$\displaystyle\left\|\mathcal{M}_{(3)}(\mathcal{S})_{b,:}-\frac{\sum_{l=1}^{L}\mathcal{M}_{(3)}(\mathcal{S})_{(z)_{l},:}\mathbf{1}(\widehat{z}_{l}=b)}{\sum_{l=1}^{L}\mathbf{1}(\widehat{z}_{l}=b)}\right\|$
$\displaystyle=$
$\displaystyle\left\|\frac{1}{\sum_{l=1}^{L}\mathbf{1}(\widehat{z}_{l}=b)}\sum_{l=1}^{L}\sum_{b^{\prime}\in[r_{3}]/b}\mathbf{1}(z_{l}=b,\widehat{z}_{l}=b^{\prime})\\{\mathcal{M}_{(3)}(\mathcal{S})_{b,:}-\mathcal{M}_{(3)}(\mathcal{S})_{b^{\prime},:}\\}\right\|$
$\displaystyle\leq$ $\displaystyle
C_{0}\frac{r_{3}}{L\Delta_{\min}}\sum_{l=1}^{L}\sum_{b^{\prime}\in[r_{3}]/b}\mathbf{1}(z_{l}=b,\widehat{z}_{l}=b^{\prime})\|\mathcal{M}_{(3)}(\mathcal{S})_{b,:}-\mathcal{M}_{(3)}(\mathcal{S})_{b^{\prime},:}\|^{2}$
$\displaystyle\leq$ $\displaystyle
C_{0}\frac{r_{3}g(\widehat{z},z)}{\Delta_{\min}}.$

Also, we know that

$\displaystyle\|\bm{W}-\widehat{\bm{W}}\|$
$\displaystyle\leq\\{\lambda_{r_{3}}(\bm{M})\\}^{-1}\cdot\|\bm{M}^{\intercal}\bm{W}-\bm{M}^{\intercal}\widehat{\bm{W}}\|$
$\displaystyle\leq\\{\lambda_{r_{3}}(\bm{M})\\}^{-1}\cdot\|\bm{M}^{\intercal}\bm{W}-\bm{M}^{\intercal}\widehat{\bm{W}}\|$
$\displaystyle\leq\\{\lambda_{r_{3}}(\bm{M})\\}^{-1}\cdot\|\bm{I}-\bm{M}^{\intercal}\widehat{\bm{W}}\|_{F}$
$\displaystyle\overset{(\ref{eq:eigen_M})}{\leq}\sqrt{\frac{r_{3}}{L}}\|\bm{I}-\bm{M}^{\intercal}\widehat{\bm{W}}\|_{F},$
(31)

where $\bm{M}^{\intercal}\bm{W}=\bm{I}$ by definition. For any $b\in[r_{3}]$,
denote $\widehat{n}_{b}=\sum_{l=1}^{L}\bm{1}\\{\widehat{z}_{l}=b\\}$. For any
$b\neq b^{\prime}$, we have

$(\bm{M}^{\intercal}\widehat{\bm{W}})_{b^{\prime}b}=\frac{\sum_{l=1}^{L}\bm{1}\\{z_{l}=b^{\prime},\widehat{z}_{l}=b\\}}{\widehat{n}_{b}},\quad\delta_{b}=\sum_{b^{\prime}\in[r_{3}]/b}(\bm{M}^{\intercal}\widehat{\bm{W}})_{b^{\prime}b}=1-(\bm{M}^{\intercal}\widehat{\bm{W}})_{bb}.$

Therefore,

$\displaystyle\|\bm{I}-\bm{M}^{\intercal}\widehat{\bm{W}}\|_{F}$
$\displaystyle=\sqrt{\sum_{b\in[r_{3}]}\left\\{\delta_{b}^{2}+\sum_{b^{\prime}\in[r_{3}]/b}(\bm{M}^{\intercal}\widehat{\bm{W}})_{b^{\prime}b}^{2}\right\\}}$
$\displaystyle\leq\sqrt{\sum_{b\in[r_{3}]}\left\\{\delta_{b}^{2}+\left(\sum_{b^{\prime}\in[r_{3}]/b}(\bm{M}^{\intercal}\widehat{\bm{W}})_{b^{\prime}b}\right)^{2}\right\\}}$
$\displaystyle\leq\sqrt{2\sum_{b\in[r_{3}]}\delta_{b}^{2}}$
$\displaystyle\leq\sqrt{2}\sum_{b\in[r_{3}]}\delta_{b}$
$\displaystyle\leq\sqrt{2}\sum_{b\in[r_{3}]}\frac{\sum_{l=1}^{L}\bm{1}\\{z_{l}\neq
b^{\prime},\widehat{z}_{l}=b\\}}{\widehat{n}_{b}}$
$\displaystyle\leq\sqrt{2}\max_{b\in[r_{3}]}(\widehat{n}_{b})^{-1}\sum_{l=1}^{L}\mathbf{1}(z_{l}\neq\widehat{z}_{l})$
$\displaystyle\lesssim\frac{r_{3}}{L}\cdot
Lh(\widehat{z},z)\lesssim\frac{r_{3}g(\widehat{z},z)}{\Delta_{\min}^{2}},$

where the last two inequalities are justified by (12). Then, we can show that

$\displaystyle\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\bm{V}\|$
$\displaystyle\leq\|\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b}\|\cdot\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\bm{V}\|_{\max}$
$\displaystyle\overset{(\ref{eq:W-What})}{\leq}C_{1}\frac{r_{3}^{3/2}g(\widehat{z},z)}{\Delta_{\min}^{2}\sqrt{L}}\cdot\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}\cdot\|\bm{V}\|_{2,\max}$
$\displaystyle\leq
C_{1}\frac{\mu_{0}r_{1}^{1/2}r_{2}^{1/2}r_{3}^{3/2}g(\widehat{z},z)\|\widehat{\Theta}-\Theta\|_{F}}{\Delta_{\min}^{2}\sqrt{NTL}}.$

By triangle inequality, it completes the proof for (8)

$\displaystyle\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta})\bm{V}\|$
$\displaystyle\leq\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\Theta)\bm{V}\|$
$\displaystyle+\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\bm{V}\|$
$\displaystyle\lesssim\frac{r_{3}g(\widehat{z},z)}{\Delta_{\min}}+\frac{\mu_{0}r_{1}^{1/2}r_{2}^{1/2}r_{3}^{3/2}g(\widehat{z},z)\|\widehat{\Theta}-\Theta\|_{F}}{\Delta_{\min}^{2}\sqrt{NTL}}.$

To prove (9),

$\displaystyle\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta})\widehat{\bm{V}}\|$
$\displaystyle\leq\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta})\bm{V}\|$
$\displaystyle+\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta})(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle\overset{(\ref{eq:error_W_V})}{\lesssim}\frac{r_{3}g(\widehat{z},z)}{\Delta_{\min}}+\frac{\mu_{0}r_{1}^{1/2}r_{2}^{1/2}r_{3}^{3/2}g(\widehat{z},z)\|\widehat{\Theta}-\Theta\|_{F}}{\Delta_{\min}^{2}\sqrt{NTL}}$
$\displaystyle+\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\Theta)(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle+\|(\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b})^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle\lesssim\frac{r_{3}g(\widehat{z},z)}{\Delta_{\min}}+\frac{\mu_{0}r_{1}^{1/2}r_{2}^{1/2}r_{3}^{3/2}g(\widehat{z},z)\|\widehat{\Theta}-\Theta\|_{F}}{\Delta_{\min}^{2}\sqrt{NTL}}$
$\displaystyle+\|\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b}\|\cdot\|\mathcal{M}_{(3)}(\Theta)\|\cdot\|\bm{V}-\widehat{\bm{V}}\|$
$\displaystyle+\|\bm{W}_{:,b}-\widehat{\bm{W}}_{:,b}\|\cdot\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}\cdot\|\bm{V}-\widehat{\bm{V}}\|_{2,\max}$
$\displaystyle\overset{(\ref{eq:W-What}),(\ref{eq:Vhat-V})}{\lesssim}\frac{r_{3}g(\widehat{z},z)}{\Delta_{\min}}+\frac{\mu_{0}r_{1}^{1/2}r_{2}^{1/2}r_{3}^{3/2}g(\widehat{z},z)\|\widehat{\Theta}-\Theta\|_{F}}{\Delta_{\min}^{2}\sqrt{NTL}}+\frac{\kappa^{2}r_{3}^{3/2}g(\widehat{z},z)\|\widehat{\Theta}-\Theta\|_{F}}{\Delta_{\min}^{2}\sqrt{L}}.$

To prove (10),

$\displaystyle\|\widehat{\bm{W}}_{:,b}^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta})(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle\leq\|\widehat{\bm{W}}_{:,b}^{\intercal}\mathcal{M}_{(3)}(\Theta)(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle+\|\widehat{\bm{W}}_{:,b}^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle\leq\|\bm{W}_{:,b}^{\intercal}\mathcal{M}_{(3)}(\Theta)(\bm{V}-\widehat{\bm{V}})\|+\|(\widehat{\bm{W}}_{:,b}^{\intercal}-\bm{W}_{:,b}^{\intercal})\mathcal{M}_{(3)}(\Theta)(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle+\|\bm{W}_{:,b}^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)(\bm{V}-\widehat{\bm{V}})\|+\|(\widehat{\bm{W}}_{:,b}^{\intercal}-\bm{W}_{:,b}^{\intercal})\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)(\bm{V}-\widehat{\bm{V}})\|$
$\displaystyle\lesssim\sqrt{r_{3}/L}\cdot\|\mathcal{M}_{(3)}(\Theta)\|\cdot\|\bm{V}-\widehat{\bm{V}}\|+\frac{r_{3}^{3/2}g(\widehat{z},z)}{\Delta_{\min}^{2}\sqrt{L}}\cdot\|\mathcal{M}_{(3)}(\Theta)\|\cdot\|\bm{V}-\widehat{\bm{V}}\|$
$\displaystyle+\sqrt{r_{3}/L}\cdot\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}\cdot\|\bm{V}-\widehat{\bm{V}}\|_{2,\max}$
$\displaystyle+\frac{r_{3}^{3/2}g(\widehat{z},z)}{\Delta_{\min}^{2}\sqrt{L}}\cdot\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}\cdot\|\bm{V}-\widehat{\bm{V}}\|_{2,\max}$
$\displaystyle\overset{(\ref{eq:Vhat-V})}{\lesssim}\sqrt{r_{3}/L}\cdot\kappa^{2}\|\widehat{\Theta}-\Theta\|_{F}+\frac{r_{3}^{3/2}g(\widehat{z},z)}{\Delta_{\min}^{2}\sqrt{L}}\cdot\kappa^{2}\|\widehat{\Theta}-\Theta\|_{F}$
$\displaystyle+\frac{\mu_{0}r_{1}^{1/2}r_{2}^{1/2}r_{3}^{1/2}}{\sqrt{NTL}}\|\widehat{\Theta}-\Theta\|_{F}+\frac{\mu_{0}r_{1}^{1/2}r_{2}^{1/2}r_{3}^{3/2}g(\widehat{z},z)}{\Delta_{\min}^{2}\sqrt{NTL}}\|\widehat{\Theta}-\Theta\|_{F}$
$\displaystyle\lesssim\kappa^{2}\sqrt{r_{3}/L}\|\widehat{\Theta}-\Theta\|_{F}.$

To prove (14), we have

$\displaystyle\|\widehat{\bm{V}}-\bm{V}\|$
$\displaystyle=\|\bm{\widehat{U}}_{1}\otimes\widehat{\bm{U}}_{2}-\bm{U}_{1}\otimes\bm{U}_{2}\|$
$\displaystyle=\|\bm{\widehat{U}}_{1}\otimes(\widehat{\bm{U}}_{2}-\bm{U}_{2})+(\bm{\widehat{U}}_{1}-\bm{U}_{1})\otimes\bm{U}_{2}\|$
$\displaystyle\leq\|\bm{\widehat{U}}_{1}\otimes(\widehat{\bm{U}}_{2}-\bm{U}_{2})\|+\|(\bm{\widehat{U}}_{1}-\bm{U}_{1})\otimes\bm{U}_{2}\|$
$\displaystyle\lesssim\left\\{\|\widehat{\bm{U}}_{1}-\bm{U}_{1}\|+\|\widehat{\bm{U}}_{2}-\bm{U}_{2}\|\right\\}\overset{\text{Lemma}\ref{lem:Davis-
Kahan}}{\lesssim}\frac{\kappa\|\widehat{\Theta}-\Theta\|_{F}}{\lambda_{\min}}.$

Since $\bm{U}_{1}$ and $\bm{U}_{2}$ are only identifiable up to rotation and
permutation, we instead focus on the upper bound of
$\min_{\bm{R}_{1}\in\mathbb{O}_{r_{1}}}\|\widehat{\bm{U}}_{1}-\bm{U}_{1}\bm{R}_{1}\|$
and
$\min_{\bm{R}_{2}\in\mathbb{O}_{r_{2}}}\|\widehat{\bm{U}}_{2}-\bm{U}_{2}\bm{R}_{2}\|$,
where $\mathbb{O}_{r}$ is the collection of all $r$-by-$r$ matrices with
orthonormal columns. Lemma B.5 is a variant of the Davis-Kahan $\sin(\Theta)$
Theorem from Yu et al. (2015), which bound the distance between subspaces
spanned by the population eigenvectors and their sample versions.

For (14) and (15), the proof is similar to Theorem 4, Yu et al. (2015):

$\displaystyle\|\bm{V}\|_{2,\max}^{2}$
$\displaystyle=\max_{j}\|e_{j}^{\intercal}(\bm{U}_{1}\otimes\bm{U}_{2})\|_{2}^{2}$
$\displaystyle\leq\max_{i,t}\|e_{i}^{\intercal}\bm{U}_{1}\|^{2}\cdot\|e_{t}^{\intercal}\bm{U}_{2}\|^{2}\text{
(Cauchy-Schwatz inequality)}$
$\displaystyle\leq\|\bm{U}_{1}\|_{2,\max}^{2}\|\bm{U}_{2}\|_{2,\max}^{2}$
$\displaystyle\leq\frac{\mu_{0}^{2}r_{1}r_{2}}{NT},$

and

$\displaystyle\|\bm{V}-\widehat{\bm{V}}\|_{2,\max}$
$\displaystyle=\max_{j}\|e_{j}^{\intercal}(\bm{U}_{1}\otimes\bm{U}_{2}-\widehat{\bm{U}}_{1}\otimes\widehat{\bm{U}}_{2})\|$
$\displaystyle=\max_{j}\|e_{j}^{\intercal}(\bm{U}_{1}\otimes\bm{U}_{2}-\bm{U}_{1}\otimes\widehat{\bm{U}}_{2}+\bm{U}_{1}\otimes\widehat{\bm{U}}_{2}-\widehat{\bm{U}}_{1}\otimes\widehat{\bm{U}}_{2})\|$
$\displaystyle\leq\max_{j}\|e_{j}^{\intercal}(\bm{U}_{1}\otimes\bm{U}_{2}-\bm{U}_{1}\otimes\widehat{\bm{U}}_{2})\|+\max_{j^{\prime}}\|e_{j^{\prime}}^{\intercal}(\bm{U}_{1}\otimes\widehat{\bm{U}}_{2}-\widehat{\bm{U}}_{1}\otimes\widehat{\bm{U}}_{2})\|$
$\displaystyle\leq\|\bm{U}_{1}\|_{2,\max}\|\widehat{\bm{U}}_{2}-\bm{U}_{2}\|_{2,\max}+\|\widehat{\bm{U}}_{2}\|_{2,\max}\|\widehat{\bm{U}}_{1}-\bm{U}_{1}\|_{2,\max}$
$\displaystyle\leq\sqrt{\frac{\mu_{0}r_{1}}{N}}\|\widehat{\bm{U}}_{2}-\bm{U}_{2}\|_{2,\max}+\sqrt{\frac{\mu_{0}r_{2}}{T}}\|\widehat{\bm{U}}_{1}-\bm{U}_{1}\|_{2,\max}.$

Under the assumption of eigengap, we are able to bound the perturbation of
individual eigenvectors:

$\|\widehat{u}_{2,i}-u_{2,i}\|_{2}\lesssim\frac{\lambda_{\max}\|\widehat{\Theta}-\Theta\|_{F}}{\min(\lambda_{i-1}^{2}-\lambda_{i}^{2},\lambda_{i}^{2}-\lambda_{i+1}^{2})}.$

Assume for any $i\in[r_{2}]$, the interval
$[\lambda_{i}-\delta_{r_{2}},\lambda_{i}+\delta_{r_{2}}]$ does not contain any
eigenvalues of $\mathcal{M}_{(2)}(\Theta)$ other than $\lambda_{i}$:

$\|\widehat{u}_{2,i}-u_{2,i}\|_{2}\lesssim\frac{\lambda_{\max}\|\widehat{\Theta}-\Theta\|_{F}}{\min(\lambda_{i-1}^{2}-\lambda_{i}^{2},\lambda_{i}^{2}-\lambda_{i+1}^{2})}\lesssim\frac{\kappa\|\widehat{\Theta}-\Theta\|_{F}}{\delta_{r_{2}}},$

which implies
$\|\widehat{\bm{U}}_{2}-\bm{U}_{2}\|_{2,\max}\leq\kappa\|\widehat{\Theta}-\Theta\|_{F}/\delta_{r_{2}}$.
Identical bounds also hold if $\widehat{u}_{2,i}$ and $u_{2,i}$ are replaced
with $\widehat{u}_{1,i}$ and $u_{1,i}$. Suppose

$\delta_{r_{1}}\geq\sqrt{\frac{N}{\mu_{0}r_{1}}}\cdot\kappa\|\widehat{\Theta}-\Theta\|_{F},\quad\delta_{r_{2}}\geq\sqrt{\frac{T}{\mu_{0}r_{2}}}\cdot\kappa\|\widehat{\Theta}-\Theta\|_{F},$

we claim that
$\|\bm{V}-\widehat{\bm{V}}\|_{2,\max}^{2}\leq\|\bm{V}\|_{2,\max}^{2}$. ∎

####  B.4.3 Proof of Lemma B.4

First, we want to prove the probability
$P\left(\langle\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\bm{V},\bm{S}_{a,:}-\bm{S}_{b,:}\rangle\leq-\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}\right)$
is negligible. We can show that

$\displaystyle
P\left(\langle\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\bm{V},\bm{S}_{a,:}-\bm{S}_{b,:}\rangle\leq-\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}\right)$
$\displaystyle\leq
P(-\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\cdot\|(\bm{S}_{a,:}-\bm{S}_{b,:})\bm{V}^{\intercal}\|_{\max}\leq-\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2})$
$\displaystyle\leq
P(\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\cdot\|\bm{S}_{a,:}-\bm{S}_{b,:}\|\cdot\|\bm{V}\|_{2,\max}\geq\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2})$
$\displaystyle\leq
P(\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\cdot\|\bm{U}_{1}\|_{2,\max}\|\bm{U}_{2}\|_{2,\max}\geq\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|)$
$\displaystyle\leq
P\left(\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|^{2}\geq\frac{1}{64}\frac{\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}}{\|\bm{U}_{1}\|_{2,\max}^{2}\|\bm{U}_{2}\|_{2,\max}^{2}}\right)$
$\displaystyle\leq
P\left(\sum_{l=1}^{L}\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|^{2}\geq\frac{1}{64}\frac{L\Delta_{\min}^{2}}{\|\bm{U}_{1}\|_{2,\max}^{2}\|\bm{U}_{2}\|_{2,\max}^{2}}\right)$
$\displaystyle\leq
P\left(\|\widehat{\Theta}-\Theta\|_{F}^{2}\geq\frac{1}{64}\frac{NTL\Delta_{\min}^{2}}{\mu_{0}^{2}r_{1}r_{2}}\right),$

where under Assumption R2):

$\|\bm{U}_{1}\|_{2,\max}^{2}\leq\frac{\mu_{0}r_{1}}{N},\quad\|\bm{U}_{2}\|_{2,\max}^{2}\leq\frac{\mu_{0}r_{2}}{T}.$

To ensure the event $\|\widehat{\Theta}-\Theta\|_{F}^{2}\leq
NTL\Delta_{\min}^{2}/(64\mu_{0}^{2}r_{1}r_{2})$ holds with high probability,
we need to have

$\displaystyle\frac{NTL\Delta_{\min}^{2}}{\mu_{0}^{2}r_{1}r_{2}}$
$\displaystyle\geq\frac{c_{0}(N\vee
T)r_{1}r_{2}r_{3}\log(N+T+L)\|\Theta\|_{\max}^{2}}{p_{\min}^{2}\max(r_{1},r_{2},r_{3})},$
(32) $\displaystyle\frac{NTL\Delta_{\min}^{2}}{\mu_{0}^{2}r_{1}r_{2}}$
$\displaystyle\geq\frac{c_{0}w_{\max}^{2}(N\vee
T)r_{1}r_{2}r_{3}\log^{2}(N+T+L)(L_{\alpha}^{2}/\gamma_{\alpha}^{2})}{w_{\min}^{2}p_{\min}^{2}\gamma_{\alpha}^{2}\max(r_{1},r_{2},r_{3})},$
(33)

under the estimation error in Theorem 5.1. Thus, we claim that

$P\left(\|\widehat{\Theta}-\Theta\|_{F}^{2}\geq\frac{1}{64}\frac{NTL\Delta_{\min}^{2}}{\mu_{0}^{2}r_{1}r_{2}}\right)\leq\frac{c}{(N+T+L)^{2}},$

under the conditions (32) and (33). Next, we show that
$P\left(\langle\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\bm{V},\tilde{\bm{S}}_{a,:}-\bm{S}_{a,:}\rangle\leq-\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}\right)$
is negligible

$\displaystyle
P\left(\langle\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\bm{V},\tilde{\bm{S}}_{a,:}-\bm{S}_{a,:}\rangle\leq-\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}\right)$
$\displaystyle\leq
P(-\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\cdot\|(\tilde{\bm{S}}_{a,:}-\bm{S}_{a,:})\bm{V}^{\intercal}\|_{\max}\leq-\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2})$
$\displaystyle\leq
P(\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\cdot\|\tilde{\bm{S}}_{a,:}-\bm{S}_{a,:}\|\cdot\|\bm{V}\|_{2,\max}\geq\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2})$
$\displaystyle\leq
P(\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\cdot\|\tilde{\bm{S}}_{a,:}-\bm{S}_{a,:}\|\cdot\|\bm{U}_{1}\|_{2,\max}\|\bm{U}_{2}\|_{2,\max}\geq\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|).$
(34)

One can observe that

$\displaystyle\|\tilde{\bm{S}}_{a,:}-\bm{S}_{a,:}\|$
$\displaystyle=\|\bm{M}_{:,a}^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\bm{V}\|$
$\displaystyle\leq\|\bm{M}_{:,a}^{\intercal}\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|\cdot\|\bm{V}\|_{\max}$
$\displaystyle\leq\|\bm{M}_{:,a}^{\intercal}\|\cdot\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}\cdot\max_{i,j}\|e_{i}^{\intercal}\bm{V}e_{j}\|$
$\displaystyle\leq\sqrt{\frac{r_{3}}{L}}\|\bm{U}_{1}\|_{2,\max}\|\bm{U}_{2}\|_{2,\max}\cdot\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}.$

Plug it back to (34), we obtain

$\displaystyle
P\left(\langle\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\bm{V},\tilde{\bm{S}}_{a,:}-\bm{S}_{a,:}\rangle\leq-\frac{1}{8}\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}\right)$
$\displaystyle\leq
P\left(\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}\geq\frac{\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}}{8\sqrt{\frac{r_{3}}{L}}\|\bm{U}_{1}\|_{2,\max}^{2}\|\bm{U}_{2}\|_{2,\max}^{2}}\right)$
$\displaystyle\leq
P\left(\sum_{l=1}^{L}\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)_{l,:}\|\|\mathcal{M}_{(3)}(\widehat{\Theta}-\Theta)\|_{F}\geq\frac{L\|\bm{S}_{a,:}-\bm{S}_{b,:}\|^{2}}{8\sqrt{\frac{r_{3}}{L}}\|\bm{U}_{1}\|_{2,\max}^{2}\|\bm{U}_{2}\|_{2,\max}^{2}}\right)$
$\displaystyle\leq
P\left(\|\widehat{\Theta}-\Theta\|_{F}^{2}\geq\frac{NTL^{3/2}\Delta_{\min}^{2}}{8\mu_{0}^{2}r_{1}r_{2}\sqrt{r_{3}}}\right),$

which is negligible implied by the signal-to-noise ratio condition since
$L\geq r_{3}$:

$\displaystyle\frac{NTL^{3/2}\Delta_{\min}^{2}}{\mu_{0}^{2}r_{1}r_{2}\sqrt{r_{3}}}$
$\displaystyle\geq\frac{NTL\Delta_{\min}^{2}}{\mu_{0}^{2}r_{1}r_{2}}\geq\frac{c_{0}L_{\alpha}^{2}(N\vee
T)r_{1}r_{2}r_{3}(\|\Theta\|_{\max}^{2}\vee\sigma^{2})\log^{2}(N+T+L)}{p_{\min}^{4}\gamma_{\alpha}^{2}\max(r_{1},r_{2},r_{3})}.$

Thus, which completes our proof of Lemma B.4.

######  Lemma B.5.

Let

$\displaystyle\lambda_{\max}$
$\displaystyle=\max\\{\|\mathcal{M}_{(1)}(\mathcal{Y})\|,\|\mathcal{M}_{(2)}(\mathcal{Y})\|,\|\mathcal{M}_{(3)}(\mathcal{Y})\|\\},$
$\displaystyle\lambda_{\min}$
$\displaystyle=\min\left[\lambda_{r_{1}}\\{\mathcal{M}_{(1)}(\mathcal{Y})\\},\lambda_{r_{2}}\\{\mathcal{M}_{(2)}(\mathcal{Y})\\},\lambda_{r_{3}}\\{\mathcal{M}_{(3)}(\mathcal{Y})\\}\right],$

and $\kappa=\lambda_{\max}/\lambda_{\min}$, then we have:

$\min_{\bm{R}_{1}\in\mathbb{O}_{r_{1}}}\|\widehat{\bm{U}}_{1}-\bm{U}_{1}\bm{R}_{1}\|_{F}\leq
C\frac{\kappa\|\widehat{\Theta}-\Theta\|_{F}}{\lambda_{\min}},\quad\min_{\bm{R}_{2}\in\mathbb{O}_{r_{2}}}\|\widehat{\bm{U}}_{2}-\bm{U}_{2}\bm{R}_{2}\|\leq
C\frac{\kappa\|\widehat{\Theta}-\Theta\|_{F}}{\lambda_{\min}}.$

The proof of Lemma B.5 is provided in Yu et al. (2015), Theorem 4.

######  Lemma B.6.

Under condition (11), we have
$r_{3}/L\lesssim|l\in[L]:(\widehat{z})_{l}=a|\lesssim r_{3}/L$ for any
$a\in[r_{3}]$. Moreover,

$\displaystyle\sqrt{L/r_{3}}\lesssim\lambda_{r_{3}}(\bm{M})\leq\|\bm{M}\|\lesssim\sqrt{L/r_{3}},$
$\displaystyle\sqrt{r_{3}/L}\lesssim\lambda_{r_{3}}(\bm{W})\leq\|\bm{W}\|\lesssim\sqrt{r_{3}/L}.$

The above inequalities also hold by replacing $\bm{M}$, $\bm{W}$ with
$\widehat{\bm{M}}$ and $\widehat{\bm{W}}$, respectively.

The proof of Lemma B.6 is provided in Han et al. (2022), Lemma 4.


∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>
11institutetext: INFN TIFPA, Via Sommarive, 14 I-38123 Trento, Italy
22institutetext: CERN, 1211 Genève 23, Switzerland 33institutetext: University
of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

# Optimising longitudinal and lateral calorimeter granularity for software
compensation in hadronic showers using deep neural networks

Coralie Neubüsere1,addr1  Jan Kieselere2,addr2  Paul Lujane3,addr3

(Received: date / Accepted: date)

###### Abstract

We investigate the effect of longitudinal and transverse calorimeter
segmentation on event-by-event software compensation for hadronic showers. To
factorize out sampling and electronics effects, events are simulated in which
a single charged pion is shot at a homogenous lead glass calorimeter, split
into longitudinal and transverse segments of varying size. As an approximation
of an optimal reconstruction, a neural network-based energy regression is
trained. The architecture is based on blocks of convolutional kernels
customized for shower energy regression using local energy densities; biases
at the edges of the training dataset are mitigated using a histogram
technique. With this approximation, we find that a longitudinal and transverse
segment size less than or equal to 0.5 and 1.3 nuclear interaction lengths,
respectively, is necessary to achieve an optimal energy measurement. In
addition, an intrinsic energy resolution of $8\%/\sqrt{E}$ for pion showers is
observed.

###### Keywords:

Machine learning calorimeters hadronic showers

###### pacs:

07.05.Mh 29.40.Vj

††journal: Eur. Phys. J. C

##  1 Introduction

Both existing high-energy physics experiments, such as those at the CERN LHC,
and future experiments at future colliders, like the Future Circular Collider
(FCC), rely heavily on the performance of hadron calorimeters and their
particle flow capabilities for measuring jet and missing transverse momentum
($p_{T}$) Sirunyan:2017ulk ; Aaboud:2017aca ; Ruan:2014paa ; Thomson:2009rp ;
Marshall:2012ry ; Marshall:2013bda ; Marshall:2015rfa ; Sefkow:2015hna ;
Tran:2017tgr . Hadron calorimeters are currently characterized not only in
terms of their intrinsic energy resolution, but by their imaging capabilities,
which allow for offline corrections using smart algorithms. Due to the diverse
composition of hadronic showers and the differences in the calorimeter
response, a correct energy measurement becomes challenging. In general, the
components of hadronic showers can be divided into electromagnetic (EM) and
hadronic parts. The hadronic part of the shower consists of particles such as
neutrinos and neutrons which are partially invisible to the detector. This can
be affected by the chosen active detector material, where, e.g., plastic
scintillators allow for neutron detection via strong interaction with the
atomic nucleus. The undetectable particles in the hadronic shower result in an
unequal detector response; that is, $e/h\neq 1$, where $e$ and $h$ are the
calorimeter response to electromagnetic and hadronic shower fractions,
respectively.

Many hadronic calorimeters currently in use and planned for future experiments
are sampling calorimeters, which consist of alternating active and passive
absorber layers CERN-LHCC-97-031 ; CERN-LHCC-96-041 ; HGCAL-TDR ;
Neubuser:2705432 . The sampling of the hadronic shower allows for tuning of
the hadronic and electromagnetic shower responses. In the past, the $e/h$
ratio has been adjusted closer to 1 by either suppressing the electromagnetic
response, e.g., by using high-$Z$ absorbers, or by enhancing the hadronic
response, using neutron-sensitive active materials. Calorimeters that have a
ratio $e/h\sim 1$ are called “compensating” calorimeters. These optimizations
in the active and passive materials often require a decreased sampling
fraction (ratio of active/passive material), which itself degrades the
calorimeter energy resolution by increasing the stochastic term $\alpha$ of

$\frac{\sigma_{E}}{\left<E\right>}=\frac{\alpha}{E}\oplus c.$ (1)

The stochastic term is dominated by the sampling fraction and frequency for
sampling calorimeters, and expresses the dependence of the calorimeter
resolution on the fluctuations of the number of particles within the hadronic
shower (following a Poisson distribution). The constant term $c$ expresses
energy-dependent uncertainties, like the fluctuations on the EM-to-hadronic
shower fraction, which is logarithmically increasing with energy, or energy
losses due to particles escaping the detector, caused by limited calorimeter
sizes. The first can be removed either by intrinsic compensation, or by an
event-by-event measurement of the EM fraction, which is called software
compensation.
Due to the cost and mechanical stability benefits, absorbers made of steel or
lead are widely in use. These materials have been found to require very small
sampling fractions in, e.g., scintillator-steel calorimeters in order to
achieve compensating behavior. Since such low sampling fractions would degrade
the performance, especially for particles at low energies ($<50$ GeV), the
solution to correct for fluctuations in the electromagnetic shower fraction is
to use software compensation techniques.
In order to allow algorithms to distinguish between the dense electromagnetic
shower core and other shower parts, such as e.g. disappearing tracks, the
granularity of the calorimeter plays a key role. The first attempt in so-
called imaging calorimetry has been made by the CALICE collaboration, which
started a R&D program of calorimeters for a future e-e+ linear collider
Israeli:2018byq ; Chefdeville:2019zzq , where the calorimeter designs have
been optimised for particle flow algorithms Marshall:2012ry . These algorithms
allow for jet energy measurements using the best suited sub-detector to
reconstruct each jet sub-particle. The prototypes of these calorimeters have
been realised with active layers made of silicon for the EM shower part and
scintillator or resistive plate chambers for the measurement of hadronic
showers. The active layers were tested and interleaved within both steel and
tungsten absorber stacks Chefdeville:2015 ; Adloff:2012gv  and achieved such
good results in testbeams Eigen:2019ccp  that the CMS collaboration decided to
adopt this concept in a full silicon-tungsten/scintillator-steel endcap
calorimeter HGCAL-TDR ; Quast:2017gnq . The developments in, e.g., silicon
photomultiplier (SiPM) technologies have been the key to measure the
scintillation light produced in calorimeter cell sizes of $3\times 3\times
0.5$ cm3 Sefkow:2018rhp . The impact of software compensation techniques on
the performance of particle flow algorithms has been studied in a specific
detector design Tran:2017tgr , and proven to provide a significant improvement
to the jet energy measurement by using a corrected calorimeter cluster which
is matched to tracks in the tracking system.
The next step towards a calorimeter design optimized for the use of software
compensation techniques is to study the necessary granularity that allows an
algorithm to determine most accurately the hadronic shower energy.

In this paper, we will discuss the performance of a software compensation
technique using a deep neural network (DNN), with a specific focus on the
dependence on the transverse and longitudinal granularity. Therefore, a
homogenous model calorimeter has been studied in full Geant4 simulations. The
performance is evaluated in terms of energy resolution and linearity for
single charged pions. The goal is to determine the minimal granularity of a
calorimeter needed to achieve the best energy measurement using DNNs. As the
choice of granularity can influence the detector design and cost, a
measurement of the impact of this choice is necessary in order to optimize the
design. Here, the DNN is utilised as a generic close-to-optimal reconstruction
algorithm that can be optimised to the granularity in an automatised fashion.

##  2 Calorimeter and dataset

The studied calorimeter is a homogeneous lead tungstate calorimeter, which
follows the EM calorimeter concept of the CMS experiment CERN-LHCC-97-033 .
However, we do not consider any passive absorber material, assuming that the
impact on the calorimeter performance of the sampling fraction and the
longitudinal and transverse segmentation are uncorrelated. The dimensions are
$1\times 1\times 2.5$ m3, which ensures complete shower containment within the
calorimeter volume and corresponds to $10\,\lambda$ and $200\,X_{0}$ of total
depth. The longitudinal and transverse segmentation is increased from no
segmentation up to $30\times 30$ segments in $x$ and $y$, and from 1 to 60
segments in the lateral direction. A list of the configurations can be found
in Table 1.

The data set consists of approximately $5\times 10^{6}$ charged pion events,
generated using the FTFP_BERT physics list of Geant4 10.04 patch 0. The
training data set comprises pions with energies sampled from a flat
distribution between 1 and 110 GeV. The test data set covers 11 discrete
energies of 5 to 105 GeV in 10 GeV steps. The Geant4 simulation has been
performed in the highest granularity, while for the tests and training of
different segmentation configurations, the same dataset has been used. For
this purpose, the energy deposits in the cells have been merged corresponding
to the tested cell sizes. This method avoids inconsistencies that are
otherwise to be expected due to the different number of surfaces and material
borders through which Geant4 propagates the particles.

Stage | Longitudinal | Depth of layers
---|---|---
segments | in cm | in ${\lambda}_{\pi}$ | in $X_{0}$
0 | 1 | 250 | 9.8 | 198
1 | 6 | 41.7 | 1.6 | 33
2 | 10 | 25 | 1.0 | 20
3 | 12 | 20.8 | 0.8 | 16
4 | 15 | 16.7 | 0.7 | 13
5 | 20 | 12.5 | 0.5 | 10
6 | 30 | 8.3 | 0.3 | 7
7 | 60 | 4.2 | 0.2 | 3
Stage | Transverse | Size of cells
segments | in cm2 | in $\lambda_{n}$ | in $X_{0}$
default | $1\times 1$ | $100\times 100$ | $3.9\times 3.9$ | $79\times 79$
A | $3\times 3$ | $33\times 33$ | $1.3\times 1.3$ | $26\times 26$
B | $5\times 5$ | $20\times 20$ | $0.8\times 0.8$ | $16\times 16$
C | $10\times 10$ | $10\times 10$ | $0.4\times 0.4$ | $8\times 8$
D | $15\times 15$ | $6.7\times 6.7$ | $0.3\times 0.3$ | $5\times 5$
E | $30\times 30$ | $3.3\times 3.3$ | $0.1\times 0.1$ | $3\times 3$
Table 1: Granularity configurations considered in this analysis.

##  3 Neural network architecture and training

At the core of the neural network architecture used here is a software
compensation block that uses convolutional neural network (CNN) layers
lecun1998gradient  to achieve local identification of the subshowers, similar
to the one introduced in Ref. Neubuser:2705432 , which is used as a subblock
in the overall model. This subblock consists of 3 parallel paths: in the first
path, the energy of all cells within the kernel range $K$ is summed up and
forwarded to the next block, while this kernel is moved with a stride of size
$K$; the second path consists of a CNN layer with the same kernel size and
$F=16$ filters; the third path contains in total three subsequent CNN layers,
out of which the first two have kernel sizes (in x, y, and depth) of
$K_{a}=(1,\ k,\ 3)$ and $K_{b}=(k,\ 1,\ 3)$, with no stride and 32 filters,
each. Here, $k$ is an adjustable parameter depending on granularity, as
described later. The final layer of this path is a CNN layer with a kernel
size of $K$ with a stride of $K$ and $F$ filters, such that the output of all
paths can be combined. This combination is done by adding the output of the
CNN layers of all paths feature by feature. All layers use a tanh activation
function. The weights of the layers in the third path are initialised with a
Gaussian distribution centred at 0 with a width of $10^{-3}$, and receive a
small $L2$ regularisation of $10^{-5}$. This structure is optimised to derive
small corrections to the simple energy sum by detecting the different shower
shape of electromagnetic subshowers.

In the final model, the input is passed through a batch normalisation layer
ioffe2015batch , normalising all inputs except for the per-cell energy. If
less than 6 calorimeter layers are present or the transverse granularity in
either direction is less than 6, the input is directly flattened and passed to
3 dense layers, the first two of which contain 128 and 64 nodes using ELU
activation elu_activation , before being finally passed to the energy
prediction layer with 1 node. In all other cases, the input is first passed
through a set of the subblocks described above before being fed through the
same structure with dense layers. These subblocks adapt to the input: if the
corresponding granularity is less than $6\times 6$ cells in the transverse
directions, a stride of $1\times 1$ is used, and the input $k$ for the kernel
size determination is set to $k=1$. Otherwise, a stride of $2\times 2$ and
$k=3$ are used in these directions. The subblock is repeated until the
dimensionality in $x$, $y$, or depth is less than or equal to 6. At this
point, the output is fed to the three final dense layers.

The model is trained using the Adam optimiser kingma2014adam  using TensorFlow
tensorflow  and Keras keras  within the DeepJetCore framework DJC . The
training consists of five steps: the first four steps use a loss function
$L_{\mathrm{calo}}$ that follows the expected calorimeter resolution:

$L_{\mathrm{calo}}=\frac{(E_{\mathrm{true}}-E_{\mathrm{pred}})^{2}}{E_{\mathrm{true}}}\mathrm{.}$
(2)

These steps are trained for 1, 19, 60, and 20 epochs with learning rates of
$10^{-4}$, $10^{-4}$, $10^{-5}$, and $10^{-5}$, and batch sizes of 256, 512,
1280, and 1280. Between the third and fourth step, the batch normalisation is
frozen.

The mean and expectation value for $E_{\mathrm{true}}$ differ at the edges of
the training sample. This typically leads to edge effects, which introduce a
bias towards higher predicted values at the low edge, and towards lower
predicted values at the high edge.

To mitigate the effect, we freeze all layers except for the last dense layers,
and introduce a loss that follows a $\chi^{2}$ distribution taking the
difference of the average predicted and truth energy in bins of
$E_{\mathrm{true}}$, and accounting for the number of samples in that bin. The
bin boundaries are randomly chosen for each batch to avoid a global bias.
Using this loss, the model is trained for another 50 epochs with a learning
rate of $10^{-5}$ and a batch size of 1280.

(a)  (b)

Figure 1: Results for a scenario with 15 longitudinal layers (stage 4) and no
transverse segmentation. (a) Energy distribution for 45 GeV pions. The width
as computed from the Gaussian fit (black line) and from the RMS are shown. (b)
Energy resolution as a function of the particle energy. The resolution is
computed two ways, using the Gaussian fit (open circles) and using the RMS
(filled squares).

##  4 Results

The energy resolution is evaluated as the ratio of the width to the most
probable value of the distribution of the reconstructed energy. These
distributions, as shown for example in Figure 1a, follow a Gaussian function.
The standard deviation can thus be extracted from a fit. This fit is limited
within 2$\sigma$ around the most probable value $\mu$, following the procedure
widely used in calorimeter performance studies. As a comparison and
validation, the energy resolution has also been evaluated from the root mean
square (RMS) and mean, which is sensitive to the tails of the distribution.
The energy resolution over the full available energy range is shown for stage
4, which corresponds to a granularity of 15 longitudinal layers, in Figure 1b.
The points are fitted following Equation 1, and the values of the stochastic
and constant term are shown in the legend. An overall 10–20% degradation in
energy resolution from the Gaussian fit to the RMS method is observed.
In the following, the energy resolutions obtained for different granularities
will refer to the results obtained from the Gaussian fit.

The results, in terms of the stochastic term $\alpha$ and constant term $c$
for all studied longitudinal and transverse granularities, are summarized in
Table 2. The theory of the different contributions to the energy resolution of
hadronic showers Fabjan:1989ti  considers that the stochastic term is in fact
a sum of two major effects,
$\alpha=\alpha_{\mathrm{int}}\oplus\alpha_{\mathrm{sampl}}$, where the first
intrinsic term is irreducible and determined by the fluctuations of the
initial energy that is transformed into ionising shower particles, and the
second is the term due to the sampling fraction. These losses are material
dependent, due to material-dependent nuclear binding energy losses, and have
been found to be on the order of 19%/11% in the ZEUS uranium/lead-scintillator
calorimeter prototypes Tiecke:1989nz .

We assume that the DNN is able to identify and re-weight the electromagnetic
and hadronic shower fractions, due to the topological differences of EM and
hadronic subshowers ($\lambda_{\pi}/X_{0}\sim 27$). Thus, we expect the
constant term to decrease. Table 2 shows the resulting measured stochastic and
constant terms (using both the Gaussian fit and the RMS to obtain the
resolution) for three different sets of scenarios: first, the different
longitudinal granularities with no transverse segmentation, the results for
which are plotted in Figure 2; second, longitudinal stage 0 with different
transverse granularities (Figure 3); and third, longitudinal stage 5 with
different transverse granularities (Figure 4). Overall, at the finest
granularities, we observe that the constant term goes to zero, while the
stochastic term decreases by approximately 50% with respect to the scenario
with no segmentation, reaching a minimum of 8%, which can be considered the
intrinsic stochastic term $\alpha_{\mathrm{int}}$.

The constant term is consistently removed as soon as the first segmentation in
transverse granularity into $3\times 3$ cells is implemented. Figure 5 shows
an event display of a 35 GeV pion shower; the bottom shows the impact of a
$3\times 3$ transverse segmentation. We can see that already at this stage, a
significant enough energy fraction of 9% (shown as
$E_{\mathrm{out}}/E_{\mathrm{tot}}$ in the legend) is found in the outer
quadrants. In comparison, the same shower is represented in 3D on the top, and
visualises the imaging power of the finest chosen granularity of the
homogeneous PbW calorimeter.

stage | stochastic term [%] | constant term [%]
---|---|---
| RMS | Gauss | RMS | Gauss
0 | 20.5 | 17.3 | 3.0 | 2.6
1 | 19.7 | 15.9 | 2.2 | 2.0
2 | 17.8 | 14.0 | 1.8 | 1.5
3 | 17.2 | 13.6 | 1.7 | 1.3
4 | 16.0 | 12.9 | 1.5 | 1.1
5 | 15.4 | 12.1 | 1.3 | 0.8
6 | 14.6 | 11.6 | 1.1 | 0.6
7 | 13.0 | 10.9 | 1.0 | 0.5
0A | 20.3 | 15.1 | 1.3 | 0
0B | 20.0 | 14.6 | 1.2 | 0
0C | 18.2 | 13.6 | 1.3 | 0
0D | 18.6 | 13.6 | 1.1 | 0
0E | 17.9 | 13.4 | 1.3 | 0
5A | 11.4 | 8.6 | 0.6 | 0
5B | 10.6 | 8.1 | 0.6 | 0
5C | 11.0 | 8.1 | 0 | 0
5D | 10.9 | 7.9 | 0 | 0
5E | 10.9 | 7.9 | 0 | 0
Table 2: Summary of energy resolution fit results. The top set shows the
different longitudinal segmentation scenarios with no transverse segmentation,
while the other two sets show two specific longitudinal stages with different
transverse segmentation scenarios, as described in Table 1.

(a)  (b)

Figure 2: Energy resolution (a) and linearity (b) for different longitudinal
granularities and no transverse segmentation. The curves correspond to the fit
with Equation 1.

(a)  (b)

Figure 3: Energy resolution (a) and linearity (b) for different transverse
granularities with 1 longitudinal layer (stage 0). The curves show the fit to
the form given in Equation 1.

(a)  (b)

Figure 4: Energy resolution (a) and linearity (b) for different transverse
granularities with 20 longitudinal layers (stage 5). The curves show the fit
to the form given in Equation 1.

(a)  (b)

Figure 5: Event display of a 35 GeV pion shower in the homogeneous PbW
calorimeter. (a) A 3D view at the finest granularity. (b) A front view of the
same shower, with a grid overlaid corresponding to the coarsest applied
transverse segmentation.

Figure 6 summarizes the energy resolution as a function of longitudinal and
transverse granularity. We observe that the behavior of the resolution as a
function of granularity exhibits the same pattern regardless of the incident
particle energy. For the transverse granularity, the resolution reaches an
optimal value at a cell size of $\approx 1\lambda_{\pi}$, and finer
segmentation does not yield any appreciable further benefit. In the
longitudinal direction, the energy resolution continues to improve as the
layer size is decreased, reaching the minimum at the finest granularity
considered ($\approx 0.2\lambda_{\pi}$ or $\approx 3X_{0}$).

Figure 7 summarizes the fitted parameters $\alpha$ and $c$ in the energy
resolution function in Equation 1, as a function of longitudinal and
transverse granularity. In the transverse direction, we observe that the
constant term goes to zero at a cell size of $\approx 1\lambda_{\pi}$
($25X_{0}$), and further decrease in the cell size does not further improve
the stochastic term $\alpha$. In the longitudinal case, a layer width in the
region 7–10 $X_{0}$ appears to offer the best balance between the obtained
resolution and the detector complexity.

(a)  (b)

Figure 6: Energy resolution as a function of the longitudinal (a) and
transverse (b) granularity. Three different particle energies are considered:
15 GeV (red circles), 55 GeV (purple squares), and 85 GeV (dark blue
triangles). In the upper plot, no transverse segmentation is used, while on
the bottom, two different longitudinal segmentations are shown: 1 layer
(dashed lines) and 20 layers (solid lines).

(a)  (b)

Figure 7: Values of the parameters $\alpha$ (black line) and $c$ (blue line)
in the energy resolution function
$\frac{\sigma_{E}}{\left<E\right>}=\frac{\alpha}{E}\oplus c$ as a function of
the longitudinal (a) and transverse (b) granularity. In the upper plot, no
transverse segmentation is used, while on the bottom, two different
longitudinal segmentations are shown: 1 layer (dashed lines) and 20 layers
(solid lines).

##  5 Conclusions

When calorimeters are designed for new high-energy physics experiments, often
the approach has been to pick a technology before optimising the
reconstruction of jet particles. From the perspective of testing various
options, this not only requires significant computing power due to the
introduced details of signal processing (digitisation) in the simulations, but
also means that the simulations are unable to answer basic questions due to
the high complexity. For example, a smaller cell size improves the spatial and
pointing resolution, which should help the particle-flow algorithm to
reconstruct the jet. However, the signal height per cell decreases, which
introduces an energy loss due to a lower signal-to-noise ratio. Thus, a high-
level optimisation becomes blind to the individual impact for each effect.
Instead, a different approach could be to first identify the necessary input
for reconstruction algorithms which allows for optimal performance, before
selecting the detector technology.

Moving towards that approach, we have defined a model calorimeter to identify
the necessary cell granularity for a DNN to perform an optimal energy
reconstruction. In this model, the impact of the sampling fraction has been
intentionally excluded. Even though we are aware that the type of chosen
active and passive material will impact the shower development, we believe
that this study can be used in order to design a future hadronic calorimeter
which allows for optimal energy measurements using DNNs.

These studies suggest that a hadronic calorimeter (with
$\lambda_{\pi}/X_{0}\sim 27$) should feature cell sizes of at most 1 nuclear
interaction length, and longitudinal layers of 7–10 $X_{0}$ thickness, in
order to allow for an optimal software compensation and thus to reach the
intrinsic stochastic term of 8%. Following this approach, one could imagine
further study to determine the optimal cell and layer sizes as a function of
the $\lambda_{\pi}/X_{0}$ ratio. However, this exceeds the scope of this
paper.

## Acknowledgments

The training of the models was performed on the GPU clusters of the CERN CMG
group.

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