$\displaystyle\mathbb{C}\mathrm{ov}\bigg{(}\varepsilon_{i,j}^{(1)}\Big{(}1+\frac{\eta_{i,j}}{p_{i,j}}\Big{)},\varepsilon_{i,j}^{(0)}\Big{(}1-\frac{\eta_{i,j}}{1-p_{i,j}}\Big{)}\bigg{)}$
$\displaystyle=\mathbb{E}\bigg{[}\varepsilon_{i,j}^{(1)}\Big{(}1+\frac{\eta_{i,j}}{p_{i,j}}\Big{)}\times\varepsilon_{i,j}^{(0)}\Big{(}1-\frac{\eta_{i,j}}{1-p_{i,j}}\Big{)}\bigg{]}$
(A.160)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\bigg{[}\Big{(}1+\frac{\eta_{i,j}}{p_{i,j}}\Big{)}\Big{(}1-\frac{\eta_{i,j}}{1-p_{i,j}}\Big{)}\bigg{]}\mathbb{E}[{\varepsilon_{i,j}^{(1)}\varepsilon_{i,j}^{(0)}}]$
(A.161)
$\displaystyle=\bigg{(}1-\mathbb{E}\bigg{[}\frac{\eta^{2}_{i,j}}{{p_{i,j}\big{(}1-p_{i,j}\big{)}}}\bigg{]}\bigg{)}\mathbb{E}[{\varepsilon_{i,j}^{(1)}\varepsilon_{i,j}^{(0)}}]$
(A.162)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}0\cdot\mathbb{E}[{\varepsilon_{i,j}^{(1)}\varepsilon_{i,j}^{(0)}}]=0,$
(A.163)

where $(a)$ follows because
$(\varepsilon^{(0)}_{i,j},\varepsilon^{(1)}_{i,j})\hskip
2.84526pt{\perp\\!\\!\\!\perp}\leavevmode\nobreak\ \eta_{i,j}$ from 2 and
$(b)$ follows because
$\mathbb{E}[\eta_{i,j}^{2}]=\mathbb{V}\mathrm{ar}(\eta_{i,j})=p_{i,j}(1-p_{i,j})$
from Eq. 3.

####  C.2.2 Proof of Lyapunov’s condition with $\omega=1$

We have

$\displaystyle\frac{\sum_{i\in[N]}\mathbb{E}\big{[}|\mathbb{Z}^{\mathrm{DR}}_{i,j}|^{3}\big{]}}{\big{(}\sum_{i\in[N]}\mathbb{V}\mathrm{ar}(\mathbb{Z}^{\mathrm{DR}}_{i,j})\big{)}^{3/2}}=\frac{1}{N^{3/2}}\cdot\frac{\sum_{i\in[N]}\mathbb{E}\big{[}|\mathbb{Z}^{\mathrm{DR}}_{i,j}|^{3}\big{]}}{\big{(}\frac{1}{N}\sum_{i\in[N]}\mathbb{V}\mathrm{ar}(\mathbb{Z}^{\mathrm{DR}}_{i,j})\big{)}^{3/2}}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{N^{3/2}}\cdot\frac{\sum_{i\in[N]}\mathbb{E}\big{[}|\mathbb{Z}^{\mathrm{DR}}_{i,j}|^{3}\big{]}}{\big{(}\overline{\sigma}_{j}\big{)}^{3/2}}$
(A.164)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\frac{1}{N^{3/2}}\cdot\frac{\sum_{i\in[N]}\mathbb{E}\big{[}|\mathbb{Z}^{\mathrm{DR}}_{i,j}|^{3}\big{]}}{c_{1}^{3/2}}$
(A.165)
$\displaystyle\stackrel{{\scriptstyle(c)}}{{\leq}}\frac{1}{N^{1/2}}\cdot\frac{c_{2}}{c_{1}^{3/2}},$
(A.166)

where $(a)$ follows by putting together Eqs. 33 and A.152, $(b)$ follows
because $\overline{\sigma}_{j}\geq c_{1}>0$ as per condition (C3), $(c)$
follows because the absolute third moments of subExponential random variables
are bounded, after noting that $\mathbb{Z}^{\mathrm{DR}}_{i,j}$ is a
subExponential random variable. Then, condition (A.150) holds for $\omega=1$
as the right hand side of Eq. A.166 goes to $0$ as $N\to\infty$.

##  Appendix D Proof of Proposition 1 (28): Finite Sample Guarantees for OI

Fix any $j\in[M]$. Recall the definitions Eqs. 5 and 12 of the parameters
$\mathrm{ATE}_{\cdot,j}$ and corresponding outcome imputation estimates
$\widehat{\mathrm{ATE}}{}_{\cdot,j}^{\,\mathrm{OI}}$. The error
$\Delta\mathrm{ATE}_{\cdot,j}^{\mathrm{OI}}=\widehat{\mathrm{ATE}}{}_{\cdot,j}^{\,\mathrm{OI}}-\mathrm{ATE}_{\cdot,j}$
can be re-expressed as

$\displaystyle\Delta\mathrm{ATE}_{\cdot,j}^{\mathrm{OI}}$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\Big{(}\widehat{\theta}_{i,j}^{(1)}-\widehat{\theta}_{i,j}^{(0)}\Big{)}-\frac{1}{N}\sum_{i\in[N]}\Big{(}\theta_{i,j}^{(1)}-\theta_{i,j}^{(0)}\Big{)}$
(A.167)
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\bigg{(}\big{(}\widehat{\theta}_{i,j}^{(1)}-\theta_{i,j}^{(1)}\big{)}-\big{(}\widehat{\theta}_{i,j}^{(0)}-\theta_{i,j}^{(0)}\big{)}\bigg{)}.$
(A.168)

Using the triangle inequality, we have

$\displaystyle\big{|}\Delta\mathrm{ATE}_{\cdot,j}^{\mathrm{OI}}\big{|}\leq\frac{1}{N}\Big{|}\sum_{i\in[N]}\big{(}\widehat{\theta}_{i,j}^{(1)}-\theta_{i,j}^{(1)}\big{)}\Big{|}+\frac{1}{N}\Big{|}\sum_{i\in[N]}\big{(}\widehat{\theta}_{i,j}^{(0)}-\theta_{i,j}^{(0)}\big{)}\Big{|}.$
(A.169)

Consider any $a\in\\{0,1\\}$. We claim that

$\displaystyle\frac{1}{N}\Big{|}\sum_{i\in[N]}\big{(}\widehat{\theta}_{i,j}^{(a)}-\theta_{i,j}^{(a)}\big{)}\Big{|}\leq\mathcal{E}\big{(}{\widehat{\Theta}}^{(a)}\big{)}.$
(A.170)

The proof is complete by putting together Eqs. A.169 and A.170.

Proof of Eq. A.170 Fix any $a\in\\{0,1\\}$. Using the Cauchy-Schwarz
inequality, we have

$\displaystyle\frac{1}{N}\Big{|}\sum_{i\in[N]}\big{(}\widehat{\theta}_{i,j}^{(1)}-\theta_{i,j}^{(1)}\big{)}\Big{|}\leq\frac{1}{N}\|{\bm{1}_{N}}\|_{2}\|{{\widehat{\Theta}}^{(1)}_{\cdot,j}-{\Theta}^{(1)}_{\cdot,j}}\|_{2}=\frac{1}{\sqrt{N}}\|{{\widehat{\Theta}}^{(1)}_{\cdot,j}-{\Theta}^{(1)}_{\cdot,j}}\|_{2}\leq\frac{1}{\sqrt{N}}|\\!|{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}|\\!|_{1,2}.$
(A.171)

The proof is complete by using the notation in Eq. 23.

##  Appendix E Proof of Proposition 1 (29): Finite Sample Guarantees for IPW

Fix any $j\in[M]$. Recall the definitions Eqs. 5 and 14 of the parameters
$\mathrm{ATE}_{\cdot,j}$ and corresponding inverse probability weighting
estimates $\widehat{\mathrm{ATE}}{}_{\cdot,j}^{\,\mathrm{IPW}}$. The error
$\Delta\mathrm{ATE}_{\cdot,j}^{\mathrm{IPW}}=\widehat{\mathrm{ATE}}{}_{\cdot,j}^{\,\mathrm{IPW}}-\mathrm{ATE}_{\cdot,j}$
can be re-expressed as

$\displaystyle\Delta\mathrm{ATE}_{\cdot,j}^{\mathrm{IPW}}$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\Big{(}\frac{y_{i,j}a_{i,j}}{\widehat{p}_{i,j}}-\frac{y_{i,j}(1-a_{i,j})}{1-\widehat{p}_{i,j}}\Big{)}-\frac{1}{N}\sum_{i\in[N]}\Big{(}\theta_{i,j}^{(1)}-\theta_{i,j}^{(0)}\Big{)}$
(A.172)
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\bigg{(}\Big{(}\frac{y_{i,j}a_{i,j}}{\widehat{p}_{i,j}}-\theta_{i,j}^{(1)}\Big{)}-\Big{(}\frac{y_{i,j}(1-a_{i,j})}{1-\widehat{p}_{i,j}}-\theta_{i,j}^{(0)}\Big{)}\bigg{)}$
(A.173)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{N}\sum_{i\in[N]}\Big{(}\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}+\mathbb{T}_{i,j}^{(\mathrm{0},\mathrm{IPW})}\Big{)},$
(A.174)

where $(a)$ follows after defining
$\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}\triangleq
y_{i,j}a_{i,j}/\widehat{p}_{i,j}-\theta_{i,j}^{(1)}$ and
$\mathbb{T}_{i,j}^{(\mathrm{0},\mathrm{IPW})}\triangleq\theta_{i,j}^{(0)}-y_{i,j}(1-a_{i,j})/1-\widehat{p}_{i,j}$
for every $(i,j)\in[N]\times[M]$. Then, we have

$\displaystyle\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}$
$\displaystyle=\frac{y_{i,j}a_{i,j}}{\widehat{p}_{i,j}}-\theta_{i,j}^{(1)}$
(A.175)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{\big{(}\theta_{i,j}^{(1)}+\varepsilon_{i,j}^{(1)}\big{)}\big{(}p_{i,j}+\eta_{i,j}\big{)}}{\widehat{p}_{i,j}}-\theta_{i,j}^{(1)}$
(A.176)
$\displaystyle=\theta_{i,j}^{(1)}\Big{(}\frac{p_{i,j}+\eta_{i,j}}{\widehat{p}_{i,j}}-1\Big{)}+\varepsilon_{i,j}^{(1)}\Big{(}\frac{p_{i,j}+\eta_{i,j}}{\widehat{p}_{i,j}}\Big{)}$
(A.177)
$\displaystyle=\frac{\theta_{i,j}^{(1)}\big{(}p_{i,j}-\widehat{p}_{i,j}\big{)}}{\widehat{p}_{i,j}}+\frac{\theta_{i,j}^{(1)}\eta_{i,j}}{\widehat{p}_{i,j}}+\frac{\varepsilon_{i,j}^{(1)}p_{i,j}}{\widehat{p}_{i,j}}+\frac{\varepsilon_{i,j}^{(1)}\eta_{i,j}}{\widehat{p}_{i,j}},$
(A.178)

where $(a)$ follows from Eqs. 1, 2, and 3. A similar derivation for $a=0$
implies that

$\displaystyle\mathbb{T}_{i,j}^{(\mathrm{0},\mathrm{IPW})}$
$\displaystyle=\theta_{i,j}^{(0)}-\frac{y_{i,j}(1-a_{i,j})}{1-\widehat{p}_{i,j}}$
(A.179)
$\displaystyle=-\frac{\theta_{i,j}^{(0)}\big{(}1-p_{i,j}-\big{(}1-\widehat{p}_{i,j}\big{)}\big{)}}{1-\widehat{p}_{i,j}}-\frac{\theta_{i,j}^{(0)}(-\eta_{i,j})}{1-\widehat{p}_{i,j}}-\frac{\varepsilon_{i,j}^{(0)}\big{(}1-p_{i,j}\big{)}}{1-\widehat{p}_{i,j}}-\frac{\varepsilon_{i,j}^{(0)}(-\eta_{i,j})}{1-\widehat{p}_{i,j}}$
(A.180)
$\displaystyle=\frac{\theta_{i,j}^{(0)}\big{(}p_{i,j}-\widehat{p}_{i,j}\big{)}}{1-\widehat{p}_{i,j}}+\frac{\theta_{i,j}^{(0)}\eta_{i,j}}{1-\widehat{p}_{i,j}}-\frac{\varepsilon_{i,j}^{(0)}\big{(}1-p_{i,j}\big{)}}{1-\widehat{p}_{i,j}}+\frac{\varepsilon_{i,j}^{(0)}\eta_{i,j}}{1-\widehat{p}_{i,j}}.$
(A.181)

Consider any $a\in\\{0,1\\}$ and any $\delta\in(0,1)$. We claim that, with
probability at least $1-6\delta$,

$\displaystyle\frac{1}{N}\Big{|}\sum_{i\in[N]}\mathbb{T}_{i,j}^{(\mathrm{a},\mathrm{IPW})}\Big{|}\leq$
$\displaystyle\frac{2}{{\bar{\lambda}}}|\\!|{\Theta}^{(a)}|\\!|_{\max}\\!\cdot\\!\mathcal{E}\big{(}{\widehat{P}}\big{)}+\frac{2\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{\ell_{1}N}}|\\!|{\Theta}^{(a)}|\\!|_{\max}+\frac{2\overline{\sigma}\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{N}}+\frac{2\overline{\sigma}m(c\ell_{\delta})}{{\bar{\lambda}}\sqrt{\ell_{1}N}}.$
(A.182)

where recall that
$m(c\ell_{\delta})=\max\big{(}c\ell_{\delta},\sqrt{c\ell_{\delta}}\big{)}$. We
provide a proof of this claim at the end of this section. Applying triangle
inequality in Eq. A.174 and using Eq. A.182 with a union bound, we obtain that

$\displaystyle\big{|}\Delta\mathrm{ATE}_{\cdot,j}^{\mathrm{IPW}}\big{|}\leq\frac{2}{{\bar{\lambda}}}|\\!|\Theta|\\!|_{\max}\\!\cdot\\!\mathcal{E}\big{(}{\widehat{P}}\big{)}+\frac{2\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{\ell_{1}N}}|\\!|\Theta|\\!|_{\max}+\frac{4\overline{\sigma}\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{N}}+\frac{4\overline{\sigma}m(c\ell_{\delta})}{{\bar{\lambda}}\sqrt{\ell_{1}N}},$
(A.183)

with probability at least $1-12\delta$. The claim in Eq. 29 follows by re-
parameterizing $\delta$.

Proof of Eq. A.182. This proof follows a very similar road map to that used
for establishing the inequality in display (A.20). Recall the partitioning of
the units $[N]$ into $\mathcal{R}_{0}$ and $\mathcal{R}_{1}$ from 4. Condition
on this partition. Now, to enable the application of concentration bounds, we
split the summation over $i\in[N]$ in the left hand side of Eq. A.182 into two
parts—one over $i\in\mathcal{R}_{0}$ and the other over
$i\in\mathcal{R}_{1}$—such that the noise terms are independent of the
estimates of ${\Theta}^{(0)},{\Theta}^{(1)},{P}$ in each of these parts as in
Eqs. 21 and 22.

Fix $a=1$ and note that
$|\sum_{i\in[N]}\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}|\leq|\sum_{i\in\mathcal{R}_{0}}\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}|+|\sum_{i\in\mathcal{R}_{1}}\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}|$.
Fix any $s\in\\{0,1\\}$. Then, Eq. A.178 and triangle inequality imply that

$\displaystyle\Big{|}\sum_{i\in\mathcal{R}_{s}}\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}\Big{|}\\!\leq\\!\Big{|}\sum_{i\in\mathcal{R}_{s}}\frac{\theta_{i,j}^{(1)}\big{(}p_{i,j}\\!-\\!\widehat{p}_{i,j}\big{)}}{\widehat{p}_{i,j}}\Big{|}\\!+\\!\Big{|}\sum_{i\in\mathcal{R}_{s}}\frac{\theta_{i,j}^{(1)}\eta_{i,j}}{\widehat{p}_{i,j}}\Big{|}+\Big{|}\sum_{i\in\mathcal{R}_{s}}\frac{\varepsilon_{i,j}^{(1)}p_{i,j}}{\widehat{p}_{i,j}}\Big{|}\\!+\\!\Big{|}\sum_{i\in\mathcal{R}_{s}}\frac{\varepsilon_{i,j}^{(1)}\eta_{i,j}}{\widehat{p}_{i,j}}\Big{|}.$
(A.184)

Next, note that the decomposition in Eq. A.184 is identical to the one in Eq.
A.23, except for the fact when compared to Eq. A.23, the first two terms in
Eq. A.184 have a factor of $\theta_{i,j}^{(1)}$ instead of
$\big{(}\widehat{\theta}_{i,j}^{(1)}\\!-\\!\theta_{i,j}^{(1)}\big{)}$. As a
result, mimicking steps used to derive Eq. A.96, we obtain the following
bound, with probability at least $1-3\delta$,

$\displaystyle\frac{1}{N}\Big{|}\sum_{i\in\mathcal{R}_{s}}\mathbb{T}_{i,j}^{(\mathrm{1},\mathrm{IPW})}\Big{|}$
$\displaystyle\\!\leq\\!\frac{1}{{\bar{\lambda}}N}|\\!|{\Theta}^{(1)}|\\!|_{1,2}|\\!|{\widehat{P}}\\!-\\!{P}|\\!|_{1,2}\\!+\\!\frac{\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{\ell_{1}}N}|\\!|{\Theta}^{(1)}|\\!|_{1,2}\\!+\\!\frac{\overline{\sigma}\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}N}|\\!|{P}|\\!|_{1,2}\\!+\\!\frac{\overline{\sigma}m(c\ell_{\delta})}{{\bar{\lambda}}\sqrt{\ell_{1}}N}|\\!|\bm{1}|\\!|_{1,2}$
(A.185)
$\displaystyle\\!\stackrel{{\scriptstyle(a)}}{{\leq}}\\!\frac{1}{{\bar{\lambda}}\sqrt{N}}|\\!|{\Theta}^{(1)}|\\!|_{\max}|\\!|{\widehat{P}}\\!-\\!{P}|\\!|_{1,2}\\!+\\!\frac{\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{\ell_{1}N}}|\\!|{\Theta}^{(1)}|\\!|_{\max}\\!+\\!\frac{\overline{\sigma}\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{N}}\\!+\\!\frac{\overline{\sigma}m(c\ell_{\delta})}{{\bar{\lambda}}\sqrt{\ell_{1}N}},$
(A.186)
$\displaystyle\\!\stackrel{{\scriptstyle(b)}}{{\leq}}\\!\frac{1}{{\bar{\lambda}}}|\\!|{\Theta}^{(1)}|\\!|_{\max}\\!\cdot\\!\mathcal{E}\big{(}{\widehat{P}}\big{)}+\frac{\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{\ell_{1}N}}|\\!|{\Theta}^{(1)}|\\!|_{\max}+\frac{\overline{\sigma}\sqrt{c\ell_{\delta}}}{{\bar{\lambda}}\sqrt{N}}+\frac{\overline{\sigma}m(c\ell_{\delta})}{{\bar{\lambda}}\sqrt{\ell_{1}N}},$
(A.187)

where $(a)$ follows because
$|\\!|{\Theta}^{(1)}|\\!|_{1,2}\leq\sqrt{N}|\\!|{\Theta}^{(1)}|\\!|_{\max}$,
$|\\!|{P}|\\!|_{1,2}\leq\sqrt{N}$ and $|\\!|\bm{1}|\\!|_{1,2}=\sqrt{N}$, and
$(b)$ follows from Eq. 23. Then, the claim in Eq. A.182 follows for $a=1$ by
using Eq. A.187 and applying a union bound over $s\in\\{0,1\\}$. The proof of
Eq. A.182 for $a=0$ follows similarly.

##  Appendix F Proofs of Propositions 2 and 3

In Section F.1, we prove Proposition 2, i.e., we show that the estimates of
${P}$, ${\Theta}^{(0)}$, and ${\Theta}^{(1)}$ generated by Cross-Fitted-MC
satisfy 4. Next, we prove Proposition 3 implying that the estimates of ${P}$,
${\Theta}^{(0)}$, and ${\Theta}^{(1)}$ generated by Cross-Fitted-SVD satisfy
the condition (C2) in Theorem 2 as long as $\sqrt{N}/M=o(1)$.

###  F.1 Proof of Proposition 2: Guarantees for Cross-Fitted-MC

Consider any matrix completion algorithm MC and any block partition
$\mathcal{P}$ of the set $[N]\times[M]$ into four blocks as in 4. Fix any
$\mathcal{I}\in\mathcal{P}$.

Consider ${\widehat{P}}$ in Eq. 40. To see why
${\widehat{P}}_{\mathcal{I}}\hskip
2.84526pt{\perp\\!\\!\\!\perp}\leavevmode\nobreak\ {W}_{\mathcal{I}}$, note
that ${A}\otimes\bm{1}^{-\mathcal{I}}$ depends only on
${W}\setminus{W}_{\mathcal{I}}$ which is independent of ${W}_{\mathcal{I}}$
from 2(c). Further, ${\widehat{P}}_{\mathcal{I}}\hskip
2.84526pt{\perp\\!\\!\\!\perp}\leavevmode\nobreak\
{E}^{(0)}_{\mathcal{I}},{E}^{(1)}_{\mathcal{I}}$ holds since
${W}\setminus{W}_{\mathcal{I}}$ is independent of
$({E}^{(0)}_{\mathcal{I}},{E}^{(1)}_{\mathcal{I}})$ from 2(b). Overall, we
conclude ${\widehat{P}}_{\mathcal{I}}\hskip
2.84526pt{\perp\\!\\!\\!\perp}\leavevmode\nobreak\
({W}_{\mathcal{I}},{E}^{(0)}_{\mathcal{I}},{E}^{(1)}_{\mathcal{I}})$.

Next, consider ${\widehat{\Theta}}^{(0)}$ and ${\widehat{\Theta}}^{(1)}$
defined in Eqs. 38 and 39, respectively. Fix any $a\in\\{0,1\\}$. To see why
${\widehat{\Theta}}^{(a)}_{\mathcal{I}}\hskip
2.84526pt{\perp\\!\\!\\!\perp}\leavevmode\nobreak\ {W}_{\mathcal{I}}$, note
that ${\widehat{\Theta}}^{(a)}_{\mathcal{I}}$ depends on
${{Y}}^{(a),\mathrm{obs}}\otimes\bm{1}^{-\mathcal{I}}$, which in turn depends
on $(i)$ ${W}\setminus{W}_{\mathcal{I}}$ and $(ii)$
${E}^{(a)}\setminus{E}^{(a)}_{\mathcal{I}}$, each of which are independent of
${W}_{\mathcal{I}}$ due to 2(c) and 2(b), respectively.

###  F.2 Proof of Proposition 3: Guarantees for Cross-Fitted-SVD

To prove this result, we first derive a corollary of Lemma A.1 in Bai and Ng,
(2021) for a generic matrix of interest $T$, such that $S=(T+H)\otimes{F}$,
and apply it to ${P}$, ${\Theta}^{(0)}\odot(\bm{1}-{P})$, and
${\Theta}^{(1)}\odot{P}$. We impose the following restrictions on $T$ and $H$.

######  Assumption 8.

There exist a constant $r_{T}\in[\min\\{N,M\\}]$ and a collection of latent
factors

$\displaystyle\widetilde{U}\in\mathbb{R}^{N\times
r_{T}}\quad\text{and}\quad\widetilde{V}\in\mathbb{R}^{M\times r_{T}},$ (A.188)

such that,

  1. (a)

$T$ satisfies the factorization: $T=\widetilde{U}\widetilde{V}^{\top}$,

  2. (b)

$|\\!|\widetilde{U}|\\!|_{2,\infty}\leq c$ and
$|\\!|\widetilde{V}|\\!|_{2,\infty}\leq c$ for some positive constant $c$, and

  3. (c)

$N^{-1}\widetilde{U}^{\top}\widetilde{U}$ and
$M^{-1}\widetilde{V}^{\top}\widetilde{V}$ are positive definite matrices.

######  Assumption 9.

The noise matrix $H$ is such that,

  1. (a)

$\\{h_{i,j}:i\in[N],j\in[M]\\}$ are zero-mean subExponential with the
subExponential norm bounded by a constant $\overline{\sigma}$,

  2. (b)

$\sum_{j^{\prime}\in[M]}\big{|}\mathbb{E}[h_{i,j}h_{i,j^{\prime}}]\big{|}\leq
c$ for every $i\in[N]$ and $j\in[M]$, and

  3. (c)

$\\{H_{i,\cdot}:i\in[N]\\}$ are mutually independent (across $i$).

The next result characterizes the entry-wise error in recovering the missing
entries of a matrix where all entries in one block are deterministically
missing (see the discussion in Section 5.1) using the TW algorithm (summarized
in Section 5.2.1). Its proof, essentially established as a corollary of Bai
and Ng, (2021, Lemma A.1), is provided in Section F.3.

######  Corollary 5.

Consider a matrix of interest $T$ that satisfies 8 and a noise matrix $H$ that
satisfies 9. Let $S\in\\{\mathbb{R},\,?\\}^{N\times M}$ be the observed matrix
as in Eq. 8. Let $\mathcal{R}_{\mathrm{obs}}\subseteq[N]$ and
$\mathcal{C}_{\mathrm{obs}}\subseteq[M]$ denote the set of rows and columns of
$S$, respectively, with all entries observed. Suppose the mask matrix ${F}$ is
such that each $i\in[N]$ belongs to $\mathcal{R}_{\mathrm{obs}}$ with
probability $1/2$ and each $j\in[M]$ belongs to $\mathcal{C}_{\mathrm{obs}}$
with probability $1/2$. Let
$\mathcal{I}=\mathcal{R}_{\mathrm{miss}}\times\mathcal{C}_{\mathrm{miss}}$
where
$\mathcal{R}_{\mathrm{miss}}\triangleq[N]\setminus\mathcal{R}_{\mathrm{obs}}$
and
$\mathcal{C}_{\mathrm{miss}}\triangleq[M]\setminus\mathcal{C}_{\mathrm{obs}}$.
Then, $\texttt{TW}_{r_{T}}$ produces an estimate $\widehat{T}_{\mathcal{I}}$
of $T_{\mathcal{I}}$ such that

$\displaystyle|\\!|\widehat{T}_{\mathcal{I}}-T_{\mathcal{I}}|\\!|_{\max}=O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)},$
(A.189)

as $N,M\to\infty$.

Given this corollary, we now complete the proof of Proposition 3. Consider the
partition $\mathcal{P}$ in step 2 of Cross-Fitted-SVD and fix any
$\mathcal{I}\in\mathcal{P}$. Recall that Cross-Fitted-SVD applies TW on
${P}\otimes\bm{1}^{-\mathcal{I}}$,
${{Y}}^{(0),\mathrm{full}}\otimes\bm{1}^{-\mathcal{I}}$, and
${{Y}}^{(1),\mathrm{full}}\otimes\bm{1}^{-\mathcal{I}}$, and note that
$\bm{1}^{-\mathcal{I}}$ satisfies the requirement on the mask matrix in
Corollary 5.

####  F.2.1 Estimating ${P}$.

Consider estimating ${P}$ using Cross-Fitted-SVD. To apply Corollary 5, we use
5 and 6 to note that ${P}$ satisfies 8 with rank parameter $r_{p}$. Then, we
use Eq. 3 and 2(b) to note that ${W}$ satisfies 9. Step 3 of Cross-Fitted-SVD
can be rewritten as
${\widehat{P}}=\texttt{Proj}_{{\bar{\lambda}}}\big{(}\overline{{P}}\big{)}$
and
$\overline{{P}}=\texttt{Cross}\allowbreak\texttt{-}\allowbreak\texttt{Fitted}\allowbreak\texttt{-}\allowbreak\texttt{MC}(\texttt{TW}_{r_{1}},{A},\mathcal{P})$
where $r_{1}=r_{p}$. Then,

$\displaystyle|\\!|{\widehat{P}}_{\mathcal{I}}-{P}_{\mathcal{I}}|\\!|_{\max}\stackrel{{\scriptstyle(a)}}{{\leq}}|\\!|\overline{{P}}_{\mathcal{I}}-{P}_{\mathcal{I}}|\\!|_{\max}\stackrel{{\scriptstyle(b)}}{{=}}O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)},$
(A.190)

where $(a)$ follows from 1 and 3, and the definition of
$\texttt{Proj}_{{\bar{\lambda}}}(\cdot)$, and $(b)$ follows from Corollary 5.
Applying a union bound over all $\mathcal{I}\in\mathcal{P}$, we have

$\displaystyle\mathcal{E}\big{(}{\widehat{P}}\big{)}\stackrel{{\scriptstyle(a)}}{{\leq}}|\\!|{\widehat{P}}-{P}|\\!|_{\max}=O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)},$
(A.191)

where $(a)$ follows from the definition of $L_{1,2}$ norm.

####  F.2.2 Estimating ${\Theta}^{(0)}$ and ${\Theta}^{(1)}$.

For every $a\in\\{0,1\\}$, we show that

$\displaystyle\mathcal{E}\big{(}{\widehat{\Theta}}^{(a)}\big{)}=O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)}.$
(A.192)

We focus on $a=1$ noting that the proof for $a=0$ is analogous. We split the
proof in two cases: (i)
$|\\!|\big{(}{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}\big{)}\odot{\widehat{P}}|\\!|_{\max}\leq|\\!|{\Theta}^{(1)}\odot\big{(}{\widehat{P}}-{P}\big{)}|\\!|_{\max}$
and (ii)
$|\\!|\big{(}{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}\big{)}\odot{\widehat{P}}|\\!|_{\max}\geq|\\!|{\Theta}^{(1)}\odot\big{(}{\widehat{P}}-{P}\big{)}|\\!|_{\max}$.

In the first case, we have

$\displaystyle{\bar{\lambda}}|\\!|{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}|\\!|_{\max}\stackrel{{\scriptstyle(a)}}{{\leq}}|\\!|\big{(}{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}\big{)}\odot{\widehat{P}}|\\!|_{\max}$
$\displaystyle\leq|\\!|{\Theta}^{(1)}\odot\big{(}{\widehat{P}}-{P}\big{)}|\\!|_{\max}$
(A.193)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}|\\!|{\Theta}^{(1)}|\\!|_{\max}|\\!|{\widehat{P}}-{P}|\\!|_{\max},$
(A.194)

where $(a)$ follows from 3 and $(b)$ follows from the definition of
$|\\!|{\Theta}^{(1)}|\\!|_{\max}$. Then,

$\displaystyle\mathcal{E}\big{(}{\widehat{\Theta}}^{(1)}\big{)}\stackrel{{\scriptstyle(a)}}{{\leq}}|\\!|{\widehat{\Theta}}^{(1)}\\!-\\!{\Theta}^{(1)}|\\!|_{\max}\stackrel{{\scriptstyle(b)}}{{\leq}}\frac{|\\!|{\Theta}^{(1)}|\\!|_{\max}}{{\bar{\lambda}}}|\\!|{\widehat{P}}\\!-\\!{P}|\\!|_{\max}\stackrel{{\scriptstyle(c)}}{{=}}\frac{|\\!|{\Theta}^{(1)}|\\!|_{\max}}{{\bar{\lambda}}}O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)},$
(A.195)

where $(a)$ follows from the definition of $L_{1,2}$ norm, $(b)$ follows from
Eq. A.194, and $(c)$ follows from Eq. A.191. Then, Eq. A.192 follows as
$1/{\bar{\lambda}}$ and $|\\!|{\Theta}^{(1)}|\\!|_{\max}$ are assumed to be
bounded.

In the second case, using Eqs. 2 and 3 to expand ${{Y}}^{(1),\mathrm{full}}$,
we have

$\displaystyle{{Y}}^{(1),\mathrm{full}}$
$\displaystyle={\Theta}^{(1)}\odot{P}+{\Theta}^{(1)}\odot\eta+\varepsilon^{(1)}\odot{P}+\varepsilon^{(1)}\odot\eta.$
(A.196)

Next, we utilize two claims proven in Sections F.2.3 and F.2.4 respectively:
${\Theta}^{(1)}\odot{P}$ satisfies 8 with rank parameter $r_{\theta_{1}}r_{p}$
and

$\displaystyle\overline{\varepsilon}^{(1)}\triangleq{\Theta}^{(1)}\odot\eta+\varepsilon^{(1)}\odot{P}+\varepsilon^{(1)}\odot\eta,$
(A.197)

satisfies 9.

Now, note that step 6 of Cross-Fitted-SVD can be rewritten as
${\widehat{\Theta}}^{(1)}={\overline{\Theta}}^{(1)}\mathrel{\vphantom{\oplus}\mathchoice{\makebox[9.99997pt]{\ooalign{$\displaystyle\vbox{\hbox{\leavevmode\resizebox{6.66666pt}{}{\leavevmode\hbox{\set@color\@imakebox[0pt][c]{$\displaystyle\bigcirc$}}}}}$\cr$\displaystyle\vbox{\hbox{\leavevmode\resizebox{2.66661pt}{}{$\displaystyle\vphantom{\oplus}{\bm{/}}$}}}$\cr}}}{\makebox[9.99997pt]{\ooalign{$\textstyle\vbox{\hbox{\leavevmode\resizebox{6.66666pt}{}{\leavevmode\hbox{\set@color\@imakebox[0pt][c]{$\textstyle\bigcirc$}}}}}$\cr$\textstyle\vbox{\hbox{\leavevmode\resizebox{2.66661pt}{}{$\textstyle\vphantom{\oplus}{\bm{/}}$}}}$\cr}}}{\makebox[7.66663pt]{\ooalign{$\scriptstyle\vbox{\hbox{\leavevmode\resizebox{4.66666pt}{}{\leavevmode\hbox{\set@color\@imakebox[0pt][c]{$\scriptstyle\bigcirc$}}}}}$\cr$\scriptstyle\vbox{\hbox{\leavevmode\resizebox{1.86662pt}{}{$\scriptstyle\vphantom{\oplus}{\bm{/}}$}}}$\cr}}}{\makebox[6.11107pt]{\ooalign{$\scriptscriptstyle\vbox{\hbox{\leavevmode\resizebox{3.33331pt}{}{\leavevmode\hbox{\set<EMAIL_ADDRESS>and
${\overline{\Theta}}^{(1)}=\texttt{Cross}\allowbreak\texttt{-}\allowbreak\texttt{Fitted}\allowbreak\texttt{-}\allowbreak\texttt{MC}(\texttt{TW}_{r_{3}},{{Y}}^{(1),\mathrm{full}},\mathcal{P})$
where $r_{3}=r_{\theta_{1}}r_{p}$. Then, from Corollary 5,

$\displaystyle|\\!|{\overline{\Theta}}^{(1)}_{\mathcal{I}}-{\Theta}^{(1)}_{\mathcal{I}}\odot{P}_{\mathcal{I}}|\\!|_{\max}=O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)}.$
(A.198)

Applying a union bound over all $\mathcal{I}\in\mathcal{P}$ and noting that
${\overline{\Theta}}^{(1)}={\widehat{\Theta}}^{(1)}\odot{\widehat{P}}$, we
have

$\displaystyle|\\!|{\widehat{\Theta}}^{(1)}\odot{\widehat{P}}-{\Theta}^{(1)}\odot{P}|\\!|_{\max}=O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)}.$
(A.199)

The left hand side of Eq. A.199 can be written as,

$\displaystyle|\\!|{\widehat{\Theta}}^{(1)}\odot{\widehat{P}}-{\Theta}^{(1)}\odot{P}|\\!|_{\max}$
$\displaystyle=|\\!|{\widehat{\Theta}}^{(1)}\odot{\widehat{P}}-{\Theta}^{(1)}\odot{\widehat{P}}+{\Theta}^{(1)}\odot{\widehat{P}}-{\Theta}^{(1)}\odot{P}|\\!|_{\max}$
(A.200)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}|\\!|\big{(}{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}\big{)}\odot{\widehat{P}}|\\!|_{\max}-|\\!|{\Theta}^{(1)}\odot\big{(}{\widehat{P}}-{P}\big{)}|\\!|_{\max}$
(A.201)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\geq}}{\bar{\lambda}}|\\!|{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}|\\!|_{\max}-|\\!|{\Theta}^{(1)}|\\!|_{\max}|\\!|{\widehat{P}}-{P}|\\!|_{\max},$
(A.202)

where $(a)$ follows from triangle inequality as
$|\\!|\big{(}{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}\big{)}\odot{\widehat{P}}|\\!|_{\max}\geq|\\!|{\Theta}^{(1)}\odot\big{(}{\widehat{P}}-{P}\big{)}|\\!|_{\max}$
and $(b)$ follows from 3 and the definition of
$|\\!|{\Theta}^{(1)}|\\!|_{\max}$. Then,

$\displaystyle\mathcal{E}\big{(}{\widehat{\Theta}}^{(1)}\big{)}\stackrel{{\scriptstyle(a)}}{{\leq}}|\\!|{\widehat{\Theta}}^{(1)}-{\Theta}^{(1)}|\\!|_{\max}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\frac{1}{{\bar{\lambda}}}|\\!|{\widehat{\Theta}}^{(1)}\odot{\widehat{P}}-{\Theta}^{(1)}\odot{P}|\\!|_{\max}+\frac{|\\!|{\Theta}^{(1)}|\\!|_{\max}}{{\bar{\lambda}}}|\\!|{\widehat{P}}\\!-\\!{P}|\\!|_{\max}$
(A.203)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\frac{1}{{\bar{\lambda}}}O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)}+\frac{|\\!|{\Theta}^{(1)}|\\!|_{\max}}{{\bar{\lambda}}}O_{p}\bigg{(}\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}\bigg{)},$
(A.204)

where $(a)$ follows from the definition of $L_{1,2}$ norm, $(b)$ follows from
Eq. A.202, and $(c)$ follows from Eqs. A.191 and A.199. Then, Eq. A.192
follows as $1/{\bar{\lambda}}$ and $|\\!|{\Theta}^{(1)}|\\!|_{\max}$ are
assumed to be bounded.

####  F.2.3 Proof that ${\Theta}^{(0)}\odot(\bm{1}-{P})$ and
${\Theta}^{(1)}\odot{P}$ satisfy 8.

Recall the factors $\overline{U}^{(0)}$ and $\overline{V}^{(0)}$ of
${\Theta}^{(0)}\odot(\bm{1}-{P})$, and $\overline{U}^{(1)}$ and
$\overline{V}^{(1)}$ of ${\Theta}^{(1)}\odot{P}$ from Section 5.3. Then, 8(a)
holds from Eq. 49. Next, we note that

$\displaystyle|\\!|\overline{U}^{(1)}|\\!|_{2,\infty}=|\\!|U*U^{(1)}|\\!|_{2,\infty}\stackrel{{\scriptstyle(a)}}{{=}}\max_{i\in[N]}\sqrt{\sum_{j\in[r_{p}]}u_{i,j}^{2}\sum_{j^{\prime}\in[r_{\theta_{1}}]}(u^{(1)}_{i,j^{\prime}})^{2}}\leq|\\!|U|\\!|_{2,\infty}|\\!|U^{(1)}|\\!|_{2,\infty}\stackrel{{\scriptstyle(b)}}{{\leq}}c,$
(A.205)

where $(a)$ follows from the definition of Khatri-Rao product (see Section 1),
and $(b)$ follows from 6. Then, ${\Theta}^{(1)}\odot{P}$ satisfies 8(b) by
using similar arguments on $\overline{V}^{(1)}$. Further,
${\Theta}^{(0)}\odot(\bm{1}-{P})$ satisfies 8(b) by noting that
$|\\!|\overline{U}|\\!|_{2,\infty}$ and $|\\!|\overline{V}|\\!|_{2,\infty}$
are bounded whenever $|\\!|U|\\!|_{2,\infty}$ and $|\\!|V|\\!|_{2,\infty}$ are
bounded, respectively. Finally, 8(c) holds from 6.

####  F.2.4 Proof that $\overline{\varepsilon}^{(1)}$ satisfies 9

Recall that
$\overline{\varepsilon}^{(1)}\triangleq{\Theta}^{(1)}\odot\eta+\varepsilon^{(1)}\odot{P}+\varepsilon^{(1)}\odot\eta$.
Then, 9(a) holds as $\overline{\varepsilon}^{(1)}$ is zero-mean from 2 and Eq.
3, and $\overline{\varepsilon}^{(1)}$ is subExponential because
$\varepsilon^{(1)}_{i,j}\eta_{i,j}$ is a subExponential random variable Lemma
3, every subGaussian random variable is subExponential random variable, and
sum of subExponential random variables is a subExponential random variable.
Next, 9(b) holds as

$\displaystyle\sum_{j^{\prime}\in[M]}\big{|}\mathbb{E}[\overline{\varepsilon}^{(1)}_{i,j}\overline{\varepsilon}^{(1)}_{i,j^{\prime}}]\big{|}\stackrel{{\scriptstyle(a)}}{{=}}\sum_{j^{\prime}\in[M]}\big{|}\mathbb{E}[\theta^{(1)}_{i,j}\theta^{(1)}_{i,j^{\prime}}\eta_{i,j}\eta_{i,j^{\prime}}+p_{i,j}p_{i,j^{\prime}}\overline{\varepsilon}^{(1)}_{i,j}\overline{\varepsilon}^{(1)}_{i,j^{\prime}}+\overline{\varepsilon}^{(1)}_{i,j}\overline{\varepsilon}^{(1)}_{i,j^{\prime}}\eta_{i,j}\eta_{i,j^{\prime}}]\big{|}\stackrel{{\scriptstyle(b)}}{{\leq}}c,$
(A.206)

where $(a)$ follows from 2, and $(b)$ follows from 2, 7, and 1, Eq. 3, Lemma
3, and because $|\\!|{\Theta}^{(1)}|\\!|_{\max}$ are bounded. Finally, 9(b)
holds from 2 and 7.

###  F.3 Proof of Corollary 5

Corollary 5 is a direct application of Bai and Ng, (2021, Lemma A.1),
specialized to our setting. Notably, Bai and Ng, (2021) make four assumptions
numbered A, B, C and D in their paper to establish the corresponding result.
It remains to establish that the conditions assumed in Corollary 5 imply the
necessary conditions used in the proof of Bai and Ng, (2021, Lemma A.1).
First, note that due to the specific sampling assumed in defining the mask
matrix in Corollary 5, Bai and Ng, (2021, Assumption D) holds immediately and
Bai and Ng, (2021, Assumption B) holds with high probability by Hoeffding’s
inequality.

It remains to show how 8 and 9 imply the remainder of their assumptions,
namely Bai and Ng, (2021, Assumptions A and C). Before doing that, note that
certain assumptions in Bai and Ng, (2021) are not actually used in their proof
of Lemma A.1 (or in the proof of other results used in that proof), namely,
the distinct eigenvalue condition in Assumption A(a)(iii), the asymptotic
normality conditions in Assumption A(c) and the asymptotic normality
conditions in Assumption C. For completeness, the remaining relevant
conditions from Bai and Ng, (2021) are collected in the following two
assumptions.

######  Assumption 10 (Strong block factors).

Consider the latent factors $\widetilde{U}\in\mathbb{R}^{N\times r_{T}}$ and
$\widetilde{V}\in\mathbb{R}^{M\times r_{T}}$ from 8. Define the following
matrices:

$\displaystyle\widetilde{U}^{\mathrm{obs}}\triangleq\widetilde{U}_{\mathcal{R}_{\mathrm{obs}}\times[r_{T}]},\qquad\widetilde{U}^{\mathrm{miss}}\triangleq\widetilde{U}_{\mathcal{R}_{\mathrm{miss}}\times[r_{T}]},\qquad\widetilde{V}^{\mathrm{obs}}\triangleq\widetilde{V}_{\mathcal{C}_{\mathrm{obs}}\times[r_{T}]},\quad\text{and}\quad\widetilde{V}^{\mathrm{miss}}\triangleq\widetilde{V}_{\mathcal{C}_{\mathrm{miss}}\times[r_{T}]},$
(A.207)

where $\mathcal{R}_{\mathrm{obs}}\subseteq[N]$ and
$\mathcal{C}_{\mathrm{obs}}\subseteq[M]$ denote the set of rows and columns of
$S$, respectively, with all entries observed, and
$\mathcal{R}_{\mathrm{miss}}\triangleq[N]\setminus\mathcal{R}_{\mathrm{obs}}$
and
$\mathcal{C}_{\mathrm{miss}}\triangleq[M]\setminus\mathcal{C}_{\mathrm{obs}}$.
Then, the matrices defined below are positive definite:

$\displaystyle\Sigma^{\widetilde{U},\mathrm{obs}}\\!\triangleq\\!\frac{\widetilde{U}^{\mathrm{obs}\top}\widetilde{U}^{\mathrm{obs}}}{|\mathcal{R}_{\mathrm{obs}}|},\\!\\!\quad\Sigma^{\widetilde{U},\mathrm{miss}}\\!\triangleq\\!\frac{\widetilde{U}^{\mathrm{miss}\top}\widetilde{U}^{\mathrm{miss}}}{|\mathcal{R}_{\mathrm{miss}}|},\\!\\!\quad\Sigma^{\widetilde{V},\mathrm{obs}}\\!\triangleq\\!\frac{\widetilde{V}^{\mathrm{obs}\top}\widetilde{V}^{\mathrm{obs}}}{|\mathcal{C}_{\mathrm{obs}}|},\\!\\!\quad\text{and}\quad\\!\\!\Sigma^{\widetilde{V},\mathrm{miss}}\\!\triangleq\\!\frac{\widetilde{V}^{\mathrm{miss}\top}\widetilde{V}^{\mathrm{miss}}}{|\mathcal{C}_{\mathrm{miss}}|}.$
(A.208)

######  Assumption 11.

The noise matrix $H$ is such that,

  1. (a)

$\max_{j\in[M]}\frac{1}{N}\sum_{j^{\prime}\in[M]}\big{|}\sum_{i\in[N]}\mathbb{E}[h_{i,j}h_{i,j^{\prime}}]\big{|}\leq
c$,

  2. (b)

$\max_{j\in[M]}\big{|}\mathbb{E}[h_{i,j}h_{i^{\prime},j}]\big{|}\leq
c_{i,i^{\prime}}$ and
$\max_{i\in[N]}\sum_{i^{\prime}\in[N]}c_{i,i^{\prime}}\leq c$,

  3. (c)

$\frac{1}{NM}\sum_{i,i^{\prime}\in[N]}\sum_{j,j^{\prime}\in[M]}\big{|}\mathbb{E}[h_{i,j}h_{i^{\prime},j^{\prime}}]\big{|}\leq
c$, and

  4. (d)

$\max_{j,j^{\prime}\in[M]}\frac{1}{N^{2}}\mathbb{E}\big{[}\big{|}\sum_{i\in[N]}\big{(}h_{i,j}h_{i,j^{\prime}}-\mathbb{E}[h_{i,j}h_{i,j^{\prime}}]\big{)}\big{|}^{4}\big{]}$.

10 is a restatement of Bai and Ng, (2021, Assumption C) (without the central
limit theorems, which are not used in Bai and Ng, (2021, Proof of Lemma A.1)
as noted above). This condition ensures a strong factor structure on the sub-
matrix corresponding to observed elements of $S$ as well as on the sub-matrix
corresponding to missing elements of $S$.

11 is a restatement of the subset of conditions from Bai and Ng, (2021,
Assumption A) necessary in Bai and Ng, (2021, proof of Lemma A.1) and it
essentially requires weak dependence in the noise across measurements and
across units. In particular, 11(a), (b), (c), and (d) correspond to Assumption
A(b)(ii), (iii), (iv), (v), respectively, of Bai and Ng, (2021). For the other
conditions in Bai and Ng, (2021, Assumption A), note that 8 above is
equivalent to their Assumption A(a)(i) and (ii) of Bai and Ng, (2021) when the
factors are non-random as in this work. Similarly, 9(a) above is analogous to
Assumption A(b)(i) of Bai and Ng, (2021). Assumption A(b)(vi) of Bai and Ng,
(2021) is implied by their other Assumptions for non-random factors as stated
in Bai, (2003).

To establish Corollary 5, it remains to establish that 10 and 11 hold, which
is done in Sections F.3.1 and F.3.2 respectively.

####  F.3.1 10 holds

We show that $\Sigma^{\widetilde{U},\mathrm{obs}}$ is positive definite. The
proof for $\Sigma^{\widetilde{U},\mathrm{miss}}$,
$\Sigma^{\widetilde{U},\mathrm{obs}}$, and
$\Sigma^{\widetilde{U},\mathrm{miss}}$ being positive definite follows
similarly. Define $\Sigma^{\widetilde{U}}\triangleq
N^{-1}\widetilde{U}^{\top}\widetilde{U}\in\mathbb{R}^{r_{T}\times r_{T}}$.
From Weyl’s inequality (Bhatia,, 2007, Theorem. 8.2), we have the following
for some $c>0$:

$\displaystyle\lambda_{\min}(\Sigma^{\widetilde{U},\mathrm{obs}})\geq\lambda_{\min}(\Sigma^{\widetilde{U}})\\!-\\!\lambda_{\max}(\Sigma^{\widetilde{U}}-\Sigma^{\widetilde{U},\mathrm{obs}})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}c\\!-\\!\lambda_{\max}(\Sigma^{\widetilde{U}}\\!-\\!\Sigma^{\widetilde{U},\mathrm{obs}})$
(A.209) $\displaystyle\geq
c\\!-\\!\big{|}\lambda_{\max}(\Sigma^{\widetilde{U}}\\!-\\!\Sigma^{\widetilde{U},\mathrm{obs}})\big{|},$
(A.210)

where $(a)$ follows from 8(c) as $\Sigma^{\widetilde{U}}$ is positive
definite. Now, it suffices to show that
$\big{|}\lambda_{\max}(\Sigma^{\widetilde{U}}\\!-\\!\Sigma^{\widetilde{U},\mathrm{obs}})\big{|}=o_{p}(1)$.

Recall that the mask matrix ${F}$ is such that each $i\in[N]$ belongs to
$\mathcal{R}_{\mathrm{obs}}$ with probability $1/2$. For every $i\in[N]$, let
$\mathds{1}_{i}$ be an indicator random variable such that $\mathds{1}_{i}=1$
if $i\in\mathcal{R}_{\mathrm{obs}}$ and $\mathds{1}_{i}=0$ if
$i\notin\mathcal{R}_{\mathrm{obs}}$. Then, we express
$\Sigma^{\widetilde{U},\mathrm{obs}}$ as follows,

$\displaystyle\Sigma^{\widetilde{U},\mathrm{obs}}=\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{\sum_{i\in[N]}\mathds{1}_{i}}.$
(A.211)

Then, we have

$\displaystyle\big{|}\lambda_{\max}(\Sigma^{\widetilde{U}}\\!-\\!\Sigma^{\widetilde{U},\mathrm{obs}})\big{|}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\bigg{|}\lambda_{\max}\bigg{(}\frac{\sum_{i\in[N]}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N}-\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{\sum_{i\in[N]}\mathds{1}_{i}}\bigg{)}\bigg{|}$
(A.212)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\bigg{|}\lambda_{\max}\bigg{(}\frac{\sum_{i\in[N]}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N}-\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N/2}\bigg{)}\bigg{|}$
(A.213)
$\displaystyle\qquad+\bigg{|}\lambda_{\max}\bigg{(}\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N/2}-\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{\sum_{i\in[N]}\mathds{1}_{i}}\bigg{)}\bigg{|},$
(A.214)

where $(a)$ follows from Eq. A.211 and the definition of
$\Sigma^{\widetilde{U}}$, and $(b)$ follows from the triangle inequality on
the operator norm after noting that the maximum eigenvalue of any symmetric
matrix coincides with its operator norm. Next, we show that each term in Eq.
A.214 is $o_{p}(1)$.

Proof that first term in Eq. A.214 is $o_{p}(1)$. We have

$\displaystyle\lambda_{\max}\bigg{(}\frac{\sum_{i\in[N]}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N}-\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N/2}\bigg{)}=\lambda_{\max}\bigg{(}\frac{1}{N}\sum_{i\in[N]}(1-2\mathds{1}_{i})\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\bigg{)}.$
(A.215)

To bound Eq. A.215, we apply Corollary 4 with

$\displaystyle
X_{i}=\frac{1}{N}(1-2\mathds{1}_{i})\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\quad\text{and}\quad
A_{i}=\frac{1}{N}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}.$
(A.216)

We note that, for every $i\in[N]$, $\mathbb{E}[X_{i}]=0$ as
$\mathbb{E}[\mathds{1}_{i}]=1/2$ and $A_{i}^{2}-X_{i}^{2}$ is positive semi-
definite as

$\displaystyle
X_{i}^{2}=\frac{1}{N^{2}}(1-2\mathds{1}_{i})^{2}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{N^{2}}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}=A_{i}^{2},$
(A.217)

where $(a)$ follows because $\mathds{1}_{i}\in\\{0,1\\}$. We claim that
$\big{|}\lambda_{\max}(\sum_{i\in[n]}A_{i}^{2})\big{|}\leq c^{2}/N$ for some
$c>0$. Then, using Corollary 4, Eq. A.215 is bounded as follows with
probability at least $1-\delta$,

$\displaystyle\lambda_{\max}\bigg{(}\frac{1}{N}\sum_{i\in[N]}(1-2\mathds{1}_{i})\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\bigg{)}\leq\frac{\sqrt{c\ell_{2\delta/r_{T}}}}{\sqrt{N}}.$
(A.218)

Therefore, the first term in Eq. A.214 is $o_{p}(1)$. It remains to bound
$\lambda_{\max}\big{(}\sum_{i\in[N]}A_{i}^{2}\big{)}$. We have

$\displaystyle\Big{|}\lambda_{\max}\Big{(}\sum_{i\in[N]}A_{i}^{2}\Big{)}\Big{|}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\bigg{|}\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}x^{\top}\Big{(}\sum_{i\in[N]}A_{i}^{2}\Big{)}x\bigg{|}$
(A.219)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\Big{|}\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}\frac{1}{N^{2}}\sum_{i\in[N]}x^{\top}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}x\Big{|}$
(A.220)
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\Big{|}\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}\frac{1}{N^{2}}\sum_{i\in[N]}\|{\widetilde{U}_{i,\cdot}}\|_{2}\cdot
x^{\top}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}x\Big{|}$ (A.221)
$\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}\frac{c}{N^{2}}\sum_{i\in[N]}|x^{\top}\widetilde{U}_{i,\cdot}|^{2}\stackrel{{\scriptstyle(e)}}{{\leq}}\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}\frac{c}{N^{2}}\sum_{i\in[N]}\|{x}\|_{2}\|{\widetilde{U}_{i,\cdot}}\|_{2}\stackrel{{\scriptstyle(f)}}{{\leq}}\frac{c^{2}}{N},$
(A.222)

where $(a)$ follows from the definition of the maximum eigenvalue of a matrix,
$(b)$ follows from Eq. A.216, $(c)$ follows because
$\widetilde{U}_{i,\cdot}\in\mathbb{R}^{r_{T}\times 1}$, $(d)$ and $(f)$ follow
from 8(b), $(e)$ follows from Cauchy-Schwarz inequality.

Proof that second term in Eq. A.214 is $o_{p}(1)$. We have

$\displaystyle\bigg{|}\lambda_{\max}\bigg{(}\frac{\sum_{i\in[N]}\\!\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N/2}\\!-\\!\frac{\sum_{i\in[N]}\\!\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{\sum_{i\in[N]}\mathds{1}_{i}}\bigg{)}\bigg{|}=\bigg{|}\lambda_{\max}\bigg{(}\frac{2}{N}\\!\\!\sum_{i\in[N]}\\!\\!\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\bigg{)}\bigg{|}\bigg{|}\frac{\sum_{i\in[N]}\\!\mathds{1}_{i}\\!-\\!N/2}{\sum_{i\in[N]}\mathds{1}_{i}}\bigg{|}.$
(A.223)

To bound Eq. A.223, we claim
$\big{|}\lambda_{\max}\big{(}\frac{1}{N}\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\big{)}\big{|}\leq
c$ for some $c>0$, and apply Corollary 2 on the vector
$(\mathds{1}_{1}-1/2,\cdots,\mathds{1}_{N}-1/2)$. We note that, for every
$i\in[N]$, $\mathds{1}_{i}-1/2$ is zero-mean and
$\textrm{subGaussian}(1/\sqrt{\ell_{1}})$ (see Example 2.5.8 in Vershynin,
(2018)). Then, with probability at least $1-\delta$,

$\displaystyle\Big{|}\sum_{i\in[N]}\mathds{1}_{i}-N/2\Big{|}\leq\sqrt{\frac{c\ell_{\delta}N}{\ell_{1}}}.$
(A.224)

Using Eq. A.224 to bound Eq. A.223, with probability at least $1-\delta$, we
have

$\displaystyle\bigg{|}\lambda_{\max}\bigg{(}\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{N/2}-\frac{\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}}{\sum_{i\in[N]}\mathds{1}_{i}}\bigg{)}\bigg{|}\leq\frac{\sqrt{c\ell_{\delta}N/\ell_{1}}}{N/2-\sqrt{c\ell_{\delta}N/\ell_{1}}}.$
(A.225)

Therefore, the first term in Eq. A.214 is $o_{p}(1)$. It remains to bound
$\big{|}\lambda_{\max}\big{(}\frac{1}{N}\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\big{)}\big{|}$.
We have

$\displaystyle\bigg{|}\lambda_{\max}\Big{(}\frac{1}{N}\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\Big{)}\bigg{|}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\bigg{|}\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}x^{\top}\Big{(}\frac{1}{N}\sum_{i\in[N]}\mathds{1}_{i}\widetilde{U}_{i,\cdot}\widetilde{U}_{i,\cdot}^{\top}\Big{)}x\bigg{|}$
(A.226)
$\displaystyle=\\!\\!\\!\\!\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}\frac{1}{N}\sum_{i\in[N]}\mathds{1}_{i}|x^{\top}\widetilde{U}_{i,\cdot}|^{2}$
(A.227)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\\!\\!\max_{x\in\mathbb{R}^{N}:\|{x}\|_{2}=1}\frac{1}{N}\sum_{i\in[N]}\mathds{1}_{i}\|{x}\|_{2}\|{\widetilde{U}_{i,\cdot}}\|_{2}\stackrel{{\scriptstyle(c)}}{{\leq}}c,$
(A.228)

where $(a)$ follows from the definition of the maximum eigenvalue of a matrix,
$(b)$ follows from Cauchy-Schwarz inequality, and $(c)$ follows from 8(b).

####  F.3.2 11 holds

First, 11(a) holds as follows,

$\displaystyle\max_{j\in[M]}\frac{1}{N}\sum_{j^{\prime}\in[M]}\Big{|}\sum_{i\in[N]}\mathbb{E}\big{[}h_{i,j}h_{i,j^{\prime}}\big{]}\Big{|}\stackrel{{\scriptstyle(a)}}{{\leq}}\max_{j\in[M]}\frac{1}{N}\sum_{i\in[N]}\sum_{j^{\prime}\in[M]}\Big{|}\mathbb{E}\big{[}h_{i,j}h_{i,j^{\prime}}\big{]}\Big{|}\stackrel{{\scriptstyle(b)}}{{\leq}}\max_{j\in[M]}\frac{1}{N}\sum_{i\in[N]}c=c,$
(A.229)

where $(a)$ follows from triangle inequality and $(b)$ follows from 9(b).
Next, from 9(a) and 9(c), we have

$\displaystyle\max_{j\in[M]}\big{|}\mathbb{E}[h_{i,j}h_{i^{\prime},j}]\big{|}=\begin{cases}0&\text{
if }i\neq i^{\prime}\\\
\max_{j\in[M]}\big{|}\mathbb{E}[h_{i,j}^{2}]\big{|}\leq c&\text{ if
}i=i^{\prime}\end{cases}$ (A.230)

Then, 11(b) holds as follows,

$\displaystyle\max_{i\in[N]}\max_{j\in[M]}\sum_{i^{\prime}\in[N]}\big{|}\mathbb{E}[h_{i,j}h_{i^{\prime},j}]\big{|}\leq
c.$ (A.231)

Next, 11(c) holds as follows,

$\displaystyle\frac{1}{NM}\sum_{i,i^{\prime}\in[N]}\sum_{j,j^{\prime}\in[M]}\big{|}\mathbb{E}[h_{i,j}h_{i^{\prime},j^{\prime}}]\big{|}\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{NM}\sum_{i\in[N]}\sum_{j,j^{\prime}\in[M]}\big{|}\mathbb{E}[h_{i,j}h_{i,j^{\prime}}]\big{|}\stackrel{{\scriptstyle(b)}}{{\leq}}\frac{1}{NM}\sum_{i\in[N]}\sum_{j\in[M]}c=c,$
(A.232)

where $(a)$ follows from 9(c) and $(b)$ follows from 9(b). Next, let
$\gamma_{i,j,j^{\prime}}\triangleq
h_{i,j}h_{i,j^{\prime}}-\mathbb{E}[h_{i,j}h_{i,j^{\prime}}]$ and fix any
$j,j^{\prime}\in[M]$. Then, 11(d) holds as follows,

$\displaystyle\frac{1}{N^{2}}\mathbb{E}\Big{[}\Big{(}\sum_{i\in[N]}\gamma_{i,j,j^{\prime}}\Big{)}^{4}\Big{]}$
$\displaystyle=\frac{1}{N^{2}}\mathbb{E}\Big{[}\Big{(}\sum_{i_{1}\in[N]}\gamma_{i_{1},j,j^{\prime}}\Big{)}\Big{(}\sum_{i_{2}\in[N]}\gamma_{i_{2},j,j^{\prime}}\Big{)}\Big{(}\sum_{i_{3}\in[N]}\gamma_{i_{3},j,j^{\prime}}\Big{)}\Big{(}\sum_{i_{4}\in[N]}\gamma_{i_{4},j,j^{\prime}}\Big{)}\Big{]}$
(A.233)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\frac{1}{N^{2}}\sum_{i\in[N]}\mathbb{E}\Big{[}\gamma_{i,j,j^{\prime}}^{4}\Big{]}+\frac{3}{N^{2}}\sum_{i\neq
i^{\prime}\in[N]}\mathbb{E}\Big{[}\gamma_{i,j,j^{\prime}}^{2}\gamma_{i^{\prime},j,j^{\prime}}^{2}\Big{]}\leq
c,$ (A.234)

where $(a)$ follows from linearity of expectation and 9(c) after by noting
that $\mathbb{E}[\gamma_{i,j,j^{\prime}}]=0$ for all
$i,j,j^{\prime}\in[N]\times[M]\times[M]$ and $(b)$ follows because
$\gamma_{i,j,j^{\prime}}$ has bounded moments due to 9(a).

##  Appendix G Data generating process for the simulations

The inputs of the data generating process (DGP) are: the probability bound
$\lambda$; two positive constants $c^{(0)}$ and $c^{(1)}$; and the standard
deviations ${\sigma_{i,j}^{(a)}}$ for every $i\in[N],j\in[M],a\in\\{0,1\\}$.
The DGP is:

  1. 1.

For positive integers $r_{p}$, $r_{\theta}$ and
$r=\max\\{r_{p},r_{\theta}\\}$, generate a proxy for the common unit-level
latent factors $U^{\textrm{shared}}\in\mathbb{R}^{N\times r}$, such that, for
all $i\in[N]$ and $j\in[r]$, $u^{\textrm{shared}}_{i,j}$ is independently
sampled from a $\texttt{Uniform}(\sqrt{\lambda},\sqrt{1-\lambda})$
distribution, with $\lambda\in(0,1)$.

  2. 2.

Generate proxies for the measurement-level latent factors
$V,V^{(0)},V^{(1)}\in\mathbb{R}^{M\times r}$, such that, for all $i\in[M]$ and
$j\in[r]$, $v_{i,j},v_{i,j}^{(0)},v_{i,j}^{(1)}$ are independently sampled
from a $\texttt{Uniform}(\sqrt{\lambda},\allowbreak\sqrt{1-\lambda})$
distribution.

  3. 3.

Generate the treatment assignment probability matrix ${P}$

$\displaystyle{P}=\frac{1}{r_{p}}U^{\textrm{shared}}_{[N]\times[r_{p}]}V_{[M]\times[r_{p}]}^{\top}.$
(A.235)

  4. 4.

For $a\in\\{0,1\\}$, run SVD on $U^{\textrm{shared}}V^{{(a)}\top}$, i.e.,

$\displaystyle\texttt{SVD}(U^{\textrm{shared}}V^{{(a)}\top})=(U^{(a)},\Sigma^{(a)},W^{(a)}).$
(A.236)

Then, generate the mean potential outcome matrices ${\Theta}^{(0)}$ and
${\Theta}^{(1)}$:

$\displaystyle{\Theta}^{(a)}=\frac{c^{(a)}\texttt{Sum}(\Sigma^{(a)})}{r_{\theta}}U^{(a)}_{[N]\times[r_{\theta}]}W^{(a)\top}_{[M]\times[r_{\theta}]},$
(A.237)

where $\texttt{Sum}(\Sigma^{(a)})$ denotes the sum of all entries of
$\Sigma^{(a)}$.

  5. 5.

Generate the noise matrices ${E}^{(0)}$ and ${E}^{(1)}$, such that, for all
$i\in[N],j\in[M],a\in\\{0,1\\}$, $\varepsilon^{(a)}_{i,j}$ is independently
sampled from a $\mathcal{N}(0,(\sigma_{i,j}^{(a)})^{2})$ distribution. Then,
determine $y^{(a)}_{i,j}$ from Eq. 2.

  6. 6.

Generate the noise matrix ${W}$, such that, for all $i\in[N],j\in[M]$,
$\eta_{i,j}$ is independently sampled as per Eq. 4. Then, determine $a_{i,j}$
and $y_{i,j}$ from Eq. 3 and Eq. 1, respectively.

In our simulations, we set $\lambda=0.05$, $c^{(0)}=1$ and $c^{(1)}=2$. In
practice, instead of choosing the values of $\sigma_{i,j}^{(a)}$ as ex-ante
inputs, we make them equal to the standard deviation of all the entries in
${\Theta}^{(a)}$ for every $i$ and $j$, separately for $a\in\\{0,1\\}$.

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spectrum becomes softer as $\kappa_{\lambda}$ increases away from the SM, but
the cross-section also grows beyond the magnitude of the excess as
$\kappa_{\lambda}$ increases much further. This slight excess also results in
the deviation of the observed limits in Figure 11 from the expected limits by
about $1\sigma$. No such excess is observed in the VBF signal region, and the
best-fit value of $\kappa_{2V}$ from the likelihood scan is found to be $1.0$.
With the values of the other modifiers ($\kappa_{V}$ and either $\kappa_{2V}$
or $\kappa_{\lambda}$, respectively) fixed to their SM value of 1, the
observed (expected) $2\sigma$ allowed range for $\kappa_{\lambda}$ is found to
be $[-3.5,11.3]$ ($[-5.4,11.4]$) and the corresponding range for $\kappa_{2V}$
is $[-0.0,2.1]$ ($[-0.1,2.1]$).

The exclusion constraints obtained using the profile likelihood ratio method
are also presented in the two-dimensional $\kappa_{\lambda}$–$\kappa_{2V}$
coupling modifier space, similarly to the $95\%$ CL constraints described
above. The excluded regions are presented in Figure 12. With both modifiers
able to float in the two-dimensional fit that combines both the ggF and VBF
signal regions, the fit converges to $\kappa_{\lambda}$ and $\kappa_{2V}$
values slightly different from the ones where the minimum is found in the fits
with a single parameter free.

In addition to constraints on the ggF and VBF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
cross-sections and the $\kappa_{\lambda}$ and $\kappa_{2V}$ coupling
modifiers, constraints for relevant coefficients can be derived from the ggF
selection of the analysis in the SMEFT and HEFT frameworks, as outlined in
Section 1. The VBF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
process was ignored for both the SMEFT and HEFT results; including the VBF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
process as a background was found to have a negligible effect on the extracted
parameter limits. The slight dependence of the
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\rightarrow\mathit{{b}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}{}\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}{}$
branching fraction on the SMEFT and HEFT coefficients is also ignored, as the
impact on the analysis sensitivity is small. Constraints on the SMEFT
coefficients are extracted by considering the $95\%$ CL exclusion of the
cross-section as a function of SMEFT parameter, as was done for the
$\kappa_{\lambda}$ and $\kappa_{2V}$ constraints discussed previously. The
extracted constraints on individual parameters in the scenario where the other
parameters are fixed to $0$ are summarized in Table 10. Limits approaching or
exceeding $\pm 4\pi$ should be interpreted with caution because of the
potential impact from effects such as missing higher-order model
contributions. The exclusion limits are also presented in two-dimensional
SMEFT coefficient subspaces. The exclusion limits for each coefficient versus
the
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
coefficient (with the remaining three coefficients fixed to $0$) are shown in
Figure 13. The upper limits on the HEFT ggF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
production cross-section in the seven benchmark models are presented in Figure
14. The spread of sensitivity between the seven benchmark models reflects the
different signal kinematics and, hence, shapes of the signal
$m_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
distributions. The different variation between observed and expected limits is
linked to the slight excess observed in the low
$m_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
region, as discussed earlier. The red crosses in Figure 14 indicate the
predicted
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
cross-sections from the respective benchmark models. As can be seen, BM3, BM5
and BM7 are observed to be excluded with more than 95% confidence. Constraints
are placed on the values of
$c_{gg\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
and
$c_{t\overline{t}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$,
with all other HEFT coefficients fixed to SM values. The observed (expected)
constraints on
$c_{gg\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
are found to be $[-0.36,0.78]$ ($[-0.42,0.75]$), while the observed (expected)
constraints on
$c_{t\overline{t}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
are found to be $[-0.55,0.51]$ ($[-0.46,0.40]$).

Table 10:  The extracted upper and lower limits on the SMEFT parameters to which the analysis is sensitive. For each parameter, the constraints are provided assuming the other parameters are fixed to $0$. The VBF $\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$ process is ignored for this result.  Parameter | Expected Constraint | Observed Constraint
---|---|---
| Lower | Upper | Lower | Upper
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$ | $-20$ | $11$ | $-22$ | $11$
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}G}$ | $-0.056$ | $0.049$ | $-0.067$ | $0.060$
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\square}$ | $-9.3$ | $13.9$ | $-8.9$ | $14.5$
$c_{t\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$ | $-10.0$ | $6.4$ | $-10.7$ | $6.2$
$c_{tG}$ | $-0.97$ | $0.94$ | $-1.12$ | $1.15$

((a))  ((b))

((c))  ((d))

Figure 13:  The observed $95\%$ CL exclusion limits on the SMEFT coefficients
in the two-dimensional spaces (a)
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}G}$
vs.
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$,
(b) $c_{tG}$ vs.
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$,
(c)
$c_{t\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$
vs.
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$,
and (d)
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}\square}$
vs.
$c_{\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}}$,
shown by the solid black lines. The dashed black line indicates the expected
$95\%$ CL exclusion limits. The shaded blue band indicates the $\pm 1\sigma$
uncertainty of the exclusion limits, while the yellow band indicates the $\pm
2\sigma$ uncertainty. The values of the other three coefficients for each plot
are fixed to $0$. The VBF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
process is ignored for this result.  Figure 14: The observed $95\%$ CL
exclusion limits on the ggF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
production cross-section in the SM and each of the seven HEFT benchmark
models, given by the solid black points. The blue and yellow bands show
respectively the $1\sigma$ and $2\sigma$ bands around the expected upper
limits, which are shown by the open circles. The predicted ggF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
production cross-section from each benchmark is indicated by a pink cross. The
VBF
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
process is ignored for this result.

##  9 Conclusion

A search for nonresonant pair production of Higgs bosons in the
$\mathit{{b}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$$\mathit{{b}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{\mkern
0.0mu\overline{\mkern
0.0mu{b}}}{}_{\mspace{-2.0mu}\scriptstyle{}}^{\mspace{0.0mu}\scriptstyle{}}}$
final state was carried out, with dedicated analyses for the ggF and VBF
production modes, using $126\text{\,}{\mathrm{fb}}^{-1}$ of
$\sqrt{s}=$13\text{\,}\mathrm{TeV}$$ $pp$ collision data collected by the
ATLAS detector at the LHC. The sensitivity of the analyses is improved
relative to previous iterations by using more sophisticated background
modeling techniques, event categorization and improved jet reconstruction and
flavor identification algorithms, in addition to the increased integrated
luminosity of the analyzed data.

No evidence of signal is found and the observed (expected) upper limit on the
cross-section for nonresonant Higgs boson pair production is determined to be
5.4 (8.1) times the Standard Model predicted cross-section at 95% confidence
level. Constraints are placed upon modifiers to the
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
and
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$VV
couplings. The observed (expected) $2\sigma$ constraints on the
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$
coupling modifier, $\kappa_{\lambda}$, are determined to be $[-3.5,11.3]$
($[-5.4,11.4]$), while the corresponding constraints for the
$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$$\mathit{{H}{}_{\mspace{-3.0mu}\scriptstyle{}}^{\mspace{-1.0mu}\scriptstyle{}}}$VV
coupling modifier, $\kappa_{2V}$, are $[-0.0,2.1]$ ($[-0.1,2.1]$). The results
are also used to derive constraints on relevant coefficients in the SM
effective field theory and the Higgs effective field theory frameworks.

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