$\displaystyle+2I_{jk}W_{2,jk}W_{3,jk}\sum_{i=1}^{n}V_{10,ijk}V_{01,ijk}$
$\displaystyle=$
$\displaystyle\widetilde{\Psi}_{jk}^{(1)}+\widetilde{\Psi}_{jk}^{(2)}+\dots+\widetilde{\Psi}_{jk}^{(5)}+\widetilde{\Psi}_{jk}^{(6)}.$

We first focus on the term $\widetilde{\Psi}_{jk}^{(1)}.$ For $a,b=0,1,2,$
define

$\displaystyle V_{ab,ijk}^{(1)}(u,v)$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}g_{ab}\big{(}h_{C},(u,v),(U_{ijl},U_{ikm})\big{)}\big{\\{}Z_{ijl}Z_{ikm}-\Sigma_{jk}(u,v)\big{\\}},$
$\displaystyle V_{ab,ijk}^{(2)}(u,v)$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}g_{ab}\big{(}h_{C},(u,v),(U_{ijl},U_{ikm})\big{)}\big{\\{}\Sigma_{jk}(u,v)-\widetilde{\Sigma}_{jk}(u,v)\big{\\}}.$

The term $\widetilde{\Psi}_{jk}^{(1)}$ can be re-expressed as

$\displaystyle\widetilde{\Psi}_{jk}^{(1)}$ $\displaystyle=$ $\displaystyle
I_{jk}W_{1,jk}^{2}\sum_{i=1}^{n}\big{\\{}V_{00,ijk}^{(1)}\big{\\}}^{2}+I_{jk}W_{1,jk}^{2}\sum_{i=1}^{n}\big{\\{}V_{00,ijk}^{(2)}\big{\\}}^{2}+2I_{jk}W_{1,jk}^{2}\sum_{i=1}^{n}V_{00,ijk}^{(1)}V_{00,ijk}^{(2)}$
(S.17) $\displaystyle=$ $\displaystyle D_{jk,1}+D_{jk,2}+D_{jk,3}.$

(a) Verification of $I_{jk}$. To show the rationale of imposing the rate
$I_{jk}$ in (10), we need to verify that

$I_{jk}\sum_{i=1}^{n}\big{(}W_{1,jk}V_{00,ijk}+W_{2,jk}V_{10,ijk}+W_{3,jk}V_{01,ijk}\big{)}^{2}\asymp
1+o_{P}(1).$ (S.18)

for each $u,v\in{\cal U}.$ Denote by
$\tilde{n}_{jk}=\sum_{i=1}^{n}L_{ij}L_{ik}.$ Recall that

$S_{ab,jk}(u,v)=\sum_{i=1}^{n}\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}g_{ab}\big{\\{}h_{C},(u,v),(U_{ijl},U_{ikm})\big{\\}}.$

It can be shown that
$S_{ab,jk}(u,v)\asymp\tilde{n}_{jk}h_{C}^{a+b}\\{1+o_{P}(1)\\}$ for
$a,b=0,1,2,$ which together with (S.12) implies that

$W_{1,jk}(u,v)\asymp\tilde{n}_{jk}^{-1}\big{\\{}1+o_{P}(1)\big{\\}},~{}~{}W_{2,jk}(u,v)\asymp
W_{3,jk}(u,v)\asymp\tilde{n}_{jk}^{-1}h_{C}^{-1}\big{\\{}1+o_{P}(1)\big{\\}}.$
(S.19)

Similarly, we can also show that

$\displaystyle\sum_{i=1}^{n}\Big{\\{}\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}g_{00}\big{(}h_{C},(u,v),(U_{ijl},U_{ikm})\big{)}\Big{\\}}^{2}$
(S.20) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}K_{h_{C}}^{2}({U_{ijl}-u})K_{h_{C}}^{2}({U_{ikm}-v})$
$\displaystyle+\sum_{i=1}^{n}\sum_{l=1}^{L_{ij}}\sum_{m^{\prime}\neq
m}^{L_{ik}}K_{h_{C}}^{2}({U_{ijl}-u})K_{h_{C}}({U_{ikm}-v})K_{h_{C}}({U_{ikm^{\prime}}-v})$
$\displaystyle+\sum_{i=1}^{n}\sum_{l\neq
l^{\prime}}\sum_{m=1}^{L_{ik}}K_{h_{C}}({U_{ijl}-u})K_{h_{C}}({U_{ijl^{\prime}}-u})K_{h_{C}}^{2}({U_{ikm}-v})$
$\displaystyle+\sum_{i=1}^{n}\sum_{l\neq l^{\prime}}^{L_{ij}}\sum_{m\neq
m^{\prime}}^{L_{ik}}K_{h_{C}}({U_{ijl}-u})K_{h_{C}}({U_{ikm}-v})K_{h_{C}}({U_{ijl^{\prime}}-u})K_{h_{C}}({U_{ikm^{\prime}}-v})$
$\displaystyle\asymp$
$\displaystyle\Big{\\{}\sum_{i=1}^{n}\big{(}L_{ij}L_{ik}h_{C}^{-2}+L_{ij}^{2}L_{ik}h_{C}^{-1}+L_{ij}L_{ik}^{2}h_{C}^{-1}+L_{ij}^{2}L_{ik}^{2}\big{)}\Big{\\}}\big{\\{}1+o_{P}(1)\big{\\}}.$

By (S.19) and (S.20), we obtain that

$W_{1,jk}^{2}\sum_{i=1}^{n}\Big{\\{}\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}g_{00}\big{(}h_{C},(u,v),(U_{ijl},U_{ikm})\big{)}\Big{\\}}^{2}\asymp
I_{jk}^{-1}\big{\\{}1+o_{P}(1)\big{\\}},$ (S.21)

which together with $\Sigma_{jk}(u,v)-\widetilde{\Sigma}_{jk}(u,v)=o_{P}(1)$
implies that $D_{jk,2}=o_{P}(1)$ and $D_{jk,3}=o_{P}(1).$ Note that
$\mathbb{E}\big{\\{}Z_{ijl}Z_{ikm}-\Sigma_{jk}(u,v)\big{\\}}^{2}$ is bounded.
Together with (S.19) and (S.20), we can also show that $D_{jk,1}(u,v)\asymp
1+o_{P}(1).$ Combining the above results yields that
$\widetilde{\Psi}_{jk}^{(1)}\asymp 1+o_{P}(1).$ In a similar fashion, we can
also show that $\widetilde{\Psi}_{jk}^{(i)}\asymp 1+o_{P}(1)$ for
$i=2,\ldots,6$ in (S.16) and hence (S.18) follows.

(b) Verification of Condition 8. To verify the uniform convergence rate in
Condition 8, we need to refine our analysis above to construct the exponential
type of tail bounds on $\widetilde{\Psi}_{jk}(u,v)-{\Psi}_{jk}(u,v)$ at each
$(u,v)\in{\cal U}^{2}$ rather than the consistency results in (a).

Consider the first term $D_{jk,1}(u,v)=(\tilde{n}_{jk}W_{1,jk})^{2}\times
I_{jk}\tilde{n}_{jk}^{-2}\sum_{i=1}^{n}\\{V_{00,ijk}^{(1)}\\}^{2}$ in (S.17).
Note that by (S.19) $\tilde{n}_{jk}|W_{1,jk}|$ is bounded with an overwhelming
probability. Suppose that $X_{ij}(\cdot)$’s are sub-Gaussian processes and
$\varepsilon_{ijl}$’s are independent sub-Gaussian errors. Since
$\\{V_{00,ijk}^{(1)}(u,v),i=1,\ldots,n\\}$ forms an independent sequence, we
can obtain the tail bound on $D_{jk,1}(u,v)-\mathbb{E}\\{D_{jk,1}(u,v)\\}$ by
calculating all $q$-th moments of
$\zeta_{ijk}=\big{\\{}V_{00,ijk}^{(1)}(u,v)\big{\\}}^{2}-\mathbb{E}\big{[}\\{V_{00,ijk}^{(1)}(u,v)\\}^{2}\big{]}$
for $q=2,3,4,\dots$ under regularity conditions. Since $\zeta_{ijk}$’s are
either sub-Gaussian or sub-exponential, we can follow the similar techniques
to prove Lemma 5 of Qiao et al. (2020) by adopting a truncation technique and
then applying Bernstein inequality (Boucheron et al., 2014) to establish a
rough exponential type of concentration inequality (i.e., the equipped tail
bound is in the same form of the exponential tail bound in (15)) for
$D_{jk,1}(u,v)$ at each $(u,v)\in{\cal U}^{2}.$ Similarly, we can also derive
the exponential type of concentration inequality for the third term
$D_{jk,3}(u,v)$ in (S.17).

Consider the second term $D_{jk,2}(u,v)$ in (S.17), which can be re-expressed
as

$D_{jk,2}(u,v)=2I_{jk}W_{1,jk}^{2}\sum_{i=1}^{n}\Big{\\{}\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}g_{ab}\big{(}h_{C},(u,v),(U_{ijl},U_{ikm})\big{)}\Big{\\}}^{2}\Big{\\{}\Sigma_{jk}(u,v)-\widetilde{\Sigma}_{jk}(u,v)\Big{\\}}^{2}.$

Note that it follows from (S.21) that
$I_{jk}W_{1,jk}^{2}\sum_{i=1}^{n}\big{\\{}\sum_{l=1}^{L_{ij}}\sum_{m=1}^{L_{ik}}g_{ab}(h_{C},(u,v),(U_{ijl},U_{ikm}))\big{\\}}^{2}$
is bounded with an overwhelming probability. Then the exponential type of
concentration bound on $D_{jk,2}(u,v)$ at each $(u,v)\in{\cal U}^{2}$ can be
obtained through the exponential type tail bound on
$\Sigma_{jk}(u,v)-\widetilde{\Sigma}_{jk}(u,v)$, which has been established in
Qiao et al. (2020), see details in proofs of its Lemmas 4 and 5 under the
sparse and dense designs, respectively.

To derive the uniform (i.e., over ${\cal U}^{2}$) concentration inequality for
$\widetilde{\Psi}_{jk}^{(1)}(u,v)$ in (S.17), we can apply the partition
technique that reduces the problem from supremum over ${\cal U}^{2}$ to the
maximum over a grid of pairs and then follow the similar developments to prove
the uniform concentration inequalities in Lemmas 4 and 5 of Qiao et al.
(2020). In a similar fashion to the above procedure, we can develop the
corresponding exponential type of uniform concentration inequality for
$\widetilde{\Psi}_{jk}^{(i)}(u,v)$ for $i=2,\dots,6.$ As a result, the
exponential type of uniform concentration inequality for
$\widetilde{\Psi}_{jk}(u,v)$ can be obtained.

The uniform convergence rate in Condition 8 is implied by the exponential type
of uniform concentration inequalities for $\widetilde{\Psi}_{jk}(u,v)$ for
each $j,k,$ which partially depend on the uniform concentration bounds on
$\widetilde{\Sigma}_{jk}(u,v)$’s. In a similar spirit to the $L_{2}$
concentration bounds on $\widetilde{\Sigma}_{jk}(u,v)$’s implied by Condition
7, we consider the uniform convergence rate of $\widetilde{\Sigma}_{jk}(u,v),$

$\max_{1\leq j,k\leq p}\underset{u,v\in{\cal
U}}{\sup}\Big{|}\widetilde{\Sigma}_{jk}(u,v)-\Sigma_{jk}(u,v)\Big{|}=O_{P}\left(\sqrt{\frac{\log
p}{n^{2\gamma_{1}}}}+h^{2}\right),$ (S.22)

which is satisfied if there exists some positive constants $c_{i}$ for
$i=6,\dots,9$ and $\gamma_{1}\in(0,1/2]$ such that for each $j,k=1,\dots,p$
and $t\in(0,1],$

$P\Big{\\{}{\sup}_{u,v\in{\cal
U}}|\widetilde{\Sigma}_{jk}(u,v)-\Sigma_{jk}(u,v)|\geq
t+c_{8}h^{2}\Big{\\}}\leq c_{7}n^{c_{9}}\exp(-{c_{6}n^{2\gamma_{1}}t^{2}}).$
(S.23)

Larger values of $\gamma_{1}$ correspond to a more frequent measurement
schedule and hence faster rate in (S.22). For sparsely sampled functional
data, it follows from Lemma 4 of Qiao et al. (2020) and the same proof
technique for $j\neq k$ that (S.23) holds by choosing $\gamma_{1}=1/2-a$ and
$c_{9}=1+2a$ with $h\asymp n^{-a}$ for some positive constant $a<1/2.$ For
densely sampled functional data, it follows from Lemma 5 of Qiao et al. (2020)
and more efforts for $j\neq k$ that (S.23) holds with the choice of
$\gamma_{1}=\min(1/2,1/3+b/6-\epsilon^{\prime}/2-2a/3)$ and
$c_{9}=\max(1,2/3-\epsilon^{\prime}-b/3+4a/3)$ for some small constant
$\epsilon^{\prime}>0$ when $h\asymp n^{-a}$ and $L\asymp n^{b}$ for some
positive constants $a,b.$

Following the proof procedure described above, we can establish exponential
type of uniform concentration inequality for $\widetilde{\Psi}_{jk}(u,v)$ for
each $j,k$ in the same form as (S.23) but with different positive constants
and in particular $\gamma_{2}\in(0,1/2],$ which will result in the uniform
convergence rate in Condition 8. It is worth mentioning that such heuristic
analysis can only help us establish uniform concentration inequalities for
$\widetilde{\Psi}_{jk}(u,v)$’s leading to the sub-optimal rate. Investigating
the corresponding optimal rate through the precise specification of the
largest values of $\gamma_{2}$ under different measurement schedules or more
generally through $n,h$ and possibly $L$ for the dense case is quite
challenging and remains an open topic to be pursued in the future.

##  Appendix D Additional empirical results

###  D.1 Simulation studies

####  D.1.1 Fully observed functional data

Figures 4 and 5 plot the heat maps of the frequency of the zeros identified
for the Hilbert–Schimidt norm of each entry of the estimated covariance
function, when $p=50,$ out of 100 simulation runs. The true nonzero patterns
of Model 1 and 2 are presented in Figures 4(a) and 5(a), respectively. Figure
6 displays the average receiver operating characteristic (ROC) curves (plots
of true positive rates versus false positive rates over a sequence of
$\lambda$ values) for both the adaptive functional thresholding and universal
functional thresholding methods. These results again demonstrate the uniform
superiority of the adaptive functional thresholding method in terms of graph
selection consistency.

(a) True

(b) Hard $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(c) Hard $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

(d) Soft $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(e) Soft $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

(f) SCAD $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(g) SCAD $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

(h) Adap. lasso $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(i) Adap. lasso $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

Figure 4:  Heat maps of the frequency of the zeros identified for the
Hilbert–Schimidt norm of each entry of the estimated covariance function (when
$p$ = 50) for Model 1 out of 100 simulation runs. White and black correspond
to 100/100 and 0/100 zeros identified, respectively.

(a) True

(b) Hard $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(c) Hard $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

(d) Soft $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(e) Soft $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

(f) SCAD $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(g) SCAD $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

(h) Adap. lasso $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$

(i) Adap. lasso $\widehat{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$

Figure 5:  Heat maps of the frequency of the zeros identified for the
Hilbert–Schimidt norm of each entry of the estimated covariance function (when
$p$ = 50) for Model 2 out of 100 simulation runs. White and black correspond
to 100/100 and 0/100 zeros identified, respectively.  Figure 6: Model 1 (top
row) and Model 2 (bottom row) for $p=50,100,150$: Comparison of the average
ROC curves for adaptive functional thresholding (solid line) and universal
functional thresholding (dotted line) over 100 simulation runs.

####  D.1.2 Partially observed functional data

Tables 8 and 9 summarize the estimation and support recovery performance of
BinLLS-based adaptive and universal functional thresholding estimators for the
setting of $p=100$ satisfying Models 1 and 2 under different measurement
schedules. The same patterns as those from Tables 5 and 6 can be observed.

Table 8:  The average (standard error) functional matrix losses for partially
observed functional scenarios and $p=100$ over 100 simulation runs.

|  | $L_{i}=11$ | $L_{i}=21$ | $L_{i}=51$ | $L_{i}=101$
---|---|---|---|---|---
Model | Method | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$
1 | Functional Frobenius norm
Hard | 11.40(0.03) | 18.34(0.01) | 9.63(0.03) | 17.80(0.01) | 8.55(0.04) | 17.51(0.01) | 8.17(0.04) | 17.42(0.01)
Soft | 12.79(0.05) | 18.33(0.01) | 11.28(0.05) | 17.71(0.02) | 10.33(0.05) | 16.68(0.07) | 10.01(0.05) | 16.06(0.07)
SCAD | 12.41(0.05) | 18.33(0.01) | 10.58(0.05) | 17.72(0.02) | 9.42(0.05) | 16.77(0.06) | 9.01(0.05) | 16.23(0.07)
Adap. lasso | 11.22(0.04) | 18.33(0.01) | 9.59(0.04) | 17.79(0.01) | 8.54(0.04) | 17.49(0.01) | 8.19(0.04) | 17.34(0.03)
Functional matrix $\ell_{1}$ norm
Hard | 5.97(0.05) | 9.41(0.01) | 5.15(0.05) | 9.35(0.01) | 4.70(0.05) | 9.33(0.01) | 4.53(0.05) | 9.32(0.01)
Soft | 7.06(0.04) | 9.41(0.01) | 6.55(0.05) | 9.34(0.01) | 6.23(0.05) | 9.19(0.02) | 6.12(0.05) | 9.02(0.03)
SCAD | 6.93(0.05) | 9.41(0.01) | 6.20(0.05) | 9.34(0.01) | 5.74(0.05) | 9.23(0.02) | 5.56(0.05) | 9.11(0.03)
Adap.lasso | 6.00(0.05) | 9.41(0.01) | 5.32(0.06) | 9.35(0.01) | 4.89(0.06) | 9.32(0.01) | 4.74(0.06) | 9.32(0.01)
2 | Functional Frobenius norm
Hard | 13.21(0.04) | 17.03(0.01) | 11.33(0.04) | 16.40(0.01) | 10.06(0.04) | 16.06(0.01) | 9.60(0.04) | 15.96(0.01)
Soft | 13.54(0.04) | 17.01(0.01) | 12.06(0.04) | 16.26(0.02) | 11.10(0.04) | 15.32(0.05) | 10.75(0.04) | 14.86(0.05)
SCAD | 13.50(0.04) | 17.01(0.01) | 11.90(0.04) | 16.26(0.02) | 10.78(0.04) | 15.35(0.05) | 10.36(0.04) | 14.93(0.05)
Adap. lasso | 12.61(0.04) | 17.01(0.01) | 10.94(0.04) | 16.39(0.01) | 9.80(0.04) | 15.99(0.02) | 9.37(0.04) | 15.81(0.03)
Functional matrix $\ell_{1}$ norm
Hard | 6.14(0.04) | 7.27(0.01) | 5.49(0.04) | 7.19(0.01) | 5.01(0.05) | 7.16(0.01) | 4.83(0.05) | 7.15(0.01)
Soft | 6.22(0.02) | 7.26(0.01) | 5.90(0.03) | 7.16(0.01) | 5.65(0.03) | 7.03(0.02) | 5.55(0.03) | 6.97(0.02)
SCAD | 6.21(0.02) | 7.26(0.01) | 5.87(0.03) | 7.16(0.01) | 5.58(0.03) | 7.04(0.02) | 5.45(0.03) | 6.99(0.02)
Adap. lasso | 5.88(0.04) | 7.26(0.01) | 5.42(0.04) | 7.19(0.01) | 5.04(0.04) | 7.15(0.01) | 4.87(0.04) | 7.14(0.01)

Table 9:  The average TPRs/ FPRs for partially observed functional scenarios
and $p=100$ over 100 simulation runs.

|  | $L_{i}=11$ | $L_{i}=21$ | $L_{i}=51$ | $L_{i}=101$
---|---|---|---|---|---
Model | Method | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{A}}$ | $\widecheck{\boldsymbol{\Sigma}}_{\scriptscriptstyle\textup{U}}$
1 | Hard | 0.57/0.00 | 0.00/0.00 | 0.62/0.00 | 0.00/0.00 | 0.65/0.00 | 0.00/0.00 | 0.66/0.00 | 0.00/0.00
Soft | 0.80/0.03 | 0.00/0.00 | 0.83/0.04 | 0.03/0.01 | 0.85/0.04 | 0.24/0.05 | 0.85/0.04 | 0.36/0.07
SCAD | 0.81/0.03 | 0.00/0.00 | 0.84/0.04 | 0.03/0.01 | 0.85/0.04 | 0.22/0.04 | 0.85/0.04 | 0.32/0.06
Adap. lasso | 0.67/0.00 | 0.00/0.00 | 0.71/0.00 | 0.00/0.00 | 0.73/0.00 | 0.00/0.00 | 0.74/0.00 | 0.01/0.00
2 | Hard | 0.48/0.00 | 0.00/0.00 | 0.57/0.00 | 0.00/0.00 | 0.65/0.00 | 0.00/0.00 | 0.68/0.00 | 0.00/0.00
Soft | 0.90/0.03 | 0.00/0.00 | 0.94/0.04 | 0.07/0.01 | 0.96/0.04 | 0.29/0.04 | 0.97/0.04 | 0.40/0.05
SCAD | 0.90/0.03 | 0.00/0.00 | 0.95/0.04 | 0.06/0.01 | 0.96/0.05 | 0.28/0.03 | 0.97/0.05 | 0.37/0.04
Adap. lasso | 0.70/0.00 | 0.00/0.00 | 0.78/0.00 | 0.00/0.00 | 0.83/0.00 | 0.02/0.00 | 0.85/0.00 | 0.03/0.00

###  D.2 Real data

Figures 7 and 8 display the pre-smoothed BOLD signal trajectories at a
selection of ROIs of subjects from the ADHD and HCP datesets, respectively.
Figures 9 and 10 plot the connectivity strengths at fluid intelligence
$\textit{gF}\leq 8$ and $\textit{gF}\geq 23$ in Fig. 2(a)–(b) and Fig.
2(c)–(d), respectively. We observe that as gF increases, the connectivity
strengths in the medial frontal and frontoparietal modules tend to increase
while those in the default mode module decrease, which is consistent with our
finding in Section 6.2.

Figure 7:  ADHD dataset: the smoothed BOLD signals at the first $5$ ROIs of
two subjects in ADHD and TDC groups respectively. The $5.73$-minute interval
with $172$ scanning points is rescaled to $[0,1].$ Figure 8: HCP dataset: the
smoothed BOLD signals at the first $5$ ROIs of one subject. The $14.40$-minute
interval with $1200$ scanning points ($14.40$ mins) is rescaled to $[0,1].$

(a) $\textit{gF}\leq 8$: the medial frontal module in Fig. 2(a)

(b) $\textit{gF}\leq 8$: the frontoparietal module in Fig. 2(a)

(c) $\textit{gF}\leq 8$: the default mode module in Fig. 2(a)

(d) $\textit{gF}\geq 23$: the medial frontal module in Fig. 2(b)

(e) $\textit{gF}\geq 23$: the frontoparietal module in Fig. 2(b)

(f) $\textit{gF}\geq 23$: the default mode module in Fig. 2(b)

Figure 9:  The connectivity strengths in Fig. 2(a)–(b) at fluid intelligence
$\textit{gF}\leq 8$ and $\textit{gF}\geq 23$. Salmon, orange and yellow nodes
represent the ROIs in the medial frontal, frontoparietal and default mode
modules, respectively. The edge color from cyan to blue corresponds to the
value of $\|\widehat{\Sigma}_{jk}^{\scriptscriptstyle\textup{A}}\|_{\cal
S}/\\{\|\widehat{\Sigma}_{jj}^{\scriptscriptstyle\textup{A}}\|_{\cal
S}\|\widehat{\Sigma}_{kk}^{\scriptscriptstyle\textup{A}}\|_{\cal S}\\}^{1/2}$
from small to large.

(a) $\textit{gF}\leq 8$: the medial frontal module in Fig. 2(c)

(b) $\textit{gF}\leq 8$: the frontoparietal module in Fig. 2(c)

(c) $\textit{gF}\leq 8$: the default mode module in Fig. 2(c)

(d) $\textit{gF}\geq 23$: the medial frontal module in Fig. 2(d)

(e) $\textit{gF}\geq 23$: the frontoparietal module in Fig. 2(d)

(f) $\textit{gF}\geq 23$: the default mode module in Fig. 2(d)

Figure 10:  The connectivity strengths in Fig. 2(c)–(d) at fluid intelligence
$\textit{gF}\leq 8$ and $\textit{gF}\geq 23$. Salmon, orange and yellow nodes
represent the ROIs in the medial frontal, frontoparietal and default mode
modules, respectively. The edge color from cyan to blue corresponds to the
value of $\|\widehat{\Sigma}_{jk}^{\scriptscriptstyle\textup{A}}\|_{\cal
S}/\\{\|\widehat{\Sigma}_{jj}^{\scriptscriptstyle\textup{A}}\|_{\cal
S}\|\widehat{\Sigma}_{kk}^{\scriptscriptstyle\textup{A}}\|_{\cal S}\\}^{1/2}$
from small to large.

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