., 36 (2):147-163) 162, 1981/82. doi 1007000F00000003
[34] Shigefumi Mori and Shigeru Mukai. Erratum: "Classification of Fano 3-folds with B2 2 2". Manuscripta Math., 110(3):407, 2003. doi 10.
[35] Harald Niederreiter and Chaoping Xing. Algebraic geometry coding theory and cryptography. Princeton University Press, Princeton, NJ, 2009.
[36] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, performaree deep learning library, 2019.

---

[37] Fabina Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg. Jake Vanderplas, Alexan- dre Passos, David Cournapeau, Matthieu Brucher, Matthieu Perrot, and Edouard Duchesnay. Scikit- learm: Machine learning in Python. Journal of Machine Learning Research, 12:2825 2830, 2011.
1555555555/1443044 2078195
[38] Miles Reid. Young person's guide to canonical singularities. In Algebraic geometry. Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 345-414, Amer. Math. Soc., Providence, RI, 1987.
[39] Adam Zsolt Wagner. Constructions in combinatorics via neural networks. ar XXXv:2104.14516 [math. 2021.
[40] Geordie Williamson. Is deep learning useful tool for the pure mathematician? math RT], 2023.
ar aiv:22304 1202
[41] Yue Wu and Jesús A De Loera. Turning mathematics problems into games: Reinforcement learning and Gröbner bases together solve integer feasibility problems. arXiv:2208.12199 12191 [cs.LG]] 2022.
12


Reviews:
[{"review_id": "CgjlNz51Ye", "reviewer": "Reviewer_9Hdp", "summary": "The paper describes an ML assisted approach to the classification of Q-Fano varieties. Checking whether the singularities of a given Q-Factorial Toric variety is very challenging by mathematically established methods, involving intensive combinatorial computations. This paper applies supervised machine learning to this problem, training a feedforward NN classifier to detect terminal varieties on a balanced dataset of 5 million 8-dimensional Q-Factorial Fano toric varieties of Picard rank 2 (there exists already a fast combinatorial formula for detecting terminality in Picard rank 1). The obtained classifier obtains a 95% test accuracy on a test set of 5 million samples from the same distribution as the training data. This is surprising because the classifier is much more efficient than known combinatorial techniques, and uses only the weight matrix corresponding to the variety. The good test performance can thus be interpreted to indicate new mathematical techniques to discover.\n\nTaking inspiration from this, the second contribution of the paper is to develop an algorithm that detects terminality, directly from the weight matrix, for Q-Factorial Fano toric varieties of Picard rank two. This is the first algorithm that uses only the weight matrix in the case of Picard rank 2, and is measured to be ~15 times faster than existing methods. The new algorithm however, remains slower than the classifier. Additionally, weight information that is used by the new algorithm is unavailable to the NN classifier, having been destroyed by the encoding and regularization. This indicates that more mathematical discovery is possible.\n\nFinally, the trained classifier is used as a tool to generate a dataset of \"probably terminal\" examples, i.e. examples that are predicted to be terminal by the classifier. This generation is much more efficient than rigorously checking terminality. A dataset of 100M probably-Q-Fano toric varieties are then investigated by performing an $\\mathbb{R}^2$ projection based on the regularized quantum period. This leads to some interesting observations about the distribution of such varieties, with the authors particularly remarking on the shape of the support and the stratification by Fano index.", "questions": "Is it reasonable to expect that a NN classifier trained and tested on a dataset with larger weight bounds would retain the surprising performance noted in the paper? Numerically probing this may be too onerous, but if the authors could describe some intuition it would help my understanding.", "limitations": "Yes", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper makes a compelling and original case for the utility of machine learning in pure mathematics. It highlights that the performance of a classifier can be an indicator of the existence of undiscovered mathematical structure: the setting here is particularly interested due to the large discrepancy between the prediction and known mathematics. It also highlights the value of an efficient classifier for data generation that may allow for numerical investigations into the underlying structures. The new algorithm for detecting terminality is of possible additional interest.\n\nThe paper also does a good job of highlighting the differences between the classifier and rigorous techniques (including the new algorithm developed here). There is also a clear discussion of the current limitations of ML assisted mathematics in this setting and general directions for future improvement.", "weaknesses": "I do not see any glaring weaknesses. The approach as currently presented has limitations: the data requirements and poor OOD performance being the primary ones, but these are openly addressed and to be expected in early works in the field. The poor OOD performance is in my opinion the biggest obstacle to the claims of the paper, raising the troubling possibility that the better performance of the classifier comes from some 'finite size effect' due to the particular weight bound.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: *The poor OOD performance is in my opinion the biggest obstacle to the claims of the paper, raising the troubling possibility that the better performance of the classifier comes from some 'finite size effect' due to the particular weight bound [...] Is it reasonable to expect that a NN classifier trained and tested on a dataset with larger weight bounds would retain the surprising performance noted in the paper? Numerically probing this may be too onerous, but if the authors could describe some intuition it would help my understanding.*\n\nA1: This is an important question. We have thought about it a lot during the course of this project, and do not have a clear view. We suggest that, in a future revision, we train and test a similar neural network classifier on a dataset with larger weight bounds, and report the results in the supplementary material. (It is not practical to complete this experiment by the end of the authors' response period.) Note, however, that one of the most valuable implications of our work -- the ability to sketch a part of the landscape of 8-dimensional Fano varieties that until now was completely out of reach -- holds regardless of whether the answer to the question is 'yes' or 'no'.\n\nOur intuition here is as follows. Given that the testing and training data are free of noise (they are created through exact mathematical calculation) and the neural network classifier is so accurate, we believe that the classifier is approximating a precise, general mathematical statement. But it is not clear to us whether this statement is about weight matrices with entries of arbitrary size or whether the underlying statement is something about weight matrices with small entries (mathematically, this would be a statement about Fano varieties with terminal singularities of bounded index).\n\nIn the first case the OOD performance drop-off would happen because the network is approximating the true statement in a way that does not generalise to higher weight bounds; this is a common phenomenon when developing and using neural network models. In the second case the OOD performance drop-off would happen because of the underlying mathematical statement that the classifier approximates. As mentioned, one of the most valuable consequences of this approach -- our sketch of part of the landscape of higher-dimensional Fano varieties that until now was inaccessible -- remains regardless of which explanation is correct."}, {"review_id": "GDxXd2DOuX", "reviewer": "Reviewer_Ukzu", "summary": "This paper applies machine learning methods to predict whether an algebraic variety is Q-Fano. Q-fano varieties are Fano varieties that are Q-factorial and have terminal singularities. Checking terminally is computational challenging. To train a machine learning classifier, the authors build a balanced dataset of 10 million eight-dimensional Q-factorial Fano toric varieties of Picard rank 2. They use an MLP to classify whether a topic variety has terminal singularities (and thus a Q-fano variety) from its weight matrix. The neural network achieves an accuracy of 95%, which leads to the speculation that these exists a simple criterion to check terminal singularities directly from the weight matrix. The authors develops a new algorithm that is 15 times faster than the naive approach but still orders of magnitude slower than the ML classifier. They then conclude that the ML results suggest that a simpler criterion exists. Facilitated by the faster ML model, the authors create and visualize a dataset of 100 million examples, which provides new insights for the patterns in the classification.\n", "questions": "- Would it be possible for the authors to comment more on the significance of the classification of Q-Fano varieties? \n- Does detecting terminal singularities have other applications beyond a criterion for Q-Fano varieties?\n", "limitations": "The authors point out limitations in Section 7. There are no potential negative societal impacts of the work.\n", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- This paper provides an excellent example that shows how ML can be used to guide research in mathematics. The authors use neural networks to accelerate the classification of Fano-varieties. The computation speed by neural networks motivates new classification algorithms and suggests room for even faster ones. The trained ML model is then used as a surrogate to generate large amount of data which would be prohibitively expensive using traditional methods, and provide insights that guides theoretical understandings. While ML for science is getting popular in recent years, ML for mathematics is still a new field but has immense potential.\n- The Q-factorial Fano dataset provided in this paper is a valuable assets for both math and ML communities. It will advance the understanding of Q-Fano classification and benefit future development on theorem discovery.\n- The last section provides a number of interesting future directions. By relating the limitations of the current method to fundamental questions in ML (out-of-distribution generalization, data efficiency, geometric deep learning, interpretability), the authors demonstrate the importance of these topics and paves way for a new area in ML research.\n", "weaknesses": "- The mathematical objects studied in this paper is not very accessible to people without a math background. As a result, the significance of results, such as the analysis of invariant of Fano-varieties section 6, is not as strong as it could be. This is perhaps due to the large amount of prerequisite for algebraic geometry, but the authors could include more introductory materials, for example, in the appendix. \n- While the AI-assisted workflow is novel, the contribution on machine learning is limited. In particular, the neural network uses multilayer perceptron, and no justification is given for the architecture choice. The theoretical result in section 5 may not be interesting to a NeurIPS audience. \n- The claim that “the ML results suggests that a simpler criterion exists” for Q-Fano classification (line 82) is not well supported. The ML model does not have 100% accuracy and does not generalize when weights matrices fall outside of the range used in training. Hence the provided results does not prove that the ML model has found a simpler and correct criterion.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: *…the authors could include more introductory materials, for example, in the appendix.*\n\nA1: We agree that including additional introductory material in an appendix would be an excellent idea, and will do so in a future revision. The planned addition will cover three points. It will contextualise the relevance of algebraic geometry with respect to other fields (e.g. coding theory, algebraic vision, tensor theory, etc.); it will motivate the importance of Q-Fano varieties (see answers A4 and A5); it will give an hands on introduction to toric varieties with examples.\n\nQ2: *While the AI-assisted workflow is novel, the contribution on machine learning is limited. In particular, the neural network uses multilayer perceptron, and no justification is given for the architecture choice. The theoretical result in section 5 may not be interesting to a NeurIPS audience.*\n\nA2: We feel that including the theoretical result is important, in order to demonstrate that the AI-assisted workflow is valuable. It also highlights the difficulty of the problem that the neural network classifier is solving.\n\nAbout the network architecture: since the use of ML models is a novel approach in this area of mathematics, it seemed natural to try more elementary methods before delving into more advanced machine learning models. We recognise that the choice of architecture is not motivated by the problem we approach. However the result is good enough to start exploring the landscape as in Section 6, which brings new mathematical insight.\n\nIn the future we would like to use network architectures that are better-suited to extracting explanations for their conclusions when applied to mathematical data. This will require improvements both in our own understanding of the techniques involved, and in the state-of-the-art for explainable models. We are hopeful that the application of AI methods to pure mathematical datasets (which are free of experimental or sampling noise) could be a fertile ground for explainable AI.\n\nQ3: *The claim that “the ML results suggests that a simpler criterion exists” for Q-Fano classification (line 82) is not well supported. The ML model does not have 100% accuracy and does not generalize when weights matrices fall outside of the range used in training. Hence the provided results does not prove that the ML model has found a simpler and correct criterion.*\n\nA3: We had meant this to read as \"We believe that ... the ML results suggest that a simpler criterion exists\" and will rephrase to avoid the ambiguity. We agree that our results are a long way from a proof of this statement.\n\nQ4: *Would it be possible for the authors to comment more on the significance of the classification of Q-Fano varieties?*\n\nA4: As mentioned in the Introduction, Q-Fano varieties are atomic pieces from which more complicated algebraic varieties are made, and so one can think of the classification of Q-Fano varieties as building a Periodic Table for geometry. Q-Fano varieties also play an important role elsewhere in mathematics, for example in the study of K-stability and the existence of Kähler–Einstein metrics. [1]\n\nIn theoretical physics, Q-Fano varieties play an essential role in constructing models of spacetime in Type II string theory -- they provide, through their ‘anticanonical sections’, the main construction of the Calabi–Yau manifolds which give geometric models of spacetime. [2,3,4]\n\nWe plan on including these remarks in the additional introduction on Fano varieties in the Supplementary materials as per A1. \n\nQ5: *Does detecting terminal singularities have other applications beyond a criterion for Q-Fano varieties?*\n\nA5: Terminal singularities are the singularities that appear in the Minimal Model Program: they are fundamental in the classification of algebraic varieties.\n\nQ-factorial terminal singularities are also important in string theory. For example, in F-theory, terminal singularities reflect the presence of localized matter states from wrapped M2-branes which are not charged under any massless gauge potential. [5]\n\nIn the toric context, having only terminal singularities means that the corresponding polytope contains no lattice points other than the origin and the vertices. These are referred to in the combinatorics literature as 1-point lattice polytopes, and are important in optimisation problems.\n\nWe plan on including these remarks in the additional introduction on Fano varieties in the Supplementary materials as per A1. \n\nREFERENCES\n\n[1] Berman RJ. K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics. Inventiones mathematicae. 2016 Mar;203(3):973-1025.\n\n[2] Candelas P, Horowitz GT, Strominger A, Witten E. Vacuum configurations for superstrings. Nuclear Physics B. 1985 Jan 1;258:46-74.\n\n[3] Greene BR. String theory on Calabi-Yau manifolds. Fields, strings and duality. 1997;543-726.\n\n[4] Polchinski J. String theory. 2005 Jun.\n\n[5] Arras P, Grassi A, Weigand T. Terminal singularities, Milnor numbers, and matter in F-theory. Journal of Geometry and Physics. 2018 Jan 1;123:71-97."}, {"review_id": "aVLSkGQuXp", "reviewer": "Reviewer_B8mu", "summary": "The authors study properties of algebraic geometric objects, four complex dimensional Fano toric varieties,\nusing mathematical data science methods, i.e. statistical analysis of a large database of examples combined\nwith mathematical domain knowledge.  They find a fairly accurate neural network classifier to identify singular\nvarieties and inspired by this propose a more efficient algorithmic test.  They also study an invariant called the\nquantum period.\n\n ", "questions": "Re section 5 and algorithm 1, the motivation is the existence of an NN predictor of singularity from the weight matrix.  \nDid anything else from the NN results enter into finding this result?  ", "limitations": "yes", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "Interesting example of mathematical data analysis for an open and actively studied problem in algebraic geometry.\nThe new algorithm for singularity testing is interesting.", "weaknesses": "The paper is very unsystematic statistically.  Since the dataset consists of mathematical objects generated by\na known sampling procedure, one could precisely characterize the ensemble being sampled in section 3 (with the\nbounds on weights given there), to help interpret the results and help compare with other (future) work.  Or\n(probably better) change the sampling to get a simple to describe ensemble.  This question is not even asked.\nIt bears on the significance of the results and the reported poor performance on out of sample test data reported in section 7.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: *The authors study properties of algebraic geometric objects, four complex dimensional Fano toric varieties…*\n\n[Small correction: all our varieties are over the complex numbers, so these are 8-dimensional complex Fano toric varieties.] \n\n*The paper is very unsystematic statistically. Since the dataset consists of mathematical objects generated by a known sampling procedure, one could precisely characterize the ensemble being sampled in section 3 (with the bounds on weights given there), to help interpret the results and help compare with other (future) work.*\n\nA1: We agree with the reviewer that characterising the ensemble being sampled in Section 3 would be highly desirable. However, this is complicated by the (infinite) symmetry group involved. The space of the objects studied (weight matrices of size $r \\times N$) carries the action of $GL(r, \\mathbb{Z})$ on the left, together with the permutation action of the symmetric group $S_N$ on the columns. Thus the group $G = GL(r,\\mathbb{Z}) \\times S_N$ acts, and any two weight matrices that differ by the action of an element of $G$ represent the same toric variety. As highly non-linear objects, the $G$-orbits are hard to describe.\n\nThe normal form for weight matrices that we propose and use in the paper (Equation 2.3) contains one representative of each $G$-orbit but, like all normal forms involving $GL(r, \\mathbb{Z})$-equivalence, is very `non-continuous'. This is exactly what makes the Geometry of Numbers (and ultimately things like the distribution of primes, which is intimately connected with the structure of $GL(n,\\mathbb{Z})$) such a challenging area of study. For example, changing a single matrix entry by plus/minus 1 will in general produce a large change in the normal form: the columns of the resulting matrix may need reordering by $S_N$, or the entire matrix may be reduced using the action of $GL(r,\\mathbb{Z})$. The topology of the space given by our chosen representatives of the orbits is, therefore, counterintuitive and definitely not Euclidean.\n\nOn top of this, the matrices we consider are all well-formed, which introduces further complexity. Indeed, well-formedness in rank 2 is already hard to describe from the viewpoint of geometry, and again depends on subtle considerations of lattice structure (that is, on the Geometry of Numbers). Whilst we agree with the referee that a precise characterisation of the ensemble is desirable, in practice this is far beyond our current geometric understanding of the problem.\n\nQ2: *Or (probably better) change the sampling to get a simple to describe ensemble.*\n\nA2: Again, we agree with the referee that this is a desirable step, but given the current state of understanding of the geometry, we do not believe any such rephrasing is possible.\n\nQ3: *Did anything else from the NN results enter into finding this result?*\n\nA3: As well as giving us confidence that there was a theoretical result waiting to be discovered, the NN also guided our formulation and proof of this result. For example, originally we thought that the result would only hold if the weight matrices contained an identity submatrix – the NN quickly persuaded us that this should not be the case. Without the NN model, we would have produced a considerably weaker statement. That said, the influence of the NN on the statement of the theoretical result was only indirect: we were unable to extract any precise statement from the network itself.\n"}, {"review_id": "VGQTqEJQAW", "reviewer": "Reviewer_VZKS", "summary": "The authors trained a feedforward neural network to detect $\\mathbb{Q}$-Fano varieties in dimension 8, among toric varieties defined by weights in a bounded region. The resulting classifier achieves a high (95%) accuracy. The authors used this classifier to build a large dataset of (probably) $\\mathbb{Q}$-Fano and visualize it. The high accuracy achieved also inspired the authors to derive a new (exact) algorithm to detect $\\mathbb{Q}$-Fano varieties.", "questions": "I have two provocative questions.\n\n1. Was Proposition 5.3 (or its proof) inspired in any way by the neural network? The authors seem to imply it, but that does not seem very likely. To motivate in finding this theoretical result, it seems that the important thing was to strongly believe that a simple classifier exists, not that it actually exists.\n\n2. Regarding the poor performance on out-of-distribution examples (Figure 4): Do you think this might mean that the trained neural network is not, in fact, good enough evidence that there is a simpler algorithm to detect $\\mathbb{Q}$-Fano varieties?\n\n\nI also found one typo:\n- Line 293: \"that\" -> \"than\"", "limitations": "The paper does a good job of recognizing the weaknesses in the closing section.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is very well written, it address a problem that is interesting and probably quite important (I am not an expert in algebraic geometry though). It has a very detailed experimental part, as well as a new theoretical result (which I haven't been able to check).", "weaknesses": "It's not very clear whether, at the end of the day, the classifier built in this work is actually going to make an impactful contribution in the area. This weakness is outweighted by the strengths, in my opinion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: *It's not very clear whether, at the end of the day, the classifier built in this work is actually going to make an impactful contribution in the area. This weakness is outweighted by the strengths, in my opinion.*\n\nA1: Understanding the classification of Q-Fano varieties is a fundamental open problem in algebraic geometry, and is the motivation behind a huge amount of important research. Techniques exist to construct certain classes (for example, the unprojection techniques in three dimensions by Reid et al in [1]) but these only scratch the surface of what is believed to exist. The ML classifier developed in this paper allows us to explore the classification at scale, giving us a \"birds eye view\" of the landscape that was previously impossible. Now that we know that this works (and it's important to emphasise how unexpected the success of this model was), it should be possible to construct similar models for higher Picard rank, non-toric varieties such as those described by Kalashnikov in [2], and produce a map of the Q-Fano classification. There is a precedent here: the highly influential Graded Ring Database of Brown, Reid, et al. [3]  provided the first coarse sketch of the landscape of Q-Fano 3-dimensional varieties via a (many-to-one) numerical invariant called the Hilbert series. Similarly to the ML model, their methods cannot guarantee that any given datapoint in their list is correct. Nevertheless, attempting to understand this (possible) classification  motivated a large amount of fundamental research, and led to significant breakthroughs in our understanding of Fano varieties.\n\nQ2: *Was Proposition 5.3 (or its proof) inspired in any way by the neural network? The authors seem to imply it, but that does not seem very likely. To motivate in finding this theoretical result, it seems that the important thing was to strongly believe that a simple classifier exists, not that it actually exists.*\n\nA2: The referee is correct in that we were unable to extract any precise statement from the structure of the neural network. The NN did, however, play two important roles in the development of Proposition 5.3. First, by testing ideas about when the neural network gave correct/incorrect answers, we were able to guide our understanding of the mathematical problem. For example, initially we believed that the presence of an identity block in the weight matrix would be crucial; the NN quickly disabused us of this. Without the NN guiding us here, we would have produced a considerably weaker statement. Second, as the referee suggests, simply knowing that this was possible was essential. The question of when a toric variety has terminal singularities is a well-studied and important one. And yet, beyond some elementary cases, no expert in the field suspected that this could be decided from the weight data alone. The standard (and, indeed, only) approach was to first construct the fan, and then to proceed via a cone-by-cone analysis. The striking and unexpected success of the NN provided us with the confidence to look for a result.\n\nQ3: *Regarding the poor performance on out-of-distribution examples (Figure 4): Do you think this might mean that the trained neural network is not, in fact, good enough evidence that there is a simpler algorithm to detect Q-Fano varieties?*\n\nA3: This is an interesting question which we debated at length whilst working on this paper, and is difficult to answer. It is connected with the two general questions: Can we ever expect a NN to recover a precise mathematical statement? And would a deep understanding of how a NN reaches its conclusions help constructing a mathematical result? It seems reasonable to us that the NN is approximating a more general, precise, mathematical statement; unfortunately because of the way these approximations are modelled within the structure of a NN, it doesn't give a good approximation to out-of-distribution input data. This is, of course, a common experience when developing and using ML models. Since the data is free of noise (it is purely mathematical) and the results are so accurate, we believe that the NN is detecting something which has a mathematical origin. Perhaps it has learnt something about the structure of 8-dimensional terminal singularities of low index? This is certainly possible, and one might expect the low-index cases to be \"easier\" to understand (although the structure in dimension 4 has only recently been understood fully; dimension 8 remains far beyond our current mathematical reach). This might explain both the efficiency of the calculation and why it fails on out-of-distribution data (which will necessarily come with higher-index singularities). We simply don't know.\n\nQ4: *Line 293: \"that\" -> \"than\"*\n\nA4: Thank you for pointing this out, this will be corrected in the revision.\n\nREFERENCES\n\n[1] Altınok S, Brown G, Reid M. Fano 3-folds, K3 surfaces and graded rings. Topology and geometry: commemorating SISTAG. 2002;314:25-53.\n\n[2] Kalashnikov E. Four-dimensional Fano quiver flag zero loci. Proceedings of the Royal Society A. 2019 May 31;475(2225):20180791.\n \n[3] Brown G, Kasprzyk A. The Graded Ring Database. Online. 2007-present."}]

Author rebuttal:
We thank the referees for their careful reading of the manuscript, and for their challenging and insightful questions. We have responded to each referee separately. In light of the referees' comments, we propose in a future revision to:
* add an expanded mathematical background section in the Supplementary Material. This will also include the background information included in answers A4 and A5 for reviewer Ukzu.
* add to the Supplementary Material a section discussing the results of training and testing a similar neural network classifier using weight matrices with a larger bound on the weights.
* change "that" to "than" on line 293.

Meta-review: All the reviewers agreed that the the paper is well-written, the subject is quite original and important and the contributions are solid. Hence I recommend an acceptance.
Decision: Accept (poster)