Var}}(\bar{X}_{k\cdot}|\widehat{W})||_{F}+||\operatorname{{\rm
Var}}(\bar{X}_{m\cdot}|\widehat{W})||_{F}+2\cdot||\mathrm{Cov}(\bar{X}_{k\cdot},\bar{X}_{m\cdot}|\widehat{W})||_{F}$
$\displaystyle\leq C_{1}\cdot\underset{k}{\text{max}}||\operatorname{{\rm
Var}}(\bar{X}_{k\cdot}|\widehat{W})||_{F}.$

This holds for all pairs $(k,m)$, and so it also holds for the maximum over
the pairs:

$\displaystyle\max_{k,m}||H_{k,m}^{+}||_{F}$ $\displaystyle\leq
C_{1}\cdot\underset{k}{\max}||\operatorname{{\rm
Var}}(\bar{X}_{k\cdot}|\widehat{W})||_{F}=C_{1}\max_{k}\|\widehat{L}^{T}\operatorname{{\rm
Var}}(X_{k\cdot})\widehat{L}\|_{F}=C_{1}K_{2}^{-1/2}\|X\|_{\widehat{W}}.$
(H.2)

Thus, with probability greater than $1-\frac{1}{2p}$,

$||\widehat{\Sigma}_{p,\widehat{W}}-\Sigma_{p,\widehat{W}}||_{\max}\leq
c_{1}\sqrt{\frac{\log
p}{nK_{2}}}\|X\|_{\widehat{W}}\leq\eta\|X\|_{\widehat{W}}$ (H.3)

for some constant $c_{1}$, where the last step is from condition (A1).

Note that conditions (A1) and (A2) together imply that

$\mathrm{MCOD}(\Sigma_{p,\widehat{W}})/\|X\|_{\widehat{W}}\geq 4\eta.$ (H.4)

In the following, we show how perfect clustering is achieved under the event
that the inequality (H.3) holds. For simplicity, we drop the subscript
$\widehat{W}$ in $\widehat{\Sigma}_{p,\widehat{W}},\Sigma_{p,\widehat{W}}$ and
$\|X\|_{\widehat{W}}$ and just write $\widehat{\Sigma}_{p},\Sigma_{p}$ and
$\|X\|$. Also, for simplicity, in the following derivation, we will denote the
$(i,j)$-th element of $\Sigma_{p}$ as $\Sigma_{p,ij}$. Define the following
quantity

$\tau:=\underset{\underset{c\neq
a,b}{a,b,c}}{\max}\big{|}\widehat{\Sigma}_{p,ac}-\widehat{\Sigma}_{p,bc}-(\Sigma_{p,ac}-\Sigma_{p,bc})\big{|}.$

Then we have:

$\displaystyle\tau$ $\displaystyle=\underset{\underset{c\neq
a,b}{a,b,c}}{\max}\big{|}\widehat{\Sigma}_{p,ac}-\Sigma_{p,ac}-(\widehat{\Sigma}_{p,bc}-\Sigma_{p,bc})\big{|}$
$\displaystyle\leq\underset{\underset{c\neq
a,b}{a,b,c}}{\max}\big{|}\widehat{\Sigma}_{p,ac}-\Sigma_{p,ac}\big{|}+\underset{\underset{c\neq
a,b}{a,b,c}}{\max}\big{|}\widehat{\Sigma}_{p,bc}-\Sigma_{p,bc}\big{|}$
$\displaystyle\leq 2\cdot||\widehat{\Sigma}_{p}-\Sigma_{p}||_{\max}$
$\displaystyle\leq 2\eta\cdot||X||,$

where the last step is from (H.3). We now want to show the following
inequality:

$\mathrm{COD}_{\widehat{\Sigma}_{p}}(a,b)-\tau\leq\mathrm{COD}_{\Sigma_{p}}(a,b)\leq\mathrm{COD}_{\widehat{\Sigma}_{p}}(a,b)+\tau.$
(H.5)

The inequality on the left holds because of the following:

$\displaystyle\mathrm{COD}_{\widehat{\Sigma}_{p}}(a,b)$
$\displaystyle=\underset{c\neq
a,b}{\max}\big{|}\widehat{\Sigma}_{p,ac}-\widehat{\Sigma}_{p,bc}\big{|}$
$\displaystyle=\underset{c\neq
a,b}{\max}\big{|}\widehat{\Sigma}_{p,ac}-\widehat{\Sigma}_{p,bc}-(\Sigma_{p,ac}-\Sigma_{p,bc})+(\Sigma_{p,ac}-\Sigma_{p,bc})\big{|}$
$\displaystyle\leq\underset{c\neq
a,b}{\max}\big{|}\Sigma_{p,ac}-\Sigma_{p,bc}\big{|}+\underset{c\neq
a,b}{\max}\big{|}\widehat{\Sigma}_{p,ac}-\widehat{\Sigma}_{p,bc}-(\Sigma_{p,ac}-\Sigma_{p,bc})\big{|}$
$\displaystyle\leq\mathrm{COD}_{\Sigma_{p}}(a,b)+\tau.$ (H.6)

With a similar technique, we can get the inequality on the right as well. Now,
for any two sets $A$ and $B$, if $A$ and $B$ indeed belong to the same cluster
(i.e., $a,b\in G_{k}$ for some $k$ and for any $a\in A$ and $b\in B$), we have
$\mathrm{COD}_{{\Sigma}_{p}}(a,b)=0$ and therefore by (H.5)

$\mathrm{COD}_{\widehat{\Sigma}_{p}}(A,B)=\max_{a\in A,b\in
B}\mathrm{COD}_{\widehat{\Sigma}_{p}}(a,b)\leq\tau\leq 2\eta\cdot||X||.$

If $A$ and $B$ are not the same clusters, there must exist $a\in A$ and $b\in
B$ with $a\in G_{k}$ and $b\in G_{j}$ for some $k\neq j$. Then

$\mathrm{COD}_{\widehat{\Sigma}_{p}}(A,B)\geq\mathrm{COD}_{\widehat{\Sigma}_{p}}(a,b)\geq\mathrm{COD}_{\Sigma_{p}}(a,b)-\tau\geq\mathrm{MCOD}(\Sigma_{p})-\tau\geq\mathrm{MCOD}(\Sigma_{p})-2\eta\cdot||X||,$

where the first inequality is from the definition of
$\mathrm{COD}_{\widehat{\Sigma}_{p}}(A,B)$, the second one is from (H.5), the
third one is from the definition of MCOD, and finally the last one is from
(H.6).

Together with (H.4), finally we show that
$\mathrm{COD}_{\widehat{\Sigma}_{p}}(A,B)\geq 2\eta\cdot||X||$, if $A$ and $B$
are not the same clusters. By taking the threshold value
$\alpha=2\eta\cdot||X||$, we attain perfect clustering using our hierarchical
algorithm. This completes the proof. ∎

###  H.2 Minimax Lower Bound

The following lemma is Birge’s Lemma applied to our specific setting, similar
to Lemma C.1 in Bunea et al. (2020). Define $M_{O}(p,q,K_{1},K_{2},\eta)$ as
in Section 3.2. For ease of notation, we will simply use $M$.

######  Lemma H.2.

For any partition estimator $\widehat{\mathcal{G}}$, and for any collection of
distinct covariance matrices $\Sigma^{(j)}\in M(p,q,K_{1},K_{2},\eta)$,

$\displaystyle\underset{\Sigma\in
M(p,q,K_{1},K_{2},\eta)}{\sup}\mathbb{P}_{\Sigma}(\widehat{\mathcal{G}}\neq\mathcal{G}^{*})$
$\displaystyle\geq\underset{j=1,...,N}{\max}\mathbb{P}_{\Sigma^{(j)}}(\widehat{\mathcal{G}}\neq\mathcal{G}^{(j)})$
$\displaystyle\geq\frac{1}{2e+1}\wedge(1-\underset{j\geq
2}{\max}\frac{\mathrm{KL}(\Sigma^{(j)},\Sigma^{(1)})}{\log(N)}).$

####  H.2.1 Proof of Theorem 3.2

Similar to the construction in Bunea et al. (2020), set our collection of
covariance matrices as follows:
$K_{1}=3,m_{p}^{(1)}=m_{p}^{(2)}=m_{p}^{(3)}=\frac{p}{3}$ (equal row cluster
size), $X=AZB^{T}+\Gamma$, where $\Gamma_{p\times q}$ has entries
$\Gamma_{ij}\underset{iid}{\sim}N(0,\sigma_{(t)}^{2})$ where
$\sigma_{(t)}^{2}$ denotes the noise variance of elements in column cluster
$t$. Note that for this construction, we assume the same noise variance within
the same column cluster. In addition, we set

$\displaystyle Z$ $\displaystyle=\begin{bmatrix}Z_{11}&...&Z_{1K_{2}}\\\
Z_{21}&...&Z_{2K_{2}}\\\ Z_{31}&...&Z_{3K_{2}}\\\
\end{bmatrix}=\begin{bmatrix}Z_{\cdot 1}&...&Z_{\cdot K_{2}}\\\
\end{bmatrix},\hskip 2.84544ptZ_{\cdot 1},...,Z_{\cdot
K_{2}}\underset{iid}{\sim}N(0,C(\epsilon)),$ $\displaystyle\text{ where
}C(\epsilon)=\begin{bmatrix}\epsilon&\epsilon-\epsilon^{2}&-\epsilon\\\
\epsilon-\epsilon^{2}&\epsilon&\epsilon\\\ -\epsilon&\epsilon&2\\\
\end{bmatrix},\leavevmode\nobreak\ \textrm{and}\leavevmode\nobreak\
A=\begin{bmatrix}1&&\\\ :&&\\\ 1&&\\\ &1&\\\ &:&\\\ &1&\\\ &&1\\\ &&:\\\
&&1\\\ \end{bmatrix}_{p\times 3},\hskip 5.69046ptB=\begin{bmatrix}1&&&&\\\
:&&&\\\ 1&&&\\\ &1&&\\\ &:&&\\\ &1&&\\\ &&:&\\\ &&&1\\\ &&&:\\\ &&&1\\\
\end{bmatrix}_{q\times K_{2}},$

where $0<\epsilon<1$ is a quantity to be specified later. We will consider
$N=(\frac{p}{3})^{2}+1$ many covariance matrices that are obtained by
switching one of the rows in the first third of the rows in $A$ and one of the
rows in the second third of the rows in $A$ (and also counting the original
matrix $A$ as well).

It follows that $\operatorname{{\rm Var}}(\text{vec}(Z))=I_{K_{2}}\otimes
C(\epsilon)$ and that

$\displaystyle\Sigma\leavevmode\nobreak\ (\in\mathbb{R}^{pq\times pq})$
$\displaystyle=\operatorname{{\rm Var}}(\text{vec}(X))$
$\displaystyle=(B\otimes A)(\operatorname{{\rm Var}}(\text{vec}(Z)))(B\otimes
A)^{T}+\operatorname{{\rm Var}}(\text{vec}(\Gamma))$ $\displaystyle=(B\otimes
A)(I_{K_{2}}\otimes C(\epsilon))(B\otimes A)^{T}+\operatorname{{\rm
Var}}(\text{vec}(\Gamma))$ $\displaystyle=\big{\\{}BB^{T}\otimes
AC(\epsilon)A^{T}\big{\\}}+\operatorname{{\rm Var}}(\text{vec}(\Gamma))$
$\displaystyle=\begin{bmatrix}1_{m_{q}^{(1)}}1_{m_{q}^{(1)}}^{T}\otimes
AC(\epsilon)A^{T}&&\\\ &...&\\\
&&1_{m_{q}^{(K_{2})}}1_{m_{q}^{(K_{2})}}^{T}\otimes AC(\epsilon)A^{T}\\\
\end{bmatrix}$ $\displaystyle\hskip
227.62204pt+\begin{bmatrix}\sigma_{(1)}^{2}I_{pm_{q}^{(1)}}&&\\\ &...&\\\
&&\sigma_{(K_{2})}^{2}I_{pm_{q}^{(K_{2})}}\\\ \end{bmatrix}.$

So we can set
$\widetilde{\Sigma}^{(t)}=\big{\\{}1_{m_{q}^{(t)}}1_{m_{q}^{(t)}}^{T}\otimes
AC(\epsilon)A^{T}\big{\\}}+\sigma^{2}_{{(t)}}I_{pm_{q}^{(t)}}$ as one of the
($pm_{q}^{(t)}\times pm_{q}^{(t)}$) blocks in the block diagonal matrix
$\Sigma\in\mathbb{R}^{pq\times pq}$.

Note that the Kullback-Leibler Divergence for $n$ iid multivariate normal
observations in $\mathbb{R}^{d}$ with mean $0$ is
$KL(\Sigma^{{}^{\prime}},\Sigma)=\frac{n}{2}\big{[}\text{tr}(\Sigma^{-1}\Sigma^{{}^{\prime}}-I_{d})-\text{log}\text{
det}(\Sigma^{-1}\Sigma^{{}^{\prime}})\big{]}$, where $\Sigma$ and
$\Sigma^{\prime}$ are two covariance matrices constructed as above. Thus,
since $\Sigma$ is a block diagonal matrix with blocks
$\widetilde{\Sigma}^{(t)}$, we have the relationship
$KL(\Sigma^{{}^{\prime}},\Sigma)=\sum_{t=1}^{K_{2}}KL(\widetilde{\Sigma}^{(t)^{\prime}},\widetilde{\Sigma}^{(t)})$,
and it suffices to just calculate
$KL(\widetilde{\Sigma}^{(t)^{\prime}},\widetilde{\Sigma}^{(t)})$ in order to
calculate $KL(\Sigma^{{}^{\prime}},\Sigma)$.

Denote the eigenvalues and eigenvectors of $C(\epsilon)$ as
$\lambda_{1},\lambda_{2},\lambda_{3}$ and $u_{1},u_{2},u_{3}$, respectively.
Also, define the $pm_{q}^{(t)}\times 1$ vector $v_{k}^{(t)}$ as
$v_{k}^{(t)}=\frac{1}{\sqrt{m_{p}m_{q}^{(t)}}}(1_{m_{q}^{(t)}}\otimes
Au_{k})$.

$\displaystyle\widetilde{\Sigma}^{(t)}$
$\displaystyle=\big{\\{}1_{m_{q}^{(t)}}1_{m_{q}^{(t)}}^{T}\otimes
AC(\epsilon)A^{T}\big{\\}}+\sigma^{2}_{(t)}I_{pm_{q}^{(t)}}=\sigma^{2}_{(t)}\big{[}\sum_{k=1}^{2}\frac{\lambda_{k}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}\cdot
v_{k}^{(t)}v_{k}^{(t)T}+I_{pm_{q}^{(t)}}\big{]},$

where we note that $\lambda_{3}=0$. Therefore,

$\displaystyle(\widetilde{\Sigma}^{(t)})^{-1}$
$\displaystyle=\frac{1}{\sigma^{2}_{(t)}}\big{[}(-\sum_{k=1}^{2}\frac{\lambda_{k}m_{p}m_{q}^{(t)}}{\lambda_{k}m_{p}m_{q}^{(t)}+\sigma^{2}_{(t)}}\cdot
v_{k}^{(t)}v_{k}^{(t)T})+I_{pm_{q}^{(t)}}\big{]}.$

Let $Q_{p\times p}$ denote a $p\times p$ perturbation matrix switching one row
in the first third of the rows in $A$ and one row in the second third of the
rows in $A$. Define $\widetilde{Q}^{(t)}=I_{m_{q}^{(t)}}\otimes Q_{p\times
p}$. Then $\widetilde{Q}^{(t)}(\widetilde{Q}^{(t)})^{T}=I_{pm_{q}^{(t)}}$
holds. Now, set $\widetilde{Q}^{(t)}v_{k}^{(t)}=v_{k}^{(t)}+\Delta_{k}^{(t)}$.
It follows that

$\displaystyle\Delta_{k}^{(t)}=\frac{1}{\sqrt{m_{p}m_{q}^{(t)}}}\cdot(1_{m_{q}^{(t)}}\otimes\begin{bmatrix}(u_{k})_{2}-(u_{k})_{1}\\\
0\\\ :\\\ 0\\\ (u_{k})_{1}-(u_{k})_{2}\\\ 0\\\ :\\\ 0\\\ 0\\\ :\\\ 0\\\
\end{bmatrix}_{p\times 1}).$

Then we can calculate
$\widetilde{\Sigma}^{(t)^{\prime}}-\widetilde{\Sigma}^{(t)}$:

$\displaystyle\widetilde{\Sigma}^{(t)^{\prime}}-\widetilde{\Sigma}^{(t)}$
$\displaystyle=\widetilde{Q}^{(t)}\widetilde{\Sigma}^{(t)}\widetilde{Q}^{(t)T}-\widetilde{\Sigma}^{(t)}$
$\displaystyle=\widetilde{Q}^{(t)}\bigg{(}\sigma^{2}_{(t)}\big{[}\sum_{k=1}^{2}\frac{\lambda_{k}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}\cdot
v_{k}^{(t)}v_{k}^{(t)T}+I_{pm_{q}^{(t)}}\big{]}\bigg{)}\widetilde{Q}^{(t)T}$
$\displaystyle\hskip
199.16928pt-\sigma^{2}_{(t)}\big{[}\sum_{k=1}^{2}\frac{\lambda_{k}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}\cdot
v_{k}^{(t)}v_{k}^{(t)T}+I_{pm_{q}^{(t)}}\big{]}$
$\displaystyle=\sigma^{2}_{(t)}\bigg{[}\big{\\{}\sum_{k=1}^{2}\frac{\lambda_{k}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}(\widetilde{Q}^{(t)}v_{k}^{(t)}v_{k}^{(t)T}\widetilde{Q}^{(t)T}-v_{k}^{(t)}v_{k}^{(t)T})\big{\\}}+\widetilde{Q}^{(t)}\widetilde{Q}^{(t)T}-I_{pm_{q}^{(t)}}\bigg{]}$
$\displaystyle=\sum_{k=1}^{2}\lambda_{k}m_{p}m_{q}^{(t)}(\widetilde{Q}^{(t)}v_{k}^{(t)}v_{k}^{(t)T}\widetilde{Q}^{(t)T}-v_{k}^{(t)}v_{k}^{(t)T})$
$\displaystyle=\lambda_{2}m_{p}m_{q}^{(t)}(v_{2}^{(t)}\Delta_{2}^{(t)T}+\Delta_{2}^{(t)}v_{2}^{(t)T}+\Delta_{2}^{(t)}\Delta_{2}^{(t)T}),$

where the last equality holds because $\Delta_{1}=0$. Now we compute
$(\widetilde{\Sigma}^{(t)})^{-1}\widetilde{\Sigma}^{(t)^{\prime}}$ in the
following way:

$\displaystyle(\widetilde{\Sigma}^{(t)})^{-1}\widetilde{\Sigma}^{(t)^{\prime}}$
$\displaystyle=I+(\widetilde{\Sigma}^{(t)})^{-1}(\widetilde{\Sigma}^{(t)^{\prime}}-\widetilde{\Sigma}^{(t)})$
$\displaystyle=I+\frac{1}{\sigma^{2}_{(t)}}\bigg{[}(-\sum_{k=1}^{2}\frac{\lambda_{k}m_{p}m_{q}^{(t)}}{\lambda_{k}m_{p}m_{q}^{(t)}+\sigma^{2}_{(t)}}\cdot
v_{k}^{(t)}v_{k}^{(t)T})+I_{pm_{q}^{(t)}}\bigg{]}\bigg{[}\lambda_{2}m_{p}m_{q}^{(t)}(v_{2}^{(t)}\Delta_{2}^{(t)T}+\Delta_{2}^{(t)}v_{2}^{(t)T}+\Delta_{2}^{(t)}\Delta_{2}^{(t)T})\bigg{]}$
$\displaystyle=I+\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}\bigg{[}\big{(}I-\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\lambda_{2}m_{p}m_{q}^{(t)}+\sigma^{2}_{(t)}}v_{2}^{(t)}v_{2}^{(t)T}\big{)}\big{(}v_{2}^{(t)}\Delta_{2}^{(t)T}+\Delta_{2}^{(t)}v_{2}^{(t)T}+\Delta_{2}^{(t)}\Delta_{2}^{(t)T}\big{)}\bigg{]}$
$\displaystyle=I+\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}F^{(t)},$

where the second to third line holds because
$v_{1}^{(t)}v_{1}^{(t)T}\Delta_{2}^{(t)}=v_{1}^{(t)}v_{1}^{(t)T}v_{2}^{(t)}=0$
and the last line holds from

$\displaystyle F^{(t)}$
$\displaystyle=\big{(}I-\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\lambda_{2}m_{p}m_{q}^{(t)}+\sigma^{2}_{(t)}}v_{2}^{(t)}v_{2}^{(t)T}\big{)}\big{(}v_{2}^{(t)}\Delta_{2}^{(t)T}+\Delta_{2}^{(t)}v_{2}^{(t)T}+\Delta_{2}^{(t)}\Delta_{2}^{(t)T}\big{)}$
$\displaystyle=\big{(}I-\widetilde{\rho}^{(t)}v_{2}^{(t)}v_{2}^{(t)T}\big{)}\big{(}v_{2}^{(t)}\Delta_{2}^{(t)T}+\Delta_{2}^{(t)}v_{2}^{(t)T}+\Delta_{2}^{(t)}\Delta_{2}^{(t)T}\big{)}$
$\displaystyle=v_{2}^{(t)}\Delta_{2}^{(t)T}(1-\widetilde{\rho}^{(t)}(1+s^{(t)}))+\Delta_{2}^{(t)}v_{2}^{(t)T}+\Delta_{2}^{(t)}\Delta_{2}^{(t)T}-\widetilde{\rho}^{(t)}s^{(t)}v_{2}^{(t)}v_{2}^{(t)T},$

where
$\widetilde{\rho}^{(t)}=\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\lambda_{2}m_{p}m_{q}^{(t)}+\sigma^{2}_{(t)}}$
and $s^{(t)}=\Delta_{2}^{(t)T}v_{2}^{(t)}$. Note that the two non-zero
eigenvalues of $F^{(t)}$, which are $\mu_{1}^{(t)}$, and $\mu_{2}^{(t)}$,
satisfy
$\mu_{1}^{(t)}+\mu_{2}^{(t)}=-\widetilde{\rho}^{(t)}s^{(t)}(2+s^{(t)})$ and
$\mu_{1}^{(t)}\mu_{2}^{(t)}=(1-\widetilde{\rho}^{(t)})s^{(t)}(2+s^{(t)})$.
Using these facts, we can finally calculate the KL divergence

$\displaystyle KL(\widetilde{\Sigma}^{(t)^{\prime}},\widetilde{\Sigma}^{(t)})$
$\displaystyle=\frac{n}{2}\bigg{[}\text{tr}\bigg{(}(\widetilde{\Sigma}^{(t)})^{-1}\widetilde{\Sigma}^{(t)^{\prime}}-I_{pm_{q}^{(t)}}\bigg{)}-\text{log}\text{
det}\bigg{(}(\widetilde{\Sigma}^{(t)})^{-1}\widetilde{\Sigma}^{(t)^{\prime}}\bigg{)}\bigg{]}$
$\displaystyle=\frac{n}{2}\big{[}\text{tr}(\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}F^{(t)})-\text{log}\text{
det}(I_{pm_{q}^{(t)}}+\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}F^{(t)})\big{]}$
$\displaystyle=\frac{n}{2}\bigg{[}\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}(-\widetilde{\rho}^{(t)}s^{(t)}(2+s^{(t)}))$
$\displaystyle\hskip
11.38092pt-\text{log}\bigg{\\{}1+\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}(-\widetilde{\rho}^{(t)}s^{(t)}(2+s^{(t)}))+\big{(}\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}\big{)}^{2}(1-\widetilde{\rho}^{(t)})s^{(t)}(2+s^{(t)})\bigg{\\}}\bigg{]}$
$\displaystyle=\frac{n}{2}\bigg{[}\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}(-\widetilde{\rho}^{(t)}s^{(t)}(2+s^{(t)}))\bigg{]},$

where the last equality holds because
$(1-\widetilde{\rho}^{(t)})\cdot\frac{\lambda_{2}m_{p}m_{q}^{(t)}}{\sigma^{2}_{(t)}}=\widetilde{\rho}^{(t)}$.
Now, plugging in $s^{(t)}=-\frac{2\epsilon^{2}}{m_{p}(2+\epsilon^{2})}$, we
get:

$\displaystyle KL(\widetilde{\Sigma}^{(t)^{\prime}},\widetilde{\Sigma}^{(t)})$
$\displaystyle=2n\widetilde{\rho}^{(t)}\cdot\frac{m_{q}^{(t)}\lambda_{2}}{\sigma^{2}_{(t)}}\frac{\epsilon^{2}}{2+\epsilon^{2}}(1-\frac{\epsilon^{2}}{m_{p}(2+\epsilon^{2})})\leq
2n\epsilon^{2}\frac{m_{q}^{(t)}}{\sigma^{2}_{(t)}},$

where we use the fact that $\lambda_{2}=2+\epsilon^{2}$. Thus,

$KL(\Sigma^{{}^{\prime}},\Sigma)\leq
2n\epsilon^{2}\sum_{t=1}^{K_{2}}\frac{m_{q}^{(t)}}{\sigma^{2}_{(t)}}.$ (H.7)

In the following, we will show an lower bound for the rate of
$\frac{\mathrm{MCOD}(\Sigma_{p,W_{O}})}{||X||_{W_{O}}}$. We will first compute
$\mathrm{MCOD}(\Sigma_{p,W_{O}})$,

$\displaystyle\mathrm{COD}_{\Sigma_{p,W_{O}}}(u,v)$
$\displaystyle=\underset{w\neq
u,v}{{\text{max}}}\frac{1}{K_{2}}\bigg{|}\Big{[}A\mathbb{E}(ZZ^{T})A^{T}+\mathbb{E}(\Gamma
B(B^{T}B)^{-2}B^{T}\Gamma^{T})\Big{]}_{uw}$ $\displaystyle\hskip
85.35826pt-\Big{[}A\mathbb{E}(ZZ^{T})A^{T}+\mathbb{E}(\Gamma
B(B^{T}B)^{-2}B^{T}\Gamma^{T})\Big{]}_{vw}\bigg{|}$
$\displaystyle=\underset{w\neq
u,v}{{\text{max}}}\frac{1}{K_{2}}\bigg{|}\big{[}A\mathbb{E}(ZZ^{T})A^{T}\big{]}_{uw}-\big{[}A\mathbb{E}(ZZ^{T})A^{T}\big{]}_{vw}\bigg{|}$
$\displaystyle\hskip 142.26378pt\text{ (}\mathbb{E}(\Gamma
B(B^{T}B)^{-2}B^{T}\Gamma^{T})\leavevmode\nobreak\ \text{is diagonal)}$
$\displaystyle=\underset{w\neq
u,v}{{\text{max}}}\frac{1}{K_{2}}\bigg{|}\sum_{t=1}^{K_{2}}\mathbb{E}\big{\\{}(Z_{r(u)t}-Z_{r(v)t})Z_{r(w)t}\big{\\}}\bigg{|}$
$\displaystyle=\underset{w\neq
u,v}{{\text{max}}}\bigg{|}\mathbb{E}\big{\\{}(Z_{r(u)1}-Z_{r(v)1})Z_{r(w)1}\big{\\}}\bigg{|}.$

and from the definition of $C(\epsilon)$, it is apparent that the minimum of
the maximum difference between rows is $2\epsilon$, that is

$\displaystyle\mathrm{MCOD}(\Sigma_{p,W_{O}})=2\epsilon.$ (H.8)

Let us choose

$\displaystyle\epsilon=\sqrt{\bigg{(}\frac{\log(\frac{p}{3})}{n}\bigg{)}\cdot\bigg{(}\frac{1}{\sum_{t=1}^{K_{2}}\frac{m_{q}^{(t)}}{\sigma^{2}_{(t)}}}\bigg{)}\cdot\bigg{(}\frac{2e}{2e+1}\bigg{)}}.$
(H.9)

The choice of $\epsilon$ will be explained later.

Furthermore, we can show that

$\displaystyle||X||_{p,W_{O}}$
$\displaystyle=\frac{1}{\sqrt{K_{2}}}\cdot\sqrt{4K_{2}+\sum_{t=1}^{K_{2}}\bigg{[}\frac{4\sigma^{2}_{(t)}}{|[t]|}+\frac{\sigma^{4}_{(t)}}{|[t]|^{2}}\bigg{]}}.$

So, getting rid of the term $\frac{4\sigma^{2}_{(t)}}{|[t]|}$, and using
$\epsilon$ in (H.9) and $m_{q}^{(t)}=\frac{q}{K_{2}}$,
$\sigma_{(t)}=\sigma=O(\sqrt{\frac{q}{K_{2}}})$, we get the following
inequalities:

$\displaystyle\frac{\mathrm{MCOD}(\Sigma_{p,W_{O}})}{||X||_{p,W_{O}}}\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\
\frac{2\epsilon}{\frac{1}{\sqrt{K_{2}}}\cdot\sqrt{4K_{2}+\sum_{t=1}^{K_{2}}\bigg{[}\frac{4\sigma^{2}_{(t)}}{|[t]|}+\frac{\sigma^{4}_{(t)}}{|[t]|^{2}}\bigg{]}}}$
$\displaystyle=\leavevmode\nobreak\
\frac{2\sqrt{\bigg{(}\frac{\log(\frac{p}{3})}{n}\bigg{)}\cdot\bigg{(}\frac{1}{\sum_{t=1}^{K_{2}}\frac{m_{q}^{(t)}}{\sigma^{2}_{(t)}}}\bigg{)}\cdot\bigg{(}\frac{2e}{2e+1}\bigg{)}}}{\frac{1}{\sqrt{K_{2}}}\cdot\sqrt{4K_{2}+\sum_{t=1}^{K_{2}}\bigg{[}\frac{4\sigma^{2}_{(t)}}{|[t]|}+\frac{\sigma^{4}_{(t)}}{|[t]|^{2}}\bigg{]}}}$
$\displaystyle=\leavevmode\nobreak\ c\cdot\sqrt{\frac{\log
p}{n}}\cdot\frac{1}{\sqrt{4\sum_{t=1}^{K_{2}}\frac{q}{K_{2}\sigma^{2}}\leavevmode\nobreak\
+\leavevmode\nobreak\
(\sum_{t=1}^{K_{2}}\frac{q}{K_{2}\sigma^{2}})(\sum_{t=1}^{K_{2}}[\frac{4\sigma^{2}}{q}+\frac{K_{2}\sigma^{4}}{q^{2}}])}}$
$\displaystyle=\leavevmode\nobreak\ c\cdot\sqrt{\frac{\log
p}{n}}\cdot\frac{1}{\sqrt{\frac{4q}{\sigma^{2}}+(\frac{q}{\sigma^{2}})(\frac{4K_{2}\sigma^{2}}{q}+\frac{K_{2}^{2}\sigma^{4}}{q^{2}})}}$
$\displaystyle\geq\leavevmode\nobreak\ c\cdot\sqrt{\frac{\log
p}{n}}\cdot\frac{1}{\sqrt{3}\sqrt{\frac{4q}{\sigma^{2}}\bigvee
4K_{2}\bigvee\frac{\sigma^{2}K_{2}^{2}}{q}}}$
$\displaystyle=\leavevmode\nobreak\ c^{\prime\prime}\cdot\sqrt{\frac{\log
p}{nK_{2}}}$

where for our construction we use the noise variance setting of
$\sigma=O\big{(}\sqrt{m_{q}}\big{)}=O\Big{(}\sqrt{\frac{q}{K_{2}}}\Big{)}$ in
the last equality. This ensures that the lower bound is as tight as possible.
This implies that our constructed $\Sigma$ belongs to the parameter space
$M_{O}(p,q,K_{1},K_{2},\eta)$ for some $\eta$ s.t. $\eta\leq
c^{\prime\prime}\sqrt{\frac{\log p}{nK_{2}}}$.

Finally, we are ready to invoke Lemma H.2. Recall that
$KL(\Sigma^{{}^{\prime}},\Sigma)$ is upper bounded in (H.7). Since we choose
$\epsilon$ in (H.9), we obtain

$1-\frac{KL(\Sigma^{{}^{\prime}},\Sigma)}{\log[(p/3)^{2}]}\geq
1-\frac{2n\epsilon^{2}\sum_{t=1}^{K_{2}}\frac{m_{q}^{(t)}}{\sigma^{2}_{(t)}}}{\log[(p/3)^{2}]}=\frac{1}{2e+1}.$

As a result, Lemma H.2 implies

$\underset{\Sigma\in
M_{I}(p,q,K_{1},K_{2},\eta)}{\sup}\mathbb{P}_{\Sigma}(\widehat{\mathcal{G}}\neq\mathcal{G}^{*})\geq\frac{1}{2e+1},$

for any $\widehat{\mathcal{G}}$, which completes the proof of Theorem 3.2.

####  H.2.2 Proof for Theorem E.1 (Minimax Lower Bound with a Perturbed
$\bar{B}$)

###### Proof.

To study the lower bound over $M_{\bar{W}}$, we follow the same construction
as in the proof for Theorem 3.2. We emphasize that all the parts leading up to
calculating the KL - divergence is the same regardless of the value of
$\bar{B}$ as the KL-divergence is related to the model on the population
level. Since the true column clustering structure $B$ does not change, the
derivations up to this point remain unchanged. The part that does indeed
change is the calculation of
$\frac{\text{MCOD}(\Sigma_{p,\bar{W}})}{||X||_{\bar{W}}}$ as both the MCOD
value and $||X||_{\bar{W}}$ depend on $\bar{B}$. We can show that

$\displaystyle\Sigma_{p,\bar{W}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\mathbb{E}(X\bar{W}X^{T})\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\mathbb{E}\Big{[}(AZB^{T}+\Gamma)\bar{W}(BZ^{T}A^{T}+\Gamma^{T})\Big{]}$
$\displaystyle=\leavevmode\nobreak\
\mathbb{E}\big{(}AZB^{T}\bar{W}BZ^{T}A^{T}\big{)}+\mathbb{E}\big{(}\Gamma\bar{W}\Gamma^{T}\big{)}$
$\displaystyle=\leavevmode\nobreak\
\frac{1}{s}\mathbb{E}\Big{[}AZB^{T}\bar{B}(\bar{B}^{T}\bar{B})^{-2}\bar{B}^{T}Z^{T}A^{T}\Big{]}+\mathbb{E}(\Gamma\bar{W}\Gamma^{T})$
$\displaystyle=\leavevmode\nobreak\
\frac{1}{s}\mathbb{E}\big{(}AZGG^{T}Z^{T}A^{T}\big{)}+\mathbb{E}\big{(}\Gamma\bar{W}\Gamma^{T}\big{)},$

where $G=B^{T}\bar{B}(\bar{B}^{T}\bar{B})^{-1}$. Thus

$\displaystyle\text{COD}_{\Sigma_{p,\bar{W}}}(u,v)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \underset{w\neq
u,v}{\max}\Big{|}\big{(}\Sigma_{p,\bar{W}}\big{)}_{uw}-\big{(}\Sigma_{p,\bar{W}}\big{)}_{vw}\Big{|}$
$\displaystyle=\leavevmode\nobreak\ \frac{1}{s}\underset{w\neq
u,v}{\max}\Bigg{|}\mathbb{E}\bigg{\\{}\sum_{t_{1},t_{2}}^{K_{2}}Z_{r(u)t_{1}}Z_{r(w)t_{2}}\big{(}GG^{T}\big{)}_{t_{1}t_{2}}-Z_{r(v)t_{1}}Z_{r(w)t_{2}}\big{(}GG^{T}\big{)}_{t_{1}t_{2}}\bigg{\\}}\Bigg{|}$
$\displaystyle\leavevmode\nobreak\ \hskip 85.35826pt\text{since
}\mathbb{E}(\Gamma\widehat{W}\Gamma^{T})\text{ is diagonal}$
$\displaystyle=\leavevmode\nobreak\ \frac{1}{s}\underset{w\neq
u,v}{\max}\Bigg{|}\sum_{t_{1},t_{2}}^{K_{2}}\big{(}GG^{T}\big{)}_{t_{1}t_{2}}\cdot\mathbb{E}\Big{\\{}Z_{r(u)t_{1}}Z_{r(w)t_{2}}-Z_{r(v)t_{1}}Z_{r(w)t_{2}}\Big{\\}}\Bigg{|}$
$\displaystyle=\leavevmode\nobreak\ \frac{1}{s}\underset{w\neq
u,v}{\max}\Bigg{|}\sum_{t=1}^{K_{2}}\big{(}GG^{T}\big{)}_{tt}\cdot\mathbb{E}\Big{\\{}Z_{r(u)t}Z_{r(w)t}-Z_{r(v)t}Z_{r(w)t}\Big{\\}}\Bigg{|}$
$\displaystyle\leavevmode\nobreak\ \hskip 85.35826pt\text{since in our
construction, }Z_{\cdot 1},\leavevmode\nobreak\ Z_{\cdot
2},\leavevmode\nobreak\ Z_{\cdot 3}\text{ are i.i.d.}$
$\displaystyle=\leavevmode\nobreak\ \frac{1}{s}\underset{w\neq
u,v}{\max}\Bigg{|}\sum_{t=1}^{K_{2}}\big{(}GG^{T}\big{)}_{tt}\cdot\mathbb{E}\Big{\\{}Z_{r(u)1}Z_{r(w)1}-Z_{r(v)1}Z_{r(w)1}\Big{\\}}\Bigg{|}$
$\displaystyle=\leavevmode\nobreak\ \frac{1}{s}\underset{w\neq
u,v}{\max}\Bigg{|}\big{|}\big{|}G\big{|}\big{|}_{F}^{2}\cdot\mathbb{E}\Big{\\{}Z_{r(u)1}Z_{r(w)1}-Z_{r(v)1}Z_{r(w)1}\Big{\\}}\Bigg{|}.$

Thus, we have

$\text{MCOD}(\Sigma_{p,\bar{W}})\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{1}{s}\cdot\big{|}\big{|}G\big{|}\big{|}_{F}^{2}\cdot 2\epsilon$

where the $\epsilon$ is from our construction. Note that all of the entries of
$G$ are non-negative and all the column sums are 1. Thus, every entry in $G$
is $\leq 1$, and the entries in $G$ add up to $s$. We can conclude that
$||G||_{F}^{2}\leq s$. Thus, $\frac{1}{s}||G||_{F}^{2}\leq 1$. It is apparent
that the MCOD value potentially decreases with a weight $\bar{W}$ with an
incorrect estimated column cluster structure. When $\bar{B}=B$, then
$G=I_{K_{2}}$ and $s=K_{2}$, and the MCOD value reduces to $2\epsilon$. Also,
when $\bar{B}=I_{q}$, then $G=B^{T}$, $s=q$, and the MCOD value reduces to
$2\epsilon$ as well. It appears that these two are the exception, as in other
cases, the MCOD value becomes strictly smaller than $2\epsilon$. For example,
when $\bar{B}=\mathbf{1}_{q}$ (when we put all the columns in the same column
cluster) we have $s=1$, $G=\frac{1}{K_{2}}\cdot\mathbf{1}_{K_{2}}$ and
$\frac{1}{s}||G||_{F}^{2}=\frac{1}{K_{2}}$ which is strictly smaller than 1.
The MCOD value becomes $\frac{2\epsilon}{K_{2}}$.

Now let’s look at $||X||_{\bar{W}}$:

$\displaystyle||X||_{\bar{W}}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\sqrt{K_{2}}\cdot\underset{a}{\max}\Big{|}\Big{|}\text{Var}(X_{a\cdot}\widehat{L})\Big{|}\Big{|}_{F}$
$\displaystyle=\leavevmode\nobreak\
\sqrt{K_{2}}\cdot\underset{a}{\max}\Big{|}\Big{|}\text{Var}(Z_{r(a)\cdot}B^{T}\widehat{L})+\text{Var}(\Gamma_{a\cdot}\widehat{L})\Big{|}\Big{|}_{F}$
$\displaystyle=\leavevmode\nobreak\
\sqrt{K_{2}}\cdot\underset{a}{\max}\Big{|}\Big{|}\widehat{L}^{T}B\mathbb{E}(Z_{r(a)\cdot}^{T}Z_{r(a)\cdot})B^{T}\widehat{L}+\text{Var}(\Gamma_{a\cdot}\widehat{L})\Big{|}\Big{|}_{F}$
$\displaystyle=\leavevmode\nobreak\
\sqrt{K_{2}}\cdot\underset{a}{\max}\Big{|}\Big{|}\widehat{L}^{T}B\mathbb{E}(Z_{r(a)1}^{2})\cdot
I_{K_{2}}B^{T}\widehat{L}+\text{Var}(\Gamma_{a\cdot}\widehat{L})\Big{|}\Big{|}_{F}$
$\displaystyle\leavevmode\nobreak\ \hskip 85.35826pt\text{since in our
construction, }Z_{\cdot 1},\leavevmode\nobreak\ Z_{\cdot
2},\leavevmode\nobreak\ Z_{\cdot 3}\text{ are i.i.d.}$
$\displaystyle=\leavevmode\nobreak\
\sqrt{K_{2}}\cdot\underset{a}{\max}\Big{|}\Big{|}\mathbb{E}(Z_{r(a)1}^{2})\cdot\frac{1}{s}G^{T}G+\text{Var}(\Gamma_{a\cdot}\widehat{L})\Big{|}\Big{|}_{F}$
$\displaystyle\leavevmode\nobreak\ \hskip 85.35826pt\text{since
}\widehat{L}=\frac{1}{\sqrt{s}}\bar{B}(\bar{B}^{T}\bar{B})^{-1}\text{ and
}G=\sqrt{s}B^{T}\widehat{L}$ $\displaystyle=\leavevmode\nobreak\
\sqrt{K_{2}}\cdot\underset{a}{\max}\Big{|}\Big{|}\frac{2}{s}G^{T}G+\text{Var}(\Gamma_{a\cdot}\widehat{L})\Big{|}\Big{|}_{F}$
$\displaystyle\leavevmode\nobreak\ \hskip 85.35826pt\text{since all the
entries in }\frac{1}{s}G^{T}G\text{ and
}\text{Var}(\Gamma_{a\cdot}\widehat{L})\text{ are nonnegative }$ (we can take
the maximum diagonal value in $C(\epsilon)$.)
$\displaystyle\leq\leavevmode\nobreak\
\frac{\sqrt{K_{2}}}{s}\cdot\underset{a}{\max}\sqrt{8\big{|}\big{|}G^{T}G\big{|}\big{|}_{F}^{2}+2\sum_{r=1}^{s}\Big{(}\frac{\sum_{j\in\widehat{[r]}}\sigma^{2}_{aj}}{|\widehat{[r]}|^{2}}\Big{)}^{2}}$
where $\widehat{[r]}$ denotes the r-th estimated column cluster and
$\sigma_{aj}^{2}=\text{Var}(\Gamma_{aj})$ $\displaystyle=\leavevmode\nobreak\
\frac{\sqrt{K_{2}}}{s}\cdot\sqrt{8\big{|}\big{|}G^{T}G\big{|}\big{|}_{F}^{2}+\Big{(}\frac{q}{K_{2}}\Big{)}^{2}\cdot
2\sum_{r=1}^{s}\frac{1}{|\widehat{[r]}|^{2}}}$ since we assume
$\sigma_{aj}^{2}=\sigma^{2}=m_{q}=\frac{q}{K_{2}}$ in our construction
$\displaystyle\leq\leavevmode\nobreak\
c\cdot\frac{\sqrt{K_{2}}}{s}\cdot\sqrt{\big{|}\big{|}G^{T}G\big{|}\big{|}_{F}^{2}+\Big{(}\frac{q}{K_{2}}\Big{)}^{2}\cdot\frac{s}{\bar{m}_{q}^{2}}}.$
where $\bar{m}_{q}^{2}$ denotes the smallest estimated column cluster size

Finally, we can derive the lower bound rate for
$\frac{\text{MCOD}(\Sigma_{p,\bar{W}})}{||X||_{\bar{W}}}$:

$\displaystyle\frac{\text{MCOD}(\Sigma_{p,\bar{W}})}{||X||_{\bar{W}}}\leavevmode\nobreak\
$ $\displaystyle\geq\leavevmode\nobreak\ c^{\prime}\cdot
2\epsilon\cdot\frac{\frac{1}{s}||G||_{F}^{2}}{\frac{\sqrt{K_{2}}}{s}\cdot\sqrt{\big{|}\big{|}G^{T}G\big{|}\big{|}_{F}^{2}+\Big{(}\frac{q}{K_{2}}\Big{)}^{2}\cdot\frac{s}{\bar{m}_{q}^{2}}}}$
$\displaystyle=\leavevmode\nobreak\ c^{\prime\prime}\cdot\sqrt{\frac{\log
p}{nK_{2}}}\cdot\frac{||G||_{F}^{2}}{\sqrt{\big{|}\big{|}G^{T}G\big{|}\big{|}_{F}^{2}+\Big{(}\frac{q}{K_{2}}\Big{)}^{2}\cdot\frac{s}{\bar{m}_{q}^{2}}}}\cdot\frac{1}{\sqrt{\sum_{t=1}^{K_{2}}\frac{m_{q}^{(t)}}{\sigma^{2}_{(t)}}}}$
$\displaystyle=c^{\prime\prime}\cdot\sqrt{\frac{\log
p}{nK_{2}}}\cdot\frac{||G||_{F}^{2}}{\sqrt{K_{2}}\cdot\sqrt{\big{|}\big{|}G^{T}G\big{|}\big{|}_{F}^{2}+\Big{(}\frac{q}{K_{2}}\Big{)}^{2}\cdot\frac{s}{\bar{m}_{q}^{2}}}}.$
since we assume $m_{q}^{(t)}=m_{q}=\sigma^{2}=\sigma^{2}_{(t)}$ in our
construction.

This completes the proof of the lower bound. ∎

####  H.2.3 Proof of Theorem F.2 (Joint Lower Bound)

###### Proof.

The extension from considering just one direction (either the rows or columns)
is quite trivial.

$\displaystyle\mathbb{P}_{\Sigma}\Big{(}(\widehat{\mathcal{G}}^{(r)},\widehat{\mathcal{G}}^{(c)})\neq(\mathcal{G}^{(r)},\mathcal{G}^{(c)})\Big{)}\leavevmode\nobreak\
$ $\displaystyle=\leavevmode\nobreak\
\mathbb{P}_{\Sigma}\Big{(}\\{\widehat{\mathcal{G}}^{(r)}\neq\mathcal{G}^{(r)}\\}\cup\\{\widehat{\mathcal{G}}^{(c)}\neq\mathcal{G}^{(c)}\\}\Big{)}$
$\displaystyle=\leavevmode\nobreak\
\mathbb{P}_{\Sigma}\Big{(}\widehat{\mathcal{G}}^{(r)}\neq\mathcal{G}^{(r)}\Big{)}\leavevmode\nobreak\
+\leavevmode\nobreak\
\mathbb{P}_{\Sigma}\Big{(}\widehat{\mathcal{G}}^{(c)}\neq\mathcal{G}^{(c)}\Big{)}$
$\displaystyle\hskip 56.9055pt\leavevmode\nobreak\ -\leavevmode\nobreak\
\mathbb{P}_{\Sigma}\Big{(}\\{\widehat{\mathcal{G}}^{(r)}\neq\mathcal{G}^{(r)}\\}\cap\\{\widehat{\mathcal{G}}^{(c)}\neq\mathcal{G}^{(c)}\\}\Big{)}$
$\displaystyle\geq\leavevmode\nobreak\
\mathbb{P}_{\Sigma}\Big{(}\widehat{\mathcal{G}}^{(r)}\neq\mathcal{G}^{(r)}\Big{)}$
$\displaystyle\geq\leavevmode\nobreak\ \frac{1}{7},$

where the last line holds from the minimax lower bound construction for
Theorem 3.2. The only thing left to check for the proof is that the
constructed $\Sigma$ in the previous minimax lower bound proof is actually an
element of both parameter spaces - $M_{O,(r)}$ and $M_{O,(c)}$ \- so that the
supremum logic is valid. We will show that the $\Sigma$ constructed in the
minimax lower bound proof for the row clusters also satisfies
$\text{MCOD}(\Sigma_{q,W_{O,(c)}})/\|X\|_{W_{O,(c)}}\geq\eta_{(c)}$ for
$\eta_{(c)}\asymp\sqrt{\frac{\log q}{nK_{1}}}$. We have

$\displaystyle\Sigma_{q,W_{O,(c)}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\mathbb{E}(X^{T}W_{O,(c)}X)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\mathbb{E}\Big{[}\big{(}BZ^{T}A^{T}W_{O,(c)}AZB^{T}\big{)}+\big{(}\Gamma^{T}W_{O,(c)}\Gamma\big{)}\Big{]}$
$\displaystyle=\leavevmode\nobreak\
\mathbb{E}\Big{[}\big{(}BZ^{T}A^{T}A(A^{T}A)^{-2}A^{T}AZB^{T}\big{)}/K_{1}+\Gamma^{T}W_{O,(c)}\Gamma\Big{]}$
$\displaystyle=\leavevmode\nobreak\
\frac{1}{K_{1}}\mathbb{E}(BZ^{T}ZB^{T})\leavevmode\nobreak\
+\leavevmode\nobreak\ \mathbb{E}(\Gamma^{T}W_{O,(c)}\Gamma)$
$\displaystyle\text{COD}_{\Sigma_{q,W_{O,(c)}}}(u,v)\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \underset{w\neq
u,v}{\max}\bigg{|}\Big{[}\Sigma_{q,W_{O,(c)}}\Big{]}_{uw}\leavevmode\nobreak\
-\leavevmode\nobreak\ \Big{[}\Sigma_{q,W_{O,(c)}}\Big{]}_{vw}\bigg{|}$
$\displaystyle=\leavevmode\nobreak\ \underset{w\neq
u,v}{\max}\bigg{|}\frac{1}{K_{1}}\Big{[}B\mathbb{E}(Z^{T}Z)B^{T}\Big{]}_{uw}\leavevmode\nobreak\
-\leavevmode\nobreak\
\frac{1}{K_{1}}\Big{[}B\mathbb{E}(Z^{T}Z)B^{T}\Big{]}_{vw}\bigg{|}$
$\displaystyle=\leavevmode\nobreak\ \frac{1}{K_{1}}\underset{w\neq
u,v}{\max}\bigg{|}\mathbb{E}\Big{(}Z_{\cdot c(u)}^{T}Z_{\cdot
c(w)}\leavevmode\nobreak\ -\leavevmode\nobreak\ Z_{\cdot c(v)}^{T}Z_{\cdot
c(w)}\Big{)}\bigg{|}$ $\displaystyle=\leavevmode\nobreak\
\frac{1}{K_{1}}\underset{w\neq
u,v}{\max}\Bigg{|}\sum_{t=1}^{K_{1}}\mathbb{E}\Big{(}Z_{t,c(u)}Z_{t,c(w)}\leavevmode\nobreak\
-\leavevmode\nobreak\ Z_{t,c(v)}Z_{t,c(w)}\Big{)}\Bigg{|}$
$\displaystyle=\leavevmode\nobreak\
\begin{cases}\frac{1}{K_{1}}\underset{1\leq u\leq
q}{\max}\Big{|}\sum_{t=1}^{K_{1}}\mathbb{E}\big{(}Z_{t,c(u)}^{2}\big{)}\Big{|}&\text{(If
$u\underset{\mathcal{G}}{\nsim}v$)}\\\ 0&\text{(If
$u\underset{\mathcal{G}}{\sim}v$)}\end{cases}$ (Note the columns of $Z$ are
independent in our construction.) $\displaystyle=\leavevmode\nobreak\
\begin{cases}\frac{1}{K_{1}}\Big{|}\sum_{t=1}^{K_{1}}\mathbb{E}\big{(}Z_{t,1}^{2}\big{)}\Big{|}&\text{(If
$u\underset{\mathcal{G}}{\nsim}v$)}\\\ 0&\text{(If
$u\underset{\mathcal{G}}{\sim}v$)}\end{cases}$ (Again, from column
independence of $Z$ in our construction.)
$\displaystyle\text{MCOD}(\Sigma_{q,W_{O,(c)}})\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\underset{u\underset{\mathcal{G}}{\nsim}v}{\min}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{COD}_{\Sigma_{q,W_{O,(c)}}}(u,v)$
$\displaystyle=\leavevmode\nobreak\
\frac{1}{K_{1}}\Bigg{|}\sum_{t=1}^{K_{1}}\mathbb{E}\Big{(}Z_{t,1}^{2}\Big{)}\Bigg{|}$
$\displaystyle=\leavevmode\nobreak\ \frac{2\epsilon+2}{3}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{(From our construction with $C(\epsilon)$.)}$
$\displaystyle\geq\leavevmode\nobreak\ \frac{2}{3}$
$\displaystyle||X||_{W_{O,(c)}}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\ \sqrt{K_{1}}\cdot\underset{1\leq b\leq
q}{\max}\leavevmode\nobreak\ \Big{|}\Big{|}L^{T}_{(c)}\text{Var}(X_{\cdot
b})L_{(c)}\Big{|}\Big{|}_{F}$ $\displaystyle=\leavevmode\nobreak\
\sqrt{K_{1}}\cdot\underset{1\leq b\leq q}{\max}\leavevmode\nobreak\
\Big{|}\Big{|}\text{Var}\big{(}L^{T}_{(c)}X_{\cdot
b}\big{)}\Big{|}\Big{|}_{F}$

$\displaystyle=\leavevmode\nobreak\ \sqrt{K_{1}}\cdot\underset{1\leq b\leq
q}{\max}\leavevmode\nobreak\
\Big{|}\Big{|}\text{Var}\big{(}\frac{1}{\sqrt{K_{1}}}Z_{\cdot
c(b)}\big{)}\leavevmode\nobreak\ +\leavevmode\nobreak\
\text{Var}\big{(}L_{(c)}^{T}\Gamma_{\cdot b}\big{)}\Big{|}\Big{|}_{F}$
$\displaystyle=\leavevmode\nobreak\ \sqrt{K_{1}}\cdot\underset{1\leq b\leq
q}{\max}\leavevmode\nobreak\
\Big{|}\Big{|}\frac{1}{K_{1}}\mathbb{E}\big{(}Z_{\cdot c(b)}Z_{\cdot
c(b)}^{T}\big{)}\leavevmode\nobreak\ +\leavevmode\nobreak\
\text{Var}\big{(}L_{(c)}^{T}\Gamma_{\cdot b}\big{)}\Big{|}\Big{|}_{F}$
$\displaystyle=\leavevmode\nobreak\
\sqrt{K_{1}}\Big{|}\Big{|}\frac{1}{K_{1}}C(\epsilon)\leavevmode\nobreak\
+\leavevmode\nobreak\ \frac{\sigma^{2}}{p}I_{K_{1}}\Big{|}\Big{|}_{F}$
$\displaystyle=\leavevmode\nobreak\
\sqrt{K_{1}}\cdot\Bigg{|}\Bigg{|}\begin{bmatrix}\frac{\epsilon}{K_{1}}+\frac{\sigma^{2}}{p}&\frac{\epsilon-\epsilon^{2}}{K_{1}}&-\frac{\epsilon}{K_{1}}\\\
\frac{\epsilon-\epsilon^{2}}{K_{1}}&\frac{\epsilon}{K_{1}}+\frac{\sigma^{2}}{p}&\frac{\epsilon}{K_{1}}\\\
-\frac{\epsilon}{K_{1}}&\frac{\epsilon}{K_{1}}&\frac{2}{K_{1}}+\frac{\sigma^{2}}{p}\end{bmatrix}\Bigg{|}\Bigg{|}_{F}$
$\displaystyle\leq\leavevmode\nobreak\
\sqrt{K_{1}}\cdot\sqrt{4\cdot\Big{(}\frac{2}{K_{1}}\Big{)}^{2}\cdot
K_{1}^{2}}$ $\displaystyle=\leavevmode\nobreak\ 4\sqrt{K_{1}}$
$\displaystyle\frac{\text{MCOD}(\Sigma_{q,W_{O,(c)}})}{||X||_{W_{O,(c)}}}\leavevmode\nobreak\
$ $\displaystyle\geq\leavevmode\nobreak\
c\cdot\frac{(2/3)}{\sqrt{K_{1}}}=c^{\prime}\cdot\sqrt{\frac{1}{K_{1}}}$
$\displaystyle>>\leavevmode\nobreak\ \sqrt{\frac{\log
q}{nK_{1}}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{since
$\log q=o(n)$}.$

We have shown that the $\Sigma$ constructed for the minimax lower bound proof
for the rows is also an element of $M_{O,(c)}$. The same argument can be made
symmetrically to show that the $\Sigma$ constructed for the minimax lower
bound proof for the columns is also an element of $M_{O,(r)}$. Thus, the proof
for joint minimax optimality is complete. ∎

###  H.3 Proofs for the Matrix Normal Model

####  H.3.1 Proof of Proposition 4.1

###### Proof.

For any rows $w,x$ of $\Sigma_{p,W_{O}}$, without loss of generality, we can
assume they belong to clusters $i,h$. Then

$\displaystyle\mathrm{COD}_{\Sigma_{p,W_{O}}}(w,x)$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{K_{2}}\bigg{|}\sum_{l=1}^{K_{2}}\mathbb{E}\big{\\{}(Z_{il}-Z_{hl})Z_{cl}\big{\\}}\bigg{|}$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{K_{2}}\bigg{|}\sum_{l=1}^{K_{2}}(U_{ic}-U_{hc})V_{ll}\bigg{|}$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{K_{2}}\big{|}U_{ic}-U_{hc}\big{|}\bigg{|}\sum_{l=1}^{K_{2}}V_{ll}\bigg{|}$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{K_{2}}\big{|}U_{ic}-U_{hc}\big{|}\mathrm{tr}(V).$

Looking at the cluster separation condition first, we need this COD value to
be greater than $c_{o}\eta\cdot||X||_{W_{O}}=c_{0}\cdot c_{1}\sqrt{\frac{\log
p}{nK_{2}}}\cdot||X||_{W_{O}}$ for all $w,x$ (where $c_{1}$ is a universal
constant and $c_{0}$ is an arbitrary constant $\geq 4$.). We know that
$||X||_{W_{O}}\leq\frac{1}{\sqrt{K_{2}}}\underset{a}{\max}||\operatorname{{\rm
Var}}(Z_{r(a)\cdot})||_{F}+\sqrt{K_{2}}\underset{a}{\max}||\operatorname{{\rm
Var}}(\Gamma_{a\cdot}L)||_{F}$ from the definition of $||X||_{W_{O}}$. In the
matrix normal setting, we have

$\displaystyle\operatorname{{\rm Var}}(Z_{r(a)\cdot})$
$\displaystyle=\begin{bmatrix}U_{r(a)r(a)}\cdot V_{11}&U_{r(a)r(a)}\cdot
V_{12}&...&...&\\\ U_{r(a)r(a)}\cdot V_{21}&U_{r(a)r(a)}\cdot
V_{22}&...&...&\\\ :&:&&&\\\ \end{bmatrix}=U_{r(a)r(a)}\cdot V$

and so, $||\operatorname{{\rm
Var}}(Z_{r(a)\cdot})||_{F}=U_{r(a)r(a)}\cdot||V||_{F}$ and we have

$\underset{a}{\max}||\operatorname{{\rm
Var}}(Z_{r(a)\cdot})||_{F}=||\text{diag}(U)||_{\max}\cdot||V||_{F}\leq||\text{diag}(U)||_{\max}K_{2}^{1/2}C_{\max}.$

It is also easily shown that

$\max_{a}||\operatorname{{\rm
Var}}(\Gamma_{a\cdot}L)||_{F}=\max_{a}\frac{1}{K_{2}}\sqrt{\sum_{t=1}^{K_{2}}\frac{(\sum_{j\in[t]}\sigma^{2}_{aj})^{2}}{|[t]|^{4}}}=C^{1/2}_{K}/K_{2}^{1/2}.$

Rearranging with algebra gives us the cluster separation condition.

Now, let’s consider the stability condition. Again, we need only consider rows
in different row clusters when looking at the $\mathrm{MCOD}$ value. That is
why we assume $i\neq h$. Now for $\mathrm{MCOD}(\Sigma_{p,\widehat{W}_{O}})$,
we would need to look at the following equivalent expressions:

$\displaystyle\mathrm{COD}_{\Sigma_{p,\widehat{W}_{O}}}(w,x)$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{s}\bigg{|}\mathbb{E}\bigg{(}\sum_{l=1}^{K_{2}}(Z_{il}-Z_{hl})Z_{cl}(\sum_{j=1}^{s}\frac{|G_{l}\cap\widehat{G}_{j}|^{2}}{|\widehat{G}_{j}|^{2}})\bigg{|}\widehat{B}\bigg{)}$
$\displaystyle\hskip 56.9055pt+\mathbb{E}\bigg{(}\sum_{l^{\prime}\neq
l^{\prime\prime}}^{K_{2}}(Z_{il^{\prime}}-Z_{hl^{\prime}})Z_{cl^{\prime\prime}}(\sum_{j=1}^{s}\frac{|G_{l^{\prime}}\cap\widehat{G}_{j}|}{|\widehat{G}_{j}|}\cdot\frac{|G_{l^{\prime\prime}}\cap\widehat{G}_{j}|}{|\widehat{G}_{j}|})\bigg{|}\widehat{B}\bigg{)}\bigg{|}$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{s}\bigg{|}\sum_{l=1}^{K_{2}}(U_{ic}-U_{hc})V_{ll}(\sum_{j=1}^{s}\frac{|G_{l}\cap\widehat{G}_{j}|^{2}}{|\widehat{G}_{j}|^{2}})$
$\displaystyle\hskip 56.9055pt+\sum_{l^{\prime}\neq
l^{\prime\prime}}^{K_{2}}(U_{ic}-U_{hc})V_{l^{\prime}l^{\prime\prime}}(\sum_{j=1}^{s}\frac{|G_{l^{\prime}}\cap\widehat{G}_{j}|}{|\widehat{G}_{j}|}\cdot\frac{|G_{l^{\prime\prime}}\cap\widehat{G}_{j}|}{|\widehat{G}_{j}|})\bigg{|}$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{s}\big{|}U_{ic}-U_{hc}\big{|}\bigg{|}\sum_{l=1}^{K_{2}}V_{ll}(\sum_{j=1}^{s}\frac{|G_{l}\cap\widehat{G}_{j}|^{2}}{|\widehat{G}_{j}|^{2}})$
$\displaystyle\hskip 56.9055pt+\sum_{l^{\prime}\neq
l^{\prime\prime}}^{K_{2}}V_{l^{\prime}l^{\prime\prime}}(\sum_{j=1}^{s}\frac{|G_{l^{\prime}}\cap\widehat{G}_{j}|}{|\widehat{G}_{j}|}\cdot\frac{|G_{l^{\prime\prime}}\cap\widehat{G}_{j}|}{|\widehat{G}_{j}|})\bigg{|}$
$\displaystyle=\underset{1\leq c\leq
K_{1}}{{\text{max}}}\frac{1}{s}\big{|}U_{ic}-U_{hc}\big{|}\big{|}\textrm{tr}(V^{T}\mathbb{G}\mathbb{G}^{T})\big{|},$

where for simplicity we drop the upperscript $(c)$ in $G_{l}$ and
$\widehat{G}_{j}$.

Note that both $\mathrm{COD}_{\Sigma_{p,\widehat{W}_{O}}}(w,x)$ and
$\mathrm{COD}_{\Sigma_{p,W_{O}}}(w,x)$ are influenced by $i$ and $h$ ($i\neq
h$) only through the common $|U_{ic}-U_{hc}|$ term, and so the $(i,h)$ pair
that gives the minimum of the two COD expressions is the same. Thus, the
following holds:

$\frac{\mathrm{MCOD}(\Sigma_{p,\widehat{W}_{O}})}{\mathrm{MCOD}(\Sigma_{p,W_{O}})}=\frac{|\textrm{tr}(V^{T}\mathbb{G}\mathbb{G}^{T})|}{|\mathrm{tr}(V)|}\cdot\frac{K_{2}}{s}.$

Thus,

$\Bigg{\\{}\frac{\mathrm{MCOD}(\Sigma_{p,\widehat{W}_{O}})}{\|X\|_{\widehat{W}_{O}}}\Bigg{\\}}\Bigg{/}\Bigg{\\{}\frac{\mathrm{MCOD}(\Sigma_{p,W_{O}})}{\|X\|_{W_{O}}}\Bigg{\\}}=\frac{|\textrm{tr}(V^{T}\mathbb{G}\mathbb{G}^{T})|}{|\mathrm{tr}(V)|}\cdot\frac{K_{2}}{s}\cdot\frac{||X||_{W_{O}}}{||X||_{\widehat{W}_{O}}}>\frac{4}{c_{0}}$

needs to hold. By Von Neumann’s trace inequality, we have
$|\textrm{tr}(V^{T}\mathbb{G}\mathbb{G}^{T})|\geq|\textrm{tr}(V)\cdot\lambda_{\text{min}}(GG^{T})|$,
and so it suffices to have:

$|\lambda_{\text{min}}(GG^{T})|\cdot\frac{K_{2}}{s}\cdot\frac{||X||_{W_{O}}}{||X||_{\widehat{W}_{O}}}>\frac{4}{c_{0}}.$

To show the above inequality, we need to further lower bound
$\frac{||X||_{W_{O}}}{||X||_{\widehat{W}_{O}}}$,

$\displaystyle\frac{||X||_{W_{O}}}{||X||_{\widehat{W}_{O}}}$
$\displaystyle=\frac{\underset{a}{\max}\leavevmode\nobreak\
||\frac{1}{K_{2}}\operatorname{{\rm Var}}(Z_{r(a)\cdot})+\operatorname{{\rm
Var}}(\Gamma_{a\cdot}L)||_{F}}{\underset{a}{\max}\leavevmode\nobreak\
||\frac{1}{s}\operatorname{{\rm Var}}(Z_{r(a)\cdot}G)+\operatorname{{\rm
Var}}(\Gamma_{a\cdot}\widehat{L})||_{F}}$
$\displaystyle\geq\frac{\underset{a}{\max}\leavevmode\nobreak\
\sqrt{\frac{1}{K_{2}^{2}}||\operatorname{{\rm
Var}}(Z_{r(a)\cdot})||_{F}^{2}+||\operatorname{{\rm
Var}}(\Gamma_{a\cdot}L)||_{F}^{2}}}{\underset{a}{\max}\leavevmode\nobreak\
\frac{1}{s}||G^{T}\operatorname{{\rm
Var}}(Z_{r(a)\cdot})G||_{F}+\underset{a}{\max}\leavevmode\nobreak\
||\operatorname{{\rm Var}}(\Gamma_{a\cdot}\widehat{L})||_{F}}.$

For the term $||G^{T}\operatorname{{\rm Var}}(Z_{r(a)\cdot})G||_{F}^{2}$ in
the denominator, we have

$\displaystyle||G^{T}\operatorname{{\rm Var}}(Z_{r(a)\cdot})G||_{F}^{2}$
$\displaystyle=\sum_{i=1}^{K_{2}}\lambda_{i}\\{(G^{T}\operatorname{{\rm
Var}}(Z_{r(a)\cdot})G)^{T}(G^{T}\operatorname{{\rm Var}}(Z_{r(a)\cdot})G)\\}$
(H.10) $\displaystyle\leq\text{min}(s,K_{2})\cdot||G^{T}\operatorname{{\rm
Var}}(Z_{r(a)\cdot})G||_{\text{op}}^{2}$ (H.11)
$\displaystyle\leq\min(s,K_{2})\cdot||G||^{4}_{\text{op}}\cdot||\operatorname{{\rm
Var}}(Z_{r(a)\cdot})||^{2}_{\text{op}},$

where (H.10) holds from the fact that $||A||_{F}^{2}=tr(A^{T}A)=\text{ the sum
of the eigenvalues of }A^{T}A$, and (H.11) holds from the fact there can be at
most min$(s,K_{2})$ non-zero eigenvalues of $G^{T}\operatorname{{\rm
Var}}(Z_{r(a)\cdot})G$ because rank($G^{T}\operatorname{{\rm
Var}}(Z_{r(a)\cdot})G$)$\leq$ min{rank($G$), rank($\operatorname{{\rm
Var}}(Z_{r(a)\cdot})$)} and rank($G$) $\leq$ min($s,K_{2}$). Using the similar
argument, we also show that

$||\operatorname{{\rm
Var}}(Z_{r(a)\cdot})||_{F}^{2}\geq||\text{diag}(U)||^{2}_{\min}\cdot||V||^{2}_{F}\geq
C^{2}_{\min}K_{2}||\text{diag}(U)||^{2}_{\min}.$

Finally, noting that $||G||_{\text{op}}^{2}=|\lambda_{\max}(GG^{T})|$, and

$\max_{a}||\operatorname{{\rm
Var}}(\Gamma_{a\cdot}L)||_{F}=C^{1/2}_{K}/K_{2}^{1/2},\leavevmode\nobreak\
\leavevmode\nobreak\ \max_{a}||\operatorname{{\rm
Var}}(\Gamma_{a\cdot}\widehat{L})||_{F}=C^{1/2}_{s}/s^{1/2},$

we have

$\displaystyle\frac{||X||_{W_{O}}}{||X||_{\widehat{W}_{O}}}$
$\displaystyle\geq\frac{\sqrt{\frac{1}{K_{2}}C^{2}_{\min}\cdot||\text{diag}(U)||^{2}_{\min}+C_{K}/K_{2}}}{\underset{a}{\max}\leavevmode\nobreak\
\frac{1}{s}\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|\cdot||\operatorname{{\rm
Var}}(Z_{r(a)\cdot})||_{\text{op}}+C^{1/2}_{s}/s^{1/2}}$
$\displaystyle\geq\frac{\sqrt{\frac{1}{K_{2}}C^{2}_{\min}\cdot||\text{diag}(U)||^{2}_{\min}+C_{K}/K_{2}}}{\frac{1}{s}\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|\cdot||\text{diag}(U)||_{\max}\cdot
C_{\max}+C^{1/2}_{s}/s^{1/2}}.$

So, combining with the above inequality, we need to show the following
condition:

$\displaystyle\frac{K^{1/2}_{2}|\lambda_{\min}(GG^{T})|\sqrt{C^{2}_{\min}\cdot||\text{diag}(U)||^{2}_{\min}+C_{K}}}{\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|\cdot||\text{diag}(U)||_{\max}\cdot
C_{\max}+s^{1/2}C^{1/2}_{s}}\leavevmode\nobreak\ \geq\leavevmode\nobreak\
\frac{4}{c_{0}}.$ (H.12)

To show (H.12), let’s consider the following two cases.
If
$\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|||\text{diag}(U)||_{\max}C_{\max}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ C^{1/2}_{s}s^{1/2}$ holds, then

$\displaystyle\frac{K^{1/2}_{2}|\lambda_{\min}(GG^{T})|\sqrt{C^{2}_{\min}\cdot||\text{diag}(U)||^{2}_{\min}+C_{K}}}{\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|\cdot||\text{diag}(U)||_{\max}\cdot
C_{\max}+s^{1/2}C^{1/2}_{s}}$
$\displaystyle\geq\frac{K^{1/2}_{2}|\lambda_{\min}(GG^{T})|C_{K}^{1/2}}{2s^{1/2}C^{1/2}_{s}}\geq\leavevmode\nobreak\
\frac{4}{c_{0}},$

when
$\frac{1}{\lambda_{\min}(GG^{T})}\leq\frac{c_{0}C_{K}^{1/2}K_{2}^{1/2}}{8C^{1/2}_{s}s^{1/2}}$.
In the second case
$\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|||\text{diag}(U)||_{\max}C_{\max}>C^{1/2}_{s}s^{1/2}$,
we can similarly show that

$\displaystyle\frac{K^{1/2}_{2}|\lambda_{\min}(GG^{T})|\sqrt{C^{2}_{\min}||\text{diag}(U)||^{2}_{\min}+C_{K}}}{\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|\cdot||\text{diag}(U)||_{\max}\cdot
C_{\max}+s^{1/2}C^{1/2}_{s}}$
$\displaystyle\geq\frac{K^{1/2}_{2}|\lambda_{\min}(GG^{T})|C_{\min}||\text{diag}(U)||_{\min}}{2\sqrt{\min(s,K_{2})}\cdot|\lambda_{\max}(GG^{T})|\cdot||\text{diag}(U)||_{\max}\cdot
C_{\max}}\geq\leavevmode\nobreak\ \frac{4}{c_{0}},$

when
$\frac{\lambda_{\max}(GG^{T})}{\lambda_{\min}(GG^{T})}\leq\frac{c_{0}\cdot
C_{\min}\cdot||\text{diag}(U)||_{\min}}{8\cdot
C_{\max}\cdot||\text{diag}(U)||_{\max}}\sqrt{\frac{K_{2}}{\min(s,K_{2})}}$.
This completes the proof. ∎

####  H.3.2 Proof of Proposition 4.2

###### Proof.

Recall that our definition of $G$ from Section 4 is as follows:

$\displaystyle G\leavevmode\nobreak\ =\leavevmode\nobreak\
B^{T}\widehat{B}(\widehat{B}^{T}\widehat{B})^{-1}\leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\begin{bmatrix}\frac{|G_{1}^{(c)}\cap\widehat{G}_{1}^{(c)}|}{|\widehat{G}_{1}^{(c)}|}&\frac{|G_{1}^{(c)}\cap\widehat{G}_{2}^{(c)}|}{|\widehat{G}_{2}^{(c)}|}&...&\frac{|G_{1}^{(c)}\cap\widehat{G}_{s}^{(c)}|}{|\widehat{G}_{s}^{(c)}|}\\\
:&:&&:\\\
\frac{|G_{K_{2}}^{(c)}\cap\widehat{G}_{1}^{(c)}|}{|\widehat{G}_{1}^{(c)}|}&\frac{|G_{K_{2}}^{(c)}\cap\widehat{G}_{2}^{(c)}|}{|\widehat{G}_{2}^{(c)}|}&...&\frac{|G_{K_{2}}^{(c)}\cap\widehat{G}_{s}^{(c)}|}{|\widehat{G}_{s}^{(c)}|}\end{bmatrix}$

Note that
$\widehat{G}_{1}^{(c)}=\widehat{G}_{2}^{(c)}=...=\widehat{G}_{s}^{(c)}=1$ when
$\widehat{B}=I_{q}$. It is easily seen that
$GG^{T}=\text{diag}(m_{q}^{(1)},m_{q}^{(2)},\leavevmode\nobreak\
...\leavevmode\nobreak\ ,m_{q}^{(K_{2})})$ (up to order) where $m_{q}^{(i)}$
denotes the $i$-th smallest column cluster size. Thus,
$\lambda_{\min}(GG^{T})=m_{q}$ and $\lambda_{\max}(GG^{T})=M_{q}$, where
$m_{q}$ and $M_{q}$ denote the smallest and largest column cluster sizes,
respectively.

Recall we have defined:

$\displaystyle C_{K}\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\max_{1\leq a\leq
p}\frac{1}{K_{2}}\sum_{t=1}^{K_{2}}\frac{(\sum_{j\in[t]}\sigma^{2}_{aj})^{2}}{|[t]|^{4}}$
$\displaystyle C_{s}\leavevmode\nobreak\ $ $\displaystyle=\leavevmode\nobreak\
\max_{1\leq a\leq
p}\frac{1}{s}\sum_{t=1}^{s}\frac{(\sum_{j\in[\widehat{t}]}\sigma^{2}_{aj})^{2}}{|[\widehat{t}]|^{4}}.$

Under the assumption that the noise is homogeneous and the cluster size is
balanced (i.e., for all $i,j$, $\mathbb{E}(\Gamma_{ij})=\sigma^{2}=O(1)$,
$\frac{M_{q}}{m_{q}}\leq C$ for some constant $C\geq 1$), we have that
$C_{K}=\frac{1}{K_{2}}\sum_{t=1}^{K_{2}}\frac{\sigma^{4}}{|[t]|^{2}}\geq\frac{\sigma^{4}}{M_{q}^{2}}$.
Also, by the formulation of our weight $\widehat{W}=\frac{1}{q}I_{q}$, we have
that $C_{s}=\sigma^{4}$.

In order to satisfy the stability condition in Proposition 4.1 (which is the
condition related to the effect of the initial weight), when using
$\widehat{W}=\frac{1}{q}I_{q}$, the second set of conditions, (ii), is more
relaxed. These conditions are implied by the following:

$\begin{cases}M_{q}\leavevmode\nobreak\ &\geq\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sqrt{\frac{q}{K_{2}}}\cdot\frac{\sigma^{2}}{||\text{diag}(U)||_{\max}\cdot
C_{\max}}\\\ C\leavevmode\nobreak\ &\leq\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{c_{0}}{8}\cdot\frac{C_{\min}}{C_{\max}}\cdot\frac{||\text{diag}(U)||_{\min}}{||\text{diag}(U)||_{\max}}.\end{cases}$

Assuming that
$\frac{C_{\min}}{C_{\max}}\cdot\frac{||\text{diag}(U)||_{\min}}{||\text{diag}(U)||_{\max}}$
is a constant, the latter condition will hold as long as we take a large
enough $c_{0}$ value. Recall this is an arbitrary constant $\geq 4$ that
determines the tradeoff between separation and stability. As for the former
condition, since $m_{q}^{(1)}+m_{q}^{(2)}+...+m_{q}^{(K_{2})}=q$, we have
$\frac{q}{K_{2}}\leq M_{q}$. Thus, as long as
$\sqrt{M_{q}}\geq\frac{\sigma^{2}}{||\text{diag}(U)||_{\max}\cdot C_{\max}}$,
the first condition is satisfied. ∎

##  Appendix I Additional Simulations

###  I.1 Main Simulation Results with Alternative Noise Settings

The results for the main simulations in the homogeneous and proportional noise
variance settings are shown in Figure 7 and Figure 8, respectively.

Figure 7: The ARI and the successful implementation rates of our hierarchical
clustering Algorithm 1 with $W_{I}$ (NAIVE COD), the one-step Algorithm 2
(1-STEP COD), the two-step Algorithm 3 (2-STEP COD), and competing model-based
tensor clustering methods DEEM [Mai et al. (2022)], TGMM and TEMM [Deng and
Zhang (2022)] under the homogeneous noise variance setting. Here, $p=q=100$,
$K_{1}=K_{2}=10$, and we have moderately unbalanced cluster sizes of
$(3,6,6,8,10,10,12,12,14,19)$ for the rows and the columns each. Successful
implementation rates are especially emphasized as the competing methods (DEEM,
TGMM, TEMM) tended to fail during execution for some of the runs. The ARI
rates for these methods were averaged over the successful runs, and when there
were no successes, the ARI was reported as 0 with a dashed line.

Figure 8: The ARI and the successful implementation rates of our hierarchical
clustering Algorithm 1 with $W_{I}$ (NAIVE COD), the one-step Algorithm 2
(1-STEP COD), the two-step Algorithm 3 (2-STEP COD), and competing model-based
tensor clustering methods DEEM [Mai et al. (2022)], TGMM and TEMM [Deng and
Zhang (2022)] under the proportional noise variance setting. The other
settings are the same as in Figure 7.

###  I.2 Unbalanced $p$ and $q$

Here we present the unbalanced setting of $p=100$, $q=15$, $K_{1}=10$,
$K_{2}=5$, shown in Figures 9, 10 and 11. In this case, we have long, skinny
matrices that are “unbalanced”. We find that the COD methods perform
drastically better in column clustering compared to the tensor model based
methods. In all three noise variance settings, the competing methods have 0
successful implementation rates across all $n$ values in column clustering,
whereas our COD methods perform significantly better. We believe this occurs
because in the tensor normal setting, having too small of a $q$ value is
equivalent to having too few “i.i.d. observations”, which causes their
algorithms to diverge. In contrast, for the COD algorithms, since the
threshold for perfect column clustering recovery is on the order of
$\sqrt{\frac{\log q}{nK_{1}}}$, if $q$ is relatively small and $K_{1}$ is
relatively large, this threshold decreases compared to the balanced matrix
setting and the condition for perfect column clustering recovery becomes more
relaxed. Thus, our COD based methods typically yield more accurate clustering
results compared to the competing tensor model based methods, especially when
the number of rows and columns are unbalanced.

Figure 9: The ARI and the successful implementation rates of our hierarchical
clustering Algorithm 1 with $W_{I}$ (NAIVE COD), the one-step Algorithm 2
(1-STEP COD), the two-step Algorithm 3 (2-STEP COD), and competing model-based
tensor clustering methods DEEM [Mai et al. (2022)], TGMM and TEMM [Deng and
Zhang (2022)] under the homogeneous noise variance setting. Here, $p=100$,
$q=15$, $K_{1}=10$, $K_{2}=5$, where the decreased $q$ is the notable
difference with the main simulation results. We have moderately unbalanced
cluster sizes of $(3,6,6,8,10,10,12,12,14,19)$ and $(2,2,2,3,6)$ for the rows
and the columns, respectively. Successful implementation rates are especially
emphasized as the competing methods (DEEM, TGMM, TEMM) tended to fail during
execution for some of the runs. The ARI rates for these methods were averaged
over the successful runs, and when there were no successes, the ARI was
reported as 0 with a dashed line.

Figure 10: The ARI and the successful implementation rates of our hierarchical
clustering Algorithm 1 with $W_{I}$ (NAIVE COD), the one-step Algorithm 2
(1-STEP COD), the two-step Algorithm 3 (2-STEP COD), and competing model-based
tensor clustering methods DEEM [Mai et al. (2022)], TGMM and TEMM [Deng and
Zhang (2022)] under the proportional noise variance setting. The other
settings are the same as in Figure 9.

Figure 11: The ARI and the successful implementation rates of our hierarchical
clustering Algorithm 1 with $W_{I}$ (NAIVE COD), the one-step Algorithm 2
(1-STEP COD), the two-step Algorithm 3 (2-STEP COD), and competing model-based
tensor clustering methods DEEM [Mai et al. (2022)], TGMM and TEMM [Deng and
Zhang (2022)] under the random noise variance setting. The other settings are
the same as in Figure 9.

##  Appendix J Real Data Analysis with a Competing Method

We add the real data analysis results from using the competing method DEEM
(Deng and Zhang (2022)) on the same gene-tissue data mentioned in Section 8.
The results are illustrated in Figure 12. Of the gene clustering results from
our method shown in Figure 2, DEEM was also able to cluster “O03Rik” and
“Nfat5” together and “Rac2” and “Mapkapk2” together. However, DEEM was not
able to capture the relationship between “Nfatc4” and “Ppp3r1”, which was
identified with our method. The mechanism behind the relationship is that the
“Ppp3r1” and “Nfatc4” genes are both related to calcineurin - one having to do
with how it is generated, and one having to do with something it activates.
Another possible explanation is given in Heit et al. (2006) which postulates
that calcineurin/NFAT signaling is critical in $\beta$-cell growth in the
pancreas. Either way, it signals that our method can cluster meaningful genes
that are not able to be identified with other competing methods. For the
tissues, the competing method DEEM was able to cluster the “Lung” - “Heart”
pair of vascular tissues and the “Adrenal” - “Gonads” pair of steroid
responsive tissues that did not appear in our results with COD. However, it
was not able to cluster the “Cerebrum” - “Cerebellum” - “Hippocampus” trio of
neural tissues, the steroid responsive “Adrenal” - “Thymus” pair, or the
vascular “Lung” - “Kidney” pair that were clustered together with COD. Thus,
although the competing method gives reasonably meaningful results that are
different from the results using COD, it is apparent that our method can
complement existing clustering methods and provide useful information that
wasn’t available.

Figure 12: Gene clusters and tissue clusters obtained from the competing
method DEEM (Deng and Zhang (2022)).

##  Appendix K Cluster Evaluation Metrics

For this section, the true row cluster partition will be denoted simply as
$\mathcal{G}=\\{G_{1},...,G_{K_{1}}\\},$
$\leavevmode\nobreak\ \textrm{where}\leavevmode\nobreak\ \leavevmode\nobreak\
G_{k}=\\{a:A_{ak}=1\\}$, while the estimated row cluster partition will be
denoted as
$\widehat{\mathcal{G}}=\\{\widehat{G}_{1},...,\widehat{G}_{s}\\},\leavevmode\nobreak\
\textrm{where}\leavevmode\nobreak\ \leavevmode\nobreak\
\widehat{G}_{k}=\\{a:\widehat{A}_{ak}=1\\}$.

###  K.1 Sensitivity and Specificity

Then, for any pair $1\leq j<k\leq p$, we can define

$\displaystyle TP_{jk}$ $\displaystyle=\textbf{1}\\{j,k\in G_{a}\text{ and
}j,k\in\widehat{G}_{b}\text{ for some }1\leq a\leq K_{1},1\leq b\leq s\\}$
$\displaystyle TN_{jk}$ $\displaystyle=\textbf{1}\\{j,k\not\in G_{a}\text{ and
}j,k\not\in\widehat{G}_{b}\text{ for some }1\leq a\leq K_{1},1\leq b\leq s\\}$
$\displaystyle FP_{jk}$ $\displaystyle=\textbf{1}\\{j,k\not\in G_{a}\text{ and
}j,k\in\widehat{G}_{b}\text{ for some }1\leq a\leq K_{1},1\leq b\leq s\\}$
$\displaystyle FN_{jk}$ $\displaystyle=\textbf{1}\\{j,k\in G_{a}\text{ and
}j,k\not\in\widehat{G}_{b}\text{ for some }1\leq a\leq K_{1},1\leq b\leq s\\}$

and define

$\displaystyle TP$ $\displaystyle=\sum_{1\leq j<k\leq
p}TP_{jk},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ TN=\sum_{1\leq
j<k\leq p}TN_{jk}$ $\displaystyle FP$ $\displaystyle=\sum_{1\leq j<k\leq
p}FP_{jk},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ FN=\sum_{1\leq
j<k\leq p}FN_{jk}$

Sensitivity (SN) and specificity (SP) can then be defined as follows:

$\displaystyle SN=\frac{TP}{TP+FN}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ SP=\frac{TN}{TN+FP}$

###  K.2 Adjusted Rand Index (ARI)

The following is a cross tabulation of two cluster partitions (the true
partition and the estimated partition).

|  $\widehat{G}_{1}$ |  $\widehat{G}_{2}$ |  … |  $\widehat{G}_{s}$ |  Sum
---|---|---|---|---|---
$G_{1}$ |  $n_{11}$ |  $n_{12}$ |  … |  $n_{1s}$ |  $|G_{1}|$
$G_{2}$ |  $n_{21}$ |  $n_{22}$ |  … |  $n_{2s}$ |  $|G_{2}|$
: |  : |  : |  … |  : |  :
$G_{K_{1}}$ |  $n_{K_{1}1}$ |  $n_{K_{1}2}$ |  … |  $n_{K_{1}s}$ |  $|G_{K_{1}}|$
Sum |  $|\widehat{G}_{1}|$ |  $|\widehat{G}_{2}|$ |  … |  $|\widehat{G}_{s}|$ |  $p$

The ARI is then defined as follows:

$\displaystyle\text{ARI}=\frac{\sum_{i,j}\binom{n_{ij}}{2}-\Big{[}\sum_{i}\binom{|G_{i}|}{2}\sum_{j}\binom{|\widehat{G}_{j}|}{2}\Big{]}\Big{/}\binom{p}{2}}{\frac{1}{2}\Big{[}\sum_{i}\binom{|G_{i}|}{2}+\sum_{j}\binom{|\widehat{G}_{j}|}{2}\Big{]}-\Big{[}\sum_{i}\binom{|G_{i}|}{2}\sum_{j}\binom{|\widehat{G}_{j}|}{2}\Big{]}\Big{/}\binom{p}{2}}$

Note that this is the corrected-for-chance version of the Rand index (Rand,
1971). An ARI value of 1 implies a perfect match between the two cluster
partitions.


# Precision Mass Measurement of Proton-Dripline Halo Candidate 22Al

S. E. Campbell<EMAIL_ADDRESS>G. Bollen  B. A. Brown  A. Dockery
Department of Physics and Astronomy, Michigan State University, East Lansing,
Michigan 48824, USA  Facility for Rare Isotope Beams, East Lansing, Michigan
48824, USA  K. Fossez  Department of Physics, Florida State University,
Tallahassee, Florida 32306, USA  Physics Division, Argonne National
Laboratory, Lemont, Illinois 60439, USA  C. M. Ireland  K. Minamisono  D.
Puentes  Department of Physics and Astronomy, Michigan State University, East
Lansing, Michigan 48824, USA  Facility for Rare Isotope Beams, East Lansing,
Michigan 48824, USA  A. Ortiz-Cortes  Facility for Rare Isotope Beams, East
Lansing, Michigan 48824, USA  B. J. Rickey  R. Ringle  Department of Physics
and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
Facility for Rare Isotope Beams, East Lansing, Michigan 48824, USA  S. Schwarz
C. S. Sumithrarachchi  A. C. C. Villari  Facility for Rare Isotope Beams, East
Lansing, Michigan 48824, USA  I. T. Yandow  Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
Facility for Rare Isotope Beams, East Lansing, Michigan 48824, USA

(December 18, 2023)

###### Abstract

We report the first mass measurement of the proton-halo candidate 22Al
performed with the LEBIT facility’s 9.4 T Penning trap mass spectrometer at
FRIB. This measurement completes the mass information for the lightest
remaining proton-dripline nucleus achievable with Penning traps. 22Al has been
the subject of recent interest regarding a possible halo structure from the
observation of an exceptionally large isospin asymmetry [Phys. Rev. Lett. 125
192503 (2020)]. The measured mass excess value of $\text{ME}=18\;093.6(7)$
keV, corresponding to an exceptionally small proton separation energy of
$S_{p}=99.2(1.0)$ keV, is compatible with the suggested halo structure. Our
result agrees well with predictions from sd-shell USD Hamiltonians. While USD
Hamiltonians predict deformation in 22Al ground-state with minimal $1s_{1/2}$
occupation in the proton shell, a particle-plus-rotor model in the continuum
suggests that a proton halo could form at large quadrupole deformation. These
results emphasize the need for a charge radius measurement to conclusively
determine the halo nature.

##  I Introduction

Nuclei at, and near, the boundaries of nuclear existence serve as excellent
probes for evaluating nuclear models given their exotic nature [1, 2, 3]. Halo
nuclei are one such example; seated at or near the nuclear driplines, their
mass distribution extends far beyond the compact core, creating their so-
called ‘halo’ structure. Efforts to identify, study, and model these nuclei
have provided critical evaluations of nuclear forces (see [4, 5, 6] for
reviews).

Studies of proton halo structures are often limited, compared to neutron
halos, as their formation is suppressed by the confining effect of the Coulomb
barrier. Additional complications arise from low production cross-sections for
nuclei approaching the proton dripline. Precision measurements of these nuclei
are of particular importance as the proton drip line is sufficiently well
known compared to that for neutrons, offering a unique probe of nuclear
forces.

22Al was first proposed as a proton-halo candidate resulting from an
exceptionally small proton separation energy extrapolated from systematic
trends in the mass region [7] and from calculations of isospin-symmetry
breaking effects in the sd-shell region [8]. Recent excitement pointing to a
halo structure in the first $1^{+}$ state of 22Al stems from an observed
isospin asymmetry of $\delta_{\beta}=2.09(96)$ in the Gamow-Teller $\beta^{+}$
transition from the first excited $1^{+}$ state of 22Si and with its mirror
22O $\beta^{-}$ transition [9]. This is by far the largest asymmetry reported
for low-lying states. Results, however, contain significant uncertainties
propagating from its unmeasured mass. The presence of an $s$-wave halo in 22Al
should favor deformation by increasing the proton $1s_{1/2}$—$0d_{5/2}$
quadrupole coupling. In fact, relativistic mean-field calculations [10] and
data evaluation [11] support large quadrupole deformation of about
$\beta_{2}>0.2$ in 22Al.

Due to the role of weak binding in the emergence of halo structures, precise
knowledge of the binding energy–and consequentially mass–is paramount [3, 12].
In quantum mechanics, the halo can be seen as a leakage of the valence nucleon
wave function, which itself depends exponentially on the nucleon separation
energy. Precise mass information is also required to extract nuclear charge
radii from isotope shift measurements [13]. Since the current mass value of
22Al is based purely on extrapolation of systematic trends of mass values of
nearby isotopes [7], there is a clear need for a first, precise mass
measurement of 22Al.

##  II Experimental Method and Analysis

At the Facility for Rare Isotope Beams (FRIB), the proton-rich 22Al isotope
was produced by projectile fragmentation: a 36Ar primary beam was sent through
the FRIB linear accelerator, ultimately impinging upon a 12C target of about 8
mm thickness. The cocktail beam was then sent to the Advanced Rare Isotope
Separator [14] for separation. Two aluminum degraders of thicknesses 3287
$\mu$m and 1015 $\mu$m, followed by a 1003 $\mu$m 2.67 mrad aluminum wedge
prepared the beam for stopping in the Advanced Cryogenic Gas Stopper (ACGS)
[15]. Following mass separation with a magnetic dipole, the extracted 30-keV
energy beam showed highest beta activity at A=22 and was sent to the Low
Energy Beam Ion Trap (LEBIT) facility. There, the continuous beam was
delivered to the Cooler-Buncher [16], allowing ions to cool for 10 ms. Ion
bunches were purified with a coarse time-of-flight gate before injection into
the 9.4 T Penning trap mass spectrometer [17]. In the trap, nearby
contaminants were cleaned using the Stored Waveform Inverse Fourier Transform
(SWIFT) technique [18].

LEBIT uses Penning trap mass spectrometry to determine an ion’s mass relative
to a well-known calibrant through measurements of their cyclotron frequencies,
$\nu_{c}$:

$R=\frac{\nu_{c,\textrm{calib}}}{\nu_{c}}=\frac{q_{\textrm{calib}}(m-m_{e}q)}{q(m_{\textrm{calib}}-m_{e}q_{\textrm{calib}})},$
(1)

where $m$ and $q$ are the ion atomic mass and charge state respectively.
Electron binding energies are negligible as they are orders of magnitude
smaller than the measurement precision.

Initial measurements were performed with the well-established Time-of-flight
Ion Cyclotron Resonance (TOF-ICR) technique in use at LEBIT [19, 20, 21]. The
achievable precision is determined in part by excitation duration, which
defines the frequency selectivity and scales linearly with duration. Because
of its half-life of $\approx 85$ ms [22], excitation times of 50 and 75 ms
were used for 22Al. A much longer 500 ms excitation was used for 23Na. Figure
1(a) shows one TOF-ICR resonance spectrum obtained for 22Al+.

(a)(b) Figure 1: (a) Observed 75 ms 22Al+ TOF-ICR resonance and fit,
containing approximately 350 detected ions. (b) Observed 50 ms PI-ICR beam
spot resulting from reduced cyclotron motion, containing approximately 50
detected ions. The spot is fit to a 2D (polar) Gaussian. The black arrow shows
the direction of motion.

Additional measurements were performed with the recently implemented Phase
Imaging Ion Cyclotron Resonance (PI-ICR) technique [23] at LEBIT. A ‘two-
phase’ approach is used, measuring the magnetron ($\nu_{-}$) and reduced
cyclotron ($\nu_{+}$) frequencies independently. After evolving for a duration
$t_{\textrm{acc}}$, the final phase $\phi_{\textrm{final}}$ of the ions
relative to a reference location $\phi_{\textrm{ref}}$ determines the
frequency:

$\nu_{\pm}=\frac{\left(\phi_{\textrm{final},\pm}-\phi_{\textrm{ref}}\right)+2\pi
n}{2\pi t_{\textrm{acc}}}.$ (2)

$n$ is the number of full revolutions undergone by the ions during
$t_{\textrm{acc}}$, and is known sufficiently, so long as the frequency
uncertainty is less than $1/t_{\textrm{acc}}$. The ion’s cyclotron frequency
is found by $\nu_{c}=\nu_{+}+\nu_{-}$ as described in [24]. Similar to the
restrictions for TOF-ICR, $t_{\textrm{acc}}$ was set to 50 ms for 22Al+, and
150 ms for 23Na+. Figure 1(b) shows an example of a final phase spot for 22Al+
with $t_{\textrm{acc}}=50$ ms.

A number of systematic uncertainties in $\bar{R}$ have been shown to scale
linearly with the difference in mass of the calibrant ion and the ion of
interest. This includes shifts due to magnetic field inhomogeneities,
deviations in the axial alignment of the Penning trap with the magnetic field,
and distortions of the electrostatic trapping potential [25]. Mass dependent
shifts for TOF-ICR measurements are known to add an uncertainty of
$\delta\bar{R}\approx 2\times 10^{-10}/u$ [26], which is negligible compared
to the statistical uncertainty. Additional contributions, such as those from
isobaric contaminants [27] and space charge effects [25], were mitigated by
in-trap beam purification with SWIFT and by limiting the number of ions in the
trap to less than five at a time. Furthermore, studies of non-linear temporal
magnetic field instabilities show a contribution to the uncertainty in
$\bar{R}$ by no more than $1\times 10^{-9}$ [28] over the course of one hour,
which is greater than the duration of any of the measurements performed.

Additional systematic effects were accounted for in the TOF-ICR measurements
by performing offline measurements of 23Na+ with 22Ne+ as an isobaric stand-in
for 22Al+. Both these ions’ masses are very well known. The ions were produced
using an offline plasma ion source operated with high-purity neon gas at
LEBIT. Experimental conditions were matched as closely as possible to those
with 22Al, and the total trapping times remained unchanged. All system
settings were not modified following the experiment. The systematic
uncertainty was determined to be $\delta\bar{R}_{\textrm{TOF}}=-1(2)\times
10^{-8}$.

With the increased sensitivity of PI-ICR, a number of additional systematic
effects are present. Extensive systematic investigations performed at other
facilities [29, 30, 31] show that these are dominated by many of the same as
listed above for TOF-ICR. PI-ICR specific effects are also largely determined
by the electrostatic trapping field harmonicity, magnetic field alignment, and
count rate dependent frequency shifts [32]. These are mitigated by using
smaller total trapping times, maintaining the same spot location on the MCP,
and limiting the MCP count rate to $\approx 2$ ions per shot respectively. A
magnetic field misalignment will result in an elliptical projection of the ion
motion. This was accounted for by imaging this projection and fitting the
result to an ellipse. The acquired fit is shown in Fig. 1(b). 23Na and 22Ne
were also used to calculate a systematic uncertainty for PI-ICR, maintaining
the accumulation times and angular locations respectively. The systematic
uncertainty was determined to be $\delta\bar{R}_{\textrm{PI}}=0(2)\times
10^{-8}$.

##  III Results

A total of four TOF-ICR and three PI-ICR measurements were taken over the
course of approximately 16 hours. The TOF-ICR measurements resulted in a
weighted average of $\bar{R}_{\textrm{TOF}}=1.04406876(6)$, and for the PI-ICR
measurements $\bar{R}_{\textrm{PI}}=1.044068841(14)$. A summary of these
measurements is shown in Fig. 2. Combined, these results yield
$\bar{R}=1.044068837(14)$, corresponding to a mass excess of $18\;093.6(7)$
keV and a proton separation energy of 99.2(1.0) keV.

Figure 2: Measured frequency ratios of 22Al+ relative to 23Na+. Blue points
and uncertainty region correspond to TOF-ICR measurements, red points and
uncertainty region correspond to PI-ICR measurements. Dashed black lines
indicate the combined TOF-ICR and PI-ICR result. All uncertainties are shown
to $1\sigma$.

##  IV Discussion

###  IV.1 Shell Model with USD Hamiltonians

The low-lying structure for the nuclei around 22Al can be described within the
$sd$ $(0d_{5/2},0d_{3/2},1s_{1/2})$ model space. Several “universal”
$sd$-shell (USD) Hamiltonians have been developed over the last 40 years [33],
[34], [35]. They are all obtained by starting with a set of two-body matrix
elements (TBME) obtained with nucleon-nucleon potentials renormalized to the
$sd$ model space. The singular-valued decomposition (SVD) method is then used
to constrain linear-combinations of TBME to obtain lower uncertainties for
binding energies and excitation energies. The lastest SVD-type fit from 2020
was based on Hamiltonians that included the Coulomb and isospin-dependent
strong interactions treated explicitly [35], and provide absolute binding
energies for all $sd$ shell nuclei relative to that of 16O. This resulted in
two Hamiltonians: USDC starting with the Bonn-A [36] interaction, and USDI
starting with the VS-IMSRG method [37], [38]. The rms deviation between the
experimental and calculated energies for both USDC and USDI was 140 keV. The
results for $S_{p}$(22Al) are 40 keV (USDC) and 30 keV (USDI) which are in
reasonable agreement with experiment. The calculated spectrum of the mirror
nucleus 22F is in excellent agreement with experiment (see Fig. 3).

Figure 3:  Experimental levels of 22F up to 5 MeV compared to the results
obtained from the USDC and USDI Hamiltonians. The calculated levels of 22Al
are also shown. The horizontal lines in the figure are proportional to the $J$
value for positive parity states. For the experiment shown on the left-hand
side, the levels with unknown $J$ are shown by black dots. Levels with
uncertain $J$ are indicated by the red lines ending with a black dot.

As discussed in Sec. V of [35], a special consideration for proton-rich nuclei
is the Thomas-Ehrman shift (TES). For example, the 1/2+ excited state in 17F
is shifted down by 0.376 MeV compared to its energy in the mirror nucleus 17O.
Single-particle states with low-$\ell$ values that are loosely bound (in this
case $\ell$=0 for the $1s_{1/2}$ which is bound by only 0.105 MeV in 17F) are
lowered in energy relative to those with high-$\ell$ values. The reason is
that the low-$\ell$ states are extended in size, and the kinetic energy of the
extended wavefunctions decreases. As discussed in [35], the TES depends on the
separation energy of the specific state and the related spectroscopic factors.
It is a many-body effect that cannot be corrected simply by a modification of
single-particle energies in the Hamiltonian. In the 22Al mass region there are
excited states that have a significant $\ell$=0 spectroscopic factor with a
TES as high as $-0.75$ MeV (see Fig. 12 of [35]).

A key feature for all of the USD Hamiltonians is the increase in the
$1s_{1/2}$—$0d_{5/2}$ single-particle energy gap from 0.9 MeV at $N=8$ to 4.0
MeV at $N=14$ (see Fig. 4 in [39]). This shift has been attributed to three-
body interactions between the valence neutrons and the core nucleons [40] and
it is contained in VS-IMSRG Hamiltonian for the $sd$ shell [41]. Taking into
account the TES, the proton $1s_{1/2}---0d_{5/2}$ gap changes from 0.5 MeV at
$Z$=8 to about 3.5 MeV at $Z=14$. The calculated proton $1s_{1/2}$ occupations
for the $4^{+}$ and $3^{+}$ states of 22Al are 0.29 and 0.38, respectively.

The $3^{+}$ state in the mirror nucleus 22F is only 0.072 MeV above the
$4^{+}$ ground-state. The TES from the larger $1s_{1/2}$ occupation of the
$3^{+}$ state could result in a $3^{+}$ ground-state for 22Al, as suggested by
[42, 43]. However, the calculated $\beta^{+}$ decay properties of 22Al are
more consistent with a $4^{+}$ ground-state assignment. It will be important
to measure the spin-parity and the moments of the 22Al ground-state to better
characterize the structure.

###  IV.2 Particle-plus-Rotor Model

Based on the small proton separation energy of about 0.1 MeV measured in 22Al,
one can also investigate the possible presence of a ground-state halo
structure using the particle-plus-rotor model (PRM) including couplings to
continuum states. The motivation for using this model is to investigate the
possible interplay between weak binding, which enhances the occupation of
$s$-waves, and deformation via the $s_{1/2}$—$d_{5/2}$ quadrupole coupling,
which lowers the proton $1s_{1/2}$ shell. Here, one wants to test how the
presence of deformation affects the formation of a halo structure.

We assume that only couplings between the angular momentum $j_{r}$ associated
with the rotational motion of 21Mg and the angular momentum $j$ of the weakly-
bound valence proton play a role in the particle-plus-rotor dynamic
($\vec{j}_{r}+\vec{j}=\vec{j}_{\text{PRM}}$), while the angular momentum
$J_{i}$ coming from the structure of 21Mg can be ignored for our purpose. This
approximation is justified by the fact that, except for one well-bound neutron
in $0d_{5/2}$, all nucleons in 21Mg are expected to be paired and coupled to
$0^{+}$. It follows that in this picture, the low-lying multiplets
$(1,2,3,4,5)^{+}$ and $(2,3)^{+}$ of 22Al are associated with
$j_{\text{PRM}}^{\pi}={5/2}^{+}$ and ${1/2}^{+}$, respectively, and the states
within each multiplet are degenerate.

In the PRM [44], the core-nucleon interaction is represented by a Woods-Saxon
potential with quadrupole deformation controlled by the deformation parameter
$\beta_{2}$, and the wave function is expanded in terms of channel wave
functions defined by the quantum numbers $(n,l,j,j_{r})$ where $(n,l,j)$ are
the usual shell quantum numbers for the valence particle. Each channel wave
function is expanded using the Berggren basis to account for continuum
couplings, and the coupled-channel equations obtained are solved exactly by
direct diagonalization.

The above shell model calculations using the USD family of interactions,
predict a fairly limited occupation of the $\pi 1s_{1/2}$ shell in 22Al and
are not indicative of any significant deformation. Consequently, we first
assume spherical symmetry for 21Mg ($\beta_{2}=0$) and adjust the potential to
reproduce the experimental binding energy of 22Al and use the energy spectra
of 21Mg to fix the $1s_{1/2}$ shell at about 200 keV above the $0d_{5/2}$
shell, respectively, which will be understood as Nilsson orbits for
$\beta_{2}\neq 0$. These s.p. energies differ from those in magic nuclei to
account for the absence of residual interaction in the PRM. We obtain a
diffuseness $d=0.67~{}\text{fm}$, a radius $R_{0}=3.2~{}\text{fm}$, a depth
$V_{0}=54.8~{}\text{MeV}$, a spin-orbit coupling
$V_{\text{so}}=3.62~{}\text{MeV}$, and a WS Coulomb potential radius
$R_{c}=3.0~{}\text{fm}$.

The potential is then deformed by varying $\beta_{2}$, and the ground-state
energy of 22Al together with the norms of channel wave functions are
calculated. Results are shown in Fig. 4.

Figure 4: Energy of the 21Mg+p system (top panel), and norms of channel wave
functions for $j_{\text{PRM}}={5/2}^{+}$ (middle panel) and ${1/2}^{+}$ (lower
panel) as a function of the quadrupole deformation parameter $\beta_{2}$.

Both prolate ($\beta_{2}>0$) and oblate ($\beta_{2}<0$) deformations are shown
to lower the energy, suggesting that a weakly-bound valence proton pushes the
system to deform.

For a small deformation ($|\beta_{2}|<0.1$) qualitatively compatible with
shell model calculations, the weight of the channel wave function
$(j_{r}=2)1s_{1/2}$ remains relatively low compared to other channels built on
the $0d_{5/2}$ shell. In this scenario, the presence of a $s$-wave halo would
thus be unlikely. However, for a significant prolate deformation
$\beta_{2}>0.1$, the ${1/2}^{+}$ PRM state associated with the $(2,3)^{+}$
multiplet in 22Al becomes the ground state. This state is dominated by the
$s$-wave channel and shown in Fig. 4 (lower panel) and would likely support a
halo structure. While such large quadrupole deformations are not supported by
the shell model, relativistic mean-field calculations [10] and data evaluation
[11], however, predict deformations of $\beta_{2}=0.284$ and $0.226$,
respectively. Using the latter estimate as a guess and readjusting the core-
proton potential to match the experimental binding energy by lowering the
potential depth does not affect the results significantly.

##  V Conclusions

In summary, the first precision mass measurement of 22Al is reported. The
exceptionally small proton separation energy observed at 99.2(1.0) keV agrees
with extrapolations by the 2020 Atomic Mass Evaluation as well as predictions
made with the sd-shell USD Hamiltonians. Such a small nucleon separation
energy is one such benchmark allowing for halo formation. The shell model,
reproducing the experimental result within uncertainties, suggests a very
small $1s_{1/2}$ orbital occupation. In the PRM, this small $1s_{1/2}$
occupation translates in a small deformation parameter and no halo structure.
Conversely, the observation of a halo state would indicate strong continuum-
induced deformation similar to the suspected situation in 29F [45]. Both
models are constrained by a $4^{+}$ ground state assignment for 22Al, though
it remains unmeasured [46, 22]. Ultimately, the charge radius and nuclear
spin, which can be determined from the hyperfine structure measurements by
laser spectroscopy, are needed to conclusively determine the halo nature of
the 22Al ground state and to resolve the confusion surrounding its structure.

###### Acknowledgements.

This work was conducted with the support of Michigan State University and the
National Science Foundation under Grants No. PHY-1102511, PHY-1126282,
PHY-2111185, and PHY-2238752. This material is based upon work supported by
the U.S. Department of Energy, Office of Science, Office of Nuclear Physics
and used resources of the Facility for Rare Isotope Beams (FRIB) Operations,
which is a DOE Office of Science User Facility under Award Number DE-
SC0023633, and under the FRIB Theory Alliance Award No. DE-SC0013617. Thank
you to Witek Nazarewicz for his fruitful discussions and input towards the
presented theoretical work.

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