# Complete steady gradient Yamabe solitons with positive scalar curvature are
rotationally symmetric

Shun Maeta  Department of Mathematics, Faculty of Education, and Department of
Mathematics and Informatics, Graduate School of Science and Engineering, Chiba
University, 1-33, Yayoicho, Inage, Chiba, 263-8522, Japan.
<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
###### Abstract.

In this paper, we solve the Yamabe soliton version of the Perelman conjecture.
We show that any nontrivial complete steady gradient Yamabe solitons with
positive scalar curvature are rotationally symmetric.

###### Key words and phrases:

steady gradient Yamabe solitons; Yamabe soliton version of the Perelman
conjecture; rotationally symmetric

###### 2010 Mathematics Subject Classification:

53C21, 53C25, 53C20

The author is partially supported by the Grant-in-Aid for Scientific Research
(C), No.23K03107, Japan Society for the Promotion of Science.

##  1\. Brief history and the result

The Yamabe flow is important for understanding structures in Riemannian
geometry. It was introduced by R. Hamilton [9] and further developed by S.
Brendle [1, 2]. Yamabe solitons are special solutions of the Yamabe flow, and
are expected to be the singularity models. From this perspective, Yamabe
solitons have developed rapidly in the last few decades. One of the main
problems is the Yamabe soliton version of the Perelman conjecture, that is,
any nontrivial complete steady gradient Yamabe solitons with positive scalar
curvature are rotationally symmetric. Various researchers have considered the
problem under the assumption of locally conformally flat condition. The
problem was solved under these assumptions and that the sectional curvature is
positive by Daskalopoulos and Sesum [8], the Ricci curvature is positive by
Catino, Mantegazza and Mazzieri [6], and the scalar curvature is positive by
Cao, Sun and Zhang [5]. In [5, 6], they rediscovered Tashiro’s theorem [14]
(see also [11]). As a corollary, Cao, Sun and Zhang solved the problem in the
3-dimensional case. However, the higher dimensional case is still open. In
this paper, we completely solve the Yamabe soliton version of the Perelman
conjecture.

######  Theorem 1.1.

Any nontrivial complete steady gradient Yamabe solitons with positive scalar
curvature are rotationally symmetric.

######  Remark 1.2.

It is known that any compact gradient Yamabe solitons are trivial $($[9,
10]$)$. The original Perelman conjecture [12] is that any $3$-dimensional
complete noncompact $\kappa$-noncollapsed gradient steady Ricci soliton with
positive curvature is rotationally symmetric, which was proven by S. Brendle
[3]. However, in higher dimensions, it is not well understood $($but, see for
example [4]$)$.

##  2\. Preliminary and the proof

An $n$-dimensional Riemannian manifold $(M^{n},g)$ is called a gradient Yamabe
soliton if there exists a smooth function $F$ on $M$ and a constant
$\lambda\in\mathbb{R}$, such that $\nabla\nabla F=(R-\lambda)g,$ where
$\nabla\nabla F$ is the Hessian of $F$, and $R$ is the scalar curvature on
$M$. If $F$ is constant, $M$ is called trivial. If $\lambda>0$, $\lambda=0$,
or $\lambda<0$, then the Yamabe soliton is called shrinking, steady, or
expanding.

Tashiro’s theorem ([14], see also [5, 6, 11]) is used for proving Theorem 1.1.

######  Theorem 2.1 ([14]).

A Riemannian manifold $(M^{n},g)$ which satisfies that for any smooth
functions $F$ and $\varphi$ on $M$, $\nabla\nabla F=\varphi g$ is either $(1)$
compact and rotationally symmetric, or $(2)$ rotationally symmetric and equal
to the warped product $([0,\infty),dr^{2})\times_{|\nabla
F|}(\mathbb{S}^{n-1},{\bar{g}}_{S}),$ where $\bar{g}_{S}$ is the round metric
on $\mathbb{S}^{n-1},$ or $(3)$ the warped product
$(\mathbb{R},dr^{2})\times_{|\nabla F|}\left(N^{n-1},\bar{g}\right),$ where
the scalar curvature $\bar{R}$ of $N$ satisfies

(2.1) $|\nabla F|^{2}R=\bar{R}-(n-1)(n-2)\varphi^{2}-2(n-1)g(\nabla
F,\nabla\varphi).$

######  Remark 2.2.

The potential function $F$ depends only on $r$, and $F^{\prime}(r)>0$ $($see,
for example, the proof of Theorem $1.1$ of [11]$)$. The manifold
$(M,g,F,\varphi)$ that satisfies the condition $\nabla\nabla F=\varphi g$ was
also studied by Cheeger and Colding [7].

As pointed out in [5], it follows as a corollary of Theorem 2.1 that any
nontrivial complete gradient Yamabe solitons with positive Ricci curvature are
rotationally symmetric.

###### Proof of Theorem 1.1.

To show rotational symmetry of $M$, we only have to consider $(3)$ of Theorem
2.1. By the soliton equation, Remark 2.2 and (2.1), one has

(2.2) $\rho^{\prime}\rho^{2}+(n-1)(n-2)\rho^{\prime
2}+2(n-1)\rho\rho^{\prime\prime}=\bar{R},$

where $\rho=F^{\prime}$. Since the left hand side of $\eqref{key2}$ depends
only on $r$, the scalar curvature $\bar{R}$ of $N$ is constant. Positivity of
the scalar curvature shows that $\bar{R}>0$ (which was shown in [5]). In fact,
if $\bar{R}$ is nonpositive, then $\rho^{\prime\prime}$ is nonpositive, hence
the positive function $\rho$ is concave. Therefore, $\rho$ is constant, which
cannot happen. Since $\rho^{\prime}>0$, $\rho$ is monotone increasing.
Furthermore, one can show that $\rho$ goes to infinity. Assume that $\rho$ is
bounded from above, that is, $F^{\prime}=\rho\leq c$ for some positive
constant $c$. Then, the convex function $F$ satisfies that $F\leq cr+b$ on
$\mathbb{R}$, which cannot happen.

The equation (2.2) is an autonomous second order equation and can be made into
a first order equation by using $\rho$ as a new independent variable. If
$\rho^{\prime}=G(\rho)$, then $\rho^{\prime\prime}=\dot{G}G,$ and one has

(2.3) $G\rho^{2}+(n-1)(n-2)G^{2}+2(n-1)\rho\dot{G}G=\bar{R}.$

By differentiating the equation, one has

(2.4) $\dot{G}\rho^{2}+2\rho
G+2(n-1)^{2}\dot{G}G+2(n-1)\rho\ddot{G}G+2(n-1)\rho\dot{G}^{2}=0.$

Assume that $\dot{G}>0$ at some point $\rho_{0}\in(0,+\infty)$, that is,
$\dot{G}>0$ on some open interval $\Omega=(\rho_{1},\rho_{2})(\ni\rho_{0})$.
If $\Omega=(\rho_{1},+\infty)$, by (2.3), $G\rho^{2}+(n-1)(n-2)G^{2}<\bar{R}$
on $(\rho_{1},+\infty).$ However, the left hand side goes to infinity as
$\rho\nearrow+\infty$, which cannot happen. Thus, one can assume that
$\dot{G}=0$ at $\rho_{2}$. Then, by $\eqref{2nd}$,
$2(n-1)\rho_{2}\ddot{G}G+2\rho_{2}G=0$ at $\rho_{2}$. Hence, $\ddot{G}<0$ at
$\rho_{2}$, and $G$ is monotone decreasing on $(\rho_{2},\rho_{3})$ for some
$\rho_{3}$. Iterating the same argument, one can extend $\rho_{3}$ to
$+\infty$, that is, $G$ is monotone decreasing on $(\rho_{2},+\infty)$. Hence,
if there exists such an open interval $\Omega$, it must be $(0,\rho_{2})$.
Therefore, $G$ has a maximum, say $C(=G(\rho_{2})>0)$, that is,
$F^{\prime\prime}=G\leq C$. Thus, one has $0<F^{\prime}\leq Cr+D$ on
$\mathbb{R}$, which cannot happen. We finally obtain $\Omega=\emptyset.$

Therefore, one has
$0\geq\dot{G}=\frac{\rho^{\prime\prime}(r)}{\rho^{\prime}(r)}$ for every
$r\in\mathbb{R}$, and $\rho^{\prime\prime}\leq 0$ on $\mathbb{R}$. Since the
positive smooth function $\rho$ is concave on $\mathbb{R}$, $\rho$ is
constant, which cannot happen. ∎

By the same argument, we also get the similar result for shrinking solitons:

######  Theorem 2.3.

Any nontrivial complete shrinking gradient Yamabe solitons with $R>\lambda$
are rotationally symmetric.

######  Remark 2.4.

The assumption $R>\lambda$ is optimal. In fact, for any $b\in\mathbb{R}$ and
any Riemannian manifold $(N,\bar{g}_{N})$ with constant positive scalar
curvature $\bar{R}$, $(\mathbb{R}\times
N,dr^{2}+\frac{\bar{R}}{\lambda}\,\bar{g}_{N},\sqrt{\frac{\bar{R}}{\lambda}}\,r+b)$
is a nontrivial shrinking gradient Yamabe soliton with $R=\lambda$.

Acknowledgements.
The author would like to express his gratitude to Ken Shirakawa for valuable
discussions.

Data availability statement
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

Conflict of interest
There is no conflict of interest in the manuscript.

## References

  * [1] S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom. (2005) 69, 217–278.
  * [2] S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math. (2007) 170, 541–576.
  * [3] S. Brendle, Rotational symmetry of self-similar solutions to the Ricci flow, Invent. Math. (2013) 194, 731–764.
  * [4] S. Brendle, Rotational symmetry of Ricci solitons in higher dimensions, J. Differential Geom. (2014) 97, 191–214.
  * [5] H.-D. Cao, X. Sun and Y. Zhang, On the structure of gradient Yamabe solitons, Math. Res. Lett. (2012) 19, 767–774.
  * [6] G. Catino, C. Mantegazza and L. Mazzieri, On the global structure of conformal gradient solitons with nonnegative Ricci tensor, Commun. Contemp. Math. (2012) 14, 12pp.
  * [7] J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (1996) 144, 189–237.
  * [8] P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math. (2013) 240, 346–369.
  * [9] R. Hamilton, Lectures on geometric flows, (1989), unpublished.
  * [10] S. Y. Hsu, A note on compact gradient Yamabe solitons, J. Math. Anal. Appl. (2012) 388 (2), 725–726.
  * [11] S. Maeta, Classification of gradient conformal solitons, arXiv:2107.05487[math DG]
  * [12] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv math.DG/0211159, (2002).
  * [13] P. Petersen, Riemannian Geometry. Third edition, Graduate Texts in Mathematics, 171. Springer (2016).
  * [14] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. math. Soc., 117 (1965), 251–275.


(A.162)

where $\varepsilon$ is an independent standard Gaussian model error, and
$\bm{X}=(X_{1},\dots,X_{p})^{\top}$ with $p\in\\{20,700\\}$ is either 1) a
multivariate Gaussian vector with zero mean and covariance matrix
$\Sigma=[\Sigma_{lk}]_{l,k=1}^{p}$, in which $\Sigma_{lk}=\rho^{|l-k|}$, or 2)
a multivariate t distribution with zero mean, covariance matrix $\Sigma$, and
degree of freedom $10$. In practice, we use the R package mvtnorm for sampling
$\\{\bm{Z}_{i}\\}_{i=1}^{n}$ from these multivariate distributions, while
their knockoff features $\\{\widetilde{\bm{Z}}_{i}\\}_{i=1}^{n}$ are generated
by our coordinate-wise Gaussian knockoff generator introduced in Section E
with $\\{\bm{Z}_{i}\\}_{i=1}^{n}$ given. Note that existing knockoff
generators, including ours in Section E, are all applicable for producing
knockoffs $\\{\widetilde{\bm{Z}}_{i}\\}_{i=1}^{n}$ given an arbitrary sample
$\\{\bm{Z}_{i}\\}_{i=1}^{n}$, but whether
$\\{\widetilde{\bm{Z}}_{i}\\}_{i=1}^{n}$ are good approximations of the ideal
coordinate-wise knockoffs depends on the underlying distribution of the
sample.

####  D.1.3 VD test for hypothesis (2)

The VD test statistic with break selection
$\big{|}T_{j}(\widehat{a}_{j})\big{[}\widehat{\sigma}_{j}(\widehat{a}_{j})\big{]}^{-1}\big{|}$
given in (12) is calculated and the test is established for each
$j\in\\{1,\dots,20\\}$. The empirical rejection rates over $100$ repetitions
of each simulation case are reported without adjusting for multiple tests in
Tables 3–4, where the rejection thresholds under null hypothesis (2) with test
size $\alpha\in\\{0.1,0.05,0.025\\}$ are given respectively by
$(t_{0.1},t_{0.05},t_{0.025})=(1.64,1.96,2.25)$ for the VD test, according to
Theorem 3. The multiplicity adjustment is not made in this simulation
experiment because the goal here is to illustrate our main results in Theorem
3. In Tables 3–4, only two features $X_{10}$ and $X_{15}$ indicated with
checkmarks are relevant, which means that entries for other features in the
tables are empirical wrong rejection rates. Each simulation setting including
the underlying feature index $j\in\\{1,\dots,20\\}$, values of $(n,p,\rho)$,
test size $\alpha\in\\{0.1,0.05,0.025\\}$, data generating models
(A.159)–(A.160), and test methods is indicated for each case and each panel in
the tables. In addition, the centering methods are indicated on the top of
each panel; they are random forests Breiman2001 and HBART
pratola2020heteroscedastic, which are implemented with the R packages
randomForest and rbart, respectively. In the simulation of the VD test, the
sample features $\\{\bm{X}_{i}\\}_{i=1}^{n}$ are sampled from the multivariate
Gaussian distribution with covariance matrix $\Sigma$ (see Section D.1.2 for
details). In Table 3, we also consider heuristic P-values calculated by double
GLM smyth1989generalized, smyth1999adjusted with the R package dglm for each
$j\in\\{1,\dots p\\}$. We do not report results from DGLM in Table 4 because
dglm does not output heuristic P-values for the non-linear high-dimensional
case.

The VD test with break selection requires sample splitting and a choice of
break candidates, which we introduce as follows. The simulated sample
$\mathcal{X}_{n}$ (see Section D.1.2) is split into two subsamples
$\mathcal{X}_{1}=\\{Y_{i},\bm{X}_{i},\widetilde{\bm{X}}_{i}\\}_{i=1}^{n_{1}}$
and
$\mathcal{X}_{2}=\\{V_{i},\bm{U}_{i},\widetilde{\bm{U}}_{i}\\}_{i=1}^{n_{2}}$
with $n_{1}=\lfloor\frac{2}{3}n\rceil$ and $n_{2}=n-n_{1}$. As have mentioned
in Section 2.4, sample $\mathcal{X}_{1}$ is used for constructing test
statistics, while $\mathcal{X}_{2}$ is used for selecting breaks. Our
practical implementation of the VD test uses the full sample $\mathcal{X}_{n}$
for training the regression trees estimate of mean functions for simplicity.
In addition, for each $l\in\\{1,\dots,p\\}$, we set break candidates
$(\kappa_{1,l},\kappa_{R,l})$ to respectively the first and third quartiles of
$(U_{1l},\dots,U_{n_{2}l}$) with $R=100$, $\kappa_{r,l}<\kappa_{r+1,l}$, and
evenly distributed $\kappa_{r,l}$’s.

(a) VD test + random forests (centering method) + model (A.159) with Gaussian
features
---
$(n,p,\rho)$ | $\alpha$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}$ | $X_{8}$ | $X_{9}$ |  $X_{10}$ ✓
(500, 20, 0.4) | $0.1\phantom{00}$ |  0.08 |  0.09 |  0.04 |  0.11 |  0.07 |  0.04 |  0.07 |  0.09 |  0.05 |  0.82
$0.05\phantom{0}$ |  0.03 |  0.05 |  0.03 |  0.01 |  0.01 |  0.00 |  0.03 |  0.05 |  0.01 |  0.66
$0.025\phantom{}$ |  0.01 |  0.02 |  0.01 |  0.00 |  0.00 |  0.00 |  0.01 |  0.01 |  0.01 |  0.52
$\alpha$ . | $X_{11}$ | $X_{12}$ | $X_{13}$ | $X_{14}$ |  $X_{15}$ ✓ | $X_{16}$ | $X_{17}$ | $X_{18}$ | $X_{19}$ | $X_{20}$
$0.1\phantom{0}$ . |  0.08 |  0.05 |  0.09 |  0.06 |  0.86 |  0.05 |  0.04 |  0.08 |  0.03 |  0.09
$0.05\phantom{0}$ |  0.05 |  0.00 |  0.03 |  0.00 |  0.71 |  0.01 |  0.02 |  0.02 |  0.00 |  0.03
$0.025\phantom{}$ |  0.02 |  0.00 |  0.00 |  0.00 |  0.59 |  0.00 |  0.00 |  0.00 |  0.00 |  0.00

(b) VD test + HBART (centering method) + model (A.159) with Gaussian features
---
$(n,p,\rho)$ | $\alpha$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}$ | $X_{8}$ | $X_{9}$ |  $X_{10}$ ✓
(500, 20, 0.4) | $0.1\phantom{00}$ |  0.06 |  0.13 |  0.07 |  0.11 |  0.06 |  0.03 |  0.06 |  0.06 |  0.05 |  0.88
$0.05\phantom{0}$ |  0.04 |  0.06 |  0.04 |  0.02 |  0.02 |  0.00 |  0.05 |  0.04 |  0.00 |  0.81
$0.025\phantom{}$ |  0.02 |  0.03 |  0.01 |  0.00 |  0.00 |  0.00 |  0.03 |  0.01 |  0.00 |  0.65
$\alpha$ . | $X_{11}$ | $X_{12}$ | $X_{13}$ | $X_{14}$ |  $X_{15}$ ✓ | $X_{16}$ | $X_{17}$ | $X_{18}$ | $X_{19}$ | $X_{20}$
$0.1\phantom{0}$ . |  0.10 |  0.06 |  0.06 |  0.05 |  0.89 |  0.07 |  0.06 |  0.08 |  0.03 |  0.07
$0.05\phantom{0}$ |  0.03 |  0.01 |  0.02 |  0.00 |  0.76 |  0.01 |  0.03 |  0.03 |  0.01 |  0.03
$0.025\phantom{}$ |  0.00 |  0.01 |  0.00 |  0.00 |  0.68 |  0.00 |  0.00 |  0.01 |  0.00 |  0.00

(c) Double GLM + model (A.159) with Gaussian features
---
$(n,p,\rho)$ | $\alpha$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}$ | $X_{8}$ | $X_{9}$ |  $X_{10}$ ✓
(500, 20, 0.4) | $0.1\phantom{00}$ |  0.17 |  0.14 |  0.16 |  0.23 |  0.13 |  0.18 |  0.16 |  0.16 |  0.17 |  1.00
$0.05\phantom{0}$ |  0.10 |  0.06 |  0.08 |  0.13 |  0.06 |  0.07 |  0.13 |  0.11 |  0.10 |  1.00
$0.025\phantom{}$ |  0.06 |  0.04 |  0.05 |  0.09 |  0.04 |  0.06 |  0.07 |  0.09 |  0.02 |  1.00
$\alpha$ . | $X_{11}$ | $X_{12}$ | $X_{13}$ | $X_{14}$ |  $X_{15}$ ✓ | $X_{16}$ | $X_{17}$ | $X_{18}$ | $X_{19}$ | $X_{20}$
$0.1\phantom{0}$ . |  0.23 |  0.20 |  0.14 |  0.16 |  1.00 |  0.16 |  0.12 |  0.18 |  0.12 |  0.15
$0.05\phantom{0}$ |  0.13 |  0.10 |  0.08 |  0.11 |  1.00 |  0.09 |  0.06 |  0.11 |  0.10 |  0.08
$0.025\phantom{}$ |  0.06 |  0.05 |  0.07 |  0.09 |  1.00 |  0.05 |  0.04 |  0.06 |  0.03 |  0.06

Table 3: The empirical rejection rates for hypothesis (2) for each feature
$j\in\\{1,\dots,20\\}$ at significance level $\alpha\in\\{0.1,0.05,0.025\\}$
over $100$ simulation repetitions. The centering methods for VD tests are
indicated in each panel. Features $X_{10}$ and $X_{15}$ with checkmarks are
relevant features; others are null features.

(a) VD test + RF (centering method) + model (A.160) with Gaussian features
---
$(n,p,\rho)$ | $\alpha$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}$ | $X_{8}$ | $X_{9}$ |  $X_{10}$ ✓
(700, 700, 0.6) | $0.1\phantom{00}$ |  0.06 |  0.08 |  0.03 |  0.04 |  0.06 |  0.07 |  0.04 |  0.10 |  0.08 |  0.50
$0.05\phantom{0}$ |  0.04 |  0.01 |  0.01 |  0.01 |  0.01 |  0.04 |  0.00 |  0.04 |  0.03 |  0.30
$0.025\phantom{}$ |  0.00 |  0.01 |  0.01 |  0.01 |  0.00 |  0.03 |  0.00 |  0.02 |  0.02 |  0.23
$\alpha$ . | $X_{11}$ | $X_{12}$ | $X_{13}$ | $X_{14}$ |  $X_{15}$ ✓ | $X_{16}$ | $X_{17}$ | $X_{18}$ | $X_{19}$ | $X_{20}$
$0.1\phantom{0}$ . |  0.07 |  0.09 |  0.09 |  0.10 |  0.48 |  0.08 |  0.02 |  0.09 |  0.09 |  0.09
$0.05\phantom{0}$ |  0.04 |  0.02 |  0.03 |  0.02 |  0.36 |  0.01 |  0.01 |  0.01 |  0.04 |  0.03
$0.025\phantom{}$ |  0.01 |  0.01 |  0.02 |  0.01 |  0.25 |  0.01 |  0.00 |  0.01 |  0.00 |  0.01

(b) VD test + HBART (centering method) + model (A.160) with Gaussian features
---
$(n,p,\rho)$ | $\alpha$ | $X_{1}$ | $X_{2}$ | $X_{3}$ | $X_{4}$ | $X_{5}$ | $X_{6}$ | $X_{7}$ | $X_{8}$ | $X_{9}$ |  $X_{10}$ ✓
(700, 700, 0.6) | $0.1\phantom{00}$ |  0.06 |  0.04 |  0.07 |  0.07 |  0.04 |  0.04 |  0.08 |  0.06 |  0.04 |  0.72
$0.05\phantom{0}$ |  0.01 |  0.00 |  0.02 |  0.03 |  0.02 |  0.03 |  0.01 |  0.02 |  0.00 |  0.60
$0.025\phantom{}$ |  0.00 |  0.00 |  0.01 |  0.00 |  0.01 |  0.01 |  0.00 |  0.01 |  0.00 |  0.40
$\alpha$ . | $X_{11}$ | $X_{12}$ | $X_{13}$ | $X_{14}$ |  $X_{15}$ ✓ | $X_{16}$ | $X_{17}$ | $X_{18}$ | $X_{19}$ | $X_{20}$
$0.1\phantom{0}$ . |  0.06 |  0.08 |  0.10 |  0.06 |  0.75 |  0.06 |  0.05 |  0.03 |  0.07 |  0.04
$0.05\phantom{0}$ |  0.04 |  0.02 |  0.02 |  0.01 |  0.57 |  0.01 |  0.02 |  0.02 |  0.03 |  0.01
$0.025\phantom{}$ |  0.02 |  0.00 |  0.01 |  0.00 |  0.42 |  0.00 |  0.01 |  0.00 |  0.01 |  0.00

Table 4: The empirical rejection rates for hypothesis (2) for each feature
$j\in\\{1,\dots,20\\}$ at significance level $\alpha\in\\{0.1,0.05,0.025\\}$
over $100$ simulation repetitions. The centering methods are indicated in each
panel. Features $X_{10}$ and $X_{15}$ with checkmarks are relevant features;
others are null features.

####  D.1.4 VDBP test for hypothesis (3)

The VDBP test statistic
$\big{|}T_{\widehat{l}}(\widehat{a}_{\widehat{l}})\big{[}\widehat{\sigma}_{\widehat{l}}(\widehat{a}_{\widehat{l}})\big{]}^{-1}\big{|}$
given by (13) is calculated and the test is established for each simulation
experiment. The sample splitting, R packages for random forests and HBART, and
the choice of change candidates follow those in Section D.1.3, but with
$n_{1}=\lfloor n/3\rceil$ and $n_{2}=n-n_{1}$. By Theorem 4, we set the
rejection thresholds $(t_{0.1},t_{0.05},t_{0.025})=(1.64,1.96,2.25)$ for the
VDBP test with break selection. Meanwhile, the benchmark method here is the
Breusch-Pagan test breusch1979simple, whose P-value is available from the R
package lmtest. In addition, to apply the Breusch-Pagan test to high-
dimensional data, we run the Breusch-Pagan test given a set of selected
features by Twohit chiou2020variable. Twohit is a recent approach for
selecting active features in both mean and standard deviation functions from
high-dimensional features; this method assumes model (1) with a linear mean
function and $\zeta(\cdot)$ as in Example 4. As have done for the VDBP test,
we split sample $\mathcal{X}_{n}$ into two subsamples with respective sample
sizes $n_{1}$ and $n_{2}$, where the second subsample is used for Twohit model
selection and the first one is for the Breusch-Pagan test.

We report the empirical rejection rates of each case over $100$ repetitions in
Table 5, with test size $\alpha\in\\{0.1,0.05,0.025\\}$, data generating
models (A.161)–(A.162), distribution of $\bm{X}$, feature dimensionality
$p\in\\{20,700\\}$, and the centering methods indicated for each case and each
panel; in addition, we set $n=700$ and $\rho=0.6$, and we refer to Section
D.1.2 for details of multivariate Gaussian and t distributions with covariance
parameter $\rho$. The cases (V) and (VI) are omitted in panel (c) of Table 5
because the Breusch-Pagan test does not apply to high-dimensional features. In
Table 5, we omit the cases with linear models for simplicity because the
simulation results for such cases are satisfactory for all methods.

(a) VDBP test + Random Forests (centering method)
---
Case | (I) | (II) | (III) | (IV) | (V) | (VI)
Model |  Model (A.161) |  Model (A.162)✓ |  Model (A.161) |  Model (A.162)✓ |  Model (A.161) |  Model (A.162)✓
Dist. of $\bm{X}$ | Gaussian | Gaussian | t | t | Gaussian | Gaussian
$p$ | $20$ | $20$ | $20$ | $20$ | $700$ | $700$
$\alpha=0.1$ | 0.05 | 0.89 | 0.06 | 0.47 | 0.08 | 0.35
$\alpha=0.05$ | 0.02 | 0.86 | 0.02 | 0.39 | 0.01 | 0.30
$\alpha=0.025$ | 0.01 | 0.78 | 0.00 | 0.30 | 0.00 | 0.23

(b) VDBP test + HBART (centering method)
---
Case | (I) | (II) | (III) | (IV) | (V) | (VI)
Model |  Model (A.161) |  Model (A.162)✓ |  Model (A.161) |  Model (A.162)✓ |  Model (A.161) |  Model (A.162)✓
Dist. of $\bm{X}$ | Gaussian | Gaussian | t | t | Gaussian | Gaussian
$p$ | $20$ | $20$ | $20$ | $20$ | $700$ | $700$
$\alpha=0.1$ | 0.07 | 1.00 | 0.06 | 0.89 | 0.16 | 0.86
$\alpha=0.05$ | 0.03 | 0.99 | 0.04 | 0.84 | 0.11 | 0.83
$\alpha=0.025$ | 0.02 | 0.99 | 0.01 | 0.83 | 0.03 | 0.77

(c) Breusch-Pagan test
---
Case | (I) | (II) | (III) | (IV) |  |
Model |  Model (A.161) |  Model (A.162)✓ |  Model (A.161) |  Model (A.162)✓ |  |
Dist. of $\bm{X}$ | Gaussian | Gaussian | t | t |  |
$p$ | 20 | 20 | 20 | 20 |  |
$\alpha=0.1$ | 0.20 | 0.94 | 0.90 | 0.92 |  |
$\alpha=0.05$ | 0.10 | 0.90 | 0.84 | 0.87 |  |
$\alpha=0.025$ | 0.08 | 0.85 | 0.78 | 0.77 |  |

Table 5: The empirical rejection rates for hypothesis (3) for each case at
each significance level $\alpha\in\\{0.1,0.05,0.025\\}$ over $100$ simulation
repetitions with $(\rho,n)=(0.6,700)$. The checkmarks indicate the models that
are against the null null hypotheses. The centering methods and model
selection method are indicated in each panel.

(d) Breusch-Pagan test + Twohit (model selection method)
---
Case | (I) | (II) | (III) | (IV) | (V) | (VI)
Model |  Model (A.161) |  Model (A.162)✓ |  Model (A.161) |  Model (A.162)✓ |  Model (A.161) |  Model (A.162)✓
Dist. of $\bm{X}$ | Gaussian | Gaussian | t | t | Gaussian | Gaussian
$p$ | 20 | 20 | 20 | 20 | 700 | 700
$\alpha=0.1$ | 0.29 | 0.76 | 0.62 | 0.59 | 0.26 | 0.77
$\alpha=0.05$ | 0.17 | 0.69 | 0.52 | 0.51 | 0.15 | 0.65
$\alpha=0.025$ | 0.14 | 0.58 | 0.45 | 0.42 | 0.10 | 0.55

Table 6: *

Table 5 (continued): The empirical rejection rates for hypothesis (3) for each
case at each significance level $\alpha\in\\{0.1,0.05,0.025\\}$ over $100$
simulation repetitions with $(\rho,n)=(0.6,700)$. The checkmarks indicate the
models that are against the null hypothesis. The centering methods and model
selection method are indicated in each panel.

####  D.1.5 Simulation results

From Tables 3–4, the VD test with each centering method mostly controls the
empirical wrong rejection rates to the respective target levels
$\alpha\in\\{0.1,0.05,0.025\\}$, showing the merits of using knockoff features
as the negative control of the test statistics. In contrast, the hueristic
P-values obtained by DGLM do not control the empirical wrong rejection rates,
which are frequently twice the sizes of the respective target levels. On the
other hand, the test power (for $X_{10}$ and $X_{15}$) based on VD tests
decreases as $\alpha$ decreases in Tables 3–4, which may be improved given a
larger sample. In these two experiment settings, the test power of the VD test
+ HBART outperformes the VD test + random forests, while the power of each VD
test decreases under model (A.160) when compared to their respective results
under model (A.159). Meanwhile, the heuristic P-values based on DGLM
ourperform other methods in temrs of power and always identify relevant
features $X_{10}$ and $X_{15}$ in Table 3. However, the DGLM in package dglm
does not scale to high-dimensional input features and cannot be applied to
cases with complicated mean functions such as model (A.160).

Regarding the tests for null hypothesis (3) in Table 5, except for the case
(V) in panel (b), the VDBP tests control the empirical wrong rejection rates.
Particularly, the results of case (III) in panel (a) and (b) show that VDBP
tests are robust when the feature vector has a multivariate t distribution. In
contrast, from panel (c) of Table 5, the Breusch-Pagan test does not control
the false positive errors under models with non-linear means. Its inferences
for case (III) is not reliable since the false positive errors are too high.
In panel (d), we see that the Breusch-Pagan test + Twohit is applicable to
high-dimensional cases, and its performance is the same in low- and high-
dimensional cases. However, the Breusch-Pagan test + Twohit increases the
false positive errors in case (I) when compared to the corresponding results
without Twohit, which is mainly because Twohit often screens out active
features $\\{X_{1},X_{2},X_{5}\\}$ with non-linear mean effects in this
scenario. Meanwhile, although the performance in case (III) of panel (d) is
largely improved when compared to case (III) of panel (c), the false positive
errors of the Breusch-Pagan test + Twohit are still too high to be useful in
such a scenario. On the power side, we see that the VDBP test + HBART has
satisfactory selection power and outperforms the power of the VDBP test +
random forests in Table 5, where the power of the latter test decreases when
the distribution of the feature vector becomes a multivariate t and/or the
dimensionality of $p$ increases. From panel (d), the use of Twohit maintains
the test power of the Breusch-Pagan test in the high-dimensional case.
Overall, these experiment results illustrate that the VD and VDBP tests are
appropriate for the respective null hypotheses (2) and (3) in challenging
learning situations with non-linear data generating models and high-
dimensional features.

###  D.2 Numerical experiment for hidden Markov model knockoffs

In our real data study, we generate knockoff features for mutation features in
the HIV-1 dataset with hidden Markov model (HMM) knockoffs sesia2019gene,
which was developed for generating knockoff features for single-nucleotide
polymorphisms. In this section, we perform numerical experiments to examine
the empirical performance of HMM knockoffs for generating knockoff features
given mutation samples from participants treated with PI drugs. The data
description and data preparation is given in Section 6.1. The number of
features and sample size are respectively $332$ and $846$, where missing
values of response variables in the HIV-1 dataset are not involved here
because we only use the explanatory features from the sample.

With the sample of mutation features, we simulate response variables and then
select relevant features by the VD test, and we report the average selection
performance over $100$ repetitions. Specifically, for the $b$th repetition, we
simulate $Y_{1}^{(b)},\dots,Y_{846}^{(b)}$ according to

$Y_{i}^{(b)}=\sum_{j\in S_{m}^{(b)}}0.5\times x_{ij}+\sum_{j\in
S_{sd}^{(b)}}\big{[}0.5+\bm{1}_{\\{x_{ij}>0\\}}\big{]}\times\varepsilon_{i}^{(b)},$
(A.163)

where $S_{m}^{(b)}$ and $S_{sd}^{(b)}$ with
$(|S_{m}^{(b)}|,|S_{sd}^{(b)}|)=(10,3)$ are random feature indices drawn from
the set of features with more than $60$ mutations in the sample, and
$\varepsilon_{i}^{(b)}$’s are i.i.d. standard Gaussian model errors. Only $31$
features in this sample have more than $60$ mutations. The model coefficients
in (A.163) are calibrated so the sample variance of $Y_{i}^{(b)}$’s are
comparable to the original dataset, and the mean function is linear for
simplicity. Notice that the use of lowercase $x_{ij}$’s in model (A.163)
emphasizes that the explanatory features are given by a real dataset and hence
are constant in this numerical experiment.

The VD statistics with random forests as the centering method for each feature
are calculated as in Section 6.1. Let $S_{sd}^{(b)}$ and $\widehat{S}^{(b)}$
for $b\in\\{1,\dots,100\\}$ be respectively the set of true relevant features
and the set of selected features with FDR controlled at $0.2$ at the $b$th
repetition. We report in Table 7 the empirical FDR and selection power, which
are given by

$\begin{split}\textnormal{empirical
FDR}&=0.01\times\sum_{b=1}^{100}\frac{\big{|}S_{sd}^{(b)}\texttt{\big{\backslash}}\widehat{S}^{(b)}\big{|}}{\max\\{1,|\widehat{S}^{(b)}|\\}},\\\
\textnormal{empirical
power}&=0.01\times\sum_{b=1}^{100}\frac{\big{|}S_{sd}^{(b)}\cap\widehat{S}^{(b)}\big{|}}{3}.\end{split}$
(A.164)

From Table 7, the empirical FDR of the VD test is controlled at the target
level, showing that the VD test with knockoffs produced by HMM knockoffs is a
proper tool for assessing significance of mutations with respect to the
conditional variance of drug resistance level.

Target FDR level | Empirical FDR | Empirical power |  #Samples/#Features
---|---|---|---
0.2 | 0.1225 | 0.42 | 846/332
Table 7: Binary explanatory features are from the sample of participants
treated with PI drugs; responses are generated by model (A.163). Empirical FDR
and power are given in (A.164).

##  Appendix E Coordinate-wise Gaussian knockoff generator

To generate ideal knockoffs satisfying the requirements such as Definition 1
is not easy. As a result, existing knockoff generators Barber2015,
CandesFanJansonLv2018, romano2020deep, lu2018deeppink, jordon2018knockoffgan
aim at producing good approximations of ideal knockoff features for practical
statistical inference. In this section, based on Definition 1, we implement a
coordinate-wise Gaussian knockoff generator that manufactures approximate
knockoff vectors given a sample $\\{\bm{X}_{i}=(X_{i,1},\dots
X_{i,p})^{\top}\\}_{i=1}^{n}$ of i.i.d. Gaussian feature vectors in a
coordinate-wise fashion. The R package knockoff is a closely related method
that produces such approximate Gaussian knockoff vectors, but knockoff is
based on the original knockoffs defined in Definition 2 and therefore it does
not enjoy the advantage brought by coordinate-wise knockoffs. To appreciate
the advantage, we demonstrate below the reduced sample correlations between
variables and their coordinate-wise knockoffs due to our method.

We now introduce our procedure for generating the $j$th coordinate-wise
knockoff feature. Here, we denote the generated sample knockoff features by
$\\{\widetilde{\bm{X}}_{i}=(\widetilde{X}_{i,1},\dots,\widetilde{X}_{i,p})^{\top}\\}_{i=1}^{n}$
for simplicity; the notation $\widetilde{\bm{X}}$ denotes the population ideal
knockoffs. Following the idea of (4), we introduce two sampling procedures for
sampling approximate coordinate-wise Gaussian knockoffs. 1) We sample
$\\{\widetilde{X}_{ij}\\}_{i=1}^{n}$ from
$\widehat{\mathbb{P}}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}\big{)}$ given
sample $\\{\bm{X}_{i}\\}_{i=1}^{n}$, where
$\widehat{\mathbb{P}}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}\big{)}$ denotes
the sample conditional distribution of
$\mathbb{P}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}\big{)}$ and that the joint
distribution of $(\widetilde{X}_{j},\bm{X})$ satisfies Definition 1. For this
sampling procedure, we make the correlation between $X_{j}$ and
$\widetilde{X}_{j}$ as small as possible, while keeping the covariance matrix
of $(\bm{{X}},\widetilde{X}_{j})$ non-singular. 2) We sample
$\\{\widetilde{X}_{ij}\\}_{i=1}^{n}$ from
$\widehat{\mathbb{P}}\big{(}X_{j}\in\cdot|\bm{X}_{-j}\big{)}$ given sample
$\\{\bm{X}_{i}\\}_{i=1}^{n}$, where
$\widehat{\mathbb{P}}\big{(}X_{j}\in\cdot|\bm{X}_{-j}\big{)}$ denotes the
sample conditional distribution of
$\mathbb{P}\big{(}X_{j}\in\cdot|\bm{X}_{-j}\big{)}$. The idea of the first
sampling procedure is also rooted in Barber2015, CandesFanJansonLv2018. Among
these two procedures, we implement the first sample procedure here because it
allows us to control and reduce the correlation between $X_{j}$ and
$\widetilde{X}_{j}$.

To implement the first sample procedure above, we have to estimate
$\mathbb{P}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}\big{)}$. Now, when $p\geq
n$, a nontrivial
$\widehat{\mathbb{P}}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}\big{)}$ may not
be available numerically since the sample covariance matrix is singular.
Hence, we sample the $j$th knockoff feature from
$\widehat{\mathbb{P}}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}_{\widehat{S}_{j}}\big{)}$
for some selected feature subset $\widehat{S}_{j}\subset\\{1,\dots,p\\}$ as
follows. For each $j\in\\{1,\dots,p\\}$, we screen out features that are less
correlated with the $j$th feature and use only the remaining features in
$\widehat{S}_{j}$ with $|\widehat{S}_{j}|=\min\\{p,\lfloor\underline{k}\times
n\rceil\\}$ and $j\in\widehat{S}_{j}$, where $\lfloor x\rceil$ is the closest
integer to $x\in\mathbb{R}$ and $0<\underline{k}<1$ is some tunning parameter
with default $\underline{k}=0.25$. To estimate the sample conditional
distribution
$\widehat{\mathbb{P}}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}_{\widehat{S}_{j}}\big{)}$,
we first define

$\begin{split}\widehat{\Sigma}^{(j)}(s,\widehat{S}_{j})=\left(\begin{array}[]{cc}\widehat{\Sigma}(\widehat{S}_{j})&\widehat{v}_{j}(s,\widehat{S}_{j})\\\
\widehat{v}_{j}^{\top}(s,\widehat{S}_{j})&\widehat{\Sigma}_{jj}\end{array}\right)\qquad\textnormal{
for each }s\in\mathbb{R},\end{split}$

where $\widehat{\Sigma}=[\widehat{\Sigma}_{lk}]_{l,k=1}^{p}$ is the $(p\times
p)$ sample covariance matrix of the given sample $\\{\bm{X}_{i}\\}_{i=1}^{n}$,
$\widehat{v}_{j}(s,\widehat{S}_{j})$ denotes the $\widehat{S}_{j}$ subset of
the $p$-dimensional vector
$(\widehat{\Sigma}_{1j},\dots,\widehat{\Sigma}_{(j-1),j},\widehat{\Sigma}_{jj}-s,\widehat{\Sigma}_{(j+1),j}\dots,\widehat{\Sigma}_{pj})^{\top}$,
and $\widehat{\Sigma}(\widehat{S}_{j})$ is the submatrix of $\widehat{\Sigma}$
with rows and columns in $\widehat{S}_{j}$. To minimizes the dependence
between $X_{j}$ and $\widetilde{X}_{j}$, we let

$\widehat{s}=\arg\inf_{s\in Q}|\widehat{\Sigma}_{jj}-s|\ \ \textnormal{ where
}\ \
Q=\big{\\{}s:\lambda_{\min}(\widehat{\Sigma}^{(j)}(s,\widehat{S}_{j}))\geq
0\big{\\}}\cup\\{0\\},$

where $\lambda_{\min}(A)$ is the minimum eigenvalue of matrix $A$. This is a
simple minimization problem that involves only a scalar $s$. The
$\widehat{S}_{j}\cup\\{p+1\\}$ subset of the $(p+1)$-dimensional sample mean
vector $(\widehat{q}_{1},\dots,\widehat{q}_{p},\widehat{q}_{j})$, in which
$\widehat{q}_{j}=n^{-1}\sum_{i=1}^{n}X_{ij}$, is denoted by
$\widehat{e}^{(j)}(\widehat{S}_{j}\cup\\{p+1\\})$. With
$\widehat{\Sigma}^{(j)}(\widehat{s},\widehat{S}_{j})$ and
$\widehat{e}^{(j)}(\widehat{S}_{j}\cup\\{p+1\\})$, the sample conditional
distribution
$\widehat{\mathbb{P}}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}_{\widehat{S}_{j}}\big{)}$
can be estimated for sampling $\\{\widetilde{X}_{ij}\\}_{i=1}^{n}$.
Specifically, if a $(|\widehat{S}_{j}|+1)$-dimensional random vector $\bm{u}$
has a multivariate Gaussian distribution with mean
$\widehat{e}^{(j)}(\widehat{S}_{j}\cup\\{p+1\\})$ and covariance matrix
$\widehat{\Sigma}^{(j)}(\widehat{s},\widehat{S}_{j})$, the conditional
distribution of
$\widehat{\mathbb{P}}\big{(}\widetilde{X}_{j}\in\cdot|\bm{X}_{\widehat{S}_{j}}\big{)}$
is calculated as the conditional distribution of last random variable in
$\bm{u}$ on the other variables in $\bm{u}$. See Section 8.1.3 of
petersen2008matrix for a formula for this conditional distribution, where the
Moore-Penrose pseudoinverse is used if
$\widehat{\Sigma}^{(j)}(\widehat{s},\widehat{S}_{j})$ is singular. The above
procedure is done for each $j\in\\{1,\dots,p\\}$ to get
$\\{\widetilde{\bm{X}}_{i}\\}_{i=1}^{n}$.

Next, to demonstrate the advantages of our knockoffs, we set up simulation
experiments as follows. An i.i.d. sample $\\{\bm{X}_{i}\\}_{i=1}^{n}$ of size
$n\in\\{500,700,1000\\}$ and feature dimensionality $p\in\\{20,700\\}$ are
sampled from a zero-mean multivariate Gaussian distribtion with covariance
martrix $\Sigma=[\Sigma_{lk}]_{l,k=1}^{p}$ with $\Sigma_{lk}=\rho^{|l-k|}$ for
some $\rho\in\\{0.4,0.6\\}$. The values of $(p,\rho,n)$ will be specified in
each experiment in Tables 8–9 below. The knockoffs produced by the R package
knockoff with default setting are denoted by
$\\{\widetilde{\bm{X}}_{i}^{(f)}=(\widetilde{X}_{i,1}^{(f)},\dots,\widetilde{X}_{i,p}^{(f)})^{\top}\\}_{i=1}^{n}$.

We first compare the sample correlation between a feature and its knockoff for
each construction. Given a Gaussian sample and its knockoff samples, we
calculate the sample correlation between $\\{X_{i,15}\\}_{i=1}^{n}$ and
$\\{\widetilde{X}_{i,15}^{(f)}\\}_{i=1}^{n}$, which is denoted by
$\widehat{\textnormal{Cor}}(X_{15},\widetilde{X}_{15}^{(f)})$; similarly, the
sample correlation between $\\{X_{i,15}\\}_{i=1}^{n}$ and
$\\{\widetilde{X}_{i,15}\\}_{i=1}^{n}$ is denoted by
$\widehat{\textnormal{Cor}}(X_{15},\widetilde{X}_{15})$. We repeat the
simulation of each case $10$ times and report the average correlation in Table
8, with the parameter values of each case indicated in the table. In addition,
we evaluate the difference between the sample covariance matrix of
$(\bm{X}_{-j},\widetilde{X}_{j})$ and that of
$(\bm{X}_{-j},\widetilde{X}_{j}^{(f)})$ for each $j\in\\{1,\dots,p\\}$.
Specifically, let $\check{\Sigma}=[\check{\Sigma}_{lk}]_{l,k=1}^{p}$ and
$\check{\Sigma}^{(f)}=[\check{\Sigma}_{lk}^{(f)}]_{l,k=1}^{p}$ such that for
each $1\leq l\leq p$ and $1\leq k\leq p$, $\check{\Sigma}_{lk}$ is the sample
covariance between $\\{\widetilde{X}_{il}\\}_{i=1}^{n}$ and
$\\{X_{ik}\\}_{i=1}^{n}$, and $\check{\Sigma}_{lk}^{(f)}$ is the sample
covariance between $\\{\widetilde{X}_{il}^{(f)}\\}_{i=1}^{n}$ and
$\\{X_{ik}\\}_{i=1}^{n}$. In Table 9, we report the average of squared off-
diagonal elements in $\check{\Sigma}-\check{\Sigma}^{(f)}$ for each of the
cases indicated on the top of each column. Each entry in Table 9 is the
average of results from $10$ repetitions.

Given that the R package knockoff is able to produce good approximations of
Gaussian knockoffs that satisfy Definition 2, we know that knockoff produces
good approximations of coordinate-wise Gaussian knockoffs that satisfy
Definition 1. This follows because Definition 1 is more flexible than
Definition 2, as have been discussed in Section 2.1. Now, since knockoff
produces good approximations of coordinate-wise Gaussian knockoffs that
satisfy Definition 1, the results in Table 9 show that our Gaussian knockoffs
are also good approximations of coordinate-wise Gaussian knockoffs. Meanwhile,
the results of Table 8 show that the sample correlations between
$\\{X_{i,15}\\}_{i=1}^{n}$ and $\\{\widetilde{X}_{i,15}\\}_{i=1}^{n}$ are
smaller in all cases, and the correlations can be much smaller in some cases.
We omit the details of experiments for other features for simplicity. These
numerical experiment results demonstrate the empirical advantages of
coordinate-wise knockoffs for our inferences.

$(p,\rho,n)$ | (20, 0.4 500) | (20, 0.6, 500) | (700, 0.4, 700) | (700, 0.6, 700) | (700, 0.6, 1000)
---|---|---|---|---|---
$\widehat{\textnormal{Cor}}(X_{15},\widetilde{X}_{15}^{(f)})$ | 0.20 | 0.54 | 0.20 | 0.40 | 0.97
$\widehat{\textnormal{Cor}}(X_{15},\widetilde{X}_{15})$ | -0.01 | 0.14 | 0.15 | 0.38 | 0.37
Table 8: Each entry is the average of sample correlations over $10$ repetitions, and is rounded to $2$ decimals. The values of $(p,\rho,n)$ are indicated on the top of each column for each case. $(p,\rho,n)$ | (20, 0.6, 500) | (700, 0.6, 700) | (700, 0.6, 1000)
---|---|---|---
$\frac{1}{p(p-1)}\sum_{1\leq k,l\leq p;l\not=k}(\check{\Sigma}_{lk}^{(f)}-\check{\Sigma}_{lk})^{2}$ | 0.0010 | 0.0019 | 0.0006
Table 9: Each entry is the average of results over $10$ repetitions, and is
rounded to $4$ decimals. The values of $(p,\rho,n)$ are indicated on the top
of each column for each case.

chicago references


blocks was useful for computing the coefficients of the various
$\mathcal{N}=8$ superconformal blocks which was discussed in section 2.1.

Next, we discuss the $\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)$. They can be
compactly written in Mellin space as [17]

$\begin{split}&\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)=\int\frac{ds\,dt}{(4\pi
i)^{2}}U^{\frac{s}{2}}V^{\frac{t}{2}}M(s,t)\Gamma(-\frac{s}{2})\Gamma(-\frac{t}{2})\Gamma\left(r_{2}+\frac{s+t}{2}\right)\\\
&\Gamma\left(\frac{1}{2}(-r_{1}-r_{2}+r_{3}+r_{4})-\frac{s}{2}\right)\Gamma\left(\frac{1}{2}(r_{1}-r_{2}-r_{3}+r_{4})-\frac{t}{2}\right)\Gamma\left(\frac{1}{2}(r_{1}+r_{2}+r_{3}-r_{4})+\frac{s+t}{2}\right)\,.\\\
\end{split}$ (A.9)

We would also like the position space expression of their $\log U$ terms for
the purpose of extracting the tree-level anomalous dimensions, which are given
in [75] as181818In the case at hand, the $\bar{D}$ functions involve half-
integer arguments so one has to be careful in reading off the $\log U$ pieces
by paying attention to the location of the poles of $\Gamma$ functions.
However upon inspection it is found that the arguments are such that formula
(C.8) of [75] is still applicable.

$\displaystyle\bar{D}_{r_{1},r_{2},r_{3},r_{4}}(U,V)\big{|}_{\log
U}=-e^{\frac{1}{2}i\pi(r_{1}+r_{2}-r_{3}-r_{4})}\sum_{m,n=0}^{\infty}U^{m}(1-V)^{n}\times$
$\displaystyle\frac{\Gamma(r_{1}+m)\Gamma(r_{2}+m+n)\Gamma\left(\frac{1}{2}(r_{1}+r_{2}-r_{3}+r_{4})+m\right)\Gamma\left(\frac{1}{2}(r_{1}+r_{2}+r_{3}-r_{4})+m+n\right)}{\Gamma(m+1)\Gamma(n+1)\Gamma(r_{1}+r_{2}+2m+n)\Gamma\left(\frac{1}{2}(r_{1}+r_{2}-r_{3}-r_{4}+2)+m\right)}\,.$
(A.10)

The arguments $r_{i}$ of the $\bar{D}$ functions that appear in the $22pp$
tree-level correlator is such that the overall phase is always real.

##  Appendix B Lorentzian inversion at large $c_{T}$

In this Appendix we explain how to extract the CFT-data at one-loop from the
double discontinuity (DD) of a 3d CFT using the Lorentzian inversion formula
following the similar 4d case in [37]. The function that describes the CFT-
data is given by

$\begin{split}c(\Delta,\ell)={\cal
N}(\Delta,\ell)\int_{0}^{1}dzd\bar{z}\frac{\bar{z}-z}{(z\bar{z})^{3}}G_{\ell+2,\Delta-2}(U,V)\text{dDisc}[\mathcal{G}(z,\bar{z})]\,,\end{split}$
(B.1)

where ${\cal N}(\Delta,\ell)$ is some normalization factor that we can fix by
comparing to GFFT, and $\mathcal{G}(z,\bar{z})$ is the 4-point function with
the usual $x$-dependence factored out as in (2.2). Note that the inversion
formula exchanges the roles of dimension and spin in the standard conformal
block $G_{\Delta,\ell}(U,V)$.

An intermediate operator of a given twist $t$ will produce a pole whose
residue is related to its three-point function. We can see this as follows.
Its contribution to the DD in a small $z$ expansion is given by (recall the DD
is taken in the $\bar{z}$ variable around 1)

$\text{dDisc}[\mathcal{G}(z,\bar{z})]=z^{t/2}h(\bar{z})+\cdots\,,$ (B.2)

for some function $h(\bar{z})$ that is analytic at $\bar{z}=1$. On the other
hand, the conformal block is expanded at small $z$ as in (3.15) in terms of
the leading lightcone block $g_{\Delta,\ell}(\bar{z})$ in (3.16), which for
$G_{\ell+2,\Delta-2}(U,V)$ in (B.1) gives

$\begin{split}G_{\Delta,\ell}(U,V)&=z^{2-(\Delta-\ell)/2}g_{\ell+2,\Delta-2}(\bar{z})+\dots\,,\\\
\end{split}$ (B.3)

where we dropped the superscript on the leading lightcone block for simplicity
and at leading order in $z$ we set $V=(1-\bar{z})$. Plugging these expansions
into the inversion formula we see that the integral over $z$ produces a pole
at $\Delta-\ell=t$. If we now expand at large $c_{T}$, then we will have
operators of even integer twist $t$ and spin $\ell$, which are corrected by
anomalous dimensions $\gamma_{\ell}$ that depend on $\ell$ and go to zero as
$c_{T}\to\infty$. The pole at $t$ at large $c_{T}$ then gives

$c(\Delta,\ell)=\frac{a_{\ell}}{\gamma_{\ell}+t-(\Delta-\ell)}\,,$ (B.4)

where $a_{\ell}$ denotes the OPE coefficients squared for the operator with
approximate twist $t$, and we have assumed that the leading twist operators
under consideration are non-degenerate (which will be the case for our
application). If we now expand the CFT data as
$a_{\ell}=a^{(0)}_{\ell}+c_{T}^{-1}a^{(1)}_{\ell}+\cdots$ and
$\gamma_{\ell}=c_{T}^{-1}\gamma^{(1)}_{\ell}+\cdots$ we see that at higher
orders in $c_{T}^{-1}$ we get higher order poles. At order $c_{T}^{-2}$ we
get191919We can get an analogous formula also at other 1-loop orders such as
$c_{T}^{-\frac{8}{3}}$.

$c(\Delta,\ell)=-\frac{R_{0}(t+2\ell)}{((\Delta-\ell)/2-t/2)^{3}}-\frac{R_{1}(t+2\ell)}{((\Delta-\ell)/2-t/2)^{2}}-\frac{R_{2}(t+2\ell)}{(\Delta-\ell)/2-t/2}+\cdots\,.$
(B.5)

These residues are related to the anomalous dimensions and OPE coefficients at
fixed $\ell$ as

$\begin{split}a^{(0)}_{\ell}\left(\gamma^{(1)}_{\ell}\right)^{2}&=8R_{0}(t+2\ell)\,,\\\
a^{(0)}_{\ell}\gamma^{(2)}_{\ell}+a^{(1)}_{\ell}\gamma^{(1)}_{\ell}&=4R_{1}(t+2\ell)+8\partial_{\ell}R_{0}(t+2\ell)\,,\\\
a^{(2)}_{\ell}&=2R_{2}(t+2\ell)+4\partial_{\ell}R_{1}(t+2\ell)+4\partial^{2}_{\ell}R_{0}(t+2\ell)\,.\end{split}$
(B.6)

From the inversion integral, these residues are computed as follows. To order
$O(c_{T}^{-2})$ the contribution to the DD from the operators of approximate
twist $t$ takes the form

$\text{dDisc}[G(z,\bar{z})]=z^{\frac{t}{2}}\left(h_{0}(\bar{z})\log^{2}z+h_{1}(\bar{z})\log
z+h_{2}(\bar{z})\right)+\cdots$ (B.7)

for some functions $h_{i}(\bar{z})$. We can then perform the integral over $z$
keeping the relevant poles in a small $z$ expansion to get

$\begin{split}R_{0}(t+2\ell)&=2{\cal
N}(t+2\ell)\int_{0}^{1}\frac{d\bar{z}}{\bar{z}^{2}}g_{\ell+2,t+\ell-2}(\bar{z})h_{0}(\bar{z})\,,\\\
R_{1}(t+2\ell)&={\cal
N}(t+2\ell)\int_{0}^{1}\frac{d\bar{z}}{\bar{z}^{2}}g_{\ell+2,t+\ell-2}(\bar{z})h_{1}(\bar{z})\,,\\\
R_{2}(t+2\ell)&={\cal
N}(t+2\ell)\int_{0}^{1}\frac{d\bar{z}}{\bar{z}^{2}}g_{\ell+2,t+\ell-2}(\bar{z})h_{2}(\bar{z})\,,\end{split}$
(B.8)

where the normalization coefficient ${\cal N}(t+2\ell)$ can be fixed, for
instance, by requiring the correct CFT-data is recovered for the GFFT. In
particular, the large $c_{T}$ GFFT is proportional to202020For the
$\mathcal{N}=8$ theory in the main text, the precise formula for $\langle
2222\rangle$ is
$\mathcal{G}_{\text{GFFT}}(U,V)=1+\sigma^{2}U+\tau^{2}\frac{U}{V}$.

$\begin{split}\mathcal{G}_{\text{GFFT}}(z,\bar{z})\propto
z\left(\frac{\bar{z}}{1-\bar{z}}\right)^{p}\Big{|}_{p=1}+\text{regular}\,.\end{split}$
(B.9)

We can compute the DD as explained in [30] by first letting $p$ be a general
variable, using

$\begin{split}\text{dDisc}[\left(\frac{\bar{z}}{1-\bar{z}}\right)^{p}]=2\sin^{2}(\pi
p)\left(\frac{\bar{z}}{1-\bar{z}}\right)^{p}\,,\end{split}$ (B.10)

and then computing $\bar{z}$ integral in (B.8), which has a pole at $p=1$ that
cancels the zero in (B.10) to give a finite nonzero answer for $R_{0}$ in
(B.8). We can then compare to the expected GFFT values to fix the
normalization.

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|  Hong Jin Kang is working towards the PhD degree at Singapore Management
University. He received the bachelor degree in Computer Science from National
University of Singapore. Currently, his research focuses on mining rules and
specifications for software engineering.
---|---
|  David Lo is an ACM Distinguished Member and a Professor of Computer Science
with Singapore Management University, leading the Software Analytics Research
(SOAR) group. His research interests include the intersection of software
engineering, cybersecurity and data science, encompassing socio-technical
aspects and analysis of different kinds of software artefacts, with the goal
of improving software quality and security, and developer productivity. His
work has been published in major and premier conferences and journals in the
area of software engineering, AI, and cybersecurity. By March 2021, he has
received more than 15 international research and service awards, including six
ACM SIGSOFT Distinguished Paper Awards.
---|---