FILTRA: Rethinking Steerable CNN by Filter Transform
Appendix

(LABEL:eq:irrep-cn-kernel) can be verified to follow Lemma LABEL:lm:condition as:

$\displaystyle\begin{split}&\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}(\phi+\theta_{i_{1}})=\operatorname{diag}\big{(}P(i_{1})\mathsf{K}\big{)}\beta_{k},\quad\text{c.f. \eqref{eq:cn-trivial-proof2}}\\ &=P(i_{1})\operatorname{diag}(\mathsf{K})P(i_{1})^{-1}\beta_{k}=\rho^{\mathrm{C}_{N}}_{\text{reg}}(g)\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}\psi_{0,k}(g)^{-1},\quad\text{c.f. \eqref{eq:beta-rotate0}.}\end{split}$

(27)

We can also verify this for

$$\overline{\mathsf{K}}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}=\operatorname{diag}(\overline{\mathsf{K}})\beta_{k}.$$

(28)

First note it is easy to verify that for $i_{0}=0$, i.e. $g=(0,i_{1})$, the Lemma LABEL:lm:condition holds in the same way as (LABEL:eq:cn-irrep-proof),

$\displaystyle\mathsf{K}^{\mathrm{D}_{N}}_{j,k\rightarrow\text{reg}}(\phi+\theta)$

$\displaystyle=\rho^{\mathrm{D}_{N}}_{\text{reg}}(g)\mathsf{K}^{\mathrm{D}_{N}}_{j,k\rightarrow\text{reg}}{\psi_{j,k}(g)}^{-1}.$

(29)

We then generalize (LABEL:eq:trivial-exchange) on $\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}$ and $\overline{\mathsf{K}}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}$ given a reflected action $g=(1,i_{1})$:

	$\displaystyle\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}(-\phi+\theta_{i_{1}})$	$\displaystyle=\operatorname{diag}\big{(}\mathsf{K}(-\phi+\theta_{i_{1}})\big{)}\beta_{k}=B(i_{1})\operatorname{diag}(\overline{\mathsf{K}})B(i_{1})^{-1}\beta_{k},\quad\text{c.f. \eqref{eq:beta-rotate1}}$		(30a)
		$\displaystyle=B(i_{1})\operatorname{diag}(\overline{\mathsf{K}})\beta_{k}\psi_{0,k}(g)^{-1}=-B(i_{1})\operatorname{diag}(\overline{\mathsf{K}}^{\mathrm{C}_{N}})\beta_{k}\psi_{1,k}(g)^{-1}$		(30b)
		$\displaystyle=B(i_{1})\operatorname{diag}(\mathsf{K}^{\mathrm{C}_{N}})\beta_{k}\psi_{0,k}(g)^{-1}=-B(i_{1})\operatorname{diag}(\mathsf{K}^{\mathrm{C}_{N}})\beta_{k}\psi_{1,k}(g)^{-1}.$		(30c)

Note that (30b) and (30c) both have two equivalent forms denoted with $\psi_{0,k}(g)$ and $\psi_{1,k}(g)$ respectively. Now we can show $\mathsf{K}^{\mathrm{D}_{N}}_{j,k\rightarrow\text{reg}}$ follows Lemma LABEL:lm:condition for $j=0$, $i_{0}=1$, i.e. $g=(1,i_{1})$ as:

	$\displaystyle\mathsf{K}^{\mathrm{D}_{N}}_{j,k\rightarrow\text{reg}}(-\phi+\theta_{i_{1}})=\begin{bmatrix}{\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}(-\phi+\theta_{i_{1}})}^{\top}&{\overline{\mathsf{K}}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}}(-\phi+\theta_{i_{1}})^{\top}\end{bmatrix}^{\top}$		(31a)
	$\displaystyle=\begin{bmatrix}B(i_{1})\operatorname{diag}(\overline{\mathsf{K}}^{\mathrm{C}_{N}})\beta_{k}\psi_{0,k}(g)^{-1}&B(i_{1})\operatorname{diag}(\mathsf{K}^{\mathrm{C}_{N}})\beta_{k}\psi_{0,k}(g)^{-1}\end{bmatrix}^{\top}\quad\text{c.f. \eqref{eq:irrep-exchange}}$		(31b)
	$\displaystyle=\rho^{\mathrm{D}_{N}}_{\text{reg}}(g)\begin{bmatrix}{\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}}^{\top}&{\overline{\mathsf{K}}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}}^{\top}\end{bmatrix}^{\top}\psi_{0,k}(g)^{-1}\quad\text{c.f. \eqref{eq:irrep-cn-kernel}, \eqref{eq:irrep-cn-kernel-conj}}$		(31c)
	$\displaystyle=\rho^{\mathrm{D}_{N}}_{\text{reg}}(g)\mathsf{K}^{\mathrm{D}_{N}}_{j,k\rightarrow\text{reg}}\psi_{0,k}(g)^{-1}.$		(31d)

The verification is similar for $j=1$, $i_{0}=1$, i.e. $g=(1,i_{1})$:

	$\displaystyle\mathsf{K}^{\mathrm{D}_{N}}_{j,k\rightarrow\text{reg}}(-\phi+\theta_{i_{1}})=\begin{bmatrix}{\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}(-\phi+\theta_{i_{1}})}^{\top}&-{\overline{\mathsf{K}}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}}(-\phi+\theta_{i_{1}})^{\top}\end{bmatrix}^{\top}$		(32a)
	$\displaystyle=\begin{bmatrix}-B(i_{1})\operatorname{diag}(\overline{\mathsf{K}}^{\mathrm{C}_{N}})\beta_{k}\psi_{1,k}(g)^{-1}&B(i_{1})\operatorname{diag}(\mathsf{K}^{\mathrm{C}_{N}})\beta_{k}\psi_{1,k}(g)^{-1}\end{bmatrix}^{\top}\quad\text{c.f. \eqref{eq:irrep-exchange}}$		(32b)
	$\displaystyle=\rho^{\mathrm{D}_{N}}_{\text{reg}}(g)\begin{bmatrix}{\mathsf{K}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}}^{\top}&-{\overline{\mathsf{K}}^{\mathrm{C}_{N}}_{k\rightarrow\text{reg}}}^{\top}\end{bmatrix}^{\top}\psi_{0,k}(g)^{-1}\quad\text{c.f. \eqref{eq:irrep-cn-kernel}, \eqref{eq:irrep-cn-kernel-conj}}$		(32c)
	$\displaystyle=\rho^{\mathrm{D}_{N}}_{\text{reg}}(g)\mathsf{K}^{\mathrm{D}_{N}}_{j,k\rightarrow\text{reg}}\psi_{0,k}(g)^{-1}.$		(32d)

This kernel can be verified as follows for $g=(0,i_{1})$:

	$\displaystyle\mathsf{K}^{\mathrm{C}_{N}}_{\text{reg}\rightarrow\text{reg}}(\phi+\theta_{i_{1}})=\begin{bmatrix}\rho^{\mathrm{C}_{N}}_{\text{reg}}(g)\mathsf{K}^{\mathrm{C}_{N}}_{0\rightarrow\text{reg}}\psi_{0,0}(g)^{-1},\cdots,\rho^{\mathrm{C}_{N}}_{\text{reg}}(g)\mathsf{K}^{\mathrm{C}_{N}}_{\lfloor\frac{N}{2}\rfloor\rightarrow\text{reg}}\psi_{0,\lfloor\frac{N}{2}\rfloor}(g)^{-1}\end{bmatrix}V^{-1}$		(33a)
	$\displaystyle=\rho^{\mathrm{C}_{N}}_{\text{reg}}(g)\begin{bmatrix}\mathsf{K}^{\mathrm{C}_{N}}_{0\rightarrow\text{reg}}\cdots\mathsf{K}^{\mathrm{C}_{N}}_{\lfloor\frac{N}{2}\rfloor\rightarrow\text{reg}}\end{bmatrix}D^{\mathrm{C}_{N}}V^{-1}$		(33b)
	$\displaystyle=\rho^{\mathrm{C}_{N}}_{\text{reg}}(g)\begin{bmatrix}\mathsf{K}^{\mathrm{C}_{N}}_{0\rightarrow\text{reg}}\cdots\mathsf{K}^{\mathrm{C}_{N}}_{\lfloor\frac{N}{2}\rfloor\rightarrow\text{reg}}\end{bmatrix}V^{-1}VD^{\mathrm{C}_{N}}V^{-1}=\rho^{\mathrm{C}_{N}}_{\text{reg}}(g)\mathsf{K}^{\mathrm{C}_{N}}_{\text{reg}\rightarrow\text{reg}}{\rho^{\mathrm{C}_{N}}_{\text{reg}}}^{-1}.$		(33c)