Machine Learning Regularization
for the Minimum Volume Formula of Toric Calabi-Yau 3-folds

Since the introduction of machine learning techniques in He:2017aed ; Krefl:2017yox ; Ruehle:2017mzq ; Carifio:2017bov ; Cole:2019enn ; Cole:2020gkd ; Halverson:2020trp ; Gukov:2020qaj ; Abel:2021rrj ; Krippendorf:2021uxu ; Cole:2021nnt ; Berglund:2023ztk ; Demirtas:2023fir for studying problems that occur in the context of string theory, machine learning – both supervised Bull:2018uow ; Jejjala:2019kio ; Brodie:2019dfx ; He:2020lbz ; Erbin:2020tks ; Anagiannis:2021cco ; Larfors:2022nep and unsupervised Krippendorf:2020gny ; Berman:2021mcw ; Bao:2021olg ; Seong:2023njx – has led to a variety of applications in string theory. A problem that appeared particularly suited for machine learning in 2017 Krefl:2017yox was the problem of identifying a formula for the minimum volume of Sasaki-Einstein 5-manifolds Martelli:2006yb ; Martelli:2005tp . The cone over these Sasaki-Einstein 5-manifolds is a toric Calabi-Yau 3-fold fulton ; 1997hep.th…11013L . Given that there are infinitely many toric Calabi-Yau 3-folds with corresponding Sasaki-Einstein 5-manifolds and that there is an infinite class of $4d$ $\mathcal{N}=1$ supersymmetric gauge theories associated to them via string theory Greene:1996cy ; Douglas:1997de ; Witten:1998qj ; Klebanov:1998hh ; Douglas:1996sw ; Lawrence:1998ja ; Feng:2000mi ; Feng:2001xr , this beautiful correspondence between geometry and gauge theory was identified in Krefl:2017yox as an ideal testbed for introducing machine learning for string theory.

These $4d$ $\mathcal{N}=1$ supersymmetric gauge theories corresponding to toric Calabi-Yau 3-folds are realized as worldvolume theories of D3-branes probing the Calabi-Yau singularities. Via the AdS/CFT correspondence Maldacena:1997re ; Morrison:1998cs ; Acharya:1998db , the minimum volume of the Sasaki-Einstein 5-manifolds is related to the maximized $a$-function Intriligator:2003jj ; Butti:2005vn ; Butti:2005ps that gives the central charges of the corresponding $4d$ $\mathcal{N}=1$ superconformal field theories Gubser:1998vd ; Henningson:1998gx . The proposal in Krefl:2017yox was that machine learning techniques can be used to give a formula of the minimum volume in terms of features taken from the toric diagram of the corresponding toric Calabi-Yau 3-folds. Such a formula would significantly simplify the computation of the minimum volume, which conventionally is computed by minimizing the volume function obtained from the equivariant index Martelli:2006yb ; Martelli:2005tp or Hilbert series of the toric Calabi-Yau 3-fold Benvenuti:2006qr ; Feng:2007ur .

In Krefl:2017yox , we made use of multiple linear regression gauss1823theoria ; fisher1922mathematical ; mendenhall2003second ; freedman2009statistical ; jobson2012applied and a combination of a regression model and a convolutional neural network (CNN) lecun1998gradient ; krizhevsky2012imagenet ; lecun2015deep ; schmidhuber2015deep to learn the minimum volume for toric Calabi-Yau 3-folds. As it is often the case for supervised machine learning rumelhart1986learning ; hastie2009elements , the models lacked interpretability and explainability, achieving high accuracies in estimating the minimum volume with giving only little insight into the mathematical structure and physical origin of the estimating formula.

In this work, we aim to highlight the pivotal role of regularization techniques in machine learning tikhonov1963regularization ; hastie2009elements . We demonstrate that employing regularized machine learning models can effectively address the limitations inherent in supervised machine learning, especially for problems that appear in string theory and, more broadly, for problems at the intersection of mathematics and physics. While the primary objective of regularization in machine learning is to prevent overfitting, certain versions of it can be employed to eliminate model parameters, echoing the spirit of regularization in quantum field theory.

By focusing on Least Absolute Shrinkage and Selection Operator (Lasso) regularization tibshirani1996regression for polynomial and logarithmic regression models, we identify several candidate formulas for the minimum volume of Sasaki-Einstein 5-manifolds corresponding to toric Calabi-Yau 3-folds. The discovered formulas depend either on 3 or 6 parameters that come from features of the corresponding toric diagrams fulton ; 1997hep.th…11013L – convex lattice polygons on $\mathbb{Z}^{2}$ that characterize uniquely the associated toric Calabi-Yau 3-fold. Compared to the extremely large number of parameters in the regression and CNN models used in our previous work in Krefl:2017yox , the formulas obtained in this study are both presentable, interpretable, and most importantly reusable for the computation of the minimum volume for toric Calabi-Yau 3-folds.

In this work, we concentrate on non-compact toric Calabi-Yau 3-folds $\mathcal{X}$. These geometries can be considered as cones over Sasaki-Einstein 5-manifolds $Y_{5}$ Maldacena:1997re ; Morrison:1998cs ; Acharya:1998db ; Martelli:2004wu ; Benvenuti:2004dy ; Benvenuti:2005ja ; Butti:2005sw . The toric Calabi-Yau 3-folds are fully characterized by convex lattice polygons $\Delta$ on $\mathbb{Z}^{2}$ known as toric diagrams fulton ; 1997hep.th…11013L . The associated Calabi-Yau singularities can be probed by D3-branes whose worldvolume theories form a class of $4d$ $\mathcal{N}=1$ supersymmetric gauge theories Greene:1996cy ; Douglas:1997de ; Witten:1998qj ; Klebanov:1998hh ; Douglas:1996sw ; Lawrence:1998ja ; Feng:2000mi ; Feng:2001xr .

This class of $4d$ $\mathcal{N}=1$ supersymmetric gauge theories can be represented in terms of a T-dual Type IIB brane configuration known as a brane tiling Franco:2005rj ; Hanany:2005ve ; Franco:2005sm . Table 1 summarizes the Type IIB brane configuration. Brane tilings can be illustrated in terms of bipartite graphs on a 2-torus $T^{2}$ 2003math…..10326K ; kasteleyn1967graph and encapsulate both the field theory information and the information about the associated toric Calabi-Yau geometry. Figure 1 shows an example of a brane tiling and its associated toric Calabi-Yau 3-fold, which is in this case the cone over the zeroth Hirzebruch surface $F_{0}$ hirzebruch1968singularities ; brieskorn1966beispiele ; Morrison:1998cs ; Feng:2000mi . The mesonic moduli spaces Witten:1993yc ; Benvenuti:2006qr ; Feng:2007ur ; Butti:2007jv formed by the mesonic gauge invariant operators of these $4d$ $\mathcal{N}=1$ supersymmetric gauge theories with $U(1)$ gauge groups is precisely the associated toric Calabi-Yau 3-folds. When all the gauge groups of the $4d$ $\mathcal{N}=1$ supersymmetric gauge theory are $U(N)$, then the mesonic moduli space is given by the $N$-th symmetric product of the toric Calabi-Yau 3-fold.

The gravity dual of the $4d$ worldvolume theories is Type IIB string theory on $AdS_{5}\times Y_{5}$, where $Y_{5}$ is the Sasaki-Einstein 5-manifold that forms the base of the associated toric Calabi-Yau 3-fold Maldacena:1997re ; Morrison:1998cs ; Acharya:1998db ; Martelli:2004wu ; Benvenuti:2004dy ; Benvenuti:2005ja ; Butti:2005sw . These $4d$ $\mathcal{N}=1$ supersymmetric gauge theories are known to flow at low energies to a superconformal fixed point. Under a procedure known as $a$-maximization Intriligator:2003jj ; Butti:2005vn ; Butti:2005ps , the superconformal $R$-charges of the $4d$ theory are determined. This procedure, involves the maximization of the trial $a$-charge, which takes the form

The maximization procedure gives the value of the central charge of the superconformal field theory at the conformal fixed point.

Under the AdS/CFT correspondence Maldacena:1997re ; Morrison:1998cs ; Acharya:1998db , the central charge is directly related to the minimized volume of the corresponding Sasaki-Einstein 5-manifold $Y_{5}$ Gubser:1998vd ; Henningson:1998gx . We have,

where the R-charges $R$ and as a result the volume function $V(R;Y_{5})$ can be expressed in terms of Reeb vector components $b_{i}$ of the corresponding Sasaki-Einstein 5-manifold Martelli:2006yb ; Martelli:2005tp . We can reverse the statement saying that computing the minimum volume,

is equivalent to obtaining the maximum value of the central charge $a(R;Y_{5})$. This correspondence is true for all $4d$ theories living on a stack of $N$ D3-branes probing toric Calabi-Yau 3-folds and has been checked extensively in various examples Intriligator:2003jj ; Butti:2005vn ; Butti:2005ps .

In this work, we will focus on the toric Calabi-Yau 3-folds and the corresponding Sasaki-Einstein 5-manifold $Y_{5}$, with particular emphasis on the minimum volume $V_{min}$ of the Sasaki-Einstein 5-manifolds $Y_{5}$. Building on the pioneering work of Krefl:2017yox , this work proposes the use of more advanced machine learning techniques. In particular, we introduce machine learning regularization by using the Least Absolute Shrinkage and Selection Operator (Lasso) tibshirani1996regression in order to identify an explicit formula for the minimum volume $V_{min}$ for Sasaki-Einstein 5-manifolds $Y_{5}$. We expect to be able to write the minimum volume formula in terms of features obtained from the toric diagram of the corresponding toric Calabi-Yau 3-folds. The use of machine learning regularization allows us to eliminate parameters, reducing the necessary parameters for the volume formula to a manageable amount that is interpretable, presentable and reusable.

Before discussing these machine learning techniques, let us first review in the following section the computation of the volume functions for toric Calabi-Yau 3-folds using Hilbert series.

Given $\mathcal{X}$ as a cone over a projective variety $X$, where $X$ is realized as an affine variety in $\mathbb{C}$, the Hilbert series Benvenuti:2006qr ; Feng:2007ur is the generating function for the dimension of the graded pieces of the coordinate ring

where $f_{i}$ are the defining polynomials of $X$. Accordingly, the Hilbert series takes the general form

For $4d$ $\mathcal{N}=1$ supersymmetric gauge theories given by brane tilings Franco:2005rj ; Hanany:2005ve ; Franco:2005sm , we have an associated toric Calabi-Yau 3-fold $\mathcal{X}$, which becomes the mesonic moduli space Witten:1993yc ; Benvenuti:2006qr ; Feng:2007ur ; Butti:2007jv of the $4d$ $\mathcal{N}=1$ supersymmetric gauge theory when the gauge groups are all $U(1)$. The corresponding Hilbert series is the generating function of mesonic gauge invariant operators that form the mesonic moduli space. For the purpose of the remaining discussion, we will consider the $4d$ $\mathcal{N}=1$ supersymmetric gauge theories given by brane tilings as abelian theories with $U(1)$ gauge groups.

Following the forward algorithm for brane tilings Feng:2000mi , we can use GLSM fields Witten:1993yc given by perfect matchings $p_{\alpha}$ Hanany:2005ve ; Franco:2005rj of the brane tilings in order to express the mesonic moduli space of the abelian $4d$ $\mathcal{N}=1$ supersymmetric gauge theory as the following symplectic quotient,

where ${\text{Irr}}\mathcal{F}^{\flat}$ is the largest irreducible component, also known as the coherent component, of the master space $\mathcal{F}^{\flat}$ Hanany:2010zz ; Forcella:2008bb ; Forcella:2008eh of the $4d$ $\mathcal{N}=1$ supersymmetric gauge theory. The master space is the spectrum of the coordinate ring generated by the chiral fields encoded in $p_{\alpha}$ and quotiented by the F-term relations encoded in $Q_{F}$. In (III.6), $Q_{F}$ is the $F$-term charge matrix summarizing the $U(1)$ charges originating from the $F$-terms, and $Q_{D}$ is the $D$-term charge matrix which summarizes the $U(1)$ gauge charges on perfect matchings $p_{\alpha}$.

Following the symplectic quotient description of the mesonic moduli space in (III.6), the Hilbert series can be obtained by solving the Molien integral Pouliot:1998yv ,

where $c$ is the number of perfect matchings in the brane tiling and $Q_{t}=(Q_{F},Q_{D})$ is the total charge matrix.

Martelli:2006yb ; Martelli:2005tp showed that the same Hilbert series can be obtained directly from the toric diagram $\Delta$ of the toric Calabi-Yau 3-fold $\mathcal{X}$. Given that the toric diagram $\Delta$ is a convex lattice polygon on $\mathbb{Z}^{2}$ with an ideal triangulation $\mathcal{T}(\Delta)$ into unit sub-triangles $\Delta_{i}\in\mathcal{T}(\Delta)$, the Hilbert series of the corresponding toric Calabi-Yau 3-fold $\mathcal{X}$ can be written as

where $i=1,\dots,r$ is the index for the $r$ unit triangles $\Delta_{i}\in\mathcal{T}(\Delta)$, and $j=1,2,3$ is the index for the $3$ boundary edges of each unit triangle $\Delta_{i}$. For each boundary edge $e_{j}\in\Delta_{i}$, we have a $3$-dimensional outer normal vector $\mathbf{u}_{i,j}$ whose components are assigned the following product of fugacities,

where $\mathbf{u}_{i,j}(a)$ indicates the $a$-th component of $\mathbf{u}_{i,j}$. We note that $\mathbf{u}_{i,j}$ is a $3$-dimensional vector because the defining vertices of $\Delta$ and $\Delta_{i}$ are all on a plane at height $z=1$ such that their coordinates are of the form $(x,y,1)$. As a result, the vectors $\mathbf{u}_{i,j}$ corresponding to edge $e_{j}\in\Delta_{j}$ are normal to the 3-dimensional surface given by the vectors connecting the origin $(0,0,0)$ to the two bounding vertices of $e_{j}\in\Delta_{j}$.

It is important to note that the fugacities $t_{1},t_{2},t_{3}$ in (III.9) relate to the components of normal vectors $\mathbf{u}_{i,j}$, and therefore depend on the triangulation and the particular instance in a given $GL(2,\mathbb{Z})$ toric orbit of a toric diagram on the $z=1$ plane. In comparison, the fugacities $y_{\alpha}$ in (III.7) refer to the GLSM fields $p_{\alpha}$ given by perfect matchings of the corresponding brane tiling. Since perfect matchings can be mapped directly to chiral fields in the $4d$ $\mathcal{N}=1$ supersymmetric gauge theory, the fugacities $y_{\alpha}$ in (III.7) can be mapped to fugacities counting global symmetry charges carried by chiral fields in the $4d$ theory. Because both Hilbert series from (III.7) and (III.8) refer to the same toric Calabi-Yau 3-fold $\mathcal{X}$, there exists a fugacity map between $y_{\alpha}$ and $t_{1},t_{2},t_{3}$ that identifies the two Hilbert series with each other.

For the rest of the discussion, let us consider Hilbert series for toric Calabi-Yau 3-folds $\mathcal{X}$ that are in terms of fugacities $t_{1},t_{2},t_{3}$ corresponding to coordinates of the normal vectors $\mathbf{u}_{i,j}\in\mathbb{Z}^{3}$ of the toric diagram $\Delta$. Given the Hilbert series $g(t_{i};\mathcal{X})$, we can obtain the volume function Martelli:2006yb ; Martelli:2005tp of the Sasaki-Einstein 5-manifold $Y_{5}$ using,

where $b_{i}$ are the Reeb vector components with $i=1,\dots 3$. We note that the Reeb vector ${\bf b}=(b_{1},b_{2},b_{3})$ is always in the interior of the toric diagram $\Delta$ and can be chosen such that one of its components is set to

for toric Calabi-Yau 3-folds $\mathcal{X}$. We further note that the limit in (III.10) takes the leading order in $\mu$ in the expansion for $g(t_{i}=\exp[-\mu b_{i}];\mathcal{X})$, which is shown to refer to the volume of the Sasaki-Einstein base $Y_{5}$ in Martelli:2006yb ; Martelli:2005tp .

Let us consider in the following paragraph an example of the computation of the volume function in terms of Reeb vector components $b_{i}$ for the Sasaki-Einstein base of the cone over the zeroth Hirzebruch surface $F_{0}$ hirzebruch1968singularities ; brieskorn1966beispiele ; Morrison:1998cs ; Feng:2000mi .

Using the outer normal vectors $\mathbf{u}_{i,j}$ for each of the four unit sub-triangles $\Delta_{i}$ of the toric diagram for $F_{0}$ in Figure 2(b), we can use (III.8) to write down the Hilbert series,

Using the limit in (III.10), we can derive the volume function of the Sasaki-Einstein base directly from the Hilbert series as follows,

where $b_{3}=3$. When we find the global minimum of the volume function $V(b_{i};F_{0})$, we obtain

up to 5 decimal points, which occurs at critical Reeb vector components $b_{1}^{*}=b_{2}^{*}=0$. In the remainder of this work, we will maintain a precision level of 5 decimal points for all numerical measurements.

The aim of this work is to identify an expression for the minimum volume $V_{min}$ of Sasaki-Einstein 5-manifolds $Y_{5}$ in terms of parameters that we know from the corresponding toric Calabi-Yau 3-folds $\mathcal{X}$. We refer to these parameters as features, denoted as $x_{a}$, of the toric Calabi-Yau 3-fold $\mathcal{X}$.

Assuming that we have $N_{x}$ features $x_{a}$ for a given toric Calabi-Yau 3-fold, the proposal in Krefl:2017yox states that we can write down a candidate linear function for the inverse minimum volume in terms of these features as follows,

where $\beta_{0}$ and $\beta_{a}$ are real coefficients, and $j$ labels the particular toric Calabi-Yau 3-fold $\mathcal{X}^{j}$ with its corresponding toric diagram $\Delta^{j}\in\mathbb{Z}^{2}$.

Let us refer to the inverse of the actual minimum volume obtained by volume minimization as $1/V_{min}^{j}\equiv y^{j}$ for a given toric Calabi-Yau 3-fold $\mathcal{X}^{j}$. If for a set $S$ of $N=|S|$ toric Calabi-Yau 3-folds $\mathcal{X}^{j}$, we know the actual minimum volumes $V_{min}^{j}$ via volume minimization, then we can calculate the following residual sum of squares of the difference between the inverses of the actual and the expected minimum volumes for the entire set $S$,

Here, $\mathcal{L}$ can be considered as a loss function goodfellow2016deep that evaluates the performance of the candidate function for the minimum volume in (IV.15). In multiple linear regression gauss1823theoria ; fisher1922mathematical ; mendenhall2003second ; freedman2009statistical ; jobson2012applied , as initially proposed in Krefl:2017yox , the optimization task is to minimize the loss function in (IV) for a given dataset $S$ of toric Calabi-Yau 3-folds,

In Krefl:2017yox , multiple linear regression was used to obtain a candidate minimum volume function using the following feature set,

where

corresponding respectively to the number of internal lattice points in $\Delta^{j}$, the number of boundary lattice points in $\Delta^{j}$, and the number of vertices that form the extremal corner points in $\Delta^{j}$, for a given toric Calabi-Yau 3-fold $\mathcal{X}^{j}$. Under Pick’s theorem pick1899geometrisches , these features are related as follows,

where $A$ is the area of the toric diagram $\Delta$, with the area of the smallest unit triangle in $\mathbb{Z}^{2}$ having $A=1/2$.

With a dataset $S$ of $N=15,147$ toric Calabi-Yau 3-folds, the work in Krefl:2017yox showed that the candidate linear function in (IV.15) with features given by (IV.18) is able to estimate the inverse minimum volume with an expected percentage relative error of 2.2%. In this work, we expand upon the accomplishments of Krefl:2017yox by introducing novel features that describe toric Calabi-Yau 3-folds, augmenting the datasets for toric Calabi-Yau 3-folds, and applying machine learning techniques incorporating regularization. These improvements are designed to address some of the shortcomings of the work in Krefl:2017yox as well as give explicit interpretable formulas for the minimum volume for toric Calabi-Yau 3-folds.

Using the $n$-enlarged toric diagram $\Delta_{n}^{j}$ for a given toric Calabi-Yau 3-fold $\mathcal{X}^{j}$, we can now refer to the area of $\Delta_{n}$ as $A_{n}$, the number of internal lattice points of $\Delta_{n}$ as $I_{n}$, and the number of boundary lattice points in $\Delta_{n}$ as $E_{n}$. We further note that the number of vertices $V_{n}$ corresponding to extremal corner points in $\Delta_{n}$ is the same for $V$ in $\Delta$ for all $n$, i.e. $V_{n}=V$.

In our work, we use features of a toric Calabi-Yau 3-fold $\mathcal{X}^{j}$ that are composed from members of the following set,

where $n=1,\dots,7$. These are defined through the corresponding toric diagram $\Delta^{j}$ and its corresponding $n$-enlarged toric diagram $\Delta_{n}^{j}$. Through the application of machine learning regularization, our objective is to differentiate between features that contribute to the expression for the minimum volume associated with a toric Calabi-Yau 3-fold and those that do not.

The Least Absolute Shrinkage and Selection Operator (Lasso) tibshirani1996regression is a machine learning regularization technique primarily employed to prevent overfitting in supervised machine learning. However, it can also be utilized for feature selection. In our work, the overarching goal in employing Lasso is to introduce a machine learning model capable of delivering optimal predictions for the minimum volume for toric Calabi-Yau 3-folds while using the fewest features from the training dataset. For problems such as the one considered in this work, it is quintessential to be able to obtain formulas with a small number of parameters. As a result, using Lasso is particularly suited for discovering new mathematical formulas such as the one aimed for in this work for the minimum volume for toric Calabi-Yau 3-folds.

In the following section, we give a brief overview of several regularization schemes including Lasso in the context of supervised machine learning for the minimum volume formula for toric Calabi-Yau 3-folds.

Let us review the following three regularization schemes:

• •

  L1 Regularization (Lasso). This regularization scheme also known as Least Absolute Shrinkage and Selection Operator (Lasso) tibshirani1996regression adds the following linear regularization term to the loss function of the regression model,

  $\displaystyle\Delta\mathcal{L}_{\text{L1}}=\alpha\sum_{a=1}^{N_{x}}|\beta_{a}|~{},~{}$

  (V.27)

  where $\beta_{a}$ are the real parameters of the regression model. $\alpha$ is a real regularization parameter. Increasing the value of $\alpha$ has the effect of increasing the strength of the L1 regularization.

• •

  L2 Regularization (Ridge). Another regularization scheme is known as Ridge regularization or L2 regularization hoerl1970ridge . It adds the following quadratic regularization term to the loss function of the regression model,

  $\displaystyle\Delta\mathcal{L}_{\text{L2}}=\alpha\sum_{a=1}^{N_{x}}\beta_{a}^{2}~{},~{}$

  (V.28)

  where $\beta_{a}$ are the real parameters of the regression model and $\alpha$ is again the real regularization parameter.

• •

  Elastic Net (L1 and L2). Elastic Net zou2005regularization is a combination of L1 (Lasso) and L2 (Ridge) regularization and adds the following regularization terms to the loss function,

  $\displaystyle\Delta\mathcal{L}_{\text{L1,L2}}=\alpha_{1}\sum_{a=1}^{N_{x}}|\beta_{a}|+\alpha_{2}\sum_{a=1}^{N_{x}}\beta_{a}^{2}~{},~{}$

  (V.29)

  where $\alpha_{1}$ and $\alpha_{2}$ are relative real regularization parameters that regulate the proportion of L1 regularization and L2 regularization in this regularization scheme.

Amongst these regularization schemes in supervised machine learning, we are going to mainly focus on Lasso and L1 regularization for the remainder of this work. While all three regularization schemes share the common goal of constraining the range of values for the model parameters $\beta_{a}$, it is noteworthy that only Lasso possesses the unique property of inducing sparsity among the model parameters, resulting in the complete elimination of certain parameters during the training process.

There are several arguments why Lasso enables the complete elimination of some of the model parameters and the corresponding features in the candidate function for the minimum volume $V_{min}$ for toric Calabi-Yau 3-folds. In order to illustrate this, let us consider the case with $N_{x}=2$ features $x_{1}^{j}$ and $x_{2}^{j}$, for which the L1 and L2 regularization terms take respectively the following form,

If we assume that under optimization, the regularization terms reach a value $\Delta\mathcal{L}_{\text{L1}}=\epsilon$ and $\Delta\mathcal{L}_{\text{L2}}=\epsilon$ for $\alpha>0$ and $\epsilon\in\mathbb{R}$, we can draw the parametric plots for the two regularization terms as shown in Figure 6 hastie2009elements . We can see from the plots in Figure 6 that for L1 regularization, the minimum of the total loss function is more likely achieved when one of the two parameters $\beta_{1}$ or $\beta_{2}$ approaches 0. This is in part due to the absolute values taken for the parameters in the linear L1 regularization term.

As a result, Lasso regularization is particularly suited for feature selection and parameter elimination in regression models. In our work, we employ L1 regularization to derive a formula for the minimum volume $V_{min}$ of Sasaki-Einstein 5-manifolds corresponding to toric Calabi-Yau 3-folds that is interpretable, presentable and reusable.

In this work, our aim is to apply Lasso regularization in order to identify explicit formulas for the minimum volume for toric Calabi-Yau 3-folds. By doing so, our aim is to maximize the accuracy of the formulas that we find while minimizing the number of parameters the formulas depend on, making them interpretable and readily presentable.

We recall that the optimization problem for the L1-regularized regression model is to minimize the loss function $\mathcal{L}+\Delta\mathcal{L}_{\text{L1}}$ with the L1 regularization term. As we discussed in the sections above, this optimization problem focuses on minimizing the mean squared error with a penalty for non-zero coefficients $\beta_{a}(\alpha)$, which depends on the regularization parameter $\alpha$.

Here, we note that there is an additional optimization problem regarding the maximization of the $R^{2}$-score in (VI.31) and the minimization of the number $N_{\beta_{a}(\alpha)}$ of non-zero coefficients $\beta_{a}(\alpha)$. We can formulate this additional optimization problem as follows,

where $0<N_{\beta_{a}(\alpha)}\leq N_{x}$, and the values of the coefficients $\beta_{a}(\alpha)$ and the $R^{2}(\alpha)$-score all depend on the regularization parameter $\alpha$. $\lambda$ is a positive hyperparameter that regulates how much we value sparsity of feature coefficients $\beta_{a}(\alpha)$ over the accuracy of the estimate given by $R^{2}(\alpha)$.

Figure 7 shows respectively for datasets $S_{\text{1a}}$ and $S_{\text{2a}}$ plots for the L1 regularization parameter $\alpha$ for polynomial regression against standardized coefficients $\overline{\beta}_{a}(\alpha)$, against the number of non-zero coefficients $N_{\beta_{a}(\alpha)}$, and against the $R^{2}$-score. Here, the standardized coefficients $\overline{\beta}_{a}(\alpha)$ are obtained when the training is conducted over normalized features $\overline{x}_{a}$. When the training is completed for a specific value of $\alpha$, the candidate formula for the minimum volume given by $y=1/V_{min}$ is obtained by reversing the normalization on the features, giving us the coefficients $\beta_{a}(\alpha)$ of the candidate formula. We also have Figure 8 which shows respectively for datasets $S_{\text{1a}}$ and $S_{\text{2a}}$ plots for the L1 regularization parameter $\alpha$ for logarithmic regression against the standardized coefficients $\overline{\beta}_{a}(\alpha)$, the number of non-zero coefficients $N_{\beta_{a}(\alpha)}$ and the $R^{2}$-score. Similar plots can also be obtained for datasets $S_{\text{1b}}$ and $S_{\text{2b}}$ for both L1-regularized polynomial regression and L1-regularized logarithmic regression.

Overall, the plots illustrate that the identified optimal regularization parameters $\alpha^{*}$ minimize the number of non-zero coefficients $N_{\beta_{a}(\alpha)}$ in the formula estimating the minimum volume given by $y=1/V_{min}$, as well as maximize the accuracy of the formulas measured by the $R^{2}$-score. Table 3 and Table 4 summarize respectively the most optimal candidate formulas for the minimum volume given by $y=1/V_{min}$ under L1-regularized polynomial regression and L1-regularized logarithmic regression for the four datasets in Table 2, with the corresponding optimal regularization parameters $\alpha^{*}$, the corresponding number of non-zero coefficients $N_{\beta_{a}(\alpha)}$ and the $R^{2}$-score.

A closer look reveals that for all models, the identified optimal regularization parameters $\alpha^{*}$ results in formulas that approximate the minimum volume $y=1/V_{min}$ extremely well for all the datasets $S_{\text{1a}}$, $S_{\text{1b}}$, $S_{\text{2a}}$ and $S_{\text{2b}}$. Overall, the L1-regularized logarithmic regression models seem to give more accurate results than the L1-regularized polynomial regression models with $N_{\beta_{a}(\alpha)}\leq 6$ over all datasets. In particular, L1-regularized logarithmic regression models trained on datasets $S_{\text{2a}}$ and $S_{\text{2b}}$ have $R^{2}$-scores above $0.99$, which is exceptionally high.

Having a closer look at explicit examples of toric Calabi-Yau 3-folds in the datasets reveals however that the performances of the regularized regression models can vary between different toric Calabi-Yau 3-folds. For example, focusing on the L1-regularized logarithmic regression models trained on $S_{\text{1a}}$ and $S_{\text{1b}}$, we observe that the minimum volumes given by $1/\hat{y}_{\text{1a}}^{\text{LR}}$ and $1/\hat{y}_{\text{1b}}^{\text{LR}}$ in Table 4 perform differently for toric diagrams with smaller areas $A$ compared to toric diagrams with larger areas $A$ as illustrated in Figure 9. Similar observations can be made for the L1-regularized logarithmic regression models trained on $S_{\text{2a}}$ and $S_{\text{2b}}$ as well as the L1-regularized polynomial regression models.

In summary, we can calculate the expected relative percentage errors $E[\epsilon]$ of the predicted minimum volumes given by $1/\hat{y}$ and the corresponding standard deviations $\sigma[\epsilon]$ for the L1-regularized logarithmic regression models as follows,

We note that the models trained on $S_{\text{2a}}$ and $S_{\text{2b}}$ have a larger expected relative percentage error than the ones trained on $S_{\text{1a}}$ and $S_{\text{1b}}$. This is partly due to the fact that $S_{\text{2a}}$ and $S_{\text{2b}}$ contain randomly selected toric diagrams in a $30\times 30$ lattice box in $\mathbb{Z}^{2}$ and $r=15$ circle, respectively, whereas $S_{\text{1a}}$ and $S_{\text{1b}}$ contain the full set of toric diagrams in a $5\times 5$ lattice box in $\mathbb{Z}^{2}$ and $r=3.5$ circle, respectively, as defined in Table 2.

We also note that the $R^{2}$-scores of the L1-regularized logarithmic regression models in Table 4,

are overall very high and close to $1$. Compared to the expected relative percentage errors in (VI), which measure how far off predictions of the minimum volume given by $1/\hat{y}$ are, the $R^{2}$-score is a measure of the accuracy of the trained regression model. It quantifies the proportion of the variation in $y=1/V_{min}$ that can be predicted using the features selected from the corresponding toric diagrams of the toric Calabi-Yau 3-folds.

With this work, we demonstrated that employing regularization in machine learning models can effectively address the limitations posed by supervised machine learning techniques applied to problems that occur in the context of string theory. In particular, we have shown that the minimum volume $V_{min}$ for Sasaki-Einstein 5-manifolds corresponding to toric Calabi-Yau 3-folds can be expressed by just 3 features of the associated toric diagrams $\Delta$ with an $R^{2}$-score $\geq 0.98$. These 3 features are the area $A$ of $\Delta$, the number of vertices $V$ in $\Delta$, and the number of internal points in the factor $n=3$ enlarged toric diagram $\Delta_{3}$.

The simultaneous maximization of the $R^{2}$-score and the minimization of the number surviving parameters in the candidate function for $y=1/V_{min}$ by varying the regularization strength given by the regularization parameter $\alpha$, the proposed regularized regression models in this work give far more presentable, interpretable and explainable results than our previous work in Krefl:2017yox . Above all, as suggested in Figure 9, the candidate formulas for the minimum volumes of toric Calabi-Yau 3-folds obtained in this study are concise enough to facilitate the examination of why some toric Calabi-Yau 3-folds are associated with minimum volumes that are more challenging to predict than those of certain other toric Calabi-Yau 3-folds. We plan to report on these investigations in the near future. We foresee that the application of regularization schemes to other supervised machine learning applications in string theory will open up equally promising research opportunities in the future.