$\displaystyle\max_{j,k}\bigg{|}\sum_{y}w(y)\int_{\bm{z}}z_{j}z_{k}\big{[}f^{\prime}(\widehat{\bm{\beta}}^{T}\bm{z}|y,\bm{z})-f^{\prime}(\bm{\beta}^{*T}\bm{z}|y,\bm{z})\big{]}f(y,\bm{z})d\bm{z}\bigg{|}$
$\displaystyle\leq$
$\displaystyle||\widehat{\bm{\beta}}-\bm{\beta}^{*}||_{1}M_{n}|f^{\prime\prime}|_{\infty}\max_{j,k}\sum_{y}\mathbb{E}[|Z_{j}Z_{k}||Y=y]$
$\displaystyle\lesssim$ $\displaystyle
M_{n}||\widehat{\bm{\beta}}-\bm{\beta}^{*}||_{1}.$

∎

######  Lemma 7.

Under the conditions in Theorem 1, we have

$(n\delta)^{1/2}\bigg{|}\bm{v}^{*T}\bigg{(}\nabla
R_{\delta}^{n_{(j)}}(\widehat{\bm{\beta}}^{(k)})-\nabla
R_{\delta}^{n_{(j)}}(\bm{\beta}^{*})-\nabla^{2}R_{\delta}^{n_{(j)}}(\bm{\beta}^{*})(\widehat{\bm{\beta}}^{(k)}-\bm{\beta}^{*})\bigg{)}\bigg{|}=o_{\mathbb{P}}(1),$

for $(j,k)\in\\{(1,2),(2,1)\\}$.

###### Proof of Lemma 7.

With some algebra we obtain

$\displaystyle\bigg{|}\bm{v}^{*T}\bigg{(}\nabla
R_{\delta}^{n_{(j)}}(\widehat{\bm{\beta}}^{(k)})-\nabla
R_{\delta}^{n_{(j)}}(\bm{\beta}^{*})-\nabla^{2}R_{\delta}^{n_{(j)}}(\bm{\beta}^{*})(\widehat{\bm{\beta}}^{(k)}-\bm{\beta}^{*})\bigg{)}\bigg{|}$
(C.9) $\displaystyle=$
$\displaystyle\bigg{|}\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}w(y_{i})\frac{\bm{z}_{i}^{T}\bm{v}^{*}y_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}-t)dt\bigg{|}$
$\displaystyle\leq$
$\displaystyle||\bm{v}^{*}||_{1}\bigg{|}\bigg{|}\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}w(y_{i})\frac{\bm{z}_{i}y_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}-t)dt\bigg{|}\bigg{|}_{\infty}.$

Now we start to analyze
$\bigg{|}\bigg{|}\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}w(y_{i})\frac{\bm{z}_{i}y_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}-t)dt\bigg{|}\bigg{|}_{\infty}$.
Denote
$G_{i}=w(y_{i})\frac{\bm{z}_{i}y_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}-t)dt$
and $G_{im},1\leq m\leq d$ as its coordinates. Consider the event
$A:=\\{||\widehat{\bm{\beta}}^{(k)}-\bm{\beta}^{*}||_{1}\lesssim
C\eta_{1}(n)\\}$ for some constant $C$. Notice that for each $m$

$\displaystyle\sum_{y}w(y)\int_{\bm{z}}\frac{z_{m}y}{\delta}\int_{x}\int_{\frac{x-\bm{\beta}^{*T}\bm{z}}{\delta}}^{\frac{x-\widehat{\bm{\beta}}^{(k)T}\bm{z}}{\delta}}K^{\prime\prime}(t)(\frac{x-\widehat{\bm{\beta}}^{(k)T}\bm{z}}{\delta}-t)dtf(x|y,\bm{z})dxf(y,\bm{z})d\bm{z}$
(C.10)
$\displaystyle\stackrel{{\scriptstyle(u=(x-\bm{\beta}^{*T}\bm{z})/\delta)}}{{=}}$
$\displaystyle\sum_{y}w(y)\int_{\bm{z}}z_{m}y\int_{u}\int_{u}^{u+\triangle}K^{\prime\prime}(t)(u+\triangle-t)dtf(u\delta+\bm{\beta}^{*T}\bm{z}|y,\bm{z})duf(y,\bm{z})d\bm{z}$
$\displaystyle=$
$\displaystyle\sum_{y}w(y)\int_{\bm{z}}z_{m}y\int_{u}\int_{u}^{u+\triangle}K^{\prime\prime}(t)(u+\triangle-t)dtf(\bm{\beta}^{*T}\bm{z}|y,\bm{z})duf(y,\bm{z})d\bm{z}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{y}w(y)\int_{\bm{z}}z_{m}y\int_{u}u\delta\int_{u}^{u+\triangle}K^{\prime\prime}(t)(u+\triangle-t)dtf^{\prime}(\tau
u\delta+\bm{\beta}^{*T}\bm{z}|y,\bm{z})duf(y,\bm{z})d\bm{z},$

where
$\triangle=\frac{(\widehat{\bm{\beta}}^{(k)}-\bm{\beta})^{T}\bm{z}}{\delta}$
and $\tau\in[0,1]$. Here the first step is by definition and the last step is
from the mean value theorem. Since

$\displaystyle\int_{u}\int_{u}^{u+\triangle}K^{\prime\prime}(t)(u+\triangle-t)dtdu=0,$
(C.11)

we can show that the first term on the RHS of the last step is 0. The second
term on the RHS can be bounded by $C^{\prime}\delta|\triangle|^{2}\lesssim
M_{n}^{2}\eta_{1}(n)^{2}/\delta$ for some constant $C^{\prime}$ on event $A$.
This implies that $\mathbb{E}[G_{im}|A]\lesssim
M_{n}^{2}\eta_{1}(n)^{2}/\delta$.

Now we look at its variance. For the second moment, for each
$i\in\mathcal{N}_{j},m=1,\dotso,d$, we have

$\displaystyle\mathbb{E}\bigg{[}\bigg{(}w(y_{i})\frac{z_{im}y_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\widehat{\bm{\beta}}^{(k)}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\widehat{\bm{\beta}}^{(k)}\bm{z}_{i}}{\delta}-t)dt\bigg{)}^{2}\bigg{|}A\bigg{]}$
(C.12) $\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}\sum_{y}w(y)^{2}\int_{\bm{z}}\frac{z_{m}^{2}}{\delta}\int_{u}\bigg{(}\int_{u}^{u+\triangle}K^{\prime\prime}(t)(u+\triangle-t)dt\bigg{)}^{2}f(u\delta+\bm{\beta}^{*T}\bm{z}|y,\bm{z})duf(y,\bm{z})d\bm{z}|A\bigg{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\bigg{[}\sum_{y}w(y)^{2}\int_{\bm{z}}\frac{z_{m}^{2}}{\delta}2|f|_{\infty}|K^{\prime\prime}|^{2}_{\infty}\triangle^{4}f(y,\bm{z})d\bm{z}|A\bigg{]}$
$\displaystyle\lesssim$
$\displaystyle\frac{M^{4}_{n}\eta_{1}(n)^{4}}{\delta^{5}}.$

Also we know for each $i\in\mathcal{N}_{j},m=1,\dotso,d$,
$\bigg{|}\frac{z_{im}y_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\widehat{\bm{\beta}}^{(k)}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\widehat{\bm{\beta}}^{(k)T}\bm{z}_{i}}{\delta}-t)dt\bigg{|}\lesssim\frac{M_{n}}{\delta}(\frac{M_{n}\eta_{1}(n)}{\delta})^{2}$
on $A$, and therefore

$\displaystyle
Var[G_{im}|A]\lesssim\frac{M^{4}_{n}\eta_{1}(n)^{4}}{\delta^{5}}.$ (C.13)

Therefore, applying Bernstein inequality with
$M_{n}\sqrt{\frac{\log(d)}{n\delta}}=\mathcal{O}(1)$, we can obtain that

$\displaystyle\mathbb{P}\bigg{(}\max_{m}|\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}G_{im}-\mathbb{E}G_{m}|>\frac{M_{n}^{2}\eta_{1}(n)^{2}}{\delta}\sqrt{\frac{\log(d)}{n\delta^{3}}}~{}\big{|}A\bigg{)}\leq\mathcal{O}(d^{-1}),$
(C.14)

which further implies that

$\displaystyle\mathbb{P}\bigg{(}\max_{m}|\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}G_{im}-\mathbb{E}G_{m}|>\frac{M_{n}^{2}\eta_{1}(n)^{2}}{\delta}\sqrt{\frac{\log(d)}{n\delta^{3}}}\bigg{)}$
(C.15) $\displaystyle\leq$
$\displaystyle\mathbb{P}\bigg{(}\max_{m}|\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}G_{im}-\mathbb{E}G_{m}|>\frac{M_{n}^{2}\eta_{1}(n)^{2}}{\delta}\sqrt{\frac{\log(d)}{n\delta^{3}}}~{}\big{|}A\bigg{)}+\mathbb{P}(A^{C})$
$\displaystyle=$ $\displaystyle o(1),$

where the last step follows from Assumption 6.

$\displaystyle\bigg{|}\bigg{|}\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}w(Y_{i})\frac{\bm{Z}_{i}Y_{i}}{\delta}\int_{\frac{X_{i}-\bm{\beta}^{*T}\bm{Z}_{i}}{\delta}}^{\frac{X_{i}-\widehat{\bm{\beta}}^{(k)}\bm{Z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{X_{i}-\widehat{\bm{\beta}}^{(k)}\bm{Z}_{i}}{\delta}-t)dt\bigg{|}\bigg{|}_{\infty}=\mathcal{O}_{\mathbb{P}}(\frac{M_{n}^{2}\eta_{1}(n)^{2}}{\delta}).$
(C.16)

Under the conditions of Theorem 1, the desired result holds. This completes
the proof. ∎

######  Lemma 8.

Under the conditions of Theorem 1, let $s^{\prime}=||\bm{\omega}^{*}||_{1}$
and $\xi>0$ is some constant. If
$s^{\prime}\big{(}M_{n}\eta_{1}(n)\vee\sqrt{\frac{\log(d)}{n\delta^{3}}}\vee\delta^{\ell-1}\big{)}=o(1)$,
then with probability tending to one it holds that
$\kappa_{D}(s^{\prime})\geq\kappa/\sqrt{2}$, where

$\kappa_{D}(s^{\prime})=\min\bigg{\\{}\frac{s^{\prime
1/2}(\bm{v}^{T}\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n}_{\delta}(\widehat{\bm{\beta}})\bm{v})^{1/2}}{||\bm{v}_{s^{\prime}}||_{1}}:\bm{v}\in\mathbb{R}^{d-1}\backslash\\{0\\},||\bm{v}_{s^{\prime
c}}||_{1}\leq\xi||\bm{v}_{s^{\prime}}||_{1}\bigg{\\}}.$

###### Proof.

This proof is similar to the proof of Lemma J.3 in Ning et al. (2017) so we
only give a sketch here. Firstly we have

$\displaystyle\kappa_{D}(s^{\prime})^{2}\geq\min\bigg{\\{}\frac{\bm{v}^{T}\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n}_{\delta}(\widehat{\bm{\beta}})\bm{v}}{||\bm{v}||_{2}^{2}}:\bm{v}\in\mathbb{R}^{d-1}\backslash\\{0\\},||\bm{v}_{s^{\prime
c}}||_{1}\leq\xi||\bm{v}_{s^{\prime}}||_{1}\bigg{\\}}.$ (C.17)

Similar to the proof of Theorem 1, Lemmas 4, 5, 6 and 13 together imply that

$\displaystyle\bigg{|}\frac{\bm{v}^{T}(\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n}_{\delta}(\widehat{\bm{\beta}})-\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R(\bm{\beta}^{*}))\bm{v}}{||\bm{v}||_{2}^{2}}\bigg{|}$
(C.18) $\displaystyle=$
$\displaystyle\bigg{|}\frac{\bm{v}^{T}\big{[}\frac{1}{2}\big{(}\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})+\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n_{(2)}}_{\delta}(\widehat{\bm{\beta}}^{(1)})\big{)}-\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R(\bm{\beta}^{*})\big{]}\bm{v}}{||\bm{v}||_{2}^{2}}\bigg{|}$
$\displaystyle\lesssim$ $\displaystyle
s^{\prime}(\xi+1)^{2}\big{[}M_{n}\eta_{1}(n)\vee\sqrt{\frac{\log(d)}{n\delta^{3}}}\vee\delta^{\ell-1}\big{]}=o_{\mathbb{P}}(1),$

where the inequality is because $||\bm{v}||_{1}^{2}\leq
s^{\prime}(\xi+1)^{2}||\bm{v}||_{2}^{2}$. Therefore for $n$ large enough, we
have
$|\frac{\bm{v}^{T}(\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n}_{\delta}(\widetilde{\bm{\beta}})-\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R(\bm{\beta}^{*}))\bm{v}}{||\bm{v}||_{2}^{2}}|\leq\frac{1}{2}\kappa^{2}$.
This implies $\kappa_{D}(\widetilde{s}^{*})^{2}\geq\frac{1}{2}\kappa^{2}$ with
probability tending to 1. This completes the proof. ∎

######  Lemma 9.

Under the same conditions as in Theorem 2, for $j=1,2$, it holds that

$(n\delta)^{1/2}|\bm{v}^{*T}(\nabla
R_{\delta}^{n_{(j)}}(0,\bm{\gamma}^{*})-\nabla
R_{\delta}^{n_{(j)}}(\bm{\beta}^{*}))+\theta^{*}\bm{v}^{*T}\nabla^{2}_{\cdot\theta}R(\bm{\beta}^{*})|=o_{\mathbb{P}}(1).$

###### Proof.

By definition

$\displaystyle|\bm{v}^{*T}(\nabla
R_{\delta}^{n_{(j)}}(0,\bm{\gamma}^{*})-\nabla
R_{\delta}^{n_{(j)}}(\bm{\beta}^{*}))+\theta^{*}\bm{v}^{*T}\nabla^{2}_{\cdot\theta}R(\bm{\beta}^{*})|$
(C.19) $\displaystyle\leq||\bm{v}^{*}||_{1}\bigg{[}\underbrace{||\nabla
R_{\delta}^{n_{(j)}}(0,\bm{\gamma}^{*})-\nabla
R_{\delta}^{n_{(j)}}(\bm{\beta}^{*})+\theta^{*}\nabla^{2}_{\cdot\theta}R^{n_{(j)}}_{\delta}(\bm{\beta}^{*})||_{\infty}}_{I_{1}}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\underbrace{||\theta^{*}(\nabla^{2}_{\cdot\theta}R^{n_{(j)}}_{\delta}(\bm{\beta}^{*})-\nabla^{2}_{\cdot\theta}R_{\delta}(\bm{\beta}^{*}))||_{\infty}}_{I_{2}}+\underbrace{||\theta^{*}(\nabla^{2}_{\cdot\theta}R_{\delta}(\bm{\beta}^{*})-\nabla^{2}_{\cdot\theta}R(\bm{\beta}^{*}))||_{\infty}}_{I_{3}}\bigg{]}.$

For $I_{1}$, similar to Lemma 7, we can write

$\displaystyle|\nabla R_{\delta}^{n_{(j)}}(0,\bm{\gamma}^{*})-\nabla
R_{\delta}^{n_{(j)}}(\bm{\beta}^{*})+\theta^{*}\nabla^{2}_{\cdot\theta}R^{n_{(j)}}_{\delta}(\bm{\beta}^{*})|$
(C.20) $\displaystyle=$
$\displaystyle\bigg{|}\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}w(y_{i})\frac{y_{i}\bm{z}_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\bm{\beta}_{0}^{*T}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\bm{\beta}_{0}^{*T}\bm{z}_{i}}{\delta}-t)dt\bigg{|}$
$\displaystyle:=$
$\displaystyle|\frac{1}{\mathcal{N}_{j}}\sum_{i\in\mathcal{N}_{j}}G_{i}|,$

where
$G_{i}=w(y_{i})\frac{y_{i}\bm{z}_{i}}{\delta}\int_{\frac{x_{i}-\bm{\beta}^{*T}\bm{z}_{i}}{\delta}}^{\frac{x_{i}-\bm{\beta}_{0}^{*T}\bm{z}_{i}}{\delta}}K^{\prime\prime}(t)(\frac{x_{i}-\bm{\beta}_{0}^{*T}\bm{z}_{i}}{\delta}-t)dt\in\mathbb{R}^{d}$.
Thus, similar to the derivation of Lemma 7, for each $1\leq m\leq d$, we can
show that $\mathbb{E}[G_{im}]\lesssim M_{n}^{2}\theta^{*2}/\delta$,
$|G_{im}|\lesssim M_{n}^{3}\theta^{*2}/\delta^{3}$ and $Var[G_{im}]\lesssim
M_{n}^{4}\theta^{*4}/\delta^{5}$, and thus applying Bernstein inequality
yields

$I_{1}=\mathcal{O}_{\mathbb{P}}(\frac{M_{n}^{2}\theta^{*2}}{\delta}).$

Meanwhile, Lemma 5 and Lemma 4 imply that
$I_{2}=\mathcal{O}_{\mathbb{P}}(\theta^{*}\sqrt{\frac{\log(d)}{n\delta^{3}}})$
and $I_{3}=\mathcal{O}(\theta^{*}\delta^{\ell-1})$. Combing the above results,
we obtain

$\displaystyle|\bm{v}^{*T}(\nabla
R_{\delta}^{n_{(j)}}(0,\bm{\gamma}^{*})-\nabla
R_{\delta}^{n_{(j)}}(\bm{\beta}^{*}))+\theta^{*}\bm{v}^{*T}\nabla^{2}_{\cdot\theta}R(\bm{\beta}^{*})|=\mathcal{O}_{\mathbb{P}}(\frac{||\bm{v}^{*}||_{1}\theta^{*}}{\delta}(M_{n}^{2}\theta^{*}\vee\sqrt{\frac{\log(d)}{n\delta}}\vee\delta^{\ell})).$
(C.21)

This together with the condition in Theorem 2 completes the proof. ∎

######  Lemma 10.

Let $s^{\prime}=||\bm{\omega}^{*}||_{0}$. Suppose Assumptions 1-5 hold, and
$\lambda_{\min}(\nabla^{2}_{\bm{\gamma},\bm{\gamma}}R(\bm{\beta}^{*}))\geq c$
for some constant $c>0$. If we choose $\delta\asymp(\log(d)/n)^{1/(2\ell+1)}$
and tuning parameter
$\lambda^{\prime}\asymp||\bm{\omega}^{*}||_{1}(M_{n}\eta_{1}(n)+(\log(d)/n)^{(\ell-1)/(2\ell+1)})$
and $s^{\prime}(M_{n}\eta_{1}(n)+(\log(d)/n)^{(\ell-1)/(2\ell+1)})=o(1)$, then
it holds that

$\displaystyle||\widehat{\bm{\omega}}-\bm{\omega}^{*}||_{1}\lesssim||\bm{v}^{*}||_{1}s^{\prime}(M_{n}\eta_{1}(n)+(\log(d)/n)^{(\ell-1)/(2\ell+1)}).$
(C.22)

###### Proof.

Here we focus on the rate for $\widehat{\bm{\omega}}^{(1)}$ and the result
will follow accordingly. Denote
$\widehat{\triangle}=\widehat{\bm{\omega}}^{(1)}-\bm{\omega}^{*}$. By
definition, we can show that
$||\widehat{\triangle}_{\widetilde{s}^{*c}}||_{1}\leq||\widehat{\triangle}_{\widetilde{s}^{*}}||_{1}$.

With some algebra, we will get

$\displaystyle\widehat{\triangle}^{T}\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})\widehat{\triangle}$
$\displaystyle=\widehat{\triangle}^{T}(\nabla^{2}_{\bm{\gamma}\cdot}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})\bm{v}^{*})-\widehat{\triangle}^{T}(\nabla^{2}_{\bm{\gamma}\cdot}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})\widehat{\bm{v}}^{(1)})$
(C.23)
$\displaystyle\leq||\widehat{\triangle}||_{1}||\nabla^{2}_{\bm{\gamma}\cdot}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})\bm{v}^{*}||_{\infty}+||\widehat{\triangle}||_{1}||\nabla^{2}_{\bm{\gamma}\theta}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})-\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})\widehat{\bm{\omega}}^{(1)}||_{\infty}.$

By definition, the second term is bounded by
$\lambda^{\prime}||\widehat{\triangle}||_{1}$. For the first term, recall by
definition $\nabla^{2}_{\bm{\gamma}\cdot}R(\bm{\beta}^{*})\bm{v}^{*}=0$, and
thus similar to the proof of Theorem 1, Lemma 4, 5 and 6 imply that with
probability approaching to 1

$||\nabla^{2}_{\bm{\gamma}\cdot}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})\bm{v}^{*}||_{\infty}\lesssim||\bm{v}^{*}||_{1}\bigg{[}M_{n}\eta_{1}(n)\vee\sqrt{\frac{\log(d)}{n\delta^{3}}}\vee\delta^{\ell-1}\bigg{]},$

and thus with the choice of $\lambda^{\prime}$ it holds that
$\widehat{\triangle}^{T}\nabla^{2}_{\bm{\gamma}\bm{\gamma}}R^{n_{(1)}}_{\delta}(\widehat{\bm{\beta}}^{(2)})\widehat{\triangle}\lesssim\lambda^{\prime}||\widehat{\triangle}||_{1}$.
This together with Lemma 8 implies that $||\widehat{\triangle}||_{1}\lesssim
s^{\prime}\lambda^{\prime}$ with high probability. This completes the proof. ∎

######  Lemma 11.

Under the same conditions of Theorem 1, if $U$ is a proper kernel of order
$\ell$ satisfying the same condition as $K$ in Assumption 2, and in addition
$U$ is $\ell$ times continuously differentiable and $U^{(i)}$ degenerates at
the boundary for $i=0,\dotso,\ell-1$, then when
$\sqrt{\frac{\log(d)}{nh^{2\ell+1}}}+(M_{n}\eta_{1}(n)\vee h)^{\zeta}=o(1)$,
it holds that

$|\widehat{\mu}-\bm{v}^{*T}\bm{b}^{*}|\lesssim||\bm{v}^{*}||_{1}\bigg{(}\eta_{2}(n)+\sqrt{\frac{\log(d)}{nh^{2\ell+1}}}+(M_{n}\eta_{1}(n)\vee
h)^{\zeta}\bigg{)}.$

###### Proof.

It suffices to proof the results for
$\widehat{\bm{v}}^{(1)T}\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})$.
By definition,

$\displaystyle|\bm{v}^{*T}T^{(\ell)}(\bm{\beta}^{*})-\widehat{\bm{v}}^{(1)T}\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})|\leq|(\bm{v}^{*}-\widehat{\bm{v}})^{T}\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})|+|\bm{v}^{*T}(T^{(\ell)}(\bm{\beta}^{*})-\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)}))|.$
(C.24)

We firstly look at
$||T^{(\ell)}(\bm{\beta}^{*})-\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})||_{\infty}$.
Direct calculation gives that

$\displaystyle||T^{(\ell)}(\bm{\beta}^{*})-\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})||_{\infty}$
(C.25) $\displaystyle\leq$
$\displaystyle||T^{(\ell)}(\bm{\beta}^{*})-T^{(\ell)}(\widehat{\bm{\beta}}^{(2)})||_{\infty}+||T^{(\ell)}(\widehat{\bm{\beta}}^{(2)})-\widetilde{T}_{h,U}^{(\ell)}(\widehat{\bm{\beta}}^{(2)})||_{\infty}+||\widetilde{T}_{h,U}^{(\ell)}(\widehat{\bm{\beta}}^{(2)})-\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})||_{\infty},$

where
$\widetilde{T}_{h,U}^{(\ell)}(\widehat{\bm{\beta}}^{(2)}):=\sum_{y\in\\{-1,1\\}}w(y)y\int\frac{\bm{z}}{h^{1+\ell}}\int
U^{(\ell)}\bigg{(}\frac{\widehat{\bm{\beta}}^{(2)T}\bm{z}-x}{h}\bigg{)}f(x|y,\bm{z})dxf(y,\bm{z})d\bm{z}$.
For the first term, we have

$\displaystyle||T^{(\ell)}(\bm{\beta}^{*})-T^{(\ell)}(\widehat{\bm{\beta}}^{(2)})||_{\infty}$
(C.26) $\displaystyle=$
$\displaystyle\bigg{|}\bigg{|}\sum_{y\in\\{-1,1\\}}w(y)\int
y\bm{z}(f^{(\ell)}(\bm{\beta}^{*T}\bm{z}|y,\bm{z})-f^{(\ell)}(\widehat{\bm{\beta}}^{(2)T}\bm{z}|y,\bm{z}))f(y,\bm{z})d\bm{z}\bigg{|}\bigg{|}_{\infty}$
$\displaystyle=$
$\displaystyle\mathcal{O}_{\mathbb{P}}((M_{n}\eta_{1}(n))^{\zeta}),$

where the last step follows from Assumption 1. For the second term, notice
that we can rewrite

$\displaystyle\widetilde{T}_{h,U}^{(\ell)}(\widehat{\bm{\beta}}^{(2)})$ (C.27)
$\displaystyle=$
$\displaystyle\sum_{y\in\\{-1,1\\}}w(y)y\int\frac{\bm{z}}{h}\int
U\bigg{(}\frac{\widehat{\bm{\beta}}^{(2)T}\bm{z}-x}{h}\bigg{)}f^{(\ell)}(x|y,\bm{z})dxf(y,\bm{z})d\bm{z}$
$\displaystyle=$ $\displaystyle\sum_{y\in\\{-1,1\\}}w(y)y\int\bm{z}\int
U(u)f^{(\ell)}(\widehat{\bm{\beta}}^{(2)T}\bm{z}-hu|y,\bm{z})duf(y,\bm{z})d\bm{z},$

where the first step is by repeated integration by parts and the second step
is by a change of variable. This implies that

$\displaystyle||T^{(\ell)}(\widehat{\bm{\beta}}^{(2)})-\widetilde{T}_{h,U}^{(\ell)}(\widehat{\bm{\beta}}^{(2)})||_{\infty}$
(C.28) $\displaystyle=$
$\displaystyle\bigg{|}\bigg{|}\sum_{y\in\\{-1,1\\}}w(y)y\int\bm{z}\int
U(u)\bigg{(}f^{(\ell)}(\widehat{\bm{\beta}}^{(2)T}\bm{z}|y,\bm{z})-f^{(\ell)}(\widehat{\bm{\beta}}^{(2)T}\bm{z}-hu|y,\bm{z})\bigg{)}duf(y,\bm{z})d\bm{z}\bigg{|}\bigg{|}_{\infty}$
$\displaystyle=$ $\displaystyle\mathcal{O}(h^{\zeta}).$

Finally, similar to the proof of Theorem 1, we can show that the third term on
the RHS of (LABEL:eq_bias_pilot_0) is
$\mathcal{O}_{\mathbb{P}}(\sqrt{\frac{\log(d)}{nh^{2\ell+1}}})$ based on
Lemmas 5 and 13. Putting all three terms together we have

$\displaystyle||T^{(\ell)}(\bm{\beta}^{*})-\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})||_{\infty}=\mathcal{O}_{\mathbb{P}}\bigg{(}\sqrt{\frac{\log(d)}{nh^{2\ell+1}}}+(M_{n}\eta_{1}(n)\vee
h)^{\zeta}\bigg{)},$ (C.29)

and thus
$||\widehat{T}_{h,U}^{(\ell),n_{(1)}}(\widehat{\bm{\beta}}^{(2)})||=\mathcal{O}_{\mathbb{P}}(1)$
by the condition of this lemma. Plugging this back to (C.24) gives the desired
result. This completes the proof. ∎

######  Lemma 12.

Under the same conditions of Theorem 1, if $L$ is a proper kernel of order
$\ell$ satisfying Assumption 2 and
$g^{\ell}+\sqrt{\frac{\log(d)}{ng}}+M_{n}\eta_{1}(n)=o(1)$, then we have

$\displaystyle|\widehat{\sigma}^{2}-\bm{v}^{*T}\bm{\Sigma}^{*}\bm{v}^{*}|\lesssim||\bm{v}^{*}||_{1}^{2}\bigg{(}\eta_{2}(n)+g^{\ell}+\sqrt{\frac{\log(d)}{ng}}+M_{n}\eta_{1}(n)\bigg{)}.$
(C.30)

###### Proof.

It suffices to show the convergence rate of
$\widetilde{\mu}_{K}\widehat{\bm{v}}^{(1)T}\widehat{H}_{g,K}^{n_{(1)}}(\widehat{\bm{\beta}}^{(2)})\widehat{\bm{v}}^{(1)}$.
Similar to the proof of Theorem 1, following Lemmas 4, 5, 6 and 13, we can
show that

$\displaystyle||\widehat{H}_{g,K}^{(1)}(\widehat{\bm{\beta}}^{(2)})-H(\bm{\beta}^{*})||_{\max}=\mathcal{O}_{\mathbb{P}}\bigg{(}g^{\ell}+\sqrt{\frac{\log(d)}{ng}}+M_{n}\eta_{1}(n)\bigg{)}.$
(C.31)

Since $||H(\bm{\beta}^{*})||_{\max}=\mathcal{O}(1)$, we know
$||\widehat{H}_{g,K}^{(1)}(\widehat{\bm{\beta}}^{(2)})||_{\max}=\mathcal{O}_{\mathbb{P}}(1)$
when $g^{\ell}+\sqrt{\frac{\log(d)}{ng}}+M_{n}\eta_{1}(n)=o(1)$. Now by
triangle inequality, we obtain

$\displaystyle|\widehat{\bm{v}}^{(1)T}\widehat{H}_{g,K}^{(1)}(\widehat{\bm{\beta}}^{(2)})\widehat{\bm{v}}^{(1)}-\bm{v}^{*T}H(\bm{\beta}^{*})\bm{v}^{*}|$
(C.32) $\displaystyle\leq$
$\displaystyle||\widehat{\bm{v}}^{(1)}-\bm{v}^{*}||^{2}_{1}||\widehat{H}_{g,K}^{(1)}(\widehat{\bm{\beta}}^{(2)})||_{\max}+2||\bm{v}^{*T}\widehat{H}_{g,K}^{(1)}(\widehat{\bm{\beta}}^{(2)})||_{\infty}||\widehat{\bm{v}}^{(1)}-\bm{v}^{*}||_{1}+|\bm{v}^{*T}(\widehat{H}_{g,K}^{(1)}(\widehat{\bm{\beta}}^{(2)})-H(\bm{\beta}^{*}))\bm{v}^{*}|.$

This implies that

$\displaystyle|\widehat{\sigma}^{2}-\bm{v}^{*T}\bm{\Sigma}^{*}\bm{v}^{*}|=||\bm{v}^{*}||_{1}^{2}\mathcal{O}_{\mathbb{P}}\bigg{(}\eta_{2}(n)+g^{\ell}+\sqrt{\frac{\log(d)}{ng}}+M_{n}\eta_{1}(n)\bigg{)}.$
(C.33)

This completes the proof. ∎

######  Lemma 13 (Lemma 6.1 of Chernozhukov et al. (2018)).

Let $\\{\bm{X}_{m}\\}$ and $\\{\bm{Y}_{m}\\}$ be sequences of random vectors.
If $||\bm{X}_{m}||=\mathcal{O}_{\mathbb{P}}(A_{m})$ conditional on
$\\{\bm{Y}_{m}\\}$ for a sequence of positive constants $\\{A_{m}\\}$, then
$||\bm{X}_{m}||=\mathcal{O}_{\mathbb{P}}(A_{m})$ unconditionally.

##  Appendix D Practical Considerations

In practice, in order to apply the smoothed decorrelated score test presented
in Section 2.2, we need to select the bandwidth and regularization parameter
in the initial estimators $\widehat{\bm{\omega}}$ and $\widehat{\bm{\beta}}$,
two bandwidths for the plug-in asymptotic bias and variance estimators in
(2.11) and (2.13), and the final bandwidth $\delta$ for the score test
statistic. In this section, we discuss practical considerations for the
proposed test as well as the data-driven selection procedure for $\delta$.

We use a path-following algorithm to compute the initial estimator
$\widehat{\bm{\beta}}$ and apply a two way cross-validation approach to choose
the tuning parameters $(\delta,\lambda)$ in the optimization (2.2); see Feng
et al. (2019) for details. The initial values of the path-following algorithm
used in the simulation are shown in the appendix Section E. To compute the
Dantzig estimator $\widehat{\bm{\omega}}$, we use the “flare” package in R.
The same cross-validation method can be used to select the tuning parameters
$(\delta,\lambda^{\prime})$ in (2.15). Meanwhile, we find the empirical
performance of the Dantzig estimator $\widehat{\bm{\omega}}$ is not very
sensitive to the choice of $(\delta,\lambda^{\prime})$. In the simulation and
real data analysis, to ease the computation, we set $\lambda^{\prime}=2(\log
d/n)^{1/5}$ and $\delta=1$ in (2.15). Notice that in rare cases, the Dantzig
solver may become unstable numerically when the Hessian
$\nabla^{2}R^{n}_{\delta}(\widehat{\bm{\beta}})$ is ill-conditioned. In this
case, we will firstly project it onto the cone of positive definite matrices
and then plug-in the projected matrix into (2.15) to solve
$\widehat{\bm{\omega}}$.

For the bandwidth parameters $h$ and $g$ in the plug-in bias and variance
estimators, Lemma 11 and Lemma 12 imply that the theoretical optimal order for
$h$ and $g$ are $(\log(d)/n)^{2\ell+2\zeta+1}$ (if $M_{n}\eta_{1}(n)=o(h)$
which holds under mild conditions) and $(\log(d)/n)^{2\ell+1}$, respectively.
In practice, we recommend choosing $h=c_{1}(\log(d)/n)^{2\ell^{\prime}+3}$ and
$g=c_{2}(\log(d)/n)^{2\ell^{\prime}+1}$ for some $c_{1},c_{2}>0$ (e.g.,
$c_{1}=c_{2}=2$ in our simulation), where $\ell^{\prime}$ is the order of the
kernel function $K$.

Finally, for the main bandwidth parameter $\delta$ appearing in the score test
statistic, if a pre-specified value is preferred, we recommend choosing
$\delta=cn^{-1/(2\ell^{\prime}+1)}$ for some constant $c>0$ (e.g., $c=1$).
When using our proposed data-driven bandwidth selector, notice that another
bandwidth parameter $b$, whose optimal order is $(\log(d)/n)^{1/(2\ell+2r+1)}$
(see the discussion after Theorem 3), is required for estimating the squared
bias $SB(\delta)$. Similarly, we suggest choosing
$b=c_{0}(\log(d)/n)^{1/(2\ell^{\prime}+2r^{\prime}+1)}$ for some $c_{0}$
(e.g., $c_{0}=0.5$), where $r^{\prime}$ is the order of kernel $J$.

To further automate and robustify the whole procedure with data-driven
bandwidth $\widehat{\delta}$, we provide an alternative variance estimator for
the score function which does not require an additional bandwidth parameter.
Given $\widehat{\delta}$, the numerator of the score statistic
$\widehat{U}_{n}$ can be written as

$\displaystyle\widehat{S}_{\widehat{\delta}}(0,\widehat{\bm{\gamma}})-\widehat{\delta}^{\ell}\widehat{\mu}=$
$\displaystyle\widehat{\bm{v}}^{T}\nabla
R_{\widehat{\delta}}^{n}(\widehat{\bm{\beta}}_{0})-\widehat{\delta}^{\ell}\gamma_{K,\ell}\widehat{\bm{v}}^{T}\widehat{T}_{h,U}^{\ell}(\widehat{\bm{\beta}})=D_{\widehat{\delta},h}^{n}(\widehat{\bm{\beta}},\widehat{\bm{v}}),$
(D.1)

where

$D_{\delta,h}^{n}(\bm{\beta},\bm{v})=\frac{1}{n}\sum_{i=1}^{n}w(y_{i})y_{i}\frac{\bm{v}^{T}\bm{z}_{i}}{\delta}M(x_{i},\bm{z}_{i};\bm{\beta}),$

and
$M(x_{i},\bm{z}_{i};\bm{\beta})=K\big{(}\frac{{\bm{\beta}}_{0}^{T}\bm{z}_{i}-x_{i}}{h}\big{)}-\frac{\delta^{\ell+1}}{h^{\ell+1}}U^{(\ell)}\big{(}\frac{{\bm{\beta}}^{T}\bm{z}_{i}-x_{i}}{h}\big{)}$.
To this end, we may alternatively seek to estimate the variance

$\displaystyle\operatorname{{\rm
Var}}[D_{\delta,h}^{n}(\bm{\beta},\bm{v})]=\frac{1}{n}\bigg{\\{}\mathbb{E}\Big{[}(w(Y)Y\frac{\bm{v}^{T}\bm{Z}}{\delta}M(X,\bm{Z};\bm{\beta}))^{2}\Big{]}-\mathbb{E}\Big{[}w(Y)Y\frac{\bm{v}^{T}\bm{Z}}{\delta}M(X,\bm{Z};\bm{\beta})\Big{]}^{2}\bigg{\\}}$
(D.2)

by replacing the expectations above with sample averages under
$\widehat{\delta},\widehat{\bm{\beta}}$ and $\widehat{\bm{v}}$. We can see
that this variance estimator does not require additional bandwidth parameters
as in the pilot estimator (2.13) and thus makes the procedure more convenient
in practice. In addition, we find this estimator empirically robust when the
bandwidth parameter $\delta$ is selected adaptively using the data.

##  Appendix E Additional Numerical Results

###  E.1 Computational details of the path-following algorithm

In Section 6, we apply the path-following algorithm proposed in Feng et al.
(2019) to estimate the true coefficient $\bm{\beta}$. We fix the number of
stages $N=25$, $\nu=0.25$, $\eta=0.25$ ,$\epsilon_{tgt}=0.0001$ and set
$\Omega=\\{\bm{\beta}:||\bm{\beta}||_{2}\leq 10^{3}\\}$ for each parameter in
Algorithm 2.1 and 2.2 in Feng et al. (2019).

###  E.2 Normal Q-Q plots in Section 6.1

Figure E.1 contains the Normal Q-Q plots for the test statistics obtained from
the experiments in Section 6.1 under both scenarios. We can see that the Q-Q
plots from the proposed SDS method appear close to Gaussian, while the Q-Q
plots for DS and Honest method deviate from standard Gaussian.

###  E.3 Empirical power in Section 6.1

Figure E.2 shows the empirical power under the Heteroskedastic Uniform
scenario in Section 6.1. Similar to the figures for the Heteroskedastic
Gaussian scenario, the empirical power converges to 1 as the magnitude of
$\beta^{*}$ becomes larger. In addition, as the correlation between $\bm{Z}$
becomes larger, the power also decreases.

###  E.4 Bandwidth selection in Section 6.2

Finally, we investigate whether the data-driven bandwidth is close to the
theoretically optimal bandwidth that minimizes the MSE. We focus on the
Heteroskedastic Gaussian scenario and consider the setting with
$n=800,d=50,s=10,\rho=0.2$ and set
$\beta^{*}_{1}=\dotso=\beta^{*}_{10}=1/\sqrt{10}$. We compare $M(\delta)$ with
$\widehat{M}(\delta)$, where $\widehat{M}(\delta)$ is obtained using the
implementation discussed in Section D, but with different bandwidth
$b=0.15,0.2,0.25,0.3$ for squared bias estimation. Notice that since directly
obtaining $M(\delta)$ is hard, we apply Monte Carlo integration to approximate
this quantity.

Figure E.3 shows the true $M(\delta)$ as well as $\widehat{M}(\delta)$ in each
case. The red line is the true MSE $M(\delta)$ approximated by Monte Carlo
method, and the black solid line is the average estimated MSE
$\widehat{M}(\delta)$ with different bandwidths $b$ over 100 repetitions. The
black dashed line is the one standard deviation band around
$\widehat{M}(\delta)$ over these repetitions. As we can see, for all cases,
$M(\delta)$ lies within the one standard deviation band around
$\widehat{M}(\delta)$ and the minimizer of $M(\delta)$ is very close to the
minimizer of the average $\widehat{M}(\delta)$. This experiment confirms that
the data-driven bandwidth $\widehat{\delta}$ is close to the optimal
$\delta^{*}$, provided that the bandwidth $b$ for squared-bias estimation lies
in a suitable range.

###  E.5 Additional results from ChAMP analysis

Table E.1 lists the five most significant variables using the DS and Honest
approach. Notice that the significant variables identified by the proposed
approach are also in the list for the DS approach. Honest yields very
different set of variables compared to the other two.

Figure E.1: Gaussian Q-Q plot of the test statistics under the setting
$d=1000,s=10,\rho=0.5$ from: (a, b) SDS method, (c, d) DS method, and (e, f)
Honest method.

Figure E.2: Empirical rejection rate of the proposed test under both scenarios
with $s=3$, $d=100,500,1000$ and $\rho=0.2,0.5,0.7$.

Figure E.3: The true $M(\delta)$ approximated by Monte Carlo Method (red solid line) and the average estimated $\widehat{M}(\delta)$ (black solid line) with bandwidth $b=0.15,0.20,0.25,0.30$ for squared-bias estimation over 100 repetitions. The black dashed line is one standard deviation bands around $\widehat{M}(\delta)$. Table E.1: Five most significant variables using DS and Honest approach, sorted from most significant (left) to less significant (right), with their corresponding p-values  Method |  |  | Significant variables |  |
---|---|---|---|---|---
DS | MMenB | X(WOMAC Pain Score) | WFunc_6mo | KQOL_6wk | KSymp_3mo
| 9.754e-05 | 3.402e-03 | 1.649e-02 | 1.685e-02 | 2.136e-02
Honest | ActualGrp | MMenTear | TrochDam | TrochCenLes | MMenMgt
|  $<$1e-16 |  $<$1e-16 |  $<$1e-16 |  $<$1e-16 |  $<$1e-16


# CramNet: Layer-wise Deep Neural Network
Compression with Knowledge Transfer
from a Teacher Network

Jon Hoffman

figure0em3.5em table0em3.5em

T H E U N I V E R S I T Y O F T U L S A

THE GRADUATE SCHOOL

CRAMNET: LAYER-WISE DEEP NEURAL NETWORK

COMPRESSION WITH KNOWLEDGE TRANSFER

FROM A TEACHER NETWORK

by

Jon Hoffman


# A clock stabilization system for CHIME/FRB Outriggers

J. Mena-Parra MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA
02139, USA  C. Leung MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA
02139, USA  Department of Physics, Massachusetts Institute of Technology, 77
Massachusetts Ave, Cambridge, MA 02139, USA  S. Cary Department of Astronomy,
Wellesley College, 106 Central Street, Wellesley, MA 02481, USA  MIT Kavli
Institute for Astrophysics and Space Research, Massachusetts Institute of
Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA  K. W. Masui MIT
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute
of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA  Department of
Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave,
Cambridge, MA 02139, USA  J. F. Kaczmarek National Research Council Canada,
Herzberg Astronomy and Astrophysics Research Centre, Dominion Radio
Astrophysical Observatory, PO Box 248, Penticton, British Columbia, V2A 6J9
Canada  M. Amiri Department of Physics and Astronomy, University of British
Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1 Canada  K. Bandura
Lane Department of Computer Science and Electrical Engineering, 1220 Evansdale
Drive, PO Box 6109, Morgantown, WV 26506, USA  Center for Gravitational Waves
and Cosmology, West Virginia University, Chestnut Ridge Research Building,
Morgantown, WV 26505, USA  P. J. Boyle Department of Physics, McGill
University, 3600 rue University, Montréal, QC H3A 2T8, Canada  McGill Space
Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7,
Canada  T. Cassanelli Dunlap Institute for Astronomy & Astrophysics,
University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada
David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto,
50 St. George Street, Toronto, ON M5S 3H4, Canada  J.-F. Cliche Department of
Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
M. Dobbs Department of Physics, McGill University, 3600 rue University,
Montréal, QC H3A 2T8, Canada  McGill Space Institute, McGill University, 3550
rue University, Montréal, QC H3A 2A7, Canada  V. M. Kaspi Department of
Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada
McGill Space Institute, McGill University, 3550 rue University, Montréal, QC
H3A 2A7, Canada  T. L. Landecker Dominion Radio Astrophysical Observatory,
Herzberg Research Centre for Astronomy and Astrophysics, National Research
Council Canada, PO Box 248, Penticton, BC V2A 6J9, Canada  A. Lanman
Department of Physics, McGill University, 3600 rue University, Montréal, QC
H3A 2T8, Canada  McGill Space Institute, McGill University, 3550 rue
University, Montréal, QC H3A 2A7, Canada  J. L. Sievers McGill Space
Institute, McGill University, 3550 rue University, Montréal, QC H3A 2A7,
Canada  Perimeter Institute for Theoretical Physics, 31 Caroline Street N,
Waterloo, ON N25 2YL, Canada  School of Chemistry and Physics, University of
KwaZulu-Natal, 238 Mazisi Kunene Rd, Glenwood, Durban, 4041, South Africa

###### Abstract

The Canadian Hydrogen Intensity Mapping Experiment (CHIME) has emerged as the
prime telescope for detecting fast radio bursts (FRBs). CHIME/FRB Outriggers
will be a dedicated very-long-baseline interferometry (VLBI) instrument
consisting of outrigger telescopes at continental baselines working with CHIME
and its specialized real-time transient-search backend (CHIME/FRB) to detect
and localize FRBs with 50 mas precision. In this paper we present a minimally
invasive clock stabilization system that effectively transfers the CHIME
digital backend reference clock from its original GPS-disciplined ovenized
crystal oscillator to a passive hydrogen maser. This enables us to combine the
long-term stability and absolute time tagging of the GPS clock with the short
and intermediate-term stability of the maser to reduce the clock timing errors
between VLBI calibration observations. We validate the system with VLBI-style
observations of Cygnus A over a 400 m baseline between CHIME and the CHIME
Pathfinder, demonstrating agreement between sky-based and maser-based timing
measurements at the $30$ ps rms level on timescales ranging from one minute to
up to nine days, and meeting the stability requirements for CHIME/FRB
Outriggers. In addition, we present an alternate reference clock solution for
outrigger stations which lack the infrastructure to support a passive hydrogen
maser.

Radio astronomy(1338), Radio transient sources (2008), Radio pulsars (1353),
Astronomical instrumentation(799), Very long baseline interferometry (1769)

††journal: AJ††facilities: CHIME††software: AllanTools (Wallin et al., 2018)

##  1 Introduction

Fast radio bursts (FRBs, Lorimer et al., 2007) are transient pulses of radio
light observed out to cosmological distances; both their origins and emission
mechanisms remain unclear. Even though thousands of FRB events occur over the
full sky every day (The CHIME/FRB Collaboration et al., 2021; Bhandari et al.,
2018), their detection with traditional radio telescopes is challenging due to
the randomly-occurring nature of the majority of bursts.

With its unique design optimized for rapid wide-field observations and a
powerful real-time transient-search engine (CHIME/FRB, CHIME/FRB Collaboration
et al., 2018), the Canadian Hydrogen Intensity Mapping Experiment
(CHIME111https://chime-experiment.ca, CHIME Scientific Collaboration et al.,
2021, in prep.) has become the leading facility for detection of FRBs,
detecting over 500 FRBs (The CHIME/FRB Collaboration et al., 2021) and 18 new
repeating sources (CHIME/FRB Collaboration et al., 2019a, b; Fonseca et al.,
2020) in its first year of full operations. Such an unprecedented sample of
events with a single survey has enabled detailed studies of statistical
properties of the FRB population such as fluence distribution and sky rate,
scattering time, dispersion measure (DM) distribution, spatial distribution,
burst morphology, and correlations with large-scale structure (The CHIME/FRB
Collaboration et al., 2021; Pleunis et al., 2021; Josephy et al., 2021;
Rafiei-Ravandi et al., 2021; Chawla et al., 2021).

However, except for FRBs with low dispersion measure (Michilli et al., 2020;
Bhardwaj et al., 2021a, b), CHIME/FRB’s arcminute localization precision is
insufficient to localize these bursts to their host galaxies, which is crucial
to understand their nature and unlock their potential as probes of the
intergalactic medium and large-scale structure. To overcome this limitation,
the CHIME/FRB collaboration is currently developing CHIME/FRB Outriggers, a
program to deploy CHIME-like outrigger telescopes at continental baseline
distances. CHIME and the outriggers will form a dedicated very-long-baseline
interferometry (VLBI) network capable of detecting hundreds of FRBs each year
with sub-arcsecond localization precision in near real-time, allowing for the
unique identification of FRB galaxy hosts and source environments.

Because VLBI localizes sources by precisely measuring the difference in the
arrival time of astronomical signals between independent telescopes across
far-separated sites, it is critical to use very stable local reference signals
(i.e., clocks) that allow the synchronization of VLBI stations without losing
coherence during observations and between calibrations. This is particularly
important for stationary telescopes like CHIME and the outrigger stations that
can only be calibrated when a bright radio source transits through their field
of view. The superior stability performance of hydrogen masers on short and
intermediate timescales makes them the preferred option for VLBI applications
(Matthews et al., 2018; Event Horizon Telescope Collaboration et al., 2019;
Schmittberger & Scherer, 2020). Here, we present a hardware and software clock
stabilization solution for the CHIME telescope that effectively transfers the
reference clock from its original GPS-disciplined crystal oscillator to a
passive hydrogen maser during VLBI observations, meeting the timing
requirements for FRB VLBI with CHIME/FRB Outriggers. Furthermore, this system
can be implemented without interrupting CHIME’s current observational campaign
and without modifications to the correlator or the data-analysis pipelines for
cosmology and radio transient science.

The paper is organized as follows: Section 2 describes the features of the
CHIME instrument that are relevant to its use as a VLBI station in CHIME/FRB
Outriggers. Section 3 discusses the CHIME/FRB Outriggers clock stability
requirements for FRB VLBI. Section 4 describes the hardware and software of
the stabilization system that transfers CHIME’s reference clock to a passive
hydrogen maser. Section 5 shows the results of the suite of tests that
validate the clock stabilization system with VLBI-style observations between
CHIME and the CHIME Pathfinder (an early small-scale prototype of CHIME
recently outfitted as an outrigger test-bed, Bandura et al., 2014; Leung et
al., 2021). Section 6 presents an alternate clock solution for outrigger
stations that do not have the infrastructure to support a hydrogen maser.
Section 7 presents the conclusions.

##  2 Instrument overview

A detailed description of the CHIME instrument and the CHIME/FRB project is
presented in CHIME Scientific Collaboration et al. (2021, in prep.) and
CHIME/FRB Collaboration et al. (2018). In this section we give a brief
introduction to these systems focused on the features that are relevant for
FRB VLBI. We also give an overview of CHIME/FRB Outriggers.

###  2.1 CHIME and CHIME/FRB

CHIME is a hybrid cylindrical transit interferometer located at the Dominion
Radio Astrophysical Observatory (DRAO) near Penticton, B.C., Canada. It
consists of four 20 m $\times$ 100 m cylindrical reflectors oriented north-
south and instrumented with a total of 1024 dual-polarization feeds and low-
noise receivers operating in the 400-800 MHz band. The cylinders are fixed
with no moving parts, so CHIME operates as a drift-scan instrument that
surveys the northern half of the sky every day with an instantaneous field of
view of $\sim 120^{\circ}$ north-south by $2.5^{\circ}-1.3^{\circ}$ east-west.

Although CHIME’s design was driven by its primary scientific goal to probe the
nature of dark energy by mapping the large-scale structure of neutral hydrogen
in the universe across the redshift range $0.8\leq z\leq 2.5$, its combination
of high sensitivity and large field of view also make it an excellent
instrument to study the radio transient sky. Thus, in its final stages of
commissioning, the CHIME correlator was upgraded with additional hardware and
software backends to perform additional real-time data processing operations
for pulsar timing and FRB science.

The correlator (Bandura et al., 2016a, b; Denman et al., 2020) is an FX design
(temporal Fourier transform before spatial cross-multiplication of data),
where the F-engine digitizes the 2048 analog inputs at 800 MSPS and separates
the 400 MHz input bandwidth into 1024 frequency channels with 390 kHz spectral
resolution. The F-engine also implements the corner-turn network that re-
arranges the complex-valued channelized data (also known as “baseband”) before
sending it to the X-engine that computes a variety of data products for the
different real-time scientific backends: interferometric visibilities for the
hydrogen intensity mapping backend (CHIME Scientific Collaboration et al.,
2021, in prep.), dual-polarization tracking voltage beams for the pulsar
monitoring backend (CHIME/Pulsar Collaboration et al., 2020), and high-
frequency resolution power beams for the 21-cm absorption systems backend (Yu
et al., 2014) and for the CHIME/FRB backend that triggers on highly dispersed
radio transients to search for FRBs in real time (CHIME/FRB Collaboration et
al., 2018). Additionally, a $\sim$36 s long memory buffer in the X-engine
stores baseband data (2.56 $\mu$s time resolution, 390 kHz spectral
resolution, and 4-bit real + 4-bit imaginary bit depth for the 2048 correlator
inputs) that can be saved to disk when the CHIME/FRB search pipeline detects
an FRB candidate, enabling polarization and high-time resolution analysis of
FRB events, as well as sub-arcminute localization precision (Michilli et al.,
2020). Eventually it will also enable VLBI localization with CHIME/FRB
Outriggers.

###  2.2 CHIME/FRB Outriggers

The scientific goal of CHIME/FRB Outriggers is to provide 50 mas localization
for nearly all CHIME detected FRBs with sub-hour latency. This angular
resolution is sufficient to determine galaxy hosts and source environments,
and is well matched to current best optical follow-up observations. To this
end, the CHIME/FRB collaboration is currently building outrigger telescopes at
distances ranging from hundreds to several thousands of kilometers from DRAO.
The outriggers will be small-scale versions of CHIME, each with about one
eighth of CHIME’s collecting area, the same field of view, and tilted such
that they monitor the same region of the sky as CHIME.

In contrast to traditional VLBI that is typically performed for known targets
with small fields of view and manageable data rates, the random nature of most
FRBs requires the real-time processing of massive data rates in order to
detect and localize these events in blind searches with wide fields of view.
The baseband data rate of CHIME is 6.6 Tbit/s while that of each outrigger
station will be 0.8 Tbit/s. Since such high data rates cannot be continuously
saved, the outriggers will adopt the triggered FRB VLBI approach demonstrated
in Leung et al. (2021), where each station buffers its local baseband data in
memory and only writes it to disk upon receipt of a trigger from the CHIME/FRB
real-time search pipeline over internet links. The local data of each station
is then transmitted to a central facility where the signals are correlated
together such that the outriggers operate with CHIME as an interferometric
instrument with the angular resolution of a telescope with an aperture of
thousands of kilometers.

##  3 Clock stability requirements

Accurate timing is critical for VLBI since the localization of radio sources
is ultimately derived from the relative time of arrival of signals at the
telescope stations. By synthesizing the available frequency channels it is
possible to obtain a statistical precision on the measured delay given by
(Rogers, 1970)

$\sigma_{\tau}^{\text{stat}}=\frac{1}{2\pi\cdot\text{SNR}\cdot\text{BW}_{\text{eff}}}$
(1)

where SNR is the signal-to-noise ratio of the VLBI event and
$\text{BW}_{\text{eff}}$ is the effective bandwidth. For the CHIME/FRB
detection threshold222The SNR in VLBI is related to the SNR at CHIME as
$\text{SNR}/\text{SNR}_{\text{CH}}=\sqrt{2A_{\text{O}}/A_{\text{CH}}}=1/2$
where $A_{\text{CH}}$ and $A_{\text{O}}$ are the collecting areas of CHIME and
the outrigger respectively, and the factor of $\sqrt{2}$ comes from the
difference in the detailed noise statistics of a cross-correlation compared to
an auto-correlation (Masui et al., 2015). While the CHIME/FRB real-time
detection pipeline has a detection threshold of $\sim 10$ (Michilli et al.,
2020), the SNR rises by $\sim 50\%$ through the more detailed analysis of the
saved baseband data. As such, we take the floor on the CHIME detection SNR to
be $\text{SNR}_{\text{CH}}=15$. and bandwidth (BW) this corresponds to

$\sigma_{\tau}^{\text{stat}}\approx\frac{184\text{
ps}}{\displaystyle\left(\frac{\text{SNR}}{7.5}\right)\left(\frac{\text{BW}}{400\text{
MHz}}\right)}.$ (2)

For a VLBI baseline $b$, a delay precision $\sigma_{\tau}$ corresponds to a
statistical localization uncertainty

$\sigma_{\theta}\approx\frac{c}{b}\sigma_{\tau}$ (3)

which gives $\sigma_{\theta}^{\text{stat}}\lesssim 11$ mas for a 1000 km
baseline. However, the (relative) delay measured by the interferometer
includes not only the geometric delay (which ultimately provides the source
localization) but also additional contributions that need to be accounted for
such as propagation though the troposphere and ionosphere, baseline errors,
drift between clocks of different stations (clock timing errors), and other
instrumental delays. In practice, the localization uncertainty of CHIME/FRB
Outriggers will be limited by systematic errors due to uncompensated delay
contributions, and particularly by errors in the determination of the
dispersive delay due to the ionosphere. Although the large observation
bandwidth of the instrument helps to mitigate this effect, it still represents
the most important challenge for system stability at CHIME frequencies.

Our simulations indicate that we can reliably localize FRB events to 50 mas
which, for $b=1000$ km, corresponds to a delay error budget of
$\sigma_{\tau}\approx 800$ ps. Anticipating that the ionosphere will be the
main contributor to delay errors, the clock timing error specification333This
is not a hard upper limit, but rather a reasonable reference value that
represents our goal to keep the clock timing errors well below the 800 ps
total timing error budget. has been set to $\sigma_{\tau}^{\text{clk}}\lesssim
200$ ps.

Note that for blind FRB searches, this specification must be met at all times.
Indeed, it may not always be possible to find a calibrator immediately after
the detection of an FRB for phase referencing. An additional complication is
that stationary telescopes like CHIME and the outriggers observe the sky as it
transits through their field of view and thus cannot slew towards favorable
calibrators. Therefore, it is especially important to have a reference clock
that is reliable on the timescales required to connect an FRB detection to a
calibrator observation, potentially hours later.

Although the list of steady radio sources potentially suitable for calibrating
low-frequency VLBI arrays with $\gtrsim 1000$ km baselines has significantly
increased thanks to the ongoing LOw Frequency ARray (LOFAR) Long-Baseline
Calibrator Survey (LBCS, Moldón et al., 2015; Jackson et al., 2016), during
its initial stages CHIME/FRB Outriggers will adopt a more conservative
strategy relying mainly on bright pulsars for calibration. Pulsars are
compact, can be separated from the steady radio background in the time domain,
and are sufficiently abundant to be used as the primary sources for phase
referencing. Accordingly, the backend of each outrigger will also have the
ability to form tracking baseband beams for pulsar analysis and calibration.
Recently, Cassanelli et al. (2021) demonstrated the potential of pulsars as
calibrators for CHIME/FRB Outriggers through the triggered VLBI detection of
an FRB over a $\sim 3000$ km baseline between CHIME and the Algonquin Radio
Observatory (ARO) 10-m Telescope using PSR B0531+21 in the Crab nebula for
phase referencing.

We estimate that an FRB detection can be phase referenced to an
$\text{SNR}\gtrsim 15$ pulsar444This represents the VLBI SNR after coherent
addition of the pulses within a single pulsar transit. within less than $\sim
10^{3}$ s. This timescale and the relative clock timing error specification
set the clock stability (Allan deviation) requirement to

$\sigma_{y}(10^{3}~{}\text{s})\lesssim 2\cdot 10^{-13},$ (4)

As explained in Appendix A, the Allan deviation is a measure of stability
commonly used in precision clocks and oscillators, with $\Delta
t\cdot\sigma_{y}(\Delta t)$ roughly representing the rms of clock timing
errors a time $\Delta t$ after calibration. The specification in Equation 4 is
typically obtained with hydrogen masers. However, in Section 6 we show that
even frequency references that initially do not meet this requirement can be
used as reference clocks for the outriggers by interpolating timing solutions
between calibrators.

##  4 clock stabilization system

In this section we discuss the considerations that led to the current clock
stabilization solution for CHIME, as well as the hardware and data analysis
for the maser signal.

###  4.1 Hardware/Software considerations

The CHIME F-engine is implemented using the ICE hardware, firmware and
software framework (Bandura et al., 2016a). It consists of field programmable
gate array (FPGA)-based motherboards specialized to perform the data
acquisition and channelization of the 2048 CHIME analog inputs. The ICE
motherboards are packaged in eight crates with custom backplanes that
implement the networking engine that re-organizes and sends the baseband data
to a dedicated graphics processing unit (GPU) cluster that performs the
X-engine operations. The outriggers will also use an ICE-based F-engine.

The data acquisition and signal processing of the F-engine are driven by a
single 10 MHz clock signal provided by a Spectrum Instruments TM-4D global
positioning system (GPS)-disciplined ovenized crystal oscillator. The GPS
module also generates an inter-range instrumentation group (IRIG)-B timecode
signal internally synchronized to the clock and that is used by the correlator
to time stamp the data. A low-jitter distribution system sends the clock and
time signals to each ICE backplane and motherboard and is ultimately used to
generate the analog-to-digital converter (ADC) sampling clocks. The time
stamping process is implicit: The F-engine uses the IRIG-B signal to
synchronize the start of the data acquisition to an integer second (up to the
10 ns resolution of the IRIG-B decoder in the FPGA firmware) and it also tags
each data frame with a frame counter value. The X-engine time stamps the data
by calculating the offset from the start time based on the frame counter value
and assuming a fixed 2.56 $\mu$s baseband sampling time.

Figure 1 shows the Allan deviation of the CHIME GPS clock (blue line) as
measured with the clock stabilization system described in Sections 4.2 and
4.3. The GPS disciplined crystal oscillator, being locked to a GPS time
reference determined by a vast network of atomic clocks, will eventually
surpass the stability performance of a single hydrogen maser on very long
timescales ($\Delta t\gtrsim 10^{6}$ s). On intermediate and long timescales
($\Delta t\sim 10^{3}-10^{5}$ s), including the ones of interest for CHIME/FRB
Outriggers, the CHIME clock stability is dominated by white delay noise
($\sigma_{y}(\Delta t)\propto 1/\Delta t$) corresponding to $\sim 6$ ns root
mean square (rms) timing errors. While the coherence of this frequency
standard is sufficient for CHIME’s operations as a connected interferometer
and for all its backends, the high precision needed for FRB VLBI requires the
development of a more stable clock system.

Figure 1: Allan deviation of the CHIME GPS clock and the DRAO hydrogen maser.
Blue: Allan deviation of the CHIME GPS clock as measured with the clock
stabilization system described in Section 4. A total of ten days of raw ADC
data at 30 s cadence was collected for the measurement. Dashed blue: expected
measurement error contribution to the Allan deviation obtained from
simulations of uncorrelated but time-dependent errors in the range $\sim 4-20$
ps rms (the range observed in the measured delays). Dashed green: stability
requirement from Equation 4 assuming white noise delay errors. Red:
manufacturer-specified Allan deviation of the DRAO maser. The CHIME GPS clock
does not meet the stability requirements for FRB VLBI, but the DRAO maser does
(Equation 4).

As a continuously tracking global navigation satellite system (GNSS) station
and as part of the Western Canada Deformation Array (WCDA, Chen et al., 1996)
and the Canadian Active Control System
(CACS555http://cgrsc.ca/resources/geodetic-control-networks/canadian-active-
control-system-cacs, Duval et al., 1996), DRAO is equipped with an atomic
frequency standard consisting of a T4Science pH Maser 1008 passive hydrogen
maser owned and operated by Natural Resources Canada (NRCan). The maser is
installed in a seismic vault at the DRAO site and has a primary output of 5
MHz (sine wave). It is directly connected to a low noise distribution
amplifier in the same rack that serves as electrical isolation for the maser
and also derives multiple copies of the 10 MHz reference signal (sine wave).
NRCan has approved the use of two of those signals for CHIME-related
operations.

The manufacturer-specified Allan deviation of the DRAO maser is shown in
Figure 1 (red points). The maser clearly exceeds the stability requirements
for FRB VLBI with CHIME (Equation 4). Some of the outrigger sites will also
have access to hydrogen maser frequency references with similar performance.

Although in principle the ICE system can be operated with an IRIG-B time
signal that is not phase-locked to the 10 MHz reference clock (such as an
independent maser), an important restriction to using the maser as the master
clock for the CHIME correlator is the fact that both the F-engine and X-engine
software and the scientific data analysis pipelines were developed on the
premise that the clock and IRIG-B signals are synced (e.g., to time stamp the
data), something that cannot be guaranteed if the two signals are generated by
independent systems (the maser and the GPS receiver respectively). Even if the
relative drift between clock and time signals could be tracked, both the
correlator and data analysis pipelines would need to be updated to implement
this change. As such, we had to develop a clock stabilization system that did
not impact the normal operations of CHIME, its existing real-time backends,
and the other scientific teams.

###  4.2 Maser signal conditioning

The clock stabilization system designed for FRB VLBI with CHIME keeps the
current GPS-disciplined crystal oscillator as the master clock and instead
feeds the maser signal to one of the ICE ADC daughter boards so it is
digitized by the crystal-oscillator-driven F-engine. The data is processed to
monitor the variations in the phase of the sampled maser signal which
correspond to variations in the relative delay between the maser and the
master clock. By using this information to correct phase variations in the
baseband data recorded at the time of an FRB detection, the system effectively
transfers the reference clock from the GPS disciplined oscillator to the more
stable maser signal during FRB observations. As shown in Figure 1 (dashed blue
line), the noise penalty associated with the clock transfer operation is
essentially white ($\sigma_{y}(\Delta t)\propto 1/\Delta t$) on the timescales
relevant for FRB VLBI and small ($\lesssim 20$ ps) compared to the total clock
timing error budget.

A block diagram of the maser signal path is shown in Figure 2. The first point
of access to the 10 MHz maser signal is the low noise distribution amplifier
within the seismic vault. From there, the signal is transported through $\sim
500$ m of buried coaxial cable to one of the two radio-frequency (RF)-shielded
huts that house the CHIME F-engine. We use the same type (LMR-400) of low-loss
coaxial cable used in the CHIME analog receivers and whose thermal
susceptibility has been extensively tested in the field. At the RF hut, the
cable interfaces with a ground block and the signal is then carried inside the
RF room using standard SMA cables where it is connected to an isolation
transformer to refer the next stages to the F-engine crate ground. One
complication in the digitization of the maser signal is that the ADC daughter
boards that specialize the ICE system for CHIME have a bandpass transfer
function that strongly attenuates signals below $\sim 100$ MHz. For this
reason, instead of feeding the maser signal directly to an ADC daughter board,
the signal is used to drive a low-noise sine-to-square wave signal translator
that generates 10 MHz harmonics well into the CHIME band. The output of the
translator is then filtered to the CHIME band using the same band-defining
filter amplifier (FLA) used in the CHIME receivers (Bandura et al., 2014;
CHIME Scientific Collaboration et al., 2021, in prep.). Finally, the FLA is
connected directly to one of the correlator inputs where it is digitized at
800 MSPS with an 8-bit ADC.

Figure 2: Maser signal path. The 10 MHz maser signal is transported through
$\sim 500$ m of buried coaxial cable from the seismic vault to one of the
CHIME F-engine RF huts. There, the maser signal is conditioned to a waveform
that can be digitized by the CHIME F-engine (see Section 4.2 for details).

###  4.3 Clock stabilization pipeline

The FPGA within each ICE motherboard processes the data from its digitizers
using the custom CHIME F-engine firmware (for details, see Bandura et al.,
2016a, b). Briefly, the raw ADC data from each input is passed to the
frequency channelizer module as frames of 2048 8-bit samples. The channelizer
forms the baseband data by splitting the 400 MHz input bandwidth into 1024
frequency channels, each truncated to a 4-bit real + 4-bit imaginary complex
number. Additionally, a probe sub-module within the channelizer can be
configured to periodically capture a subset of the raw ADC data that is
separately saved and typically used in CHIME for system monitoring.

By default, the CHIME F-engine software pipeline saves one raw ADC frame (2.56
$\mu$s of data) from each input every 30 s, but this cadence can be modified
before starting a data acquisition. The clock stabilization pipeline extracts
the raw ADC frames from the maser input, Fourier transforms each frame via a
Fast Fourier Transform (FFT), and separates the frequency channels
corresponding to the harmonics of the 10 MHz signal in the CHIME band. The
quality of each harmonic is assessed based on its signal-to-quantization-noise
ratio (SQNR) and its susceptibility to spurious aliased harmonics (relevant
for harmonics near the edges of the CHIME band). Low quality harmonics are
discarded.

Since the ADC that digitizes the maser signal uses the GPS clock as the
reference for sampling, the variations in the delay of the sampled maser
signal represent the delay variations of the GPS clock with respect to the
maser, the latter of which is more stable on short and intermediate
timescales. These delay variations $\Delta\tau(t)$ will induce phase
variations $\Delta\phi(t,\nu)$ in the maser harmonics of the form

$\Delta\phi(t,\nu)=2\pi\nu\Delta\tau(t)$ (5)

where $\nu$ is the harmonic frequency.

Since we are interested in the GPS clock delay variations relative to the
delay at the time of VLBI calibration, the phase of the maser harmonics is
initially referenced to the phase of a frame close to calibration time. Then
for each frame, a line described by Equation 5 is fit to the phase as a
function of harmonic frequency to recover the GPS clock delay (relative to the
maser) as a function of time. The dominant component of the recovered delay
$\Delta\tau(t)$ is a slow linear drift as function of time ($\sim 50$ ns/day)
corresponding to a constant offset of the maser frequency from 10 MHz. This
linear trend is removed from $\Delta\tau(t)$ since it is due to the maser
frequency calibration and not due to the instability of the GPS clock.

The captured raw ADC frames are only a small fraction of the available CHIME
data; thus, the times at which GPS clock delay measurements are available are
not necessarily aligned with the times of a calibration observation or an FRB
detection. This means that the GPS clock delay timestream must be interpolated
in order to find the clock contribution to the total delay measured in a VLBI
observation. We use linear interpolation to find the GPS clock delay at any
arbitrary time, a method that is motivated by the short timescale behavior of
the clock delay variations. Figure 3 shows a few examples of the behavior of
the GPS clock delay on timescales of a few seconds as measured by the clock
stabilization system. On these timescales the timing variations are dominated
by the tuning jitter generated by the algorithm that disciplines the crystal
oscillator. In essence, the algorithm works by counting the number of clock
cycles between successive GPS receiver pulse-per-second (PPS) pulses and
adjusting the crystal’s temperature to ensure ten million counts between
pulses. The size of the temperature tuning steps is progressively reduced as
the crystal oscillator frequency approaches 10 MHz. As shown in Figure 3, this
discipline procedure gives a characteristic triangle-wave shape to the tuning
jitter, and although in a perfectly-tuned oscillator the transitions should
occur every second, in practice we observe that they can take longer. Thus, as
long as the GPS clock delay is sampled at cadences below $\sim 500$ ms we can
track the tuning jitter features and a linear interpolation provides a good
approximation to the true delay at any time.

Figure 3: Three examples of the behavior of the GPS clock delay on timescales
of a few seconds as measured by the clock stabilization system with respect to
the DRAO maser. The raw ADC data cadence for this measurements was 40 ms. The
measurement errors are in the range $\sim 2-13$ ps. The characteristic
triangle wave pattern is due to the algorithm that disciplines the crystal
oscillator in the GPS unit. The algorithm works by counting the number of
clock cycles between successive GPS PPS pulses and adjusting the crystal’s
temperature to ensure ten million counts between pulses. The size of the
temperature tuning steps changes depending on the tuning history of the
oscillator.

As the current version of the F-engine control software only allows saving raw
ADC data for all the correlator inputs at the time, raw ADC data at cadences
below 10 s cannot be saved during normal telescope operations and are
restricted to operations during times scheduled for hardware maintenance and
software upgrades. A modification to the F-engine control software is ongoing
to allow saving fast-cadence raw ADC data for the maser input while keeping
the default cadence for the remaining correlator inputs, a change that does
not impact the the normal operations of the correlator and the data-analysis
pipelines.

It is also possible to process baseband data directly to extract the maser
signal and measure the GPS clock delay variations. The operation is very
similar to that of raw ADC data, except that the maser data has already been
transformed to the frequency domain by the F-engine. Since in this case most
maser harmonics do not lie exactly in the center of a frequency channel, the
pipeline selects the closest F-engine frequency channel. Then for each
selected channel, it performs an additional channelization by using an FFT
along the time domain to isolate the harmonic frequency.

Although working with baseband data is logistically convenient in cases where
we need to test the performance of the clock stabilization system (see Section
5), during regular operations the current system is designed to work mostly
with raw ADC data. This is mainly because a baseband dump for an FRB event is
typically collected in $\sim 100$ ms segments at different times for each
frequency channel in order to account for the dispersion delay of the
transient, with a total event duration lasting tens of seconds (Michilli et
al., 2020). This leaves only a few megahertz of bandwidth available at any
particular instant, making the monitoring of clock delay variations more
challenging. Furthermore, when using baseband dumps we still need to rely on
the continuously saved raw ADC data to track and correct the long-timescale
linear drift of the maser.

##  5 Validation of the clock stabilization system

We tested the reliability of clock stabilization system by installing it in
the Pathfinder telescope and comparing maser-based measurements of the CHIME-
Pathfinder relative clock drift to independent measurements obtained from
VLBI-style observations of steady radio sources.

###  5.1 The Pathfinder as an outrigger

The Pathfinder is presented in detail in Bandura et al. (2014). It is a small-
scale prototype of CHIME with identical design and field of view, and it has
the same collecting area of the outriggers under construction. The telescope
is located $\sim 400$ m from CHIME and was constructed before CHIME as a test-
bed for technology development. With the same correlator architecture as
CHIME, the Pathfinder operates as an independent connected interferometer with
its own GPS-disciplined clock. Recently, Leung et al. (2021) repurposed the
Pathfinder as an outrigger to demonstrate the feasibility of triggered FRB
VLBI for CHIME/FRB Outriggers. The Pathfinder correlator is now equipped with
a custom baseband data recorder capable of processing one quarter of the CHIME
band and programmed to write its local baseband data to disk upon receipt of a
trigger from CHIME/FRB. We also connected the additional copy of the maser
signal from the seismic vault to one of the Pathfinder correlator inputs using
a signal path identical to that of CHIME and shown in Figure 2 (except for the
transport cable which is longer for the Pathfinder setup).

###  5.2 Comparison to interferometric observations

The clock stabilization system measures the delay variations of the CHIME and
Pathfinder clocks by processing the maser data from each telescope as
described in Section 4. The delays from each clock are then interpolated to
the observation times so the relative clock drift can be tracked over
time666If the maser data comes from simultaneous baseband dumps instead of raw
ADC samples then clock delays from the two telescopes can be directly compared
without interpolation.. An independent way to measure the CHIME-Pathfinder
relative clock delay is to interferometrically track a known point source over
time using both telescopes and their independently-running backends. If we
properly account for all the other contributions to the measured
interferometric delay (geometric, ionosphere, etc.) as the source transits
through the field of view of the two telescopes, then any residual delay
should correspond to the relative drift between the two clocks. If the clock
stabilization system is robust, its measurements should agree very closely
with the interferometric measurement, which we use as a standard.

####  5.2.1 Short timescale test

We used Cygnus-A (henceforth referred to as CygA) for the VLBI-style
observations since it is the brightest radio source seen by CHIME that is
unresolved on a CHIME-Pathfinder baseline. For the first test, we programmed
the CHIME/FRB backend to trigger short baseband dumps simultaneously for CHIME
and the Pathfinder during a single CygA transit. In this way, we collected
seven 10 ms-long baseband dumps, spaced by a minute, while the source was in
the field of view. The observation was carried out in November 2020 during a
day scheduled for instrument maintenance, so we were also able to collect raw
ADC maser data at 200 ms cadence with the two telescopes.

Since the telescopes are co-located, they experience a common ionosphere,
suppressing relative ionospheric fluctuations (we see no evidence for these in
our observations). Thus, the residual delay in the interferometric visibility
after accounting for the geometric contribution gives a measurement of the
relative clock drift. The residual interferometric delays are calculated using
a procedure identical to that described in Leung et al. (2021). In summary,
each telescope is internally calibrated (to measure the directional response
of each antenna in the telescope array) using a separate observation of a
bright point source (CHIME Scientific Collaboration et al., 2021, in prep.).
Then, for each telescope and baseband dump, the data is coherently summed over
antennas to form a phased-array voltage beam towards CygA. The beamformed data
from CHIME and the Pathfinder is then cross-correlated on a frequency-by-
frequency basis to form the complex visibility. The phase of each visibility
is compensated for the geometric delay. The variance of each visibility is
found empirically by splitting each baseband dump into short time segments,
computing the visibility for each segment, and calculating the variance over
segments as function of frequency. The seven visibilities are phase-referenced
to that of the first baseband dump since we are only interested in changes of
the relative clock delay. Finally, the wideband fringe-fitting procedure to
find residual delay performs a least-mean-squares fit of a complex exponential
with linear phase and frequency-dependent amplitude to the measured
visibilities.

Figure 4: Comparison of the CHIME-Pathfinder relative clock delay inferred via
the clock stabilization system and interferometric observations from a single
transit of CygA. Top: relative clock delay (in ns) as function of time as
inferred from CygA baseband data (blue), raw ADC maser data (red) and maser
baseband data (green). Bottom: Difference between sky-based and maser-based
measurements of the relative clock delay (raw ADC maser data in red, baseband
maser data in green), demonstrating agreement between the two methods. The
large error bars for all but the last point in the raw ADC data analysis (red)
are due to current limitations of the Pathfinder raw ADC acquisition system
(see Section 5.2.1 for details). The error bar in the last red point of the
top plot ($\sim 14$ ps) is representative of the expected accuracy of the
clock stabilization system using raw ADC data.

The top panel of Figure 4 shows the resulting comparison between the CHIME-
Pathfinder relative clock delays calculated from interferometric observations
(blue) and those found through the clock stabilization system (from raw ADC
data in red, from baseband data in green). The $1\sigma$ error bars are too
small to be visible in the plot but they are in the range $\sim 17-22$ ps for
CygA measurements, $\sim 14-238$ ps for raw ADC maser data measurements, and
$\sim 18-33$ ps for baseband maser data measurements. Measurements with the
clock stabilization system show excellent agreement with the sky. This is
further highlighted in the bottom panel of Figure 4 that shows the difference
between sky-based and maser-based measurements of the relative clock delay
(raw ADC maser data in red, baseband maser data in green).

The large error bars for all but the last point in the raw ADC data analysis
(red) are dominated by the error in the measurements of the maser delay at the
Pathfinder. These can be traced back to current limitations of the Pathfinder
raw data acquisition system, which occasionally drops packets when we collect
raw ADC data at fast cadence for the reasons explained in Section 4.3. This
limitation will be solved in the next upgrade of the F-engine control
software. For the observation times that fell within sections of missing
Pathfinder raw ADC data (the longest of which was $\sim 80$ s), the delay
values were obtained by performing a smoothing spline interpolation based on
the available measurements. To estimate the uncertainty in the delay values
obtained with this method we analyzed a segment of the delay timestream for
which there were no gaps due to dropped packets, $\sim 10$ min before the sky
observations. The errors were found using a procedure similar to the one used
in Section 6 to evaluate the performance of alternate reference clocks, where
we introduce artificial gaps in the delay timestream and analyze the
statistics of an ensemble of interpolation residuals. Only the raw ADC delays
(red) for the last observation time could be measured using the default
interpolation for both telescopes (see Section 4.3). The uncertainty for this
measurement is $\sim 14$ ps, and is representative of the expected accuracy of
the clock stabilization system using raw ADC data.

####  5.2.2 Long timescale test

To test the performance of the clock stabilization system on long timescales
we collected five CygA baseband dumps, each 10 ms in duration, spaced one
minute apart, for nine days in a row for a total of 45 delay measurements. The
observations were carried out in April 2021 during normal CHIME operations so
we relied on baseband maser data for delay measurements with the clock
stabilization system for the reasons explained in Section 4.3. Both
interferometric and maser-based delays are calculated using the same procedure
as in the short timescale test described in Section 5.2.1, with the
visibilities phase-referenced to one of the observations on the fifth day.

Figure 5: Top: Comparison of the CHIME-Pathfinder relative clock delay
inferred via the clock stabilization system (red) and interferometric
observations (blue) of from multiple transits of CygA. For each transit, we
made five measurements of the relative clock delay, spaced by one minute, for
nine days in a row. Bottom: Difference between sky-based and maser-based
measurements of the relative clock delay. The two methods show excellent
agreement on short (minute) and long (many days) timescales. This indicates
that the clock stabilization system we have implemented can track clock delay
variations with better than $\sim 30$ ps rms level precision.

The results of the long timescale test are shown in Figure 5. The
interferometric measurements agree with the maser-based measurements at the
$\sim 30$ ps rms level, demonstrating that after correction with the clock
stabilization system the resulting reference clock is stable over timescales
of more than a week, and that the signal chain used to inject the maser signal
into the correlator is not a limitation for the system’s performance in
CHIME/FRB Outriggers.

##  6 A reference clock for outrigger stations without a maser

The clock stabilization system allows us to inject external reference clock
signals into radio telescopes which share the CHIME correlator architecture.
In addition to its use in the new outriggers, the system enables reference
clocks to be swapped out at existing telescopes like CHIME and legacy systems
like the Pathfinder without making major changes to the software framework or
existing scientific backends, while expanding the telescopes’ capabilities to
include VLBI.

Most outriggers will also have access to hydrogen maser frequency references
that can be used in the same way as CHIME (Section 4) to meet the stability
requirements for FRB VLBI. However, we still need to address the possibility
that certain outrigger stations may be built at locations (e.g., greenfield
land) that will lack the infrastructure to support a hydrogen maser. In this
scenario, alternate reference signals (e.g., from rubidium microwave
oscillators) can also be injected into the correlator to track and compensate
for GPS clock drifts. The performance of these oscillators is inferior to that
of a hydrogen maser, but they are still more stable than the CHIME GPS clock
on the timescales relevant for FRB VLBI. They are also less expensive and more
readily available than a maser. Even if these frequency references can
potentially be used directly as the correlator master clock since they
typically come in units that can provide GPS disciplining as well as absolute
time, it is still desirable to use them separately as free-running clocks for
short and intermediate timescale observations. Not only are they inherently
more stable than the primary CHIME clock, but they are not subject to short-
timescale tuning jitter when not locked to GPS.

Equation 4 provides a convenient way to determine whether an off-the-shelf
clock meets the requirements for FRB VLBI by simply reading the
$\sigma_{y}(10^{3}~{}\text{s})$ value from the unit’s datasheet. Passive
hydrogen masers meet and exceed this requirement. However, this specification
was derived from Equation A5 which assumes uncorrelated clock timing errors
and perfect calibration measurements. In practice, the actual clock timing
errors will depend on aspects not necessarily captured by this equation
including the detailed statistics of the delay variations, the methods used to
estimated them, the timing and accuracy of the calibration measurements, and
the technique that we use to inject the clock to our system. These aspects
become relevant when the frequency standard does not clearly exceed the
specification in Equation 4.

As part of the implementation of the clock stabilization system, Cary et al.
(2021) developed a software package with methods to determine the suitability
of precision clocks for VLBI with transit telescopes like CHIME/FRB
Outriggers. These methods take into account the details of the noise processes
that determine the stability of the clocks and simulate realistic timing
calibration scenarios. The basic input to the software is a timestream that
represents the delay variations as function of time of the clock under test.
The delay timestream data can be either from measurements or from simulations;
in the latter case the software provides tools to generate timestreams
described by combinations of power-law noise processes commonly observed in
precision clocks and oscillators including white phase modulation noise, white
frequency modulation noise, flicker frequency modulation noise, and random
walk frequency modulation noise (Allan, 1987). Similarly, the software
provides tools to generate delay timestreams from a set of Allan deviation
measurements, which is convenient to evaluate the performance of a clock based
on its manufacturer specifications. In this case, it is assumed that the delay
variations are described by a combination of power-law noise processes where
the weight of each noise component is found by fitting the Allan variance data
to a model consisting of a linear combination of the Allan variances of the
previously described noise processes.

A calibrator is parametrized by its observing time, number of clock timing
measurements, and SNR per transit. For example, for a calibrator at the
equator the observing time with CHIME is $\sim 6$ min, and with a $\sim 2$ min
integration time we would have three delay measurements per transit. The SNR
determines the uncertainty in the calibration delay measurements (see Equation
1). Given a timescale $\Delta t_{cal}$ that represents the maximum expected
time separation between calibrators, the method masks a random $\Delta
t_{cal}$-long section of the delay timestream, interpolates using a best-fit
function determined from the available calibration measurements at each end of
the masked section, and keeps the interpolation residuals. The process is
repeated a configurable number of times to obtain a statistical ensemble of
interpolation residual timestreams, each of length777In practice there is an
additional overhead equivalent to one integration. $\sim\Delta t_{cal}$. As
the default metric of the stability of the clock at $\Delta t_{cal}$
timescales, the method uses the largest value of the ensemble standard
deviation in the interval $\left[0,\Delta t_{cal}\right]$. Other metrics of
performance are available. Since throughout the paper we have used the
convention that $\Delta t$ represents the time between a calibration and an
observation, this metric can be interpreted as an estimate of the largest
clock timing rms error for $\Delta t$ up to $\sim\Delta t_{cal}/2$.

Different interpolation methods are available including linear fit, smoothing
spline, and nearest available calibrator. The fitting weights are determined
by the calibrator SNR and the level of the noise added by the timing
stabilization system.

As a candidate for outriggers without a maser, we evaluated the performance of
the EndRun Technologies Meridian II US-Rb rubidium oscillator. The unit was
installed in the Pathfinder RF room and connected to a separate input of the
correlator so we could test its performance against the DRAO maser under
conditions comparable to those of a typical outrigger. A signal conditioning
chain identical to the maser was used (signal translator + FLA). We collected
$\sim 42$ h of raw ADC data at 1 s cadence for both the maser and the Rb clock
and used the clock stabilization pipeline to extract the clock delay
variations relative to the maser. The top panel of Figure 6 shows the measured
delay variations after removing slow linear drift component due to the maser
(see Section 4.3). The bottom panel shows the corresponding Allan deviation of
the Rb clock (blue), the measurement error (dashed blue), and the
manufacturer-specified Allan deviation of the Rb clock (green points) and the
DRAO maser (red points). For $\Delta t\lesssim 40$ s the variations are
dominated by the noise associated to the timing stabilization system, which is
in the range $\sim 10-30$ ps, small compared to the $\sim 200$ ps clock timing
error budget. At longer timescales the measured Allan deviation of the Rb
oscillator is consistent with the manufacturer specification. These results
are also consistent with direct Allan deviation measurements of the Rb
oscillator performed with a phase noise analyzer, confirming that the hardware
of the stabilization system is not a limitation for clock performance in
CHIME/FRB Outriggers.

Figure 6: Top: Measured delay variations of the rubidium oscillator tested as
a candidate reference clock for outriggers without a maser. Bottom: Measured
Allan deviation of the Rb clock (blue), measurement error (dashed blue), and
manufacturer-specified Allan deviation of the Rb clock (green points) and the
DRAO maser (red points). The measured Allan deviation of the Rb clock is
consistent with the specification at intermediate and long timescales. At
short timescales the noise of the timing stabilization system dominates the
performance, but it is still small ($\sim 10-30$ ps) compared to the clock
timing error budget. This confirms that the hardware of the clock
stabilization system is not a limitation for clock performance in CHIME/FRB
Outriggers.

Note that if we rely only on the manufacturer-specified Allan deviation, the
Rb clock does not meet the requirement in Equation 4. This justifies a more
detailed analysis of the clock performance to determine whether it can still
be used as a frequency reference for the outriggers without a maser.

The measured delay timestream was analyzed with the software package described
above and in Cary et al. (2021) to evaluate the expected performance of the Rb
clock under different calibration conditions. The results are shown in Figure
7. For the analysis we assumed that the calibrator observing time was 9 min
(roughly the value at CHIME’s zenith) with two integrations per observation.
The best performance is obtained with linear interpolation between
calibrators. We tested calibrator SNRs of 10 (purple), 15 (green), 20 (red),
and $\infty$ (blue), the latter representing the case where the clock
performance is not limited by calibration errors. For comparison we also show
the projected clock timing errors for the DRAO maser using synthetic data
generated from the manufacturer-specified Allan deviation with $\text{SNR}=15$
(dashed green). Figure 7 shows that, even in the most conservative scenario
where we assume that all the calibrators have $\text{SNR}=15$, the Rb clock
timing errors stay below $225$ ps up to $\Delta t\sim 10^{3}$ s by
interpolating between timing solutions (solid green). This clock timing error
is still well below the total timing budget of $\sim 800$ ps, leaving enough
room to handle ionospheric delay errors.

Figure 7: Projected clock errors, $\sigma_{\tau}^{\text{clk}}$, of the Rb
clock as function of the time between calibrators $\Delta t_{cal}$ from
measured delay variations and simulations of realistic timing calibration
scenarios. This metric represents an estimate of the largest clock timing
error for $\Delta t$ up to $\sim\Delta t_{cal}/2$ (see Section 6 for details).
The dashed black horizontal line represents $\sigma_{\tau}^{\text{clk}}=200$
ps. Even in the most conservative scenario where we assume that all the
calibrators have $\text{SNR}=15$ (solid green), the Rb clock timing errors
stay below $225$ ps up to $\Delta t\sim 10^{3}$ s by interpolating between
timing solutions, meeting the requirements for FRB VLBI with CHIME/FRB
Outriggers.

##  7 Conclusions

We developed a clock stabilization system for CHIME/FRB Outriggers that allows
synchronization of CHIME and outrigger stations at the $\sim 200$ ps level on
short and long timescales. This meets the requirements for 50 mas localization
of FRBs detected with the CHIME/FRB real-time pipeline. Our proof-of-principle
clock transfer has demonstrated that a variety of different data products can
be used for precise time transfer from an external reference clocks into data
acquisition backends using the ICE framework. This method is minimally
invasive to existing telescopes like CHIME and the Pathfinder, expanding the
capabilities of these instruments to include VLBI without impacting their
existing scientific backends. It also allows for increased flexibility and
modularity for future systems such as those at CHIME outriggers. For
outriggers that do not have the infrastructure to support a hydrogen maser, we
demonstrated that it is still possible to meet the required clock stability
specification by using alternate reference clocks and interpolating timing
solutions between calibrations.

We acknowledge that CHIME is located on the traditional, ancestral, and
unceded territory of the Syilx/Okanagan people. We are grateful to the staff
of the Dominion Radio Astrophysical Observatory, which is operated by the
National Research Council Canada. CHIME is funded by a grant from the Canada
Foundation for Innovation (CFI) 2012 Leading Edge Fund (Project 31170) and by
contributions from the provinces of British Columbia, Québec and Ontario. The
CHIME/FRB Project is funded by a grant from the CFI 2015 Innovation Fund
(Project 33213) and by contributions from the provinces of British Columbia
and Québec, and by the Dunlap Institute for Astronomy and Astrophysics at the
University of Toronto. Additional support was provided by the Canadian
Institute for Advanced Research (CIFAR), McGill University and the McGill
Space Institute via the Trottier Family Foundation, and the University of
British Columbia. The CHIME/FRB Outriggers program is funded by the Gordon and
Betty Moore Foundation and by a National Science Foundation (NSF) grant
(2008031). FRB research at MIT is supported by an NSF grant (2008031). FRB
research at WVU is supported by an NSF grant (2006548, 2018490). J.M.P. is a
Kavli Fellow. C.L. was supported by the U.S. Department of Defense (DoD)
through the National Defense Science & Engineering Graduate Fellowship (NDSEG)
Program. V.M.K. holds the Lorne Trottier Chair in Astrophysics & Cosmology and
a Distinguished James McGill Professorship and receives support from an NSERC
Discovery Grant and Herzberg Award, from an R. Howard Webster Foundation
Fellowship from CIFAR, and from the FRQNT Centre de Recherche en Astrophysique
du Québec. J.L.S. acknowledges support from the Canada 150 programme.

##  Appendix A Clock stability and Allan Deviation

On timescales of a second and larger, clock stabilities are usually quoted in
terms of the Allan variance, or its square-root the Allan deviation (Allan,
1966). In this appendix, we show how the clock stability required for FRB VLBI
relates to the Allan deviation.

Let the “true” time be $t$, and the difference in seconds between what our
clock reads and the true time be $x(t)$. In general, we will not know what the
clock timing error $x(t)$ is for all time, but for CHIME/FRB Outriggers we
will measure it whenever we do an on-sky calibration. If we calibrate at two
points in time, we naturally would like to predict the timing error at the
midpoint between the two calibrations. In particular, if the calibration times
are $t$ and $t+2\cdot\Delta t$, then we want to estimate $x(t+\Delta t)$ given
$x(t)$ and $x(t+2\cdot\Delta t)$. The simplest assumption that we can make is
that our clock runs at a constant rate between $t$ and $t+2\cdot\Delta t$, in
which case our prediction for $x(t+\Delta t)$ will be
$\left[x(t)+x(t+2\cdot\Delta t)\right]/2$. The variance of the timing error
estimate halfway between the calibrations will then be:

$\left<\left[x(t+\Delta t)-\frac{x(t)+x(t+2\cdot\Delta
t)}{2}\right]^{2}\right>=\frac{1}{4}\left<\left[x(t)-2x(t+\Delta
t)+x(t+2\cdot\Delta t)\right]^{2}\right>.$ (A1)

The Allan variance is defined to be

$\sigma_{y}^{2}(\Delta t)\equiv\frac{1}{2\cdot\Delta
t^{2}}\left<\left[x(t)-2x(t+\Delta t)+x(t+2\cdot\Delta t)\right]^{2}\right>,$
(A2)

so we can now relate our timing error variance halfway between the
calibrations directly to the Allan variance:

$\left<\left[x(t+\Delta t)-\frac{x(t)+x(t+2\cdot\Delta
t)}{2}\right]^{2}\right>=\frac{\Delta t^{2}}{2}\sigma_{y}^{2}(\Delta t).$ (A3)

In general, and up to factors of order unity, the variance of the timing error
will be determined by the product $\Delta t^{2}\cdot\sigma_{y}^{2}(\Delta t)$,
so

$\sigma_{x}(\Delta t)\approx\Delta t\cdot\sigma_{y}(\Delta t).$ (A4)

is typically used as an approximation to the standard deviation of the clock
timing error after a time $\Delta t$ (Barnes et al., 1971; Kartaschoff, 1979;
Rogers & Moran, 1981). In practice, the actual errors will depend on the
details of the prediction algorithm and the noise processes that dominate the
stability of the clock. For the particular case of white timing noise it
follows from Equation A2 that

$\sigma_{y}^{WN}(\Delta t)=\sqrt{3}\frac{\sigma_{x}}{\Delta t}.$ (A5)

From Equation A2 the Allan deviation is dimensionless, so it is telling us the
fractional uncertainty in our clock between calibrations. Quantitatively, if
our pulsar calibrations are separated by $2\cdot\Delta t=2000$ s, and the
limit on the _differential_ timing residual at the midpoint is $200$ ps rms
(Section 3), then our Allan deviation requirement is

$\sigma_{y}(10^{3}~{}\text{s})\approx\sqrt{\frac{3}{2}}\cdot\frac{2\cdot
10^{-10}~{}\text{s}}{10^{3}~{}\text{s}}\approx 2\cdot 10^{-13}.$ (A6)

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