# Learning to extrapolate using continued fractions: Predicting the critical
temperature of superconductor materials

Pablo Moscato
College of Engineering, Science and Environment
The University of Newcastle
University Drive, Callaghan, NSW 2308, Australia
<EMAIL_ADDRESS>
https://www.newcastle.edu.au/profile/pablo-moscato
& Mohammad N. Haque
College of Engineering, Science and Environment
The University of Newcastle
University Drive, Callaghan, NSW 2308, Australia
<EMAIL_ADDRESS>
& Kevin Hunag
California Institute of Technology
1200 E California Blvd, Pasadena, CA 91125, US
<EMAIL_ADDRESS>
& Julia Sloan
California Institute of Technology
1200 E California Blvd, Pasadena, CA 91125, US
<EMAIL_ADDRESS>
& Jon Corrales de Oliveira
California Institute of Technology
1200 E California Blvd, Pasadena, CA 91125, US
<EMAIL_ADDRESS>

###### Abstract

In Artificial Intelligence we often seek to identify an unknown target
function of many variables $y=f(\mathbf{x})$ giving a limited set of instances
$S=\\{(\mathbf{x^{(i)}},y^{(i)})\\}$ with $\mathbf{x^{(i)}}\in D$ where $D$ is
a domain of interest. We refer to $S$ as the training set and the final quest
is to identify the mathematical model that approximates this target function
for new $\mathbf{x}$; with the set $T=\\{\mathbf{x^{(j)}}\\}\subset D$ with
$T\neq S$ (i.e. thus testing the model generalisation). However, for some
applications, the main interest is approximating well the unknown function on
a larger domain $D^{\prime}$ that contains $D$. In cases involving the design
of new structures, for instance, we may be interested in maximizing $f$; thus,
the model derived from $S$ alone should also generalize well in $D^{\prime}$
for samples with values of $y$ larger than the largest observed in $S$. In
that sense, the AI system would provide important information that could guide
the design process, e.g., using the learned model as a surrogate function to
design new lab experiments.

We introduce a method for multivariate regression based on iterative fitting
of a continued fraction by incorporating additive spline models. We compared
it with established methods such as AdaBoost, Kernel Ridge, Linear Regression,
Lasso Lars, Linear Support Vector Regression, Multi-Layer Perceptrons, Random
Forests, Stochastic Gradient Descent and XGBoost. We tested the performance on
the important problem of predicting the critical temperature of
superconductors based on physical-chemical characteristics.

### Impact Statement

When the samples used to test the performance are drawn only from the 10
percent of the samples (i.e., those with the highest values of the critical
temperature, greater or equal to 89 Kelvin), the new method called Spline
Continued Fraction Regression (Spln-CFR), is the top-ranked; the results also
show that Spln-CFR, is competitive with the top-performing established
techniques like XGBoost and Random Forest. However, trained with samples with
all critical temperatures < 89 Kelvin, Spln-CFR correctly reported that 108
materials had temperatures greater or equal to 89 Kelvin, followed by Linear
Support Vector Regression and multilayer perceptrons with 34 and 21. Due to
its simplicity and time efficiency (nearly the same as of Random Forests), the
approach deserves further study and extensions (e.g., with Bagging and
Boosting variants) particularly due to its already proven capacity to
approximate well outside of the range of values observed for the target value
in the training set samples.

_K_ eywords Regression $\cdot$ Continued Fractions $\cdot$ Superconducting
materials $\cdot$ Superconductivity.

##  1 Introduction

Superconductors are materials with the unique ability to conduct electrical
current with zero resistance. This incredibly useful characteristic opens up a
variety of applications for these substances. Among these usages, Magnetic
Resonance Imaging (MRI) systems are used worldwide to produce detailed images
of internal organs and tissues as a vital medical tool. As our energy demands
continue to rise with the increasing prevalence of renewable energy, solar
cars, and more, another indispensable usage could be the efficient forms of
energy transfer.

Superconductors conducting current with zero resistance greatly reduce the
amount of energy wasted as it is simply moved from one place to another.
However, a major drawback of today’s superconductors is that they are unable
to conduct current with zero resistance - their main appeal - unless cooled to
their critical temperatures ($T_{c}$). These temperatures are extremely cold
(often around -196°C), and are unique for each superconducting material [1].
Because of the importance and variation of these temperatures, predicting the
$T_{c}$ for superconductors has become a problem of great interest in the
world of material science.

Here, we use various machine learning tools and introduce a new method based
on multivariate continued fractions to build mathematical models that predict
the critical temperature for superconductors only using information hidden in
the superconductor’s characterization of the chemical structure. The ability
to accurately predict $T_{c}$ for superconductors will allow us to more easily
use them to our advantage, opening up an electric new world of possibility.

##  2 Background

###  2.1 Continued Fraction Regression

In 2019, a new approach for multivariate regression using continued fractions
was introduced in [2] and compared with a state of the art genetic programming
method for regression. A year later, this technique’s results on 354 datasets
from the physico chemical sciences were presented in [3] and compared with
some of the state-of-the-art top 10 regression techniques. The new method was
the top ranked performer in the training set in 352 out of the 354, and it was
the also first in terms of generalisation in 192, more than half of the total
of times of all other 10 methods combined. The figure of merit was the Mean
Squared Error. We named this known approach as ‘Continued Fraction
regression’, or CFR. The best existing algorithm currently utilizes a memetic
algorithm for optimizing the coefficients of a model that approximates a
target function as the convergent of a continued fraction [2, 3, 4]. Memetic
Algorithms are well-established research areas in the field of Evolutionary
Computation and the IEEE had established a Task Force in Computational
Intelligence for their study. Therefore, it is important to refer the readers
to some of the latest references and reviews on the field [5, 6, 7, 8, 9].

Some basic introduction on analytic continued fraction approximation is
perhaps necessary. A continued fraction for a real value $\alpha$ is of the
following form (1) and may be finite or infinite [10], according to $\alpha$
being a rational number or not, respectively.

$\alpha=a_{0}+\cfrac{b_{1}}{a_{1}+\cfrac{b_{2}}{a_{2}+\ldots}}$ (1)

Euler’s proved a mathematical formula that allows us to write a sum of
products as a continued fraction(2):

$\beta=a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\ldots+a_{0}a_{1}a_{2}\dots a_{n}\\\
=\cfrac{a_{0}}{1-\cfrac{a_{1}}{1+a_{1}-\cfrac{a_{2}}{1+a_{2}-\cfrac{\ddots}{\ddots\frac{a_{n-1}}{1+a_{n-1}-\frac{a_{n}}{1+a_{n}}}}}}}.$
(2)

This simple yet powerful equation reveals how infinite series can be written
as infinite continued fractions, meaning that continued fractions can be a
good general technique to approximate analytic functions thanks to the
improved optimization methods such as those provided by memetic algorithms
[4]. Indeed, CFR has already demonstrated to be an effective regression
technique on the real-world benchmark provided by the Penn Machine Learning
Database [4].

In this paper, we will use Carl Friedrich Gauss’ mathematical notation for
generalized continued fractions [11] (i.e. a compact notation where “K” stands
for the German word “Kettenbruch” which means ‘Continued Fraction’). Using
this notation, we may write the continued fraction in (1) as:

$\alpha=a_{0}+_{i=1}^{\infty}\frac{b_{i}}{a_{i}},$ (3)

thus the problem of finding an approximation of an unknown target function of
$n$ variables $\mathbf{x}$ given a training dataset of $m$ samples
$S=\\{(\mathbf{x^{(i)}},y^{(i)})\\}$ is that of finding the set of functions
$F=\\{a_{0}(x)...,b_{1}(x),...\\}$ such that a certain objective function is
minimized; i.e. we aim to find

$f(\mathbf{x})=a_{0}(\mathbf{x})+_{i=1}^{\infty}\frac{b_{i}(\mathbf{x})}{a_{i}(\mathbf{x})}.$
(4)

##  3 Methodology

###  3.1 A new approach: Continued Fractions with Splines

In previous contributions [2, 3, 4], a memetic algorithm was always employed
to find optimal continued fraction solutions. Here, we present another method
to fit continued fraction representations by iteratively fitting splines.

Splines are a regression technique that involves fitting piecewise polynomial
functions to the given data [12]. The domain is partitioned into intervals at
locations known as “knots”. Then, a polynomial model of degree $n$ is
separately fitted for each interval, generally enforcing boundary conditions
including continuity of the function as well as the continuity of the first
$(n\text{-}1)$-order derivatives at each of the knots. Splines can be
represented as a linear combination of basis functions, of which the standard
is the B-spline basis. Thus, fitting a spline model is equivalent to fitting a
linear model of basis functions. We refer to Hastie et al. [13] for the
particular definition of the B-spline basis.

First, when all the functions $b_{i}(\mathbf{x})=1$, for all $i$, we have a
simple continued fraction representation, and we can write it as:

$f(\mathbf{x})=g_{0}(\mathbf{x})+\cfrac{1}{g_{1}(\mathbf{x})+\cfrac{1}{g_{2}(\mathbf{x})+\cfrac{1}{g_{3}(\mathbf{x})+...}}}.$
(5)

Note that for a term $g_{i}(\mathbf{x})$, we say that it is at ”depth” $i$.

Finding the best values for the coefficients in the set of functions
$\\{g_{i}(\mathbf{x})\\}$, can be addressed as a non-linear optimization
problem as in [2, 3, 4]. However, despite the great performance of that
approach, we aim to introduce a faster variant that can scale well to larger
datasets such as this one.

Towards that end, and thinking about the scalability, we fit the model
iteratively by depth as follows: we first consider only the first term,
$g_{0}(\mathbf{x})$ (at depth 0), ignoring all other terms. We fit a model for
the first term using predictors $\mathbf{x}$ and the target $f(x)$. Next, we
consider only the first and second depths, with the terms $g_{0}(\mathbf{x})$
and $g_{1}(\mathbf{x})$, ignoring the rest. We then fit $g_{1}(\mathbf{x})$
using the previously fit model for $g_{0}(\mathbf{x})$. For example,
truncating the expansion at depth 1, we have that

$g_{1}(\mathbf{x})=\frac{1}{f(\mathbf{x})-g_{0}(\mathbf{x})}.$ (6)

Thus, we fit $g_{1}(\mathbf{x})$ using the predictors $\mathbf{x}$ and the
target $(f(\mathbf{x})-g_{0}(\mathbf{x}))^{-1}$. We label this target as
$y^{(1)}$. We repeat this process, fitting a new model by truncating at the
next depth by using the models fit from previous depths and iterations.

We have that at depth $i>0$, the target $y^{(i)}$ for the model
$g_{i}(\mathbf{x})$ is $(\epsilon_{i-1})^{-1}(\mathbf{x})$, where
$\epsilon_{i-1}(\mathbf{x})$ is the residual of the previous depth’s model,
$y^{(i-1)}-g_{i-1}(\mathbf{x})$.

One notable characteristic of this approach is that if any model
$g_{i}(\mathbf{x})$, $i>0$ evaluates to 0, then we will have a pole in the
continued fraction, which is often spurious. To remedy this, we modify the
structure of the fraction such that each fitted $g_{i}(\mathbf{x})$, $i>0$ is
encouraged to be strictly positive on the domain of the training data. To do
this, we add a constant $C_{i}$ to $\epsilon_{i}$ when calculating the target
$y^{(i+1)}$, where $C_{i}=|\min_{x}\epsilon_{i}|$. Thus, the targets $y^{(i)}$
for $i>0$ are all non-negative, encouraging each $g_{i}(\mathbf{x})$, $i>0$,
to be strictly positive. For example, for $g_{1}(\mathbf{x})$, we would have
that the target $y^{(1)}=(f(\mathbf{x})-g_{0}(\mathbf{x})+C_{1})^{-1}$. Of
course, we must then subtract $C_{i}$ from $g_{i-1}(\mathbf{x})$ in the final
continued fraction model.

We have found that data normalization often results in a better fit using this
approach. It is sufficient to simply divide the targets uniformly by a
constant when training and multiply by the same constant for prediction. We
denote this constant parameter norm.

A good choice of the regression model for each $g_{i}(\mathbf{x})$ is a spline
since they are well-established. For reasons stated in the next section, the
exception is the first term $g_{0}(\mathbf{x})$, which is a linear model. We
use an additive model to work with multivariate data where each term is a
spline along a dimension. That is, given $m$ predictor variables, we have that

$g_{i}(\mathbf{x})=\sum_{j=1}^{m}f_{j}(x_{j})$ (7)

for each term $g_{i}(\mathbf{x})$, $i>0$, where each function $f_{j}$ is a
cubic spline along variable $j$. That is, $f_{j}$ is a piecewise polynomial of
degree 3 and is a function of variable $j$.

We implement the splines with a penalized cubic B-spline basis. That is,
$f_{j}(\mathbf{x})=\sum_{i=1}^{k}\beta_{k}B_{k}(x_{j})$, where each $B_{i}(x)$
is one of $k$ cubic B-spline basis functions along dimension $j$ and
corresponds to one of $k$ knots. We use the following loss function
$L\left(\mathbf{B}\left(\mathbf{x},\mathbf{y},\boldsymbol{\beta}\right)\right)$,
i.e.

$L\left(\mathbf{B}\left(\mathbf{x},\mathbf{y},\boldsymbol{\beta}\right)\right)=\|\mathbf{y}-\mathbf{B}\boldsymbol{\beta}\|^{2}+\lambda\sum_{j=0}^{m}\boldsymbol{\beta}^{T}\mathbf{P_{j}}\boldsymbol{\beta}$
(8)

where $\mathbf{B}$ is the matrix of cubic B-spline basis functions for all
variables, $\boldsymbol{\beta}$ is the vector of all of the weights, and
$\mathbf{P_{j}}$ is the associated second derivative smoothing penalty matrix
for the basis for the spline $f_{j}$. This is standard for spline models [13].
The pseudocode for this approach is shown in Algorithm 1.

Input: Training data
$\mathcal{D}=\\{(\mathbf{x_{1}},f(\mathbf{x_{1}}),...,(\mathbf{x_{n}},f(\mathbf{x_{n}}))\\}$
and parameters $\lambda$, $k$, norm, and max_depth

/* Let $n$ be the number of samples; */

/* $m$ be the number of variables */

/* Let $\mathbf{X}\in\mathbb{R}^{n\times m}$ be the data matrix and
$\mathbf{y}\in\mathbb{R}^{n}$ be the vector of targets. */

1 $\texttt{knot\\_indices}=\\{\\}$

2$\mathbf{y^{(0)}}\leftarrow\mathbf{y}/\texttt{norm}$

3for $i\leftarrow$ 0, 1, …, max_depth do

4 if $i=0$ then

/* $g_{0}$ is a linear model parameterized by $\mathbf{\beta}$, and is fit
with least squares. */

5

6
$\mathbf{\beta}\leftarrow\text{argmin}_{\mathbf{\beta}}\displaystyle\|\mathbf{y^{(0)}}-\mathbf{X}\mathbf{\beta}\|^{2}$

7 else

/* Let $g_{i}$ be an additive spline model as given in equation (7),
parameterized by $\mathbf{\beta}$. For each predictor variable, the knots are
at the samples indexed by the first $k$ indices in knot_indices */

8

9 for $j\leftarrow$ 1, 2, …, $m$ do

10 $f_{j}\leftarrow$ new SplineModel()

11 for each index $p$ in knot_indices do

12 $f_{j}\leftarrow$ AssignKnotAt($\mathbf{X}[p][j]$)

13 end for

14

15 end for

16 $g_{i}=\sum_{j=1}^{m}f_{j}(x_{j})$

/* Construct the splines, and then fit with regularized least squares */

17

18 $\mathbf{B}\leftarrow$ BSplineBasisMatrix($g_{i}$.knots)

19 $\mathbf{P}_{j}\leftarrow$ BSplinePenaltyMatrix($f_{j}$: for each $f_{j}$
in $g_{i}$)

20
$\mathbf{\beta}\leftarrow\text{argmin}_{\mathbf{\beta}}\displaystyle\|\mathbf{y^{(i)}}-\mathbf{B}\mathbf{\beta}\|^{2}+\lambda\sum_{j=1}^{m}\mathbf{\beta}^{T}\mathbf{P}_{j}\mathbf{\beta}$

21 end if

22

/* Compute $\mathbf{\epsilon_{i}}$, the vector of residuals of the $i$th
model, and then compute the targets and knot locations for the next depth. */

23 $\mathbf{\epsilon}_{i}\leftarrow\mathbf{y^{(i)}}-g_{i}(\mathbf{X})$

24 $C_{i}\leftarrow|\min_{x}\mathbf{\epsilon}_{i}|$

25 $\mathbf{y^{(i+1)}}\leftarrow(\mathbf{\epsilon_{i}}+C_{i})^{-1}$

26 $\texttt{knot\\_indices}\leftarrow$ SelectKnots($\mathbf{\epsilon_{i}}$)

27 end for

28

The estimate for $f(\mathbf{\mathbf{x}})$ is,
$f^{(\texttt{max\\_depth})}(\mathbf{x})$
$=\texttt{norm}\;\cdot\left[g_{0}(\mathbf{x})-C_{0}+_{i=1}^{\texttt{max\\_depth}}\frac{1}{g_{i}(\mathbf{x})-C_{i}}\right]$

Algorithm 1 Iterative CFR using additive spline models with adaptive knot
selection

###  3.2 Adaptive knot selection

(a) depth 3 (b) depth 5

(c) depth 10 (d) depth 15

Figure 1: Examples of the fit obtained by the Spline Continued Fraction using
a dataset generated thanks to the gamma function with added noise. We present
several continued fractions with depth of 3 (a), 5 (b), 10 (c), and 15 (d). In
this example, the number of knots $k$ was chosen to be 3, $norm$ = 1, and
$\lambda$ = 0.1

.

The iterative method of fitting continued fractions also allows for an
adaptive method of selecting knot placements for the additive spline models.
For the spline model $g_{i}(\mathbf{x})$ at depth $i>0$, we use all of the
knots of the spline model $g_{i-1}(\mathbf{x})$ at depth $i-1$. Then, for each
variable, we place $k$ new knots at the unique locations of the $k$ samples
with the highest absolute error from the model $g_{i-1}(\mathbf{x})$ at depth
$i-1$. As the points with the highest error can be likely to be very close to
each other, we impose the condition that we take the samples with the highest
error, but they must have alternating signs.

That is, for $g_{i}(\mathbf{x})$, $i>0$, we select $k$ knots, with the first
knot at the location of the sample with the highest absolute error computed
from the model $g_{i-1}(\mathbf{x})$. For the rest of the knots, the
$j^{\text{th}}$ knot is selected at the sample’s location with the next
highest absolute error than the sample used for the $(j-1)^{\text{th}}$ knot.
Nevertheless, only if the sign of the (non-absolute) error of that sample is
different from the sign of the (non-absolute) error of the sample used for the
$(j-1)^{\text{th}}$ knot. Otherwise, we move on to the next highest absolute
error sample, and so on, until we fulfill this condition. This knot selection
procedure is shown in Algorithm 2. Note that we let $g_{0}$ be a linear model
as there is no previous model to obtain the knot locations from.

Input: $\epsilon_{i}$

/* Given the vector of residuals $\mathbf{\epsilon_{i}}$ of the spline model
at depth $i$, select the knot placements for the next spline model at depth
$i+1$ */

1

/* Sort by indices of highest absolute error */

2

3$\texttt{abs\\_error}\leftarrow$
elementWiseAbsoluteValue($\mathbf{\epsilon_{i}}$)

4$\texttt{highest\\_error\\_indices}\leftarrow$ argsortDecreasing(abs_error)

/* Take the top $k$ highest order indices, such that each error term has
opposite sign of the last */

5

6$\texttt{current\\_sign}\leftarrow null$

7$\texttt{knots\\_added}\leftarrow 0$

8for each i in highest_error_indices do

9 if $\texttt{knots\\_added}\geq k$ then

10 break

11 end if

12 if sign($\epsilon_{i}$[i] $\neq\texttt{current\\_sign}$) then

13

14 $\texttt{current\\_sign}\leftarrow$ sign($\epsilon_{i}$)

15 knot_indices.append($\epsilon_{i}$[i])

16 $\texttt{knots\\_added}\leftarrow\texttt{knots\\_added}+1$

17 end if

18

19 end for

20return knot_indices

Algorithm 2 SelectKnots (Adaptive Knot Selection)

The goal of using additive spline models with the continued fraction is to
take advantage of the continued fraction representation’s demonstrated ability
to approximate general functions (see the discussion on the relationship with
Padé approximants in [4]). The fraction’s hierarchical structure allows for
the automatic introduction of variable interactions, which is not included in
the additive models individually that constitute the fraction. The iterative
approach to fitting allows for a better algorithm for knot selection.

An example of this algorithm modeling the well-known gamma function (with
standard normally distributed noise added) is demonstrated in Fig. 1d. Here,
we showed how the fitting to gamma is affected by different values of depths
(3, 5, 10, 15) in Spline Continued Fraction. As desired, it is evident from
the figure that Spline Continued Fraction with more depth fits better with the
data.

##  4 Experimental Design

###  4.1 Data and Study Method

We used the superconductivity dataset, also used by Hamidieh [1], from the UCI
Machine Learning
repository111https://archive.ics.uci.edu/ml/datasets/Superconductivty+Data.
The website contains two files. In this work, we have only used the train.csv
file, which contains information of 21263 superconductors along with the
critical temperature and a total of 81 attributes for each of them.

We conducted two main studies to see the generalization capabilities of many
regression algorithms. We denote them as the Out-of-Sample and Out-of-Domain,
respectively. For the Out-of-Sample study, the data is randomly partitioned
into 2/3rds training data and 1/3rd test data. Each model was fit on the
training data, and the RMSE is calculated on the separated test portion of the
data.

For the Out-of-Domain study, the data was partitioned such that the training
samples are always extracted from the set of samples with the lowest 90% of
critical temperatures. For the test set, the samples come from the highest 10%
of critical temperatures. It turned out the lowest 90% have critical
temperatures < 89 K, whereas the highest 10% have temperatures greater or
equals to 89 K that range from 89 K to 185 K (we highlight that the range of
variation of the test set is more than the one of the training set making the
generalization a challenging task). For each of the 100 repeated runs of Out-
of-Domain test, we have randomly taken 1/2 of the training set (from lowest
90% of the observed value) to train the models and the same ratio from the
test data (from 10% of the highest actual value) to estimate the model
performance. This said the Out-of-Domain study allows us to see the capacity
of several regression models in “predicting” on a set of materials that have
higher critical temperatures, meaning that generalization, in this case, is
strictly connected with the extrapolation capacity of the fitted models. We
executed both the Out-of-Sample and Out-of-Domain tests for 100 times to help
us validate our conclusions with statistical results.

The Spline Continued Fraction model had a depth of 5, five knots per depth, a
normalization constant of $1000$, and a regularization parameter $\lambda$ of
0.5. These parameters resulted from one-dimensional non-linear model fitting
to problems like the gamma function with noise (already discussed in Fig. 1d)
and others such as fitting the function $f(x)=\sin(x)/x$. The parameters were
selected empirically using these datasets, and no problem-specific tuning on
the superconductivity datasets was conducted.

The final model was then iteratively produced by beginning at a depth of 1 and
increasing the depth by one until the error was greater than the one observed
for a previous depth (which we considered as a proxy for overfitting the
data).

To evaluate the performance of the Spline Continued Fraction (Spln-CFR)
introduced in this paper with other state-of-the-art regression methods, we
used a set of 11 regressors form two popular Python libraries (XGBoost [14]
and Scikit-learn machine learning library [15]). The name of the regression
methods are listed as follows:

  * •

AdaBoost (ada-b)

  * •

Gradient Boosting (grad-b)

  * •

Kernel Ridge (krnl-r)

  * •

Lasso Lars (lasso-l)

  * •

Linear Regression (l-regr)

  * •

Linear SVR (l-svr)

  * •

MLP Regressor (mlp)

  * •

Random Forest (rf)

  * •

Stochastic Gradient Descent (sgd-r)

  * •

XGBoost (xg-b)

The XGBoost code is available as an open-source
package222https://github.com/dmlc/xgboost. The parameters of the XGBoost model
were the same as used in Hamidieh (2018) [1]. We kept the parameters of other
machine learning algorithms the same as Scikit defaults.

All executions of the experiments were performed on an Intel®
Core${}^{\text{TM}}$ i7-9750H hex-core based computer with hyperthreading and
16GB of memory. The machine was running on Windows 10 operating system. We
used Python v3.7 to implement the Spline Continued Fraction using pyGAM [16]
package. All experiments were executed under the same Python runtime and
computing environment.

###  4.2 Results

Table 1: Results form the 100 runs of the proposed Spline Continued Fraction and ten regression methods all trained on the dataset, with median of Root Mean Squared Error (RMSE) and standard deviation as the uncertainty of error.  Regressor | Median RMSE Score $\pm Std$
---|---
Out-of-Sample | Out-of-Domain
Spln-CFR | 10.989 $\pm$ 0.382 | 36.327 $\pm$ 1.187
xg-b | 9.474 $\pm$ 0.190 | 37.264 $\pm$ 0.947
rf | 9.670 $\pm$ 0.197 | 38.074 $\pm$ 0.751
grad-b | 12.659 $\pm$ 0.178 | 39.609 $\pm$ 0.619
l-regr | 17.618 $\pm$ 0.187 | 41.265 $\pm$ 0.466
krnl-r | 17.635 $\pm$ 0.163 | 41.427 $\pm$ 0.464
mlp | 19.797 $\pm$ 5.140 | 41.480 $\pm$ 9.640
ada-b | 18.901 $\pm$ 0.686 | 47.502 $\pm$ 0.743
l-svr | 26.065 $\pm$ 7.838 | 47.985 $\pm$ 1.734
lasso-l | 34.234 $\pm$ 0.267 | 74.724 $\pm$ 0.376
sgd-ra | N.R. | N.R.
aThe Stochastic Gradient Descent Regressor (sgd-r), without parameter
estimation, predicted unreasonable high values and the predicted error measure
is extreme. Hence, we are not reporting (N.R.) the performance of sgd-r and
have omitted it form further analysis.

Table 1 presents the results of the regression methods along-with with those
of the Spline Continued Fraction approach for both of Out-of-Sample and Out-
of-Domain studies. The median RMSE value obtained from 100 runs is taken as
the Out-of-Sample RMSE estimate.

For each of the 100 repeated runs of Out-of-Domain test, we estimate the model
performance via the Out-of-Domain RMSE score. The median RMSE score obtained
from this test performance is reported in Table 1 as Out-of-Domain RMSE. We
also report on some other descriptive statistics like, for instance, the
number of times that the regressor correctly predicted a material to have a
critical temperature greater or equal to 89 K.

(a) Heatmap for Out-of-Sample test (b) Critical Diagram plot for Out-of-Sample
test

Figure 2: Statistical Comparison of the regressors for the Out-of-Sample test.
a) Heatmap showing the significance levels of $p$-values obtained by the
Friedman Post-hoc Test and b) Critical difference (CD) plot showing the
statistical significance of rankings achieved by the regression methods.

(a) Heatmap for Out-of-Domain test (b) Critical Diagram plot for Out-of-Domain
test

Figure 3: Statistical Comparison of the regressors for the Out-of-Domain Test.
a) Heatmap showing the significance levels of $p$-values obtained by the
Friedman Post-hoc Test and b) Critical difference (CD) plot showing the
statistical significance of rankings achieved by the regression methods.

###  4.3 Out-of-Sample Test

For the Out-of-Sample testing, XGBoost achieved the lowest error (median RMSE
score of 9.47) among the 11 regression methods. The three closest regression
methods to XGBoost are Random Forest (median RMSE of 9.67), Spline Continued
Fraction (median RMSE of 10.99) and Gradient Boosting (median RMSE of 12.66).
The Stochastic Gradient Descent, without parameter estimation, performed the
worst among all regression methods used in the experiment and due to the
unreasonable high error observed in the runs we have omitted it from further
analysis.

####  4.3.1 Statistical Significance Testing on the Results obtained for Out-
of-Sample test

To evaluate the significance in results obtained by different regression
methods for Out-of-Sample, we applied a Friedman test for repeated measure
[17] for the 100 runs. Here, we computed the ranking of the methods for each
of the runs based on the RMSE score obtained in the test distribution of the
Out-of-Sample settings. It will help us determine if the experiment’s
techniques are consistent in terms of their performance. The statistical test
found $p$-value = $1.9899\text{\times}{10}^{-183}$ which “rejected” the null
hypothesis “all the algorithms perform the same” and we proceeded with the
post-hoc test.

We applied Friedman’s post-hoc test on the ranking of 10 regressors computed
for the test RMSE scores obtained for 100 runs of Out-of-Sample test. In Fig.
2 (a) the $p$-values obtained for the test are plotted as a heatmap. It is
noticeable that there exist ‘no significant differences’ (NS) in performances
of Spline Continued Fraction (Spnl-CFR) with: rf and grad-b.

Additionally, we generated the Critical Difference (CD) diagram proposed in
[18] to visualize the differences among the regressors for their median
ranking. The CD plot used Nyemeni post-hoc test and placed the regressors on
the $x$-axis of their median ranking. It then computes the critical difference
of rankings between them and connects those which are closer than the critical
difference with a horizontal line denoting them as statistically ‘non-
significant’.

We plot the CD graph, in Fig. 2 (b), using the implementation from Orange data
mining toolbox [19] in Python. The Critical Difference (CD) is found to be
$1.25$. We can see that the xg-b ranked 1 among the regressors with ‘no
significant difference’ with 2 ranked rf. The median ranking of the proposed
Spline Continued Fraction is ranked 3 with ‘no significant differences’ in the
performance rankings of rf and grad-b.

###  4.4 Out-of-Domain Test

For the task of Out-of-Domain prediction, the Spline Continued Fraction
regressor exhibited the best performance (median RMSE score of 36.3) among all
regression methods used in the experiment (in Table 1). Three closest
regressors to the proposed Spline Continued Fraction method are XGBoost
(median RMSE=37.3), Random Forest (median RMSE=38.1) and Gradient Boosting
(median RMSE=39.6).

####  4.4.1 Statistical Significance Testing on the Results obtained for Out-
of-Domain test

To test the significance of the results obtained by different regression
methods for Out-of-Domain test, we employed the same statistical test used for
Out-of-Sample (in Sec. 4.3.1). The test returned a $p$-value =
$1.2065\text{\times}{10}^{-156}$ which “rejected” the null hypothesis and we
proceeded with the post-hoc test.

The $p$-values obtained for the post-hoc test are plotted as a heatmap in Fig.
3 (a) for Out-of-Domain test. It is noticeable that there exist ‘no
significant differences’ (NS) in performances of Spline Continued Fraction
(Spnl-CFR) with Random Forest (rf) and XGBoost (xg-b). There is also no
significance difference in performance ranking of Linear Regression (l-regr)
with mlp, l-svr, krnl-r and grad-b.

We plot the Critical Difference (CD) graph, in Fig. 3 (b), for Out-of-Domain
test. The Critical Difference (CD) is $1.3898$. From the critical difference
plot; it is evident that the top three methods in Out-of-Domain prediction are
Spline Continued Fraction, XGBoost and Random Forest. We can see that the
average ranking of Spln-CFR is very close to 2, which is the best-ranking
performance among the 10 regressors. There is no significant difference of
Spline Continued Fraction with the 2 best ranked method, XGBoost (xg-b with
average ranking is between 2 and 3), in Out-of-Domain predictions.

####  4.4.2 Runtime Required by the methods for out-of-domain test

Figure 4: Run-time (in seconds) required for model building and predicting by
the regressors for 100 runs of the Out-of-Domain test, where samples with the
lowest 90% of critical temperatures were drawn to be the training data, with
an equal number of samples constitute the test data (but these were withdrawn
from the top 10% highest critical temperatures).

Fig. 4 shows the running time required by each of the regression methods (in
s) for the 100 runs of Out-of-Domain test. We can see that the Linear
Regression (50 percentile runtime of 0.02 s and maximum runtime 0.158 s) and
Lasso lars (50 percentile 0.013 s and maximum of 0.027 s) required lowest
running times. XGBoost (xg-b) required the most amount of CPU time (50
percentile runtime of 55.33 s and maximum 79.05 s). On the other hand, Random
Forest and the proposed Spline Continued Fraction Regression required nearly
similar running time (50 percentile runtime of 36.88 s and 41.65 s for rf and
Spln-CFR, respectively) for the Out-of-Domain test.

##  5 Discussion

(a) Linear Regression (b) XGBoost (c) Spline Continued Fraction

Figure 5: Out-of-Sample Test results showing Predicted vs actual temperatures
for entire data with regression models trained on the training data. a)
Results replicate Linear Regression outcome from Hamidieh, b) XGBoost and b)
Spline Continued Fraction model.

To illustrate on the performance of models in the Out-of-Sample study, we
employed Linear regression, XGBoost and Spline Continued Fraction on the
training set and plotted the prediction vs actual temperatures for the entire
dataset (in Fig. 5). We show that we were able to reproduce the result of the
Out-of-Sample test from Hamidieh [1] Fig. 5 (a), with RMSE of 17.7. The Out-
of-Sample model for Spline Continued Fraction and XGBoost model are used to
predict the critical temperature for the entire dataset. Together, the figures
show that Spln-CFR performed better in modelling Out-of-Sample critical
temperatures than that of Linear Regression, particularly for the larger
temperatures.

(a) Linear Regression (b) XGBoost (c) Spline Continued Fraction

Figure 6: Out-of-Domain Test results showing Predicted vs actual temperatures
of the samples for the highest 10% critical temperatures, where a model is
fitted using the samples with the lowest 90% critical temperatures. We have
shown the $x$-axis values up to 145 K which only left an extreme value (185 K)
out of the visual area. Results of Out-of-Domain test for a) Linear Regression
with RMSE of 41.3 b) XGBoost with RMSE of 36.3 and c) Spline Continued
Fraction model’s with RMSE of 34.8.

Fig. 6 shows actual vs predicted critical temperature for the Out-of-Domain
test for Linear Regression, XGBoost and Spline Continued Fraction models. We
recall that in Out-of-Domain settings, we trained each of the models with the
samples from the bottom 90% of the observed temperature (which is < 89 K). We
measured the samples’ testing performance with the top 10% of the observed
critical temperatures (containing 2126 samples in the test set).

Table 2: Predicted vs. Actual critical temperatures for the materials with the
top 20 predicted temperatures in the Out-of-Domain study, i.e. the one in
which the lowest 90% of critical temperature samples were used for drawing the
training data. The average values of the critical temperatures ($\bar{x}$),
the average relative error ($\bar{\eta}$), and the Root mean squared error
(RMSE) of these materials for the top 20 predictions (which are not
necessarily the same since they depend on the models) are shown in the last
rows.

| Spln-CFR | xg-b | rf | grad-b | mlp | l-regr | l-svr | krnl-r | ada-b | lasso-l
---|---|---|---|---|---|---|---|---|---|---
| y | pred | y | pred | y | pred | y | pred | y | pred | y | pred | y | pred | y | pred | y | pred | y | pred
| 92.00 | 114.14 | 89.20 | 89.64 | 91.19 | 87.89 | 89.50 | 83.44 | 109.00 | 100.81 | 98.00 | 91.59 | 112.00 | 94.81 | 98.00 | 91.02 | 89.50 | 58.63 | 89.00 | 27.06
| 90.00 | 109.69 | 94.20 | 89.19 | 89.90 | 87.88 | 89.90 | 83.44 | 124.90 | 100.31 | 112.00 | 89.14 | 100.00 | 93.49 | 112.00 | 88.67 | 89.50 | 58.63 | 89.00 | 27.06
| 111.00 | 108.54 | 89.88 | 88.69 | 90.00 | 87.88 | 90.50 | 83.44 | 114.00 | 99.70 | 105.00 | 87.53 | 132.60 | 93.49 | 105.00 | 86.84 | 89.70 | 58.63 | 89.00 | 27.06
| 93.50 | 108.01 | 89.93 | 88.34 | 90.20 | 87.88 | 91.50 | 83.44 | 128.40 | 99.59 | 117.00 | 87.06 | 105.00 | 92.94 | 117.00 | 86.65 | 89.80 | 58.63 | 89.00 | 27.06
| 99.00 | 106.50 | 90.00 | 88.15 | 90.90 | 87.88 | 90.00 | 83.42 | 127.40 | 99.53 | 100.00 | 85.92 | 115.00 | 92.93 | 100.00 | 85.88 | 89.80 | 58.63 | 89.00 | 27.06
| 105.60 | 105.01 | 90.10 | 88.15 | 91.00 | 87.88 | 91.80 | 83.42 | 127.80 | 99.53 | 132.60 | 85.92 | 111.00 | 92.90 | 132.60 | 85.88 | 89.90 | 58.63 | 89.00 | 27.06
| 113.00 | 104.35 | 91.00 | 88.15 | 92.00 | 87.88 | 90.00 | 82.22 | 130.10 | 98.76 | 115.00 | 85.50 | 110.00 | 92.84 | 115.00 | 85.51 | 90.00 | 58.63 | 89.00 | 27.06
| 113.00 | 103.95 | 91.30 | 88.15 | 92.20 | 87.88 | 89.50 | 79.29 | 128.50 | 98.55 | 111.00 | 84.97 | 106.70 | 92.54 | 111.00 | 84.46 | 90.00 | 58.63 | 89.00 | 27.06
| 106.60 | 103.95 | 96.10 | 88.15 | 92.40 | 87.88 | 90.00 | 79.29 | 128.40 | 98.45 | 132.00 | 84.96 | 126.90 | 91.73 | 132.00 | 84.42 | 90.50 | 58.63 | 89.00 | 27.06
| 128.70 | 103.92 | 90.00 | 88.10 | 92.50 | 87.88 | 91.00 | 79.29 | 128.80 | 98.45 | 110.00 | 84.31 | 117.00 | 91.73 | 110.00 | 84.38 | 91.50 | 58.63 | 89.00 | 27.06
| 91.80 | 102.10 | 91.40 | 88.10 | 92.74 | 87.88 | 91.80 | 79.29 | 131.40 | 98.33 | 106.70 | 83.95 | 126.80 | 91.30 | 106.70 | 82.97 | 100.00 | 58.63 | 89.00 | 27.06
| 108.00 | 101.56 | 92.60 | 87.82 | 92.80 | 87.88 | 92.30 | 79.29 | 128.80 | 98.10 | 126.90 | 82.72 | 115.00 | 90.84 | 95.00 | 82.64 | 108.00 | 58.63 | 89.00 | 27.06
| 92.00 | 101.32 | 91.60 | 87.53 | 93.00 | 87.88 | 90.00 | 78.85 | 128.70 | 93.96 | 105.00 | 82.63 | 95.00 | 90.80 | 105.00 | 82.01 | 110.00 | 58.63 | 89.00 | 27.06
| 90.00 | 101.19 | 93.00 | 87.53 | 93.00 | 87.88 | 91.60 | 78.85 | 130.30 | 93.94 | 95.00 | 82.62 | 121.60 | 90.80 | 107.00 | 81.88 | 110.90 | 58.63 | 89.00 | 27.06
| 105.10 | 100.50 | 93.80 | 87.49 | 93.05 | 87.88 | 89.10 | 78.79 | 131.30 | 93.93 | 107.00 | 82.47 | 100.00 | 90.78 | 126.90 | 81.82 | 114.00 | 58.63 | 89.00 | 27.06
| 130.30 | 100.35 | 89.90 | 87.48 | 93.20 | 87.88 | 89.20 | 78.79 | 122.00 | 91.96 | 105.00 | 82.41 | 107.00 | 90.78 | 105.00 | 81.51 | 114.00 | 58.63 | 89.00 | 27.06
| 93.00 | 100.24 | 90.00 | 87.48 | 93.40 | 87.88 | 89.40 | 78.79 | 123.50 | 91.64 | 126.80 | 82.12 | 90.00 | 90.63 | 90.00 | 81.40 | 116.00 | 58.63 | 89.10 | 27.06
| 91.50 | 100.00 | 90.20 | 87.48 | 93.50 | 87.88 | 89.40 | 78.79 | 121.00 | 90.69 | 98.50 | 82.03 | 96.00 | 90.49 | 126.80 | 81.24 | 122.50 | 58.63 | 89.10 | 27.06
| 91.50 | 99.18 | 90.90 | 87.48 | 91.80 | 87.75 | 89.40 | 78.79 | 115.00 | 90.14 | 112.00 | 82.03 | 128.70 | 90.48 | 117.00 | 80.89 | 127.00 | 58.63 | 89.10 | 27.06
| 116.00 | 98.39 | 91.00 | 87.48 | 92.10 | 87.69 | 89.50 | 78.79 | 110.00 | 90.01 | 117.00 | 81.83 | 130.30 | 90.26 | 121.60 | 80.87 | 130.90 | 58.63 | 89.10 | 27.06
$\bar{x}$ | 103.08 | 103.64 | 91.31 | 88.03 | 92.044 | 87.86 | 90.27 | 80.49 | 124.47 | 96.32 | 111.63 | 84.59 | 112.33 | 91.83 | 111.68 | 84.05 | 102.68 | 58.63 | 89.02 | 27.06
$\bar{\eta}$ | 0.1085 | 0.036 | 0.0453 | 0.1083 | 0.224 | 0.2351 | 0.1733 | 0.2389 | 0.4187 | 0.696
RMSE | 13.6023 | 3.7753 | 4.3261 | 10.0078 | 28.9783 | 29.3265 | 23.9282 | 30.2426 | 46.2473 | 61.96

Another set of observed results are interesting for discussion and might be
relevant for future research directions. In Table 2, we report the top 20
predicted vs, actual ($y$) temperatures for all ten regression methods for
Out-of-Domain test of a single run. The last row of the table shows the
average of the corresponding (actual) critical temperature for the materials
with the highest 20 predicted values by each of the models. Interestingly,
XGBoost’s top 20 predictions of the critical temperatures are all below 90 K
(in the range of 87.48 to 89.64 K). Similarly, Random Forest’s top 20
predictions are in the range of 87.69 to 87.89 K. The top 20 predicted
critical temperatures by the Linear Regression are in the range of 81.83 to
91.59 K. In contrast, the top 20 predicted critical temperature by Spline
Continued Fraction varies from 98.39 to 114.14 K, which, in comparison, has
the highest starting and ending values among all regressors. We also reported
the average temperature ($\bar{x}$), average relative errors ($\bar{\eta}$)
and RMSE score computed for the top 20 predictions. XGBoost has showed the
lowest value for both $\bar{\eta}$ (0.036) and RMSE ($3.775$) among 10
regressors. In terms of those scores, the proposed Spln-CFR is 4 position.
However, if we look at the average of predictions, Spln-CFR has the highest
average prediction temperatures for the top 20 predictions in Out-of-Domain
tests.

Since all the actual critical temperatures of the test set in Out-of-Domain
settings are $\geq 89$ K, it is relevant to evaluate for how many of these
samples each regression method was able to predict above that value. Here, we
considered the predicted value as P = critical temperature value $\geq 89$ K
(denoted as ‘P’, for positive) and N = critical temperature value < 89 K
(denoted as ‘N’, for negative). In Table 3, we reported the number of samples
for which each of the methods predicted a temperature value in the P and N
category for the whole testing set of Out-of-Domain test. It is found that
only six regression methods predicted the critical temperature being $\geq 89$
K for at least one sample. Both Linear Regression and XGBoost predicted two
sample’s temperature with critical temperature $\geq 89$ K. Kernel Ridge
predicted only one sample’s value within that range. MLP Regressor and Linear
SVR predicted it for 21 and 34 samples, respectively. The proposed Spline
Continued Fraction predicted 108 sample’s value $\geq 89$ K, which is the best
among all regression methods used in the experiments.

Table 3: Number of times the methods predicted a critical temperature value $T_{c}\geq 89$ K (denoted as ‘P’, for positive) and $T_{c}<89$ K (denoted as ‘N’ for Negative) for Out-of-Domain test. Regressor | Out-of-domain predicted critical temperature, $T_{c}$
---|---
P ($T_{c}$ $\geq 89$ K) | N ($T_{c}$ < 89 K)
Spln-CFR | 108 | 2018
xg-b | 2 | 2124
rf | 0 | 2126
grad-b | 0 | 2126
mlp | 21 | 2105
l-regr | 2 | 2124
l-svr | 34 | 2092
krnl-r | 1 | 2125
ada-b | 0 | 2126
lasso-l | 0 | 2126
Table 4: Inter-rater agreement between the pairs of regressor methods where the resulting models were able to predict at least one positive temperature value ($T_{c}\geq 89$ K). Rater 1 | Rater 2 | Value of Kappa ($\kappa$) | Level of Agreement
---|---|---|---
Spln-CFR | xg-b | -0.001851 | No Agreement
Spln-CFR | mlp | 0.030476 | None to Slight
Spln-CFR | l-regr | 0.016365 | None to Slight
Spln-CFR | l-svr | 0.104988 | None to Slight
Spln-CFR | krnl-r | -0.000933 | No Agreement
xg-b | mlp | -0.001721 | No Agreement
xg-b | l-regr | -0.000942 | No Agreement
xg-b | l-svr | -0.001780 | No Agreement
xg-b | krnl-r | -0.000628 | No Agreement
mlp | l-regr | -0.001721 | No Agreement
mlp | l-svr | 0.208516 | Fair
mlp | krnl-r | -0.000899 | No Agreement
l-regr | l-svr | 0.053874 | None to Slight
l-regr | krnl-r | 0.666457 | Substantial
l-svr | krnl-r | -0.000915 | No Agreement

We look at the consensus between regression methods in Out-of-Domain
prediction. Only five regressors (Spln-CFR, xg-b, mlp, l-regr, l-svr and
krnl-r) which were able to predict at least one positive value (critical
temperature $\geq 89$ K). We computed pairwise inter-rater agreement
statistics, Cohen’s kappa [20], for those regression methods. We tabulated the
value of Kappa($\kappa$) ordered by highest to lowest and outlined the level
of agreement in Table 4. We can see that most of the cases, there is either
“No” (9 cases) or “None to Slight” (4 cases) agreement exists between the
pairs of regressors. We witness such behaviour in the agreement between the
pairs formed with Spln-CFR and each of the other five methods. MLP Regressor
and Linear SVR have “Fair” agreement in the predictions. We witnessed the
highest value $\kappa=0.67$ for Linear Regression and Kernel Ridge, which
yields a “Substantial” agreement.

###  5.1 Extrapolation Capability of the Regressors in General

As all of the results presented in this work are for a special case of finding
models for the extrapolation of the critical temperature of superconductors,
we included more robust experimental outcomes with a set of six datasets used
in [2]. This additional test will help us to evaluate the extrapolation
capabilities of the regressors in other problem domains.

Jerome Friedman proposed a Multivariate Adaptive Regression Splines (MARS)
algorithm in [21] which aggregates multiple linear regression models
throughout the range of target values. We used the implementation of the MARS
algorithm from py-earth Python library 333https://contrib.scikit-learn.org/py-
earth/content.html#multivariate-adaptive-regression-splines. We included a
comparison of MARS with Spln-CFR and other regressors for extrapolation
capability.

Here, the samples from each of the datasets were sorted based on the target
value. Then we split it into the out-of-domain setting by taking samples with
lower 90% target values as train and higher 10% target value as a test. We
uniformly at random took half of the samples from the out-of-domain train to
build the model and the same ratio from the out-of-domain test sets for
prediction for each of the 100 independent runs. We applied min-max
normalization on the train set and used the same distribution to normalize the
test set.

Table 5: Number of times the prediction by a method in the range of out-of-domain threshold in test sets for 100 repeated runs on six datasets form [2] Regressor | in Range | Regressor | in Range | Regressor | in Range
---|---|---|---|---|---
Spln-CFR | 13560 | grad-b | 1227 | ada-b | 0
MARS | 3716 | mlp | 1158 | lasso-l | 0
l-regr | 2594 | xg-b | 826 | rf | 0
l-svr | 2045 | krnl-r | 735 | sgd-r | 0
Figure 7: Critical difference (CD) plot showing the statistical significance
of rankings achieved by the regression methods for 100 runs on the six
datasets form [2].

We have analyzed their performance statistically (in Fig. 7) and found that
MARS has a median ranking of 5, and statistically significantly different from
only krnl-r, sgd-r and lasso-l. However, the proposed Spln-CFR has achieved
the first rank among all the methods with a median ranking between two to
three. The predictions by each model are de-normalized to count the number of
predictions above the threshold (the maximum target value in training portion
of data) in out-of-domain settings. We show the complete outcome in Table 5.
These counts show that the Spln-CFR has the highest number of predictions
(13560) followed by MARS (3716) and l-regr (2594) are in the range. These
results demonstrate the strength of the regressors for their extrapolation
capability.

##  6 Conclusion and Future Work

We give a brief summary of some of the results observed on this new technique:

  * •

For the Out-of-Sample study, the median RMSE obtained for 100 independent
runs, the proposed Spln-CFR is in the top three methods (in Table 1).

  * •

For the statistical test of Out-of-Sample rankings, Spln-CFR is statistically
similar to the 2 ranked method (Random Forest) in Fig. 2 (b).

  * •

For Out-of-Domain median RMSE obtained for 100 runs, the proposed Spln-CFR is
the top method (ranked 1 in Table 1).

  * •

For the statistical test of Out-of-Domain rankings, in Fig. 3 (b), Spln-CFR is
the best method (median ranking is close to 2) and statistically similar to
the second best regressor, XGBoost (with a median ranking between 2 and 3).

  * •

Spln-CFR correctly predicted that 108 unique materials have critical
temperature values that are greater than or equal to 89 K in Out-of-Domain
test (close to twice the number of all other regression methods tested
combined which was 60) (Table 3).

Table 2 also reveals interesting characteristics of all methods that deserve
further consideration as an area of research. First, note that the 20 top
materials for each of the methods are not necessarily the same, although some
intersection obviously may exist. In the Out-of-Domain study, the top 20
predicted critical temperature values by Spln-CFR were all above 98.9 K (with
18 being above 100 K). The average RMSE critical temperature on this set
(103.64 K) is nearly the same as the one predicted (103.08 K). The RMSE of
xg-b, however, is nearly three times smaller, but the method’s top predictions
are materials with relatively smaller values (average of 91.31 K). We
observed, for the collected information of materials in the dataset, the top
suggestions of critical temperatures in superconductors are closer to the
measured temperature, at least on the average, by the Spln-CFR. Therefore, the
usage of Spln-CFR as a surrogate model to explore the possibility of testing
the superconductivity in materials may bring better returns.

Interestingly, we have also observed a similar behavior of xg-b with other
multivariate regression techniques, but also important differences worth
noting. For instance, Linear Regression, perhaps the simplest scheme of them
all, has an interesting behavior: the top 20 highest predictions are all in
the range $[81.83,91.59]$ K while the actual values are in the interval
$[98.00,132.60]$ K. For the multi-layer perceptron method (mlp), the top 20
highest predictions are all in the range $[90.01,100.81]$ K, yet true values
are in the interval $[109.00,131.40]$ K. These means that trained using the
MSE, these techniques could still give valuable information about materials
that could be prioritized for testing if we better consider the ranking given
to several materials and have less concerned about the predicted value.

Overall, the results show the limitations of the current dataset. One possible
limitation is the lack of other useful molecular descriptors that can bring
important problem domain knowledge about the structure of the materials and
their properties. In addition, it is also possible that a careful
“segmentation” of the different materials is necessary. In some sense, the
results of the experiments presented here may help the AI community reflect on
how to do these analyses and motivate the researchers to work in closer
collaboration with superconductivity specialists to provide other molecular
descriptors.

We actually often compare the inherent difficulties in prediction in this
dataset to other areas on which some of us have been working extensively (like
the prediction of survivability in breast cancer using transcriptomic data).
In both cases, without separating the training samples into meaningful
subgroups, the models obtained generalised poorly.

This said, one of the reasons that our continued fraction based method may be
doing just a bit better in the generalisation test in our Out-of-Domain study,
is that there might be some structural similarities in the set of compounds
used to define the “last” continued fraction approximation, then, indirectly
perhaps, from the molecular descriptors present some useful information which
the continued fraction representation has exploited. We will investigate this
hypothesis in a future publication where we aim to include more relevant
problem-domain information, in collaboration with specialists, to benefit from
the structure and known properties of the actual compounds.

In terms of future research on the algorithm we propose here, it is clear that
Spln-CFR is already a promising approach that has some obvious extensions
worth considering in the future, for instance, the inclusion of bagging and
boosting techniques which can improve the Out-of-Sample performance. In
addition, we consider that learning with modifications of the MSE in the
training set may lead to better performance for the Out-of-Domain scenario,
and we plan to conduct further research in that area as well.

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### Author Biographies

Pablo Moscato obtained his B.S. in Physics from National University of La
Plata in 1987, Argentina, before completing a PhD at the University of
Campinas, Brazil, in 2001. In 1988, while at the Caltech, he developed with
M.G. Norman the first memetic algorithm for the Traveling Salesman Problem and
championed memetic computing since then.

Over the past three decades, he introduced several new computer science
methods that lead to advances in personalised medicine, including the
discovery of biomarkers and hallmarks of cancer, Alzheimer and other human
diseases. Due to his lifetime achievements in interdisciplinary research and
in memetic computing in particular, he was nominated to the Rotary STAR
(Science, Technology, Aerospace, Robotics) 2018 Awards in the categories of
“Health and Medical” and “Knowledge Sharing”. These awards are annually given
“in recognition to outstanding scientific and technological achievements with
significant humanitarian benefit”.

He has close to 300 peer-reviewed scientific publications including the edited
books “Handbook of Memetic Algorihtms” (2012) and “Business and Consumer
Analytics: New Ideas” (2019).

He was an Australian Research Council (ARC) Future Fellow (2012-2016) and the
Founding Director of the University of Newcastle’s Priority Research Centre
(PRC) for Bioinformatics, Biomarker Discovery and Information-Based Medicine
(2007-2015) and the Newcastle Bioinformatics Initiative (2003-2006). He was
also Director of the Information-based Medicine Program of the Hunter Medical
Research Institute. He is currently a Professor of Data Science at The
University of Newcastle, Australia.

Mohammad Nazmul Haque (M’12–SM’21) received the B.Sc and M.Sc in Computer
Science & Engineering from Daffodil International University (DIU), Dhaka,
Bangladesh in 2006 and 2011, respectively. In 2017, he received the Ph.D.
degree in Computer Science from the University of Newcastle, Australia. This
author became a Member (M) of IEEE in 2012 and a Senior Member (SM) in 2021.

He is currently working as a Postdoctoral Research Associate at the College of
Engineering, Science & Environment at the University of Newcastle, Australia,
since 2017. He was a Lecturer and Senior Lecturer of Daffodil International
University, Dhaka, Bangladesh from 2009 to 2012, Lecturer at Daffodil
Institute of IT, Dhaka, Bangladesh from 2007 to 2009 and Software Engineer at
KMC e-Technology, Dhaka, Bangladesh from 2006 to 2007. He possesses
interdisciplinary research experiences in data analytics and biomarker
discovery and has worked with various data sources (e.g. gene expression,
images, consumer behaviour, disease datasets). His current research interests
include Data Analytics, Evolutionary Computing, Machine Learning, Artificial
Intelligence, Image Processing and Business Analytics.

Dr Haque is a member of the Australian Computer Society (ACS) and the
Association for Computing Machinery (ACM).

|  Kevin Huang is an undergraduate student at the California Institute of
Technology (Caltech) pursuing B.S. in Computer Science. He is interested in
fundamental machine learning research, including in optimization theory for
deep neural networks. In Summer 2020, as part of Caltech’s Summer
Undergraduate Research Fellowship program, he worked on research in memetic
algorithms under the supervision of Prof. Moscato.
---|---
|  Julia Sloan is originally from Boston, Massachusetts and lived there until
she moved to LA as a Caltech undergrad student. She now majors in Computer
Science but initially entered as a Chemistry student, a field she is still
very fond of. Julia has previously done research on novel battery
architectures with the See Group at Caltech. In the future, Julia hopes to
apply her technical Computer Science knowledge to problems in the area of
climate change and the environment.
|  Jonathon Corrales de Oliveira is from Miami, Florida and is currently an
undergraduate student at the California Institute of Technology (Caltech)
pursuing a B.S. in Computer Science. In the Summer of 2020, as part of
Caltech’s Summer Undergraduate Research Fellowship program, he worked on
research in memetic algorithms under the supervision of Prof. Moscato. Jon is
entering his second year of undergraduate study and plans to continue
exploring different fields within CS.


# Notes on no black hole theorem in three dimensional gravity

Pujian Mao and Weicheng Zhao

Notes on no black hole theorem in three dimensional gravity

Pujian Mao and Weicheng Zhao

Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

Abstract: In this paper, we revisit the no black hole theorem in three
dimensional (3D) gravity in the Newman-Penrose formalism in a generalized
Newman-Unti gauge. After adapting the well established 4D NP formalism and its
gauge and coordinates system to 3D gravity, we show that the no black hole
theorem is manifest in the NP equations. We further study in detail the
horizon properties of the 3D charged rotating solutions and confirm that a
black hole solution requires a negative cosmological constant.

##  1 Introduction

Einstein gravity in three dimensions has no propagating degree of freedom.
Hence, it is very surprising that three dimensional Einstein gravity with
negative cosmological constant admits black hole solutions [1], which has
received significant attentions from the holographic perspective, see, e.g.,
[2, 3, 4]. It is shown that the 3D Bañados-Teitelboim-Zanelli (BTZ) black hole
is obtained by the identification of points of anti-de Sitter space [5]. Then
an intensely curious point is that if a similar construction can arise black
hole solution for the case of a positive cosmological constant or the
vanishing cosmological constant case. This question was perfectly answered by
Ida with a no black hole theorem in 3D [6]. The proof in [6] can be summarized
as follows. Assuming that there is a black hole in the sense of the existence
of an apparent horizon in the spacetime. By definition, the expansion of a
null congruence of future directed ingoing null vector orthogonal to the
horizon is always positive in $(+,-,-)$ signature. While the expansion for the
outgoing null vector changes sign at the horizon and should be negative
outside the horizon. However, the author of [6] found a particular direction
to deform the apparent horizon. The expansion of the outgoing null vector is
positive at the deformed horizon in case that the stress tensor satisfies the
dominant energy condition. That means there is a trapped region outside the
apparent horizon which is in contradiction with the assumption of the
existence of an apparent horizon.

In this work, we revisit the 3D no black hole theorem in the context of the
Newman-Penrose (NP) formalism [7]. The adapted 3D NP formalism consists of two
null basis vectors and one spacelike basis vector. The advantage of the NP
formalism is that one can simply choose the two null basis vectors as the
outgoing and ingoing null direction, which are orthogonal to the horizon, to
investigate the horizon property. We construct the coordinates system in the
well established Newman-Unti (NU) gauge [8] with slight generalization. We
find that the 3D no black hole theorem is simply encoded in the near horizon
solution space. There is no need to introduce the particular direction to show
that it is still a trapped region outside the horizon. As a concrete example,
we investigate in details of the horizon property of the 3D charged rotating
solution [9, 10]. Our results show that a black hole requires negative
cosmological constant. For the positive and vanishing cosmological constant
cases, there is a cosmological horizon.

This paper is organized as follows. In section 2, we first adapt the NP
formalism to 3D and introduce the generalized NU gauge for the 3D NP system.
We show that the 3D no black hole theorem is simply encoded in the NP
equations in a transparent manner. Parallelly, we review the derivation of [6]
with adaptions to our conventions. Then, we present the full near horizon
solution space of 3D pure Einstein gravity. The solution space is in a closed
form. In section 3, we transform the 3D charged rotating solution in the NU
gauge. Section 4 presents the detailed analysis of the horizon property of the
3D charged rotating solution in the cases of a negative, vanishing, and
positive cosmological constant. In section 5, we comment on the flat limit of
the 3D charged rotating solution and show that the spinning parameter of the
solution with vanishing cosmological constant is simply introduced by a pure
coordinates transformation. We conclude in the last section.

##  2 NP formalism and no black hole theorem in 3D

###  2.1 3D NP formalism

We follow the construction in [11] to adapt 3D theory in an NP-like formalism.
Note that we will take the standard NP signature $(+,-,-)$ in this work. The
tetrad system consists of two null vectors $n$ and $n^{\prime}$ and a
spacelike vector $m$. The basis vectors have the orthogonality conditions

$n\cdot m=n^{\prime}\cdot m=0$ (1)

and are normalized as

$n\cdot n^{\prime}=1,\;\;\;\;m\cdot m=-1.$ (2)

The spacetime metric is obtained from

$g_{\mu\nu}=n_{\mu}n^{\prime}_{\nu}+n^{\prime}_{\mu}n_{\nu}-m_{\mu}m_{\nu}.$
(3)

The directional operators are defined as

$D=n^{\mu}\nabla_{\mu},\qquad
D^{\prime}=n^{\prime\mu}\nabla_{\mu},\qquad\delta=m^{\mu}\nabla_{\mu}.$ (4)

Below, we list all the quantities and relations for the 3D NP system.

Spin coefficients

Definition:

$\displaystyle\kappa=\Gamma_{311}=m\cdot
Dn,\qquad\epsilon=\Gamma_{211}=n^{\prime}\cdot
Dn,\qquad\tau^{\prime}=\Gamma_{321}=m\cdot Dn^{\prime},$
$\displaystyle\tau=\Gamma_{312}=m\cdot
D^{\prime}n,\qquad\epsilon^{\prime}=\Gamma_{122}=n\cdot
D^{\prime}n^{\prime},\qquad\kappa^{\prime}=\Gamma_{322}=m\cdot
D^{\prime}n^{\prime},$ (5) $\displaystyle\rho=\Gamma_{313}=m\cdot\delta
n,\qquad\rho^{\prime}=\Gamma_{323}=m\cdot\delta
n^{\prime},\qquad\beta=\Gamma_{213}=n^{\prime}\cdot\delta n.$

Implicitly, one has $\beta^{\prime}=-\beta$.

Properties:

$\displaystyle Dn=-\kappa m+\epsilon n,\qquad D^{\prime}n=-\tau
m-\epsilon^{\prime}n,\qquad\delta n=-\rho m+\beta n,$ $\displaystyle
Dn^{\prime}=-\tau^{\prime}m-\epsilon n^{\prime},\qquad
D^{\prime}n^{\prime}=-\kappa^{\prime}m+\epsilon^{\prime}n^{\prime},\qquad\delta
n^{\prime}=-\rho^{\prime}m-\beta n^{\prime},$ (6) $\displaystyle
Dm=-\tau^{\prime}n-\kappa n^{\prime},\qquad D^{\prime}m=-\kappa^{\prime}n-\tau
n^{\prime},\qquad\delta m=-\rho^{\prime}n-\rho n^{\prime}.$

Geodesic deviation:

$\nabla_{\mu}n_{\nu}=-\kappa n^{\prime}_{\mu}m_{\nu}+\epsilon
n^{\prime}_{\mu}n_{\nu}-\tau
n_{\mu}m_{\nu}-\epsilon^{\prime}n_{\mu}n_{\nu}+\rho m_{\mu}m_{\nu}-\beta
m_{\mu}n_{\nu}.$ (7)

NP equations

Curvature tensor:

$\displaystyle D\epsilon^{\prime}+D^{\prime}\epsilon$
$\displaystyle=-2\epsilon\epsilon^{\prime}+\beta(\tau^{\prime}-\tau)+\tau\tau^{\prime}-\kappa\kappa^{\prime}+2(R_{12}-\frac{1}{4}R),\qquad[R_{1212}]$
(8) $\displaystyle D\beta-\delta\epsilon$
$\displaystyle=\beta\rho+\kappa\epsilon^{\prime}+\kappa\rho^{\prime}-\epsilon\tau^{\prime}-\epsilon\beta-\rho\tau^{\prime}-R_{13},\qquad[R_{1213}]$
(9) $\displaystyle D^{\prime}\beta+\delta\epsilon^{\prime}$
$\displaystyle=\tau\rho^{\prime}+\epsilon^{\prime}\tau+\beta\rho^{\prime}-\beta\epsilon^{\prime}-\epsilon\kappa^{\prime}-\rho\kappa^{\prime}+R_{23},\qquad[R_{1223}]$
(10) $\displaystyle D\rho-\delta\kappa$
$\displaystyle=\rho^{2}+\epsilon\rho-2\kappa\beta-\kappa\tau-\kappa\tau^{\prime}-R_{11},\qquad[R_{1313}]$
(11) $\displaystyle D^{\prime}\rho-\delta\tau$
$\displaystyle=\rho\rho^{\prime}-\kappa\kappa^{\prime}-\tau^{2}-\rho\epsilon^{\prime}+(R_{12}-\frac{1}{2}R),\qquad[R_{1323}]$
(12) $\displaystyle D\rho^{\prime}-\delta\tau^{\prime}$
$\displaystyle=\rho\rho^{\prime}-\kappa\kappa^{\prime}-\tau^{\prime
2}-\epsilon\rho^{\prime}+(R_{12}-\frac{1}{2}R),\qquad[R_{2313}]$ (13)
$\displaystyle D^{\prime}\rho^{\prime}-\delta\kappa^{\prime}$
$\displaystyle=2\beta\kappa^{\prime}+\rho^{\prime
2}-\tau\kappa^{\prime}-\tau^{\prime}\kappa^{\prime}+\epsilon^{\prime}\rho^{\prime}-R_{22},\qquad[R_{2323}]$
(14) $\displaystyle D\kappa^{\prime}-D^{\prime}\tau^{\prime}$
$\displaystyle=\rho^{\prime}\tau-2\kappa^{\prime}\epsilon-\rho^{\prime}\tau^{\prime}+R_{23},\qquad[R_{2312}]$
(15) $\displaystyle D\tau-D^{\prime}\kappa$
$\displaystyle=2\kappa\epsilon^{\prime}+\rho\tau-\rho\tau^{\prime}-R_{13},\qquad[R_{1312}]$
(16)

Commutation relation:

$\displaystyle DD^{\prime}-D^{\prime}D=-\epsilon
D^{\prime}+\epsilon^{\prime}D+(\tau-\tau^{\prime})\delta,$ (17) $\displaystyle
D\delta-\delta D=-\kappa D^{\prime}-(\beta+\tau^{\prime})D+\rho\delta,$ (18)
$\displaystyle D^{\prime}\delta-\delta
D^{\prime}=-\kappa^{\prime}D+(\beta-\tau)D^{\prime}+\rho^{\prime}\delta.$ (19)

###  2.2  Bondi-Weyl gauge conditions and coordinates system

In 4D NP formalism, a special coordinates system is constructed by introducing
a family of null hypersurfaces labelled by the retarded time coordinate $u$
[7], see also [8, 12]. Then one null basis vector is chosen to be orthogonal
to the null hypersurface. Since the hypersurface is null, the basis vector is
tangent to a family of null geodesics on the hypersurface. It is convenient to
choose $u$, the affine parameter $r$, and two other parameters that label the
geodesics on each $u=$constant hypersurface as the coordinates.

Here we will follow a similar construction for the near horizon coordinates
system. However, we will choose the advanced time coordinate $v$ which allows
us to study both sides of the horizon [13], since the retarded time coordinate
$u$ does not cover the black hole region but the white hole region. The other
two coordinates are the affine parameter $r$ and one remaining coordinate
$\phi$ to label each geodesic. Following the 4D treatment [7, 8], in this
coordinates system, we can choose for the null basis

$n=-\frac{\partial}{\partial r}.$ (20)

The minus sign indicates that $n$ is an ingoing null vector. Since $n$ is the
normal vector of the $v=$constant null hypersurface, we should have that

$n=W(v,r,\phi)dv.$ (21)

This will yield the remaining two null basis vectors with the following forms

$n^{\prime}=\frac{1}{W}\frac{\partial}{\partial v}+U\frac{\partial}{\partial
r}-\frac{N}{W}\frac{\partial}{\partial\phi},\qquad
m=\omega\frac{\partial}{\partial
r}-\frac{1}{\Omega}\frac{\partial}{\partial\phi},$ (22)

and

$n^{\prime}=(UW+\omega\Omega N)dv-dr+\omega\Omega d\phi,\qquad
m=\Omega(Ndv+d\phi).$ (23)

where $U,N,\omega,\Omega$ are arbitrary functions of the coordinates. We will
further use the freedom from the rotation of the basis vectors to set
$n^{\prime}$ and $m$ parallelly propagated along $n$. In terms of the spin
coefficients, those conditions yield

$\kappa=\epsilon=\tau^{\prime}=0.$ (24)

One can further use the freedom from the rotation of the basis vectors and a
combined coordinates transformation to set $n$ being a gradient field as was
chosen in [11]. However, we will not take this option in this work as it is in
some sense more convenient to have exact solutions in this relaxed gauge. We
refer to this gauge condition as Bondi-Weyl gauge following the terminology in
[14].

###  2.3  No black hole theorem

In Bondi-Weyl gauge

The two components of the curvature tensor which are relevant to the no black
hole theorem are

$\displaystyle D\rho-\delta\kappa$
$\displaystyle=\rho^{2}+\epsilon\rho-2\kappa\beta-\kappa\tau-\kappa\tau^{\prime}-R_{11},$
(25) $\displaystyle D\rho^{\prime}-\delta\tau^{\prime}$
$\displaystyle=\rho\rho^{\prime}-\kappa\kappa^{\prime}-\tau^{\prime
2}-\epsilon\rho^{\prime}+(R_{12}-\frac{1}{2}R).$ (26)

In the Bondi-Weyl gauge, they are reduced to

$\displaystyle D\rho$ $\displaystyle=\rho^{2}-R_{11},$ (27) $\displaystyle
D\rho^{\prime}$ $\displaystyle=\rho\rho^{\prime}+(R_{12}-\frac{1}{2}R).$ (28)

We assume that there is an apparent horizon at $r=0$ and all the fields are
given in a near horizon series expansion

$\phi=\sum_{a=0}^{\infty}\phi_{a}(v,\phi)r^{a}.$ (29)

Moreover, this type of series expansion is valid on both sides of the horizon,
namely $r>0$ outside the horizon while $r<0$ inside the horizon. An apparent
horizon requires that $\rho_{0}>0$, $\rho^{\prime}_{0}=0$, and
$\rho^{\prime}_{1}<0$. Einstein equation in NP formalism is in the form

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=-T_{\mu\nu}.$ (30)

Thus, it is clear that if $-R_{12}+\frac{1}{2}R=T_{12}+\Lambda\geq 0$,
equation (26) yields that $\rho^{\prime}_{1}\geq 0$. This is in contradiction
with the assumption that an apparent horizon is located at $r=0$. Hence, we
can conclude that a black hole solution requires a negative cosmological
constant if the stress tensor satisfies the dominant energy condition [6].

In gauge conditions and coordinates system of [6]

In [6], the no black hole theorem was obtained from a different gauge choice.
In a 3D spacetime $(M,g)$, a spacelike hypersurface $\Sigma$ is selected. It
is assumed that there is a trapped region on $\Sigma$ with an apparent horizon
$H$ as the outer boundary of the trapped region on $H$. Then, the frame bases
are chosen as follows. $m$ is a unit tangent vector of the apparent horizon
$H$, $n$ and $n^{\prime}$ are future directed outgoing and ingoing null
vectors orthogonal to $H$. Note that we chose a different convention that $n$
is ingoing in previous discussion. Using boost transformation of $n$ and
$n^{\prime}$, it is possible to arrange that $n-n^{\prime}$ lies in $\Sigma$.
Moreover, it is assumed that $e^{f}(n-n^{\prime})$ and $e^{h}m$ form holonomic
bases on $\Sigma$ for some functions $f$ and $h$. Those requirements can be
achieved in the coordinates $(t,r,\phi)$ with $\phi$ being periodic and
$(r,\phi)$ spanning the spacelike hypersurface $\Sigma$. The conditions of
holonomic bases yield that

$e^{f}(n-n^{\prime})=\frac{\partial}{\partial r},\qquad
e^{h}m=\frac{\partial}{\partial\phi}.$ (31)

In terms of the spin coefficients, these conditions are indicating that

$\delta f=\kappa-\tau+\beta=\kappa^{\prime}-\tau^{\prime}-\beta.$ (32)

Condition $n-n^{\prime}$ lying in $\Sigma$ and the orthogonality conditions of
the bases require that

$n-n^{\prime}=-2e^{f}dr,\qquad n+n^{\prime}=2e^{c}dt,\qquad
m=e^{a}dt-e^{h}d\phi,$ (33)

and

$n+n^{\prime}=e^{-c}\frac{\partial}{\partial
t}+e^{-h}e^{a-c}\frac{\partial}{\partial\phi},$ (34)

for some functions $a$ and $c$. Hence, the basis vectors are given by

$\displaystyle n=\frac{1}{2}\left(e^{-c}\frac{\partial}{\partial
t}+e^{-f}\frac{\partial}{\partial
r}+e^{-h}e^{a-c}\frac{\partial}{\partial\phi}\right),$ (35) $\displaystyle
n^{\prime}=\frac{1}{2}\left(e^{-c}\frac{\partial}{\partial
t}-e^{-f}\frac{\partial}{\partial
r}+e^{-h}e^{a-c}\frac{\partial}{\partial\phi}\right),$ (36) $\displaystyle
m=e^{-h}\frac{\partial}{\partial\phi}.$ (37)

The no black hole theorem resides in the near horizon relation

$(D-D^{\prime})\rho=\delta(\kappa-\tau)+(\kappa-\tau)^{2}-R_{11}-(R_{12}-\frac{1}{2}R),$
(38)

where we used the fact that $\rho=0$ at the horizon and the relation

$\kappa-\tau+\beta=\kappa^{\prime}-\tau^{\prime}-\beta.$ (39)

A special choice of $f$ which sets $\delta(\delta f-\beta)=0$ leads to

$(D-D^{\prime})\rho=(\kappa-\tau)^{2}+T_{11}+T_{12}+\Lambda,$ (40)

A positive cosmological constant and the dominant energy condition yield that
the next-to-leading order of the expansion $\rho$ in the near horizon
expansion is always positive, which is the statement of no black hole theorem
[6].

###  2.4  Solution space for standard Einstein gravity

The near horizon solution space for 3D pure Einstein gravity with cosmological
constant is first derived in [15] in the form of near horizon expansion and in
[14] with closed form, see also [16, 17, 18] for several extensions. Here we
present a self-contained derivation in the NP formalism. The solution of the
spin coefficients from the radial equations in the Bondi-Weyl gauge is given
by

$\begin{split}&\rho=\frac{1}{r+A},\quad\rho_{0}=\frac{1}{A},\quad\rho^{\prime}=\frac{\rho^{\prime}_{0}A}{r+A}+\Lambda
r-\frac{r^{2}}{2(r+A)}\Lambda,\\\
&\beta=\frac{\beta_{0}A}{r+A},\quad\tau=\frac{\tau_{0}A}{r+A},\quad\epsilon^{\prime}=\beta_{0}\tau_{0}A(1-\frac{A}{r+A})+\epsilon^{\prime}_{0}-\Lambda
r,\\\ &\kappa^{\prime}=\kappa^{\prime}_{0}-\frac{1}{2}A\tau_{0}\left[\Lambda
r+\frac{A(\Lambda A-2\rho^{\prime}_{0})}{r+A}-\Lambda
A+2\rho^{\prime}_{0}\right],\end{split}$ (41)

where quantities with subscript $0$ are integration constants with respect to
$r$. The commutation relations of the null bases lead to the following
equations

$\begin{split}&D\frac{1}{W}=0,\quad DU=-\epsilon^{\prime}+\omega\tau,\quad
D\frac{N}{W}=\frac{\tau}{\Omega},\quad D\omega=\beta+\rho\omega,\quad
D\frac{1}{\Omega}=\frac{\rho}{\Omega},\\\ &D^{\prime}\omega-\delta
U=\kappa^{\prime}+(\beta-\tau)U+\rho^{\prime}\omega,\quad-D^{\prime}\frac{1}{\Omega}+\delta\frac{N}{W}=-(\beta-\tau)\frac{N}{W}-\frac{\rho^{\prime}}{\Omega},\\\
&-\delta\frac{1}{W}=(\beta-\tau)\frac{1}{W}.\end{split}$ (42)

The solutions of the first line in (LABEL:commutator) are

$\begin{split}&\Omega=\frac{\Omega_{0}}{A}(r+A),\qquad W=W_{0},\\\
&N=N_{0}+\frac{W_{0}\tau_{0}A}{\Omega_{0}}(\frac{A}{r+A}-1),\qquad\omega=\frac{\omega_{0}A}{r+A}-\frac{\beta_{0}Ar}{r+A},\\\
&U=-\frac{1}{2}\Lambda
r^{2}+(\beta_{0}\tau_{0}A+\epsilon^{\prime}_{0})r+\frac{\omega_{0}\tau_{0}A^{2}+\beta_{0}\tau_{0}A^{3}}{r+A}+U_{0}-(\omega_{0}\tau_{0}A+\beta_{0}\tau_{0}A^{2}).\end{split}$
(43)

The solution to the last line of the commutation relation (LABEL:commutator)
is

$\tau_{0}=\beta_{0}-\frac{\partial_{\phi}W_{0}}{\Omega_{0}W_{0}}.$ (44)

The second line of the commutation relation (LABEL:commutator) and the
remaining three relations from the curvature tensor determine only the time
evolution of the integration constants $\beta_{0}$, $\rho_{0}$,
$\rho^{\prime}_{0}$, $\omega_{0}$, $\Omega_{0}$. The evolution relations are
just tedious without further information about the solution space. We checked
in Mathematica and will not present in full details here.

##  3 3D charged rotating solution in NU gauge

The equations of motion for Einstein-Maxwell theory in the convention of the
NP formalism are

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\frac{1}{\ell^{2}}g_{\mu\nu}-\left(2F_{\mu\rho}{F_{\nu}}^{\rho}-\frac{1}{2}g_{\mu\nu}F^{2}\right)=0,\qquad\Lambda=-\frac{1}{\ell^{2}},$
(45)

and

$\nabla_{\nu}F^{\mu\nu}=0.$ (46)

The stress tensor of this case is

$T_{\mu\nu}=-\left(2F_{\mu\rho}{F_{\nu}}^{\rho}-\frac{1}{2}g_{\mu\nu}F^{2}\right).$
(47)

The metric of the 3D charged rotating solution is [10]

$ds^{2}=H^{2}F^{2}dT^{2}-\frac{dR^{2}}{F^{2}}-R^{2}(d\varphi-N^{\varphi}dT)^{2},\\\
$ (48)

and the gauge field is

$A=-\frac{Q\log r}{\sqrt{1-\frac{\omega^{2}}{\ell^{2}}}}\left(dT-\omega
d\varphi\right),$ (49)

where

$\displaystyle f^{2}=\frac{r^{2}}{\ell^{2}}-M-Q^{2}\log r^{2},$ $\displaystyle
R^{2}=\frac{r^{2}-f^{2}\omega^{2}}{1-\frac{\omega^{2}}{\ell^{2}}}=r^{2}+Y+X\log
r^{2},\quad Y=\frac{\omega^{2}M}{1-\frac{\omega^{2}}{\ell^{2}}},\quad
X=\frac{\omega^{2}Q^{2}}{1-\frac{\omega^{2}}{\ell^{2}}},$ $\displaystyle
F^{2}=(\frac{dR}{dr})^{2}f^{2},\quad
H^{2}=\frac{r^{2}}{R^{2}}(\frac{dr}{dR})^{2}$ $\displaystyle
N^{\varphi}=\frac{\omega(\frac{r^{2}}{\ell^{2}}-f^{2})}{(1-\frac{\omega^{2}}{\ell^{2}})R^{2}}=\frac{\omega(M+Q^{2}\log
r^{2})}{(1-\frac{\omega^{2}}{\ell^{2}})R^{2}},$

where $\omega$, $M$ and $Q$ are constants which are related to angular
velocity, mass, and charge parameters. In $(T,r,\varphi)$ coordinates, the
metric is given by

$ds^{2}=\frac{r^{2}f^{2}}{R^{2}}dT^{2}-\frac{dr^{2}}{f^{2}}-R^{2}(d\varphi-N^{\varphi}dT)^{2}.\\\
$ (50)

Via coordinates transformation,

$\displaystyle
dv=dT+\frac{Rdr}{rf^{2}},\;\;\;\;d\phi=d\varphi+\frac{N^{\varphi}R}{rf^{2}}dr,$
(51)

the metric becomes

$ds^{2}=\frac{r^{2}f^{2}}{R^{2}}dv^{2}-2\frac{r}{R}dvdr-R^{2}(d\phi-N^{\varphi}dv)^{2},$
(52)

and

$A=-\frac{Q\log r}{\sqrt{1-\frac{\omega^{2}}{\ell^{2}}}}\left(dv-\omega
d\phi\right)+\frac{Q\log r}{\sqrt{1-\frac{\omega^{2}}{\ell^{2}}}}(1-\omega
N^{\varphi})\frac{1}{HF^{2}}\frac{dR}{dr}dr.$ (53)

The last term in the gauge field can be eliminated by a pure gauge
transformation. Literally, this solution is not yet in the Bondi-Weyl gauge,
because the metric element $g_{vr}$ still has radial dependence.111The radial
dependence in the basis vector (21) is eliminated by commutation relation
(LABEL:commutator). In the metric formalism, it is part of the gauge condition
[15, 14, 16]. This can be achieved by a pure radial coordinate transformation,
namely defining $d\tilde{r}=\frac{r}{R}dr$. In the $(v,\tilde{r},\phi)$
coordinates, the 3D charged rotating solution is in the Bondi-Weyl gauge.
Actually, the solution is even simpler, it is now written in the NU gauge.

##  4 Horizon of 3D charged rotating solution

The apparent horizon is defined to be the outermost marginally trapped
surface. Let us choose two null directions $n^{\prime}$ being outgoing and $n$
being incoming following the choice of the Bondi-Weyl gauge,

$n^{\prime}=\frac{1}{2}\frac{r^{2}f^{2}}{R^{2}}dv-d\tilde{r},\qquad n=dv.$
(54)

The expansions of the two null directions can be computed directly. They are
given by

$\displaystyle\rho^{\prime}=-\frac{r^{2}f^{2}\frac{dR}{d\tilde{r}}}{2R^{3}}=-\frac{r^{2}f^{2}\frac{dR}{dr}\frac{dr}{d\tilde{r}}}{2R^{3}}=-\frac{r^{2}f^{2}}{2R^{2}}\frac{dR}{dr}\frac{1}{r},$
(55)
$\displaystyle\rho=\frac{\frac{dR}{d\tilde{r}}}{R}=\frac{\frac{dR}{dr}\frac{dr}{d\tilde{r}}}{R}=\frac{dR}{dr}\frac{1}{r}.$
(56)

Since $\rho^{\prime}$ and $\rho$ are scalar fields, they are given by the same
value in different coordinates. So we will use coordinate $r$ to characterize
the location of the horizon which is much simpler. The property of the horizon
depends on the cosmological constant. We will discuss that separately for
negative, zero, and positive cosmological constant.

###  4.1 Negative cosmological constant

For the negative cosmological constant case, it is proven that the 3D charged
rotating solution is a black hole solution for some particular regions of the
mass, charge, and angular velocity parameters [10]. For completeness, we will
present a self-contained discussion on this issue which will be useful for the
discussions of other choices of the cosmological constant. The expansions of
the two null directions now are

$\displaystyle\rho^{\prime}=-\frac{r^{2}f^{2}}{2R^{2}}\frac{dR}{dr}\frac{1}{r}=\frac{1}{2R^{3}}(r^{2}+\frac{\omega^{2}Q^{2}}{1-\frac{\omega^{2}}{\ell^{2}}})(-\frac{r^{2}}{\ell^{2}}+M+Q^{2}\log
r^{2}),$ (57)
$\displaystyle\rho=\frac{1}{r^{2}R}(r^{2}+\frac{\omega^{2}Q^{2}}{1-\frac{\omega^{2}}{\ell^{2}}}).$
(58)

Those expressions involve

$R=\sqrt{r^{2}+\frac{\omega^{2}}{1-\frac{\omega^{2}}{\ell^{2}}}(M+Q^{2}\log
r^{2})},$ (59)

a square root function, in the denominator, so it must be positive.
Alternatively, this condition defines the region where the coordinate $r$ is
valid for describing the spacetime. We see that $R^{2}$ is a monotonically
increasing function of $r$. Suppose that $r=r_{0}$ is the only root of
$R^{2}=0$, we should restrict ourselves in the region $r>r_{0}$. The
expression of the gauge field in (53) requires that
$1-\frac{\omega^{2}}{\ell^{2}}>0$. The reason for such a requirement is that
the rotating parameter $\omega$ is introduced from a boost transformation with
such assumption [10]. We will further comment on the boost transformation in
the next section. The function $-f^{2}=-\frac{r^{2}}{\ell^{2}}+M+Q^{2}\log
r^{2}$ has the maximum value at $r_{m}=Q^{2}\ell^{2}$. To guarantee that
$-f^{2}=0$ has a root, the mass and charge parameter must satisfy the
condition

$-Q^{2}+M+Q^{2}\log(Q^{2}\ell^{2})>0.$ (60)

Then $-f^{2}=0$ will have two roots. Supposing that $r=r_{i}$ and $r=r_{h}$
are the smaller and larger roots respectively, it is clear that
$r_{0}<r_{i}<r_{h}$. When $r>r_{h}$, we have $\rho^{\prime}<0$ and $\rho>0$,
it is the normal region. While, both expansions $\rho$ and $\rho^{\prime}$ are
larger than zero between the two horizons, namely $r_{i}<r<r_{h}$. This is the
black hole region and $r=r_{h}$ defines the apparent horizon of the black
hole. Correspondingly, the stress tensor should satisfy that
$T_{12}-\frac{1}{\ell^{2}}<0$. The $T_{12}$ component of the stress tensor is

$T_{12}=\frac{Q^{2}(\omega
N^{\varphi}-1)^{2}\frac{dr}{d\tilde{r}}}{(1-\frac{\omega^{2}}{\ell^{2}})r^{2}}=\frac{Q^{2}}{(1-\frac{\omega^{2}}{\ell^{2}})R^{2}}.$
(61)

On the horizon it reduces to

$T_{12}=\frac{Q^{2}}{r_{h}^{2}}.$ (62)

The fact $r_{h}^{2}>r_{m}^{2}=Q^{2}\ell^{2}$ ensures that
$T_{12}-\frac{1}{\ell^{2}}<0$.

###  4.2 Zero cosmological constant

In the $\Lambda=0$ case, substituting $\ell=\infty$ into the 3D charged
rotating solution, the expansions of the null vectors $n^{\prime}$, $n$ are
given by

$\displaystyle\rho^{\prime}=-\frac{r^{2}f^{2}}{2R^{2}}\frac{dR}{dr}\frac{1}{r}=\frac{1}{2R^{3}}(r^{2}+\omega^{2}Q^{2})(M+Q^{2}\log{r^{2}}),$
(63) $\displaystyle\rho=\frac{1}{r^{2}R}(r^{2}+\omega^{2}Q^{2}).$ (64)

Suppose that $r=r_{0}$ is the root of $R^{2}=r^{2}+\omega^{2}(M+Q^{2}\log
r^{2})=0$, we should restrict ourselves in the region $r>r_{0}$. There is only
one root from $M+Q^{2}\log{r^{2}}=0$, which is located at
$r_{h}=e^{-\frac{M}{2Q^{2}}}$ and it is obvious that $r_{0}<r_{h}$. This is a
cosmological horizon as both expansions $\rho$ and $\rho^{\prime}$ are larger
than zero when $r>r_{h}$. Hence this is the trapped region of the spacetime.
While the spacetime is regular when $r_{0}<r<r_{h}$. This is in consistency
with the no black hole theorem presented in previous section. Correspondingly,
the $T_{12}$ component of the stress tensor is larger than zero at $r_{h}$,

$T_{12}=\frac{Q^{2}}{R^{2}}\bigg{|}_{r=r_{h}}.$ (65)

The presence of Maxwell fields does exclude a black hole formation in the case
of $\Lambda=0$ [6].

###  4.3 Positive cosmological constant

In the case of a positive cosmological constant, all $\ell^{2}$ terms should
be replaced by $-\ell^{2}$, and the expansions of null vectors $n^{\prime}$
and $n$ can be directly calculated as

$\displaystyle\rho^{\prime}=-\frac{r^{2}f^{2}}{2R^{2}}\frac{dR}{dr}\frac{1}{r}=\frac{1}{2R^{3}}(r^{2}+\frac{\omega^{2}Q^{2}}{1+\frac{\omega^{2}}{\ell^{2}}})(\frac{r^{2}}{\ell^{2}}+M+Q^{2}\log{r^{2}}),$
(66)
$\displaystyle\rho=\frac{1}{r^{2}R}(r^{2}+\frac{\omega^{2}Q^{2}}{1+\frac{\omega^{2}}{\ell^{2}}}).$
(67)

Since both
$R^{2}=r^{2}+\frac{\omega^{2}}{1+\frac{\omega^{2}}{\ell^{2}}}(M+Q^{2}\log
r^{2})$ and $-f^{2}=\frac{r^{2}}{\ell^{2}}+M+Q^{2}\log{r^{2}}$ are
monotonically increasing functions of $r$, they both can only have one root.
Suppose that $r=r_{0}$ and $r=r_{h}$ are the roots of the two functions
respectively, it is obvious that $r_{0}<r_{h}$. Both expansions are positive
when $r>r_{h}$, while $\rho>0$ and $\rho^{\prime}<0$ in the region
$r_{0}<r<r_{h}$. So $r_{h}$ is a cosmoligical horizon which is similar to the
$\Lambda=0$ case. The $T_{12}$ component of the stress tensor is larger than
zero at $r_{h}$,

$T_{12}=\frac{Q^{2}}{(1+\frac{\omega^{2}}{\ell^{2}})R^{2}}\bigg{|}_{r=r_{h}}.$
(68)

##  5 Comment on flat limit of 3D charged rotating solution

The 3D black hole of standard Einstein gravity with a negative cosmological
constant arises from identification of points of AdS3 spacetime. The static
and rotating black hole are obtained from identification along different
directions [5]. The static BTZ black hole with the metric

$ds^{2}=-f^{2}dt^{2}+\frac{dr^{2}}{f^{2}}+r^{2}d\theta^{2},\quad
f^{2}=\frac{r^{2}}{\ell^{2}}-m,$ (69)

arises from the identification $\theta\rightarrow\theta+2k\pi$ [5], where $k$
is an integer. The rotating BTZ black hole

$ds^{2}=-N^{2}dT^{2}+\frac{dR^{2}}{N^{2}}+R^{2}(d\varphi-N^{\varphi}dT)^{2},$
(70)

where

$N^{2}=\frac{R^{2}}{\ell^{2}}-M+\frac{J^{2}}{4R^{2}},\quad
N^{\varphi}=\frac{J}{2R^{2}}.$ (71)

arises from the identification $\varphi\rightarrow\varphi+2k\pi$ [5]. Before
the identification, the two solutions are both describing AdS3 spacetime and
they are connected by a boost transformation [10], see also [19, 20, 21, 22],

$t=\frac{T-\omega\varphi}{\sqrt{1-\frac{\omega^{2}}{\ell^{2}}}},\quad\theta=\frac{\varphi-\frac{\omega}{\ell^{2}}T}{\sqrt{1-\frac{\omega^{2}}{\ell^{2}}}}.$
(72)

In the new coordinates $(T,R,\varphi)$, it is the metric of the rotating BTZ
(70) with the following identifications,

$R^{2}=\frac{r^{2}-f^{2}\omega^{2}}{1-\frac{\omega^{2}}{\ell^{2}}},\qquad
M=m\frac{1+\frac{\omega^{2}}{\ell^{2}}}{1-\frac{\omega^{2}}{\ell^{2}}},\qquad
J=\frac{2\omega m}{1-\frac{\omega^{2}}{\ell^{2}}}.$ (73)

The boost transformation can be implemented as a generic trick to create
rotating solutions from static ones in 3D. In practice, one should first
unwrap the identification of the static black hole solution. The second step
is to apply the boost transformation to obtain the rotating metric before
identification. The last step is to choose a new Killing direction to identify
points to recover the rotating black hole. This is precisely the way that the
authors of Ref. [10] found the charged rotating black hole.

When taking the flat limit $\ell\rightarrow\infty$, it is obvious from the
boost transformation (72) that $\theta=\varphi$. Hence, the two solutions are
obtained from identification along the same direction. They are just ordinary
coordinates transformation of each other, see, for instance, in [23] for more
details of the non-charged case. For the charged case, it is the same boost
transformation to build the rotating black hole in [10]. Thus, the rotating
parameter of the flat limit of the 3D charged rotating solution is again
introduced from pure coordinates transformation which does not describe an
intrinsic rotation of the source.

##  6 Conclusion

In this paper, we present an alternative derivation of the 3D no black hole
theorem in the NP formalism. We show that, in the well established NU gauge
choice, the no black hole theorem is transparent from the NP equations. The
existence of an apparent horizon requires a negative cosmological constant.
The near horizon solution space of pure Einstein gravity is derived in a self-
contained way in the NP formalism. The solution space is in a closed form
which is consistent with previous results [14, 17]. As a consistency check, we
study in detailed the horizon property of the 3D charged rotating solution [9,
10]. A black hole solution only exists in the case of a negative cosmological
constant. For the cases of vanishing or positive cosmological constant, there
will be a cosmological horizon. We examine the flat limit of the 3D charged
rotating solution and show that the spinning parameter is from a pure
coordinates transformation which is the same as the non-charged case [23].

## Acknowledgments

The authors thank Xiaoning Wu and Run-Qiu Yang for valuable discussions. The
authors thank Professor G. Clement for informing us that the 3D charged
rotating solution was first derived in [9]. This work is supported in part by
the National Natural Science Foundation of China (NSFC) under Grants No.
11905156 and No. 11935009.

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