$\displaystyle\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\frac{\mu_{1}((\tilde{K}^{\gamma}_{>k})^{-1})}{\mu_{n}((\tilde{K}^{\gamma}_{>k})^{-1})^{2}}\frac{\mu_{1}(\psi_{\leq
k}\hat{S}^{*}_{n}\hat{S}_{n}\psi^{*}_{\leq k})}{\mu_{k}(\psi_{\leq
k}\hat{S}^{*}_{n}\hat{S}_{n}\psi^{*}_{\leq k})^{2}}\|\phi_{\leq k}f^{*}_{\leq
k}\|_{\Lambda^{\leq k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}\Bigg{)}.$

###### Proof.

Similar as variance, by lemma A.4 we can bound $\leq k$ and $>k$ separately,
for brevity we define the error vector
$\xi:=\phi(\hat{f}(\hat{S}_{n}\mathcal{A}f^{*})-f^{*})\in\mathbb{R}^{\infty}$,
by lemma A.3 we can bound $\|\xi_{\leq k}\|_{\Sigma^{1-\beta^{\prime}}}$ and
$\|\xi_{>k}\|_{\Sigma^{1-\beta^{\prime}}}$ separately.

We first discuss $\|\xi_{\leq k}\|_{\Sigma^{1-\beta^{\prime}}}$, by lemma A.2,
we have

$\displaystyle\phi_{\leq k}\hat{f}(\hat{S}_{n}\mathcal{A}f^{*})+\phi_{\leq
k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}\hat{f}(\hat{S}_{n}\mathcal{A}f^{*})_{\leq
k}=\phi_{\leq k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}f^{*}.$
(7)

By definition of $\xi$, we have $\xi_{\leq k}=\phi_{\leq
k}(\hat{f}-f^{*})=\phi_{\leq k}\hat{f}_{\leq k}-\phi_{\leq k}f^{*}_{\leq k}$,
so we have $\phi_{\leq k}\hat{f}=\xi_{\leq k}+\phi_{\leq k}f^{*}_{\leq k}$.

$\displaystyle\text{LHS of (\ref{eq:biasproof})}=$ $\displaystyle\ \xi_{\leq
k}+\phi_{\leq k}f^{*}_{\leq k}+\phi_{\leq k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k}\Lambda_{\mathcal{A}}^{\leq k}(\xi_{\leq k}+\phi_{\leq k}f^{*}_{\leq k})$
$\displaystyle=$ $\displaystyle\ \xi_{\leq k}+\phi_{\leq k}f^{*}_{\leq
k}+\phi_{\leq k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k}\Lambda_{\mathcal{A}}^{\leq k}\xi_{\leq k}$ $\displaystyle+$
$\displaystyle\underbrace{\phi_{\leq k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k}\Lambda_{\mathcal{A}}^{\leq k}\phi_{\leq k}f^{*}_{\leq k}}_{\text{(*)}}.$

And

$\displaystyle\text{RHS of (\ref{eq:biasproof})}=$ $\displaystyle\ \phi_{\leq
k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}(\phi_{\leq
k}^{*}\Lambda^{\leq k}_{\mathcal{A}}\phi_{\leq k}f^{*}_{\leq
k}+\phi_{>k}^{*}\Lambda^{>k}_{\mathcal{A}}\phi_{>k}f^{*}_{>k})$
$\displaystyle=$ $\displaystyle\ \underbrace{\phi_{\leq k}\mathcal{A}_{\leq
k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi_{\leq
k}^{*}\Lambda^{\leq k}_{\mathcal{A}}\phi_{\leq k}f^{*}_{\leq k}}_{\text{(*)}}$
$\displaystyle+$ $\displaystyle\phi_{\leq k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi_{>k}^{*}\Lambda^{>k}_{\mathcal{A}}\phi_{>k}f^{*}_{>k}.$

The two (*) terms get cancelled out, therefore

$\displaystyle\xi_{\leq k}$ $\displaystyle+\phi_{\leq k}\mathcal{A}_{\leq
k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k}\Lambda_{\mathcal{A}}^{\leq k}\xi_{\leq k}$ $\displaystyle=\phi_{\leq
k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{>k}\Lambda_{\mathcal{A}}^{>k}\phi_{>k}f^{*}_{>k}-\phi_{\leq
k}f^{*}_{\leq k}.$

We multiply $\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1-\beta-\beta^{\prime}/2}}$ in both sides and
since $\|\xi_{\leq k}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1-\beta-\beta^{\prime}/2}}}\geq 0$,

$\displaystyle\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1-\beta-\beta^{\prime}/2}}\phi_{\leq
k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k}\Lambda_{\mathcal{A}}^{\leq k}\xi_{\leq k}$ $\displaystyle\leq$
$\displaystyle\ \xi_{\leq k}^{T}\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1-\beta-\beta^{\prime}/2}}\phi_{\leq
k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{>k}\Lambda_{\mathcal{A}}^{>k}\phi_{>k}f^{*}_{>k}-\xi_{\leq
k}^{T}\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1-\beta-\beta^{\prime}/2}}\phi_{\leq k}f^{*}_{\leq
k}.$

LHS is the quadratic term w.r.t. $\xi_{\leq k}$ and RHS is the linear term
w.r.t. $\xi_{\leq k}$, similar to Variance case, we lower bound LHS and upper
bound RHS respectively.

LHS $\displaystyle=\overbrace{\xi_{\leq k}^{T}}^{1\times
k}\overbrace{\Lambda^{\leq k}_{\Sigma^{-\beta^{\prime}/2}}}^{k\times
k}\overbrace{\phi_{\leq k}\hat{S}_{n}^{*}}^{k\times
n}\overbrace{(\tilde{K}_{>k}^{\gamma})^{-1}}^{n\times
n}\overbrace{\hat{S}_{n}\phi^{*}_{\leq k}}^{n\times
k}\overbrace{\Lambda_{\mathcal{A}}^{\leq k}}^{k\times k}\overbrace{\xi_{\leq
k}}^{k\times 1}$ $\displaystyle=\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\Sigma^{(1-\beta^{\prime})/2}}\psi_{\leq
k}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k}\Lambda^{\leq k}_{\mathcal{A}\Sigma^{1/2}}\xi_{\leq k}.$

Since $(1-\beta^{\prime})+\beta^{\prime}/2=(1-\beta^{\prime})/2+1/2$, it can
be lower bounded by

$\displaystyle\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})\ (\xi_{\leq
k}^{T}\Lambda^{\leq k}_{\Sigma^{1-\beta^{\prime}}}\xi_{\leq
k})\mu_{k}\left(\psi_{\leq k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq
k}\right)\mu_{k}\left(\Lambda^{\leq
k}_{\mathcal{A}\Sigma^{\beta^{\prime}/2}}\right)$ $\displaystyle=\|\xi_{\leq
k}\|_{\Lambda^{\leq
k}_{\Sigma^{1-\beta^{\prime}}}}^{2}\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})\mu_{k}\left(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k}\right)\mu_{k}\left(\Lambda^{\leq
k}_{\mathcal{A}\Sigma^{\beta^{\prime}/2}}\right).$

Next we upper bound RHS, first we bound the first term in RHS

First term in RHS $\displaystyle=\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1-\beta-\beta^{\prime}/2}}\phi_{\leq
k}\mathcal{A}_{\leq k}\Sigma_{\leq
k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{>k}\Lambda_{\mathcal{A}}^{>k}\phi_{>k}f^{*}_{>k}$
$\displaystyle=\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\Sigma^{-\beta^{\prime}/2}}\phi_{\leq
k}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{>k}\Lambda_{\mathcal{A}}^{>k}\phi_{>k}f^{*}_{>k}.$

Since $(1-\beta^{\prime})/2-1/2=-\beta^{\prime}/2$, it equals to

$\displaystyle\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\Sigma^{(1-\beta^{\prime})/2}}\Lambda^{\leq k}_{\Sigma^{-1/2}}\phi_{\leq
k}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}$
$\displaystyle=\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\Sigma^{(1-\beta^{\prime})/2}}\psi_{\leq
k}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}$
$\displaystyle\leq\|\xi_{\leq k}\|_{\Lambda^{\leq
k}_{\Sigma^{(1-\beta^{\prime})}}}\mu_{1}((\tilde{K}_{>k}^{\gamma})^{-1})\sqrt{\mu_{1}(\underbrace{\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k}}_{k\times
k})}\|\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|.$

Then we bound the second term in RHS.

Second term in RHS $\displaystyle=\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1-\beta-\beta^{\prime}/2}}\phi_{\leq k}f_{\leq
k}^{*}=\xi_{\leq k}^{T}\Lambda^{\leq
k}_{\Sigma^{(1-\beta^{\prime})/2}}\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1/2-\beta}}\phi_{\leq k}f^{*}_{\leq k}$
$\displaystyle\leq\|\xi_{\leq k}\|_{\Lambda^{\leq
k}_{\Sigma^{1-\beta^{\prime}}}}\|\phi_{\leq k}f^{*}_{\leq k}\|_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}.$

Therefore, gather the terms we have

$\displaystyle\|\xi_{\leq k}\|_{\Lambda^{\leq
k}_{\Sigma^{1-\beta^{\prime}}}}^{2}\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})\mu_{k}\left(\Lambda^{\leq
k}_{\mathcal{A}^{1/2}\Sigma^{\beta^{\prime}/4}}\psi_{\leq
k}\hat{S}_{n}^{*}(\tilde{K}_{>k}^{\gamma})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k}\Lambda^{\leq k}_{\mathcal{A}^{1/2}\Sigma^{\beta^{\prime}/4}}\right)$
$\displaystyle\leq$ $\displaystyle\ \ \|\xi_{\leq k}\|_{\Lambda^{\leq
k}_{\Sigma^{(1-\beta^{\prime})}}}\mu_{1}((\tilde{K}_{>k}^{\gamma})^{-1})\sqrt{\mu_{1}(\underbrace{\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k}}_{k\times
k})}\|\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|$ $\displaystyle+\ \|\xi_{\leq
k}\|_{\Lambda^{\leq k}_{\Sigma^{1-\beta^{\prime}}}}\|\phi_{\leq k}f^{*}_{\leq
k}\|_{\Lambda^{\leq k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}.$

So

$\displaystyle\|\xi_{\leq k}\|_{\Lambda^{\leq k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle\leq\frac{\mu_{1}((\tilde{K}_{>k}^{\gamma})^{-1})}{\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})}\frac{\sqrt{\mu_{1}(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k})}}{\mu_{k}\left(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k}\right)\mu_{k}\left(\Lambda^{\leq
k}_{\mathcal{A}\Sigma^{\beta^{\prime}/2}}\right)}\|\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|$
$\displaystyle+\frac{\|\phi_{\leq k}f^{*}_{\leq k}\|_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})\mu_{k}\left(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k}\right)\mu_{k}\left(\Lambda^{\leq
k}_{\mathcal{A}\Sigma^{\beta^{\prime}/2}}\right)}.$

By $\|a+b\|^{2}\leq 2(\|a\|^{2}+\|b\|^{2})$, we can bound $\|\xi_{\leq
k}\|^{2}_{\Sigma^{1-\beta^{\prime}}}$ by

$\displaystyle
2\Bigg{(}\frac{\mu_{1}((\tilde{K}_{>k}^{\gamma})^{-1})^{2}}{\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})^{2}}\frac{\mu_{1}(\overbrace{\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k}}^{k\times
k})}{\mu_{k}(\psi_{\leq k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq
k})^{2}\mu_{k}(\Lambda^{\leq
k}_{\mathcal{A}^{2}\Sigma^{\beta^{\prime}}})}\|\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|^{2}$
$\displaystyle+\frac{\|\phi_{\leq k}f^{*}_{\leq k}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{\mu_{k}(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k})^{2}\mu_{k}(\Lambda^{\leq
k}_{\mathcal{A}^{2}\Sigma^{\beta^{\prime}}})}\Bigg{)}.$

Now we discuss the $>k$ case, which is more complicated, we bound it by three
quantities by the fact that $(A+B+C)^{2}\leq 3(A^{2}+B^{2}+C^{2})$ and bound
them respectively as follows

$\displaystyle\|\phi_{>k}f^{*}_{>k}-\phi_{>k}\mathcal{A}_{>k}\Sigma_{>k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}f^{*}\|_{\Lambda_{\Sigma^{1-\beta^{\prime}}}^{>k}}^{2}$
$\displaystyle\leq$ $\displaystyle
3(\|\phi_{>k}f^{*}_{>k}\|_{\Lambda_{\Sigma^{1-\beta^{\prime}}}^{>k}}^{2}+\|\phi_{>k}\mathcal{A}_{>k}\Sigma_{>k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|_{\Lambda_{\Sigma^{1-\beta^{\prime}}}^{>k}}^{2}+\|\phi_{>k}\mathcal{A}_{>k}\Sigma_{>k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}_{\leq
k}f^{*}_{\leq k}\|_{\Lambda_{\Sigma^{1-\beta^{\prime}}}^{>k}}^{2}).$

We first bound the second term

$\displaystyle\|\phi_{>k}\mathcal{A}_{>k}\Sigma_{>k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|_{\Lambda_{\Sigma^{1-\beta^{\prime}}}^{>k}}^{2}$
$\displaystyle\leq$ $\displaystyle\
\|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\|\phi_{>k}\mathcal{A}_{>k}\Sigma_{>k}^{\beta-1}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|^{2}$
$\displaystyle=$ $\displaystyle\ \|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\|\Lambda^{>k}_{\mathcal{A}\Sigma^{\beta-1}}\phi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\phi_{>k}^{*}\Lambda_{\mathcal{A}}^{>k}\phi_{>k}f^{*}_{>k}\|^{2}$
$\displaystyle\leq$ $\displaystyle\
\|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\mu_{1}[(\tilde{K}^{\gamma})^{-1}]^{2}\|\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|^{2}\mu_{1}(\underbrace{\hat{S}_{n}\phi_{>k}^{*}\Lambda^{>k}_{\mathcal{A}^{2}\Sigma^{2(\beta-1)}}\phi_{>k}\hat{S}^{*}_{n}}_{n\times
n})$ $\displaystyle\leq$ $\displaystyle\
\|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\mu_{1}[(\tilde{K}_{>k}^{\gamma})^{-1}]^{2}\|\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|^{2}\mu_{1}(\underbrace{\hat{S}_{n}\phi_{>k}^{*}\Lambda^{>k}_{\mathcal{A}^{2}\Sigma^{2(\beta-1)}}\phi_{>k}\hat{S}^{*}_{n}}_{n\times
n}).$

The last inequality is by
$\mu_{1}((\tilde{K}_{>k}^{\gamma})^{-1})\geq\mu_{1}((\tilde{K}^{\gamma})^{-1})$.

Then we move on to bound the third term, that is, we want to bound

$\displaystyle\|\phi_{>k}\mathcal{A}_{>k}\Sigma^{\beta-1}_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\mathcal{A}_{\leq
k}f^{*}_{\leq k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle=$ $\displaystyle\
\|\Lambda^{>k}_{\mathcal{A}\Sigma^{\beta-1}}\phi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k}\Lambda^{\leq k}_{\mathcal{A}}\phi_{\leq k}f^{*}_{\leq
k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}.$

First we deal with $(\tilde{K}^{\gamma})^{-1}(\hat{S}_{n}\phi_{\leq k}^{*})$
first, we can write it as

$(\tilde{K}^{\gamma})^{-1}(\hat{S}_{n}\phi_{\leq
k}^{*})=(\tilde{K}_{>k}^{\gamma}+(\hat{S}_{n}\phi_{\leq k}^{*})\Lambda^{\leq
k}_{\mathcal{A}^{2}\Sigma^{\beta-1}}(\phi_{\leq
k}\hat{S}_{n}^{*}))^{-1}(\hat{S}_{n}\phi_{\leq k}^{*}),$

then apply A.6 with $A=\tilde{K}_{>k}^{\gamma}$, $U=\hat{S}_{n}\phi_{\leq
k}^{*}$, $C=\Lambda^{\leq k}_{\mathcal{A}^{2}\Sigma^{\beta-1}}$, $V=\phi_{\leq
k}\hat{S}_{n}^{*}$, we have it equal to

$(\tilde{K}_{>k}^{\gamma})^{-1}(\hat{S}_{n}\phi_{\leq
k}^{*})(I_{k}+\Lambda^{\leq k}_{\mathcal{A}^{2}\Sigma^{\beta-1}}(\phi_{\leq
k}\hat{S}_{n}^{*})(\tilde{K}_{>k}^{\gamma})^{-1}(\hat{S}_{n}\phi_{\leq
k}^{*}))^{-1}.$

Then we sub. the identity above to obtain

$\displaystyle\|\Lambda^{>k}_{\mathcal{A}\Sigma^{\beta-1}}\phi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k}\Lambda^{\leq k}_{\mathcal{A}}\phi_{\leq k}f_{\leq
k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$ $\displaystyle=$
$\displaystyle\
\|\Lambda^{>k}_{\Sigma^{(1-\beta^{\prime})/2}}\Lambda^{>k}_{\mathcal{A}\Sigma^{\beta-1}}\phi_{>k}\hat{S}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\phi_{\leq
k}^{*}(I_{k}+\Lambda^{\leq k}_{\mathcal{A}^{2}\Sigma^{\beta-1}}\phi_{\leq
k}\hat{S}^{*}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\phi^{*}_{\leq
k})^{-1}\Lambda^{\leq k}_{\mathcal{A}}\phi_{\leq k}f^{*}_{\leq k}\|^{2}$
$\displaystyle=$ $\displaystyle\
\|\Lambda^{>k}_{\mathcal{A}\Sigma^{(-\beta^{\prime}+2\beta-1)/2}}\phi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\phi_{\leq
k}^{*}(\Lambda^{\leq k}_{\mathcal{A}^{2}\Sigma^{\beta-1/2}}(\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{-\beta}}+\psi_{\leq
k}\hat{S}^{*}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k})\Lambda^{\leq k}_{\Sigma^{1/2}})^{-1}\Lambda^{\leq
k}_{\mathcal{A}}\phi_{\leq k}f^{*}_{\leq k}\|^{2}$ $\displaystyle=$
$\displaystyle\
\|\Lambda^{>k}_{\mathcal{A}\Sigma^{(-\beta^{\prime}+2\beta-1)/2}}\phi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\phi_{\leq
k}^{*}\Lambda^{\leq k}_{\Sigma^{-1/2}}(\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{-\beta}}+\psi_{\leq
k}\hat{S}^{*}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k})^{-1}\Lambda^{\leq k}_{\mathcal{A}^{-2}\Sigma^{1/2-\beta}}\Lambda^{\leq
k}_{\mathcal{A}}\phi_{\leq k}f^{*}_{\leq k}\|^{2}$ $\displaystyle=$
$\displaystyle\
\|\underbrace{\Lambda^{>k}_{\mathcal{A}\Sigma^{(-\beta^{\prime}+2\beta)/2}}\psi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma}_{>k})^{-1/2}}_{(1)}\underbrace{(\tilde{K}^{\gamma}_{>k})^{-1/2}}_{(2)}\underbrace{\hat{S}_{n}\psi_{\leq
k}^{*}}_{(3)}\underbrace{(\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{-\beta}}+\psi_{\leq
k}\hat{S}^{*}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k})^{-1}}_{(4)}\underbrace{\Lambda^{\leq
k}_{\mathcal{A}^{-1}\Sigma^{1/2-\beta}}\phi_{\leq k}f^{*}_{\leq
k}}_{(5)}\|^{2}.$

Above can be bounded by

$\displaystyle\underbrace{\|(\tilde{K}^{\gamma}_{>k})^{-1/2}\hat{S}_{n}\psi_{>k}^{*}\Lambda^{>k}_{\mathcal{A}^{2}\Sigma^{-\beta^{\prime}+2\beta}}\psi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma}_{>k})^{-1/2}\|}_{(1)}\underbrace{\mu_{1}((\tilde{K}^{\gamma}_{>k})^{-1})}_{(2)}$
$\displaystyle\underbrace{\mu_{1}(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi_{\leq
k}^{*})}_{(3)}\underbrace{\mu_{1}((\psi_{\leq
k}\hat{S}^{*}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k})^{-1})^{2}}_{(4)}\underbrace{\|\phi_{\leq k}f^{*}_{\leq k}\|_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}_{(5)}.$

For (1) it can be upper bounded by

$\displaystyle\|(\tilde{K}^{\gamma}_{>k})^{-1/2}\hat{S}_{n}\psi_{>k}^{*}\Lambda^{>k}_{\mathcal{A}^{2}\Sigma^{-\beta^{\prime}+2\beta}}\psi_{>k}\hat{S}_{n}^{*}(\tilde{K}^{\gamma}_{>k})^{-1/2}\|$
$\displaystyle\leq$
$\displaystyle\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\|I_{n}-n\gamma_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\|$
$\displaystyle\leq$
$\displaystyle\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|,$

where the last transition is by the fact that
$I_{n}-n\gamma_{n}(\tilde{K}^{\gamma}_{>k})^{-1}$ is PSD matrix with norm
bounded by 1 for $\gamma_{n}\geq 0$.

For (4), it can be upper bounded by

$\displaystyle\mu_{1}((\psi_{\leq
k}\hat{S}^{*}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k})^{-1})^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{\mu_{k}((\psi_{\leq
k}\hat{S}^{*}_{n}(\tilde{K}^{\gamma}_{>k})^{-1}\hat{S}_{n}\psi^{*}_{\leq
k}))^{2}}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\mu_{k}((\psi_{\leq
k}\hat{S}^{*}_{n}\hat{S}_{n}\psi^{*}_{\leq
k}))^{2}\mu_{n}((\tilde{K}^{\gamma}_{>k})^{-1})^{2}}.$

Therefore, the third term overall can be bounded by

$\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\frac{\mu_{1}((\tilde{K}^{\gamma}_{>k})^{-1})}{\mu_{n}((\tilde{K}^{\gamma}_{>k})^{-1})^{2}}\frac{\mu_{1}(\psi_{\leq
k}\hat{S}^{*}_{n}\hat{S}_{n}\psi^{*}_{\leq k})}{\mu_{k}(\psi_{\leq
k}\hat{S}^{*}_{n}\hat{S}_{n}\psi^{*}_{\leq k})^{2}}\|\phi_{\leq k}f^{*}_{\leq
k}\|_{\Lambda^{\leq k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}.$

We gather all the terms then we get the desired bound. ∎

######  Lemma E.2 (Simplified Upper bound for bias using concentration).

There exists some absolute constant $c,c^{\prime},C_{2}>0$ s.t. for any
$k\in\mathbb{N}$ with $c\beta_{k}k\log(k)\leq n$, it holds w.p. at least
$1-\delta-8\exp(-\frac{c^{\prime}}{\beta_{k}^{2}}\frac{n}{k})$, the bias can
be upper bounded as:

$\displaystyle B\leq C_{2}{\big{(}}$
$\displaystyle\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})$
$\displaystyle+\
\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}\|\phi_{\leq k}f_{\leq
k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+\
\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle+\ \|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\frac{1}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda_{\Sigma}^{>k}})(p_{k+1}^{2}\lambda_{k+1}^{2\beta-1})$
$\displaystyle+\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})}\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}{\big{)}}.$

###### Proof.

Recall that from E.1 we have

$\displaystyle B\leq$ $\displaystyle\
3\Bigg{(}\frac{\mu_{1}((\tilde{K}_{>k}^{\gamma})^{-1})^{2}}{\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})^{2}}\frac{\mu_{1}(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k})}{\mu_{k}(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k})^{2}\mu_{k}(\Lambda^{\leq
k}_{\mathcal{A}^{2}\Sigma^{\beta^{\prime}}})}\|\hat{S}_{n}\mathcal{A}_{>k}f^{*}_{>k}\|^{2}$
$\displaystyle+$ $\displaystyle\ \frac{\|\phi_{\leq k}f^{*}_{\leq
k}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})^{2}\mu_{k}(\psi_{\leq
k}\hat{S}_{n}^{*}\hat{S}_{n}\psi^{*}_{\leq k})^{2}\mu_{k}(\Lambda^{\leq
k}_{\mathcal{A}^{2}\Sigma^{\beta^{\prime}}})}$ $\displaystyle+$
$\displaystyle\
\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle+$ $\displaystyle\ \|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\mu_{1}[(\tilde{K}_{>k}^{\gamma})^{-1}]^{2}\|\hat{S}_{n}\mathcal{A}_{>k}f_{>k}\|^{2}\mu_{1}(\underbrace{\hat{S}_{n}\psi_{>k}^{*}\Lambda^{>k}_{\mathcal{A}^{2}\Sigma^{2\beta-1}}\psi_{>k}\hat{S}^{*}_{n}}_{n\times
n})$ $\displaystyle+$
$\displaystyle\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\frac{\mu_{1}((\tilde{K}^{\gamma}_{>k})^{-1})}{\mu_{n}((\tilde{K}^{\gamma}_{>k})^{-1})^{2}}\frac{\mu_{1}(\psi_{\leq
k}\hat{S}^{*}_{n}\hat{S}_{n}\psi^{*}_{\leq k})}{\mu_{k}(\psi_{\leq
k}\hat{S}^{*}_{n}\hat{S}_{n}\psi^{*}_{\leq k})^{2}}\|\phi_{\leq k}f^{*}_{\leq
k}\|_{\Lambda^{\leq k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}\Bigg{)}.$

We first apply
$\mu_{1}((\tilde{K}_{>k}^{\gamma})^{-1})=\frac{1}{n\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})}$
and
$\mu_{n}((\tilde{K}_{>k}^{\gamma})^{-1})=\frac{1}{n\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})}$
, also apply concentration inequalities using Lemma B.3, Lemma B.4 and Lemma
C.2 , then w.p. at least
$1-\delta-8\exp(-\frac{c}{\beta_{k}^{2}}\frac{n}{k})$, we can obtain bound
like this

$\displaystyle{\big{(}}$
$\displaystyle\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}\frac{c_{1}n}{c_{2}^{2}n^{2}p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}(\frac{1}{\delta}n\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})$
$\displaystyle+\
\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}\|\phi_{\leq k}f_{\leq
k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{c_{1}^{2}p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+\
\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle+\ \|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\frac{1}{n^{2}\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}(\frac{1}{\delta}n\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda_{\Sigma}^{>k}})(np_{k+1}^{2}\lambda_{k+1}^{2\beta-1})$
$\displaystyle+\|\Lambda^{>k}_{-\beta^{\prime}+\beta}\|\frac{n^{2}\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}}{n\mu_{n}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})}\frac{c_{2}n}{c_{1}^{2}n^{2}}\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}{\big{)}}.$

This can be upper bounded by

$\displaystyle C_{2}{\big{(}}$
$\displaystyle\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})$
$\displaystyle+\
\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}\|\phi_{\leq k}f_{\leq
k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+\
\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle+\ \|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\frac{1}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda_{\Sigma}^{>k}})(p_{k+1}^{2}\lambda_{k+1}^{2\beta-1})$
$\displaystyle+\|\Lambda^{>k}_{-\beta^{\prime}+\beta}\|\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})}\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}{\big{)}}$

where $C_{2}>0$ is some constant only depends on $c_{1},c_{2}$. ∎

######  Theorem E.3 (Bound on bias).

There exists some absolute constant $C_{2},c,c^{\prime}>0$ s.t. for any
$k\in\mathbb{N}$ with $c\beta_{k}k\log(k)\leq n$, it holds w.p. at least
$1-\delta-8\exp(-\frac{c^{\prime}}{\beta_{k}^{2}}\frac{n}{k})$, the bias can
be further bounded as

$\displaystyle B\leq$ $\displaystyle
C_{2}\frac{\rho_{k,n}^{3}}{\delta}(\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}+\
\|\phi_{\leq k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+$ $\displaystyle\
\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}).$

###### Proof.

We refer result from previous lemma E.2.

$\displaystyle B\leq C_{2}{\big{(}}$
$\displaystyle\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})$
$\displaystyle+\
\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}\|\phi_{\leq k}f_{\leq
k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+\
\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle+\ \|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\frac{1}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda_{\Sigma}^{>k}})(p_{k+1}^{2}\lambda_{k+1}^{2\beta-1})$
$\displaystyle+\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})}\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}{\big{)}}.$

Note that by definition of $\rho_{k,n}$ (refer to Definition 3.3), we have a
following estimations:

$\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}=\frac{(\mu_{1}(\frac{1}{n}\tilde{K}_{>k})+\gamma_{n})^{2}}{(\mu_{n}(\frac{1}{n}\tilde{K}_{>k})+\gamma_{n})^{2}}\leq\rho_{k,n}^{2},$
$\displaystyle\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}=$
$\displaystyle\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}$
$\displaystyle\leq$
$\displaystyle\rho_{k,n}^{2}(\frac{1}{n}\operatorname{tr}(\frac{1}{n}\tilde{K}^{\gamma}_{>k}))^{2}\leq\rho_{k,n}^{2}(\gamma_{n}+\frac{1}{n}\sum_{j=1}^{n}\sum_{i>k}\lambda_{i}^{\beta}p_{i}^{2}\psi_{i}(x_{j})^{2})^{2}$
$\displaystyle\leq$
$\displaystyle\rho_{k,n}^{2}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2},$
$\frac{\|\Lambda^{>k}_{\mathcal{A}^{2}\Sigma^{\beta}}\|}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k})}\leq\rho_{k,n}$

and

$\displaystyle\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})}=\frac{\|\Lambda^{>k}_{\mathcal{A}^{2}\Sigma^{\beta}}\|}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k})}\|\Lambda^{>k}_{\mathcal{A}^{-2}\Sigma^{-\beta^{\prime}}}\|\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}$
$\displaystyle\leq$
$\displaystyle\rho_{k,n}^{3}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}\|\Lambda^{>k}_{\mathcal{A}^{-2}\Sigma^{-\beta^{\prime}}}\|.$

We bound first and forth term first

$\displaystyle\frac{\mu_{1}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})+\|\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}\|\
\frac{1}{\mu_{n}(\frac{1}{n}\tilde{K}_{>k}^{\gamma})^{2}}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda_{\Sigma}^{>k}})(p_{k+1}^{2}\lambda_{k+1}^{2\beta-1})$
$\displaystyle\leq$
$\displaystyle(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})(\rho_{k,n}^{2}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}+\frac{\|\Lambda^{>k}_{\mathcal{A}^{4}\Sigma^{2\beta}}\|}{\mu_{n}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}}p_{k+1}^{2}\lambda_{k+1}^{2\beta-1}\|\Lambda^{>k}_{\mathcal{A}^{-4}\Sigma^{1-\beta^{\prime}-2\beta}}\|)$
$\displaystyle\leq$
$\displaystyle\rho_{k,n}^{2}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})(\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}+p_{k+1}^{2}\lambda_{k+1}^{2\beta-1}\|\Lambda^{>k}_{\mathcal{A}^{-4}\Sigma^{1-\beta^{\prime}-2\beta}}\|).$

Since two terms here have the same order, we can just bound it by

$c_{1}\rho_{k,n}^{2}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$

where $c_{1}$ is some constant.

Next we bound the second and fifth term

$\displaystyle\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}+\|\Lambda^{>k}_{\Sigma^{-\beta^{\prime}+\beta}}\|\frac{\mu_{1}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})^{2}}{\mu_{n}(\frac{1}{n}\tilde{K}^{\gamma}_{>k})}\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}$
$\displaystyle\leq$ $\displaystyle\|\phi_{\leq k}f_{\leq
k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}(\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}\rho_{k,n}^{2}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}+\rho_{k,n}^{3}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}\|\Lambda^{>k}_{\mathcal{A}^{-2}\Sigma^{-\beta^{\prime}}}\|).$

We know $\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$ and
$\|\Lambda^{>k}_{\mathcal{A}^{-2}\Sigma^{-\beta^{\prime}}}\|$ are of the same
order, and $\rho_{k,n}\geq 1$ by its definition, therefore, the second term
would be dominated by the fifth term. So we can bound it by

$c_{2}\rho_{k,n}^{3}\|\phi_{\leq k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}.$

Therefore, the final bound becomes

$\displaystyle
C_{2}(c_{1}\rho_{k,n}^{2}(\frac{1}{\delta}\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}})\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}+c_{2}\rho_{k,n}^{3}\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+$
$\displaystyle\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}})$
$\displaystyle\leq$ $\displaystyle
C_{2}^{\prime}\frac{\rho_{k,n}^{3}}{\delta}(\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}+\|\phi_{\leq
k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+$
$\displaystyle\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}),$

$C_{2}^{\prime}$ is w.r.t. $C_{2},c_{1},c_{2}$, and we finally just take
$C_{2}=C_{2}^{\prime}$ to finish the proof. ∎

##  Appendix F Applications

###  F.1 Regularized Case

######  Theorem F.1 (Regularized case, Proof of Theorem 4.1).

Let the kernel and target function satisfies Assumption 2.2,
$\gamma_{n}=\Theta(n^{-\gamma})$, and $\gamma<2p+\beta\lambda$, $2p+\lambda
r>0$ and $r>\beta^{\prime}$ then for any $\delta>0$, it holds w.p.
$1-\delta-O(\frac{1}{\log(n)})$ that

$\displaystyle
V=\sigma_{\varepsilon}^{2}O(n^{\max\\{\frac{\gamma(1+2p+\lambda\beta^{\prime})}{2p+\lambda\beta},0\\}-1}),B\leq\frac{1}{\delta}\cdot\tilde{O}_{n}(n^{\frac{\gamma}{2p+\beta\lambda}(\max\\{{\lambda(\beta^{\prime}-r)},{-2p+\lambda(\beta^{\prime}-2\beta)}\\})}).$

###### Proof.

We use the two lemmas D.3, E.3 for upper bounding bias and variance in this
proof, there exists some absolute constants $c,c^{\prime}>0$, first we need to
pick $k$ s.t. $c\beta_{k}k\log(k)\leq n$, then the two lemmas will
simultaneously hold w.p. at least
$1-\delta-16\exp(-\frac{c^{\prime}}{\beta_{k}^{2}}\frac{n}{k})$. With
regularization, we can pick $k$ large enough s.t. the concentration
coefficient $\rho_{k,n}=o(1)$, to achieve so, we want
$\mu_{1}(\frac{1}{n}\tilde{K}_{>k})=O(\gamma_{n})$. By Lemma F.6, we can show
w.p. at least
$1-4\frac{r_{k}}{k^{4}}\exp(-\frac{c^{\prime}}{\beta_{k}}\frac{n}{r_{k}})$

$\mu_{1}(\frac{1}{n}\tilde{K}_{>k})=O_{n}(p_{k+1}^{2}\lambda_{k+1}^{\beta})=O_{n}(k^{-2p-\beta\lambda})=O_{n}(\gamma_{n})=O_{n}(n^{-\gamma}).$
(8)

This can be achieved by setting $k(n)=\lceil
n^{\frac{\gamma}{2p+\beta\lambda}}\rceil$, note that we have
$\frac{\gamma}{2p+\beta\lambda}<1$, therefore, $k(n)=O(\frac{n}{\log(n)})$ and
the lemmas can be used for sufficient large $n$.

We combine the probability of both D.3, E.3 and 8 hold:

$1-\delta-16\exp{\big{(}}-\frac{c^{\prime}}{\beta_{k}^{2}}\frac{n}{k}\big{)}-O(\frac{1}{k^{3}})\exp(-\Omega(\frac{n}{k}))=1-\delta-O(\frac{1}{n})$

where we use the fact that
$\frac{c^{\prime}}{\beta_{k}^{2}}\frac{n}{k}=\Omega(\log(n))$ since
$k(n)=O(\frac{n}{\log(n)})$.

Then now we can assume D.3, E.3 and 8 hold, and we provide the bound on
variance and bias respectively.

By Theorem D.3 and we sub. $p_{i}=\Theta(i^{-p})$,
$\lambda_{i}=\Theta(i^{-\lambda})$,
$\|\Sigma_{>k}\|=p_{k+1}^{2}\lambda_{k+1}^{\beta}=\Theta((k+1)^{-\beta\lambda-2p})=\Theta(k^{-\beta\lambda-2p})$,

$\displaystyle V\leq$ $\displaystyle
C_{1}\sigma_{\varepsilon}^{2}\rho_{k,n}^{2}{\Big{(}}\frac{{\sum_{i\leq
k}p_{i}^{-2}\lambda_{i}^{-\beta^{\prime}}}}{{n}}+\frac{\sum_{i>k}p_{i}^{2}\lambda_{i}^{-\beta^{\prime}+2\beta}}{n{\|\tilde{\Sigma}_{>k}\|^{2}}}{\Big{)}}$
$\displaystyle=$
$\displaystyle\sigma_{\varepsilon}^{2}O(1)O(\frac{\max\\{k^{1+2p+\lambda\beta^{\prime}},1\\}}{n},\frac{k^{1-2p+\lambda(\beta^{\prime}-2\beta)}}{nk^{-2\beta\lambda-4p}})=\sigma_{\varepsilon}^{2}\tilde{O}(\frac{\max\\{k^{1+2p+\lambda\beta^{\prime}},1\\}}{n}).$

We substitute $k$ with $\lceil n^{\frac{\gamma}{2p+\beta\lambda}}\rceil$ to
obtain the final bound

$V=\sigma_{\varepsilon}^{2}O(n^{\max\\{\frac{\gamma(1+2p+\lambda\beta^{\prime})}{2p+\lambda\beta},0\\}-1}).$

For bias, recall that by Theorem E.3, we have

$\displaystyle B\leq$ $\displaystyle
C_{2}\frac{\rho_{k,n}^{3}}{\delta}(\|\phi_{>k}\mathcal{A}_{>k}f_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+$ $\displaystyle\ \|\phi_{\leq k}f_{\leq
k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}\frac{1}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}$
$\displaystyle+$ $\displaystyle\
\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}).$

By
$\operatorname{tr}(\tilde{\Sigma}_{>k})=\sum_{i>k}p_{i}^{2}\lambda_{i}^{\beta}=O(k\lambda_{k}^{\beta}p_{k}^{2})=O(k\gamma_{n})$,
then

$(\gamma_{n}+\frac{\beta_{k}\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})^{2}=O((\gamma_{n}+\frac{n}{k}\gamma_{n})^{2})=O(\gamma_{n}^{2})=O(k^{-4p-2\lambda\beta})$

Recall that

$\frac{\|\phi_{\leq k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}}{p_{k}^{2}\lambda_{k}^{\beta^{\prime}}}=\tilde{O}(k^{\max\\{1+4p-\lambda(1-\beta^{\prime}-2\beta)-2r^{\prime},2p+\lambda\beta^{\prime}\\}}).$

Therefore, the second term’s bound is

$O(k^{\max\\{1-2r-\lambda(1-\beta^{\prime}),-2p+\lambda(\beta^{\prime}-2\beta)\\}}).$

Since $2p+\lambda r>0$ and $r>\beta^{\prime}$, we have
$2p+2r^{\prime}+\lambda>1$, and $2r^{\prime}+(1-\beta^{\prime})\lambda>1$, We
can quote Lemma F.5 for the remaining terms, so the third term’s bound is

$O(k^{1-2r^{\prime}-(1-\beta^{\prime})\lambda}).$

First term’s bound is the same as the second

$O(k^{\max\\{1-2r^{\prime}-\lambda(1-\beta^{\prime}),-2p+\lambda(\beta^{\prime}-2\beta)\\}}).$

So we sub. $k=\lceil n^{\frac{\gamma}{2p+\beta\lambda}}\rceil$ to obtain

$B\leq\frac{1}{\delta}\cdot\tilde{O}_{n}(n^{\frac{\gamma}{2p+\beta\lambda}(\max\\{1-2r^{\prime}-\lambda(1-\beta^{\prime}),-2p+\lambda(\beta^{\prime}-2\beta)\\})}).$

And we substitute $r^{\prime}=\frac{1-\lambda(1-r)}{2}$ to obtain the final
bound

$B=O(n^{\frac{\gamma}{2p+\beta\lambda}(\max\\{{\lambda(\beta^{\prime}-r)},{-2p+\lambda(\beta^{\prime}-2\beta)}\\})}).$

∎

###  F.2 Interpolation Case

######  Theorem F.2 (Interpolation case, proof of Theorem 4.2).

Let the kernel and target function satisfies Assumption 2.2,
$2p+\beta\lambda>0$, $2p+\lambda r>0$ and $r>\beta^{\prime}$, then for any
$\delta>0$ it holds w.p. at least $1-\delta-O(\frac{1}{\log(n)})$ that

$\displaystyle
V\leq\sigma_{\varepsilon}^{2}\rho_{k,n}^{2}\tilde{O}(n^{\max\\{2p+\lambda\beta^{\prime},-1\\}}),B\leq\frac{\rho_{k,n}^{3}}{\delta}\tilde{O}(n^{\max\\{{\lambda(\beta^{\prime}-r)},{-2p+\lambda(\beta^{\prime}-2\beta)}\\}\\}}),$

where $\rho_{k,n}=\tilde{O}(n^{2p+\beta\lambda-1})$, when features are well-
behaved i.e. subGaussian it can be improved to $\rho_{k,n}=o(1)$.

###### Proof.

Same as regularized case, we use the two theorems D.3, E.3 for upper bounding
bias and variance in this proof, there exists some absolute constants
$c,c^{\prime}>0$, first we need to pick $k$ s.t. $c\beta_{k}k\log(k)\leq n$,
then the two lemmas will simultaneously hold w.p. at least
$1-\delta-16\exp(-\frac{c^{\prime}}{\beta_{k}^{2}}\frac{n}{k})$. Since
$\beta_{k}=o(1)$ we know it can be upper bounded by $C_{0}$ for some
$C_{0}>0$. Similar to [61], we let $k:=k(n):=\frac{n}{\max\\{cC0,1\\}\log n}$
and we also let $k^{\prime}:=k^{\prime}(n)=n^{2}\log^{4}(n)$. So the
probability of those theorems hold become $1-\delta-O(\frac{1}{n})$.

In this case, $\rho_{k,n}$ cannot be regularized to $o(1)$ if the features are
not well-behaved, we compute its bound first, which requires bounding
$\mu_{1}(\frac{1}{n}\tilde{K}_{>k})$ and $\mu_{n}(\frac{1}{n}\tilde{K}_{>k})$
respectively.

We apply Lemma C.6 by setting $\delta=\log n$, then w.p. $1-\frac{1}{\log(n)}$
we have

$\displaystyle\mu_{n}(\frac{1}{n}\tilde{K}_{>k})\geq$
$\displaystyle\alpha_{k}(1-\frac{1}{\log
n}\sqrt{\frac{n^{2}}{\operatorname{tr}(\tilde{\Sigma}_{>k^{\prime}})^{2}/\operatorname{tr}(\tilde{\Sigma}^{2}_{>k^{\prime}})}})\frac{\operatorname{tr}(\tilde{\Sigma}_{>k^{\prime}})}{n}$
$\displaystyle=$ $\displaystyle\ \Omega((1-\log
n\sqrt{\frac{1}{\log^{4}n}})\frac{\operatorname{tr}(\tilde{\Sigma}_{>k^{\prime}})}{n})$
$\displaystyle=$ $\displaystyle\
\Omega(\frac{(k^{\prime})^{1-2p-\beta\lambda}}{n})$ $\displaystyle=$
$\displaystyle\ \Omega(\frac{(n^{2}\log^{4}n)^{1-2p-\beta\lambda}}{n})$
$\displaystyle=$ $\displaystyle\ \tilde{\Omega}(n^{1-4p-2\beta\lambda}).$

Note that the first equality is because we have
$\operatorname{tr}(\tilde{\Sigma}_{>k^{\prime}})^{2}/\operatorname{tr}(\tilde{\Sigma}^{2}_{>k^{\prime}})=\frac{(\sum_{i>k^{\prime}}p_{i}^{2}\lambda_{i}^{\beta})^{2}}{\sum_{i>k^{\prime}}p_{i}^{4}\lambda_{i}^{2\beta}}=\frac{{k^{\prime}}^{2-2p-\lambda\beta}}{{k^{\prime}}^{1-2p-\lambda\beta}}=k^{\prime}=n^{2}\log^{4}(n)$,
$\tilde{\Omega}$ means we neglect logarithmic terms.

For $\mu_{1}(\frac{1}{n}\tilde{K}_{>k})$ term by Lemma F.6, we have w.p.
$1-O(\frac{1}{k^{3}})\exp(-\Omega(\frac{n}{k}))$

$\mu_{1}(\frac{1}{n}\tilde{K}_{>k})=O(p_{k+1}^{2}\lambda_{k+1}^{\beta})=O(k^{-2p-\beta\lambda})=\tilde{O}(n^{-2p-\beta\lambda}).$
(9)

So we have Eq. 9, Lemma C.6, Theorem D.3, E.3 all hold simultaneously hold
with probability $1-\delta-O(\frac{1}{\log(n)})$.

Therefore, we have $\rho_{k,n}=\tilde{O}(n^{2p+\beta\lambda-1})$.

Recall from Lemma D.3 that

$\displaystyle V\leq$ $\displaystyle\
C_{1}\sigma_{\varepsilon}^{2}\rho_{k,n}^{2}{\Big{(}}\frac{{\sum_{i\leq
k}p_{i}^{-2}\lambda_{i}^{-\beta^{\prime}}}}{{n}}+\frac{\sum_{i>k}p_{i}^{2}\lambda_{i}^{-\beta^{\prime}+2\beta}}{n{\|\tilde{\Sigma}_{>k}\|^{2}}}{\Big{)}}$
$\displaystyle=$ $\displaystyle\
\sigma_{\varepsilon}^{2}\rho_{k,n}^{2}O(\frac{\max\\{k^{1+2p+\lambda\beta^{\prime}},1\\}}{n}+\frac{k^{1-2p+\lambda(\beta^{\prime}-2\beta)}}{nk^{-2\beta\lambda-4p}})$
$\displaystyle=$ $\displaystyle\
\sigma_{\varepsilon}^{2}\rho_{k,n}^{2}\tilde{O}(\frac{\max\\{k^{1+2p+\lambda\beta^{\prime}},1\\}}{n}).$

So we sub. $k=\tilde{\Theta}(n)$ and the final bound of variance is

$V\leq\sigma_{\varepsilon}^{2}\rho_{k,n}^{2}\tilde{O}(n^{\max\\{2p+\lambda\beta^{\prime},-1\\}}).$

For bias, similar to the regularized case, the bound is

$\frac{1}{\delta}\rho_{k,n}^{3}O(k^{\max\\{1-2r^{\prime}-\lambda(1-\beta^{\prime}),-2p+\lambda(\beta^{\prime}-2\beta)\\}}).$

The main difference is the choice of $k$, since $k=\tilde{\Theta}(n)$, the
final bound is

$\frac{1}{\delta}\rho_{k,n}^{3}O(n^{\max\\{1-2r^{\prime}-\lambda(1-\beta^{\prime}),-2p+\lambda(\beta^{\prime}-2\beta)\\}}).$

Note that if the features are well-behaved, then $\rho_{k,n}$ can be improved
to $o(1)$. ∎

###  F.3 Lemmas for substituting polynomial decay

######  Lemma F.3.

Let $a\in\mathbb{R}$, $1<k\in\mathbb{N}$, then

$\sum_{i\leq k}i^{-a}\leq\begin{cases}1+\frac{1}{1-a}k^{1-a}\ \ &a<1\\\
1+\log(k)\ \ &a=1\\\ 1+\frac{1}{a-1}\ \ &a>1.\end{cases}$

Therefore, $\sum_{i\leq k}i^{-a}=\tilde{O}(\max\\{k^{-a+1},1\\})$

###### Proof.

We know that, for $a<1$

$\sum_{i\leq k}i^{-a}\leq 1+\int_{1}^{k}x^{-a}\
dx=1+\frac{1}{1-a}(k^{1-a}-1)\leq 1+\frac{1}{1-a}k^{1-a}.$

For $a=1$

$\sum_{i\leq k}i^{-a}\leq 1+\int_{1}^{k}x^{-a}\ dx=1+\log(k).$

For $a>1$

$\sum_{i\leq k}i^{-a}\leq 1+\int_{1}^{\infty}x^{-a}\ dx=1+\frac{1}{a-1}.$

∎

######  Lemma F.4.

Let $a\in\mathbb{R}$, $1<k\in\mathbb{N}$, then

$\sum_{i>k}i^{-a}\in\begin{cases}\infty\ \ &a\leq 1\\\
[\frac{1}{a-1}(k+1)^{-a+1},(k+1)^{-a}+\frac{1}{a-1}(k+1)^{-a+1}]\ \
&a>1.\end{cases}$

Therefore, $\sum_{i>k}{i^{-a}}$ is $O(k^{-a+1})$ if $a>1$, otherwise it
diverges to infinity

###### Proof.

We know that,

$\int_{k+1}^{\infty}x^{-a}\
dx\leq\sum_{i>k}i^{-a}\leq(k+1)^{-a}+\int_{k+1}^{\infty}x^{-a}\ dx.$

If $a<1$ then $\int_{k+1}^{\infty}x^{-a}=\infty$ which implies the series
diverge, otherwise, $\int_{k+1}^{\infty}x^{-a}=\frac{1}{a+1}(k+1)^{-a+1}$

∎

######  Lemma F.5.

Assume $[\phi f^{*}]_{i}=\Theta(i^{-r^{\prime}})$, $\Sigma$’s polynomial
decaying eigenvalues satisfy $\lambda_{i}=\Theta(i^{-\lambda})$ ($\lambda$ >
0), and $\mathcal{A}$’s eigenvalue is $\Theta(i^{-p})$ ($p<0$), then

$\displaystyle\|\phi_{>k}\mathcal{A}_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}}$
$\displaystyle=\Theta\left(\frac{1}{k^{2p+2r^{\prime}+\lambda-1}}\right)\text{
if }2p+2r^{\prime}+\lambda>1;$ $\displaystyle\|\phi_{\leq k}f_{\leq
k}^{*}\|^{2}_{\Lambda^{\leq k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}$
$\displaystyle=\tilde{O}(\max\\{k^{1+2p-\lambda(1-2\beta)-2r^{\prime}},1\\});$
$\displaystyle\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}$
$\displaystyle=\Theta\left(\frac{1}{k^{2r^{\prime}+(1-\beta^{\prime})\lambda-1}}\right)\text{
if }2r^{\prime}+(1-\beta^{\prime})\lambda>1,$

where $r^{\prime}=\frac{1-\lambda(1-r)}{2}$.

###### Proof.

We know from F.4 that,

$\|\phi_{>k}\mathcal{A}_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma}}=\sum_{i>k}[\phi
f^{*}]_{i}^{2}\cdot
p_{i}^{2}\lambda_{i}=\sum_{i>k}\Theta\left(\frac{1}{i^{2p+2r^{\prime}+\lambda}}\right)=\Theta\left(\frac{1}{k^{2p+2r^{\prime}+\lambda-1}}\right)\text{
if }2p+2r^{\prime}+\lambda>1.$

Similarly, using F.3

$\|\phi_{\leq k}f_{\leq k}^{*}\|^{2}_{\Lambda^{\leq
k}_{\mathcal{A}^{-2}\Sigma^{1-2\beta}}}=\sum_{i\leq k}[\phi
f^{*}]_{i}^{2}\cdot p_{i}^{2}\lambda_{i}^{1-2\beta}=\sum_{i\leq
k}\Theta\left(\frac{1}{i^{2r^{\prime}-2p+\lambda(2\beta-1)}}\right)=\tilde{O}(\max\\{k^{1+2p-\lambda(1-2\beta)-2r^{\prime}},1\\}).$

Using F.4 again, we’ll have

$\|\phi_{>k}f^{*}_{>k}\|^{2}_{\Lambda^{>k}_{\Sigma^{1-\beta^{\prime}}}}=\sum_{i>k}[\phi
f^{*}]_{i}^{2}\cdot\lambda_{i}^{\beta^{\prime}-1}=\sum_{i>k}\Theta\left(\frac{1}{i^{2r^{\prime}+(1-\beta^{\prime})\lambda}}\right)=\Theta\left(\frac{1}{k^{2r^{\prime}+(1-\beta^{\prime})\lambda-1}}\right)\text{
if }2r^{\prime}+(1-\beta^{\prime})\lambda>1.$

∎

######  Lemma F.6.

Assume $\Sigma$’s polynomial decaying eigenvalues satisfy
$\lambda_{i}=\Theta(i^{-\lambda})$ ($\lambda$ > 0), and $\mathcal{A}$’s
eigenvalue is $\Theta(i^{-p})$. And we suppose
$\frac{\beta_{k}k\log(k)}{n}=o(1),\beta_{k}=o(1)$.

Then it holds w.p. at least $1-O(\frac{1}{k^{3}})\exp(-\Omega(\frac{n}{k}))$
that

$\mu_{1}(\frac{1}{n}\tilde{K}_{>k})=O(\lambda_{k+1}^{\beta}p_{k+1}^{2})=O(k^{-2p-\beta\lambda}).$

###### Proof.

We use C.6 then there exists absolute constant $c,c^{\prime}>0$ s.t. it holds
w.p. at least
$1-4\frac{r_{k}}{k^{4}}\exp(-\frac{c^{\prime}}{\beta_{k}}\frac{n}{r_{k}})$
that

$\displaystyle\mu_{1}(\frac{1}{n}\tilde{K}_{>k})\leq$ $\displaystyle
c(\lambda_{k+1}^{\beta}p_{k+1}^{2}+\beta_{k}\log(k+1)\frac{\operatorname{tr}(\tilde{\Sigma}_{>k})}{n})$
$\displaystyle=$ $\displaystyle\
O(\lambda_{k+1}^{\beta}p_{k+1}^{2}(1+\beta_{k}\log(k+1)\frac{k}{n}))$
$\displaystyle=$ $\displaystyle O(\lambda_{k+1}^{\beta}p_{k+1}^{2}),$

where $\tilde{\Sigma}:=\mathcal{A}^{2}\Sigma^{\beta}$,
$r_{k}:=\frac{\operatorname{tr}(\tilde{\Sigma}_{>k})}{p_{k+1}^{2}\lambda_{k+1}^{\beta}}$.
The last inequality is because $\frac{k\log(k+1)}{n}=o(1)$.

Now we bound the probability of this holds, we can derive
$r_{k}=\frac{k^{1-2p-\lambda\beta}}{(k+1)^{-2p-\lambda\beta}}=\Theta(k)$,
$1-4\frac{r_{k}}{k^{4}}\exp(\frac{-c^{\prime}}{\beta_{k}}\frac{n}{r_{k}})=1-O(\frac{1}{k^{3}})\exp(-\Omega(\frac{n}{k}))$.
∎


# A Novel 3D Non-stationary Localization-assisted ISAC Channel Model

Runruo Yang1,2, Yang Wu1, Jie Huang1,2*, Cheng-Xiang Wang1,2*
1National Mobile Communications Research Laboratory, School of Information
Science and Engineering,
Southeast University, Nanjing 210096, China.
2Purple Mountain Laboratories, Nanjing 211111, China.
*Corresponding Authors: Jie Huang and Cheng-Xiang Wang
Email: {yang_rr, wu_yang, j_huang<EMAIL_ADDRESS>
###### Abstract

Integrated sensing and communication (ISAC) has attracted wide attention as an
emerging application scenario for the sixth generation (6G) wireless
communication system. In this paper, a novel three-dimensional (3D) non-
stationary localization-assisted ISAC geometry-based stochastic model (GBSM)
is proposed. The locations of the first-bounce scatterer and last-bounce
scatterer in the communication channel can be estimated by the particle filter
with the assistance of backscattering sensing. The important channel
statistical properties of the proposed channel model are simulated and
compared with the ray tracing (RT) results, including the delay spread,
azimuth angle of departure/arrival (AAoD/AAoA) spread, and elevation angle of
departure/arrival (EAoD/EAoA) spread. The simulation results of the proposed
channel model show a good agreement with the RT results, which proves the
correctness of the proposed channel model. Utilizing the localization
parameters of scatterers, the proposed ISAC channel model can better map the
real environment.

###### Index Terms:

6G, localization-assisted, ISAC, GBSM, particle filter

##  I Introduction

With the development of wireless communication technology, the 6G wireless
communication system requires both high data rate communication and high-
precision sensing/localization to realize the mapping of the physical world
and the digital world [1, 2]. For the reason that sensing and communication
have commonalities in terms of utilized frequency bands, waveform design, and
hardware platform [3, 4, 5], ISAC has received extensive attention as an
emerging 6G application scenario. The channel model plays an essential role in
system design, resource allocation, and performance optimization [6].
Therefore, in order to realize the mutual assistance between sensing and
communication of ISAC system, it is necessary to study the ISAC channel.

The localization method realized on the channel model, and it has been studied
in the literatures. In [7], a millimeter wave (mmWave) beam domain
communication channel model was proposed, and the maximum-likelihood (ML)
classifier was utilized for receiver (Rx) localization. In [8], a multipath
components (MPC) tracking algorithm based on the GBSM was proposed, and made a
comparison between the simulation results and the measurement data. In [9], a
method for tracking user motion using multipath phase information by extended
Kalman filter was proposed. Compared with the measured data in line-of-sight
(LoS) scenario, this method was proved to be able to track the user motion
trajectory well. However, scatterers in the environment cannot be located by
these methods. In [10], a map-free localization method for indoor scenario was
proposed, and the estimated transmitter (Tx) and clusters locations were
compared with measurement results. However, this method can only be used to
locate Tx and scatterers in the LoS path or one-bounce path, and it does not
apply to scenarios with multi-bounce paths. In [11], an indoor environment
mapping method based on sensing backscattering channel was proposed. However,
the LoS path was required between the target and transceiver, and scatterers
can not be located. Localization methods mentioned above can only locate the
user or single bounce clusters. For multi-bounce clusters within MPCs, the
location of the scatterer cannot be estimated by the above methods.

Channel models have also been studied in many literatures. In [6], a 6G
pervasive channel model (6GPCM) was proposed, in which positions of scatterers
were randomly generated by ellipsoid Gaussian distribution. A 3D non-
stationary unmanned aerial vehicle (UAV) multiple-input multiple-output (MIMO)
channel model was proposed in [12]. Angle parameters were modelled by von
Mises-Fisher distribution. In [13], a mobile-to-mobile two-ring non-stationary
channel model was proposed, which assumed that the scatterers were regularly
distributed on the Tx and Rx rings. In [14, 15], the 3D non-stationary UAV
MIMO GBSMs were proposed, in which scatterers were distributed on the cylinder
surface at Rx side. A high-speed train channel model was proposed in [16], and
the scatterers were located on the ellipse with Tx and Rx as the foci.
However, in the above channel models, the locations of scatterers are
characterized by specific random distributions or the scatterers are
distributed on the surface of a specific geometry. The spatial positions of
the scatterers generated by the above methods can not reflect the real
positions of the objects in the physical environment.

In this paper, a novel 3D non-stationary localization-assisted ISAC channel
model is proposed. Through the combination of communication and sensing, the
locations of multi-bounce scatterers in the channel can be determined with the
help of the particle filter. Considering the non-stationarity of the channel
model, the locations of scatterers will change with time, and the resampling
process of the particle filter can reflect this property. The process of
resampling also implies the birth-death of scatterers in the channel model.

The remainder of this paper is organized as follows. In Section II, a novel 3D
non-stationary localization-assisted ISAC channel model is proposed. In
Section III, the channel statistical properties are analyzed, including the
delay spread, AAoD/AAoA spread, and EAoD/EAoA spread. Section IV presents the
results and analysis. Finally, the paper is concluded in Section V.

##  II A Novel 3D Localization-assisted ISAC GBSM

###  II-A Channel Model Description

The ISAC channel model is composed of a mono-static sensing channel and a
communication channel. There is an ISAC base station (BS) at Tx, and the Rx is
a communication user. A transmit antenna array and a receive antenna array are
both equipped at the ISAC BS. The ISAC BS can transmit sensing and
communication signals, and receive the sensing echo signal. The Rx possesses
an antenna array for receiving communication signals. Since the communication
transmitting and receiving arrays are located at different positions, the
communication channel can also be seen as a bi-static channel. Locations of
the first-bounce (last-bounce) scatterers within the channel model can be
determined by delay, AAoD (AAoA), and EAoD (EAoA) parameters, which are
indispensable parameters of the channel model. Fig. 1 depicts the proposed
ISAC channel model.

Figure 1: ISAC channel model.

####  II-A1  Mono-static Sensing Channel Model

The ISAC BS transmits pilot symbols or pseudo-noise (PN) sequences to sense
the obstacles in the physical environment distributed around the ISAC BS.
These obstacles will be abstracted as scatterers in the sensing channel model.
The scatterer will be sensed by the ISAC BS, if there is a LoS path between
the ISAC BS and the scatterer. The distance, angles, and velocity of the
scatterer can be obtained by the sensing channel. The mono-static sensing echo
channel impulse response (CIR) is given by

$\displaystyle{h^{mo}}\left({t,\tau}\right)=$
$\displaystyle\sum\limits_{i=1}^{I\left(t\right)}{\sum\limits_{j=1}^{{J_{i}}}{\sqrt{{G_{{j_{i}}}}\left(t\right)}}}{e^{j2\pi{f_{c}}{\tau_{{j_{i}}}}\left(t\right)}}{e^{j2\pi{f_{D,{j_{i}}}}\left(t\right){\tau_{{j_{i}}}}\left(t\right)}}$
(1) $\displaystyle
A\left({{\alpha_{{j_{i}}}}\left(t\right),{\beta_{{j_{i}}}}\left(t\right)}\right)\delta\left({\tau-{\tau_{{j_{i}}}}\left(t\right)}\right)$

where
${{G}_{{{j}_{i}}}}\left(t\right)={{\lambda}^{2}}{{{\varepsilon}_{RCS,{{j}_{i}}}}}/{64{{\pi}^{3}}{{d}_{{{j}_{i}}}^{4}}{{\left(t\right)}}}$
is the complex channel gain, ${{\varepsilon}_{RCS,{{j}_{i}}}}$ is the radar
cross section (RCS) of the $j$th scatterer in the $i$th cluster, $\lambda$ is
the wavelength, ${{d}_{{{j}_{i}}}}\left(t\right)$ is the distance between ISAC
BS and the $j$th scatterer, ${{f}_{c}}$ represents the carrier frequency,
${{\tau}_{{{j}_{i}}}}\left(t\right)$ represents the delay, and
${{f}_{D,{{j}_{i}}}}\left(t\right)$ is Doppler shift of the $j$th scatterer.
The steering vector of ISAC BS is represented as
$A\left({{\alpha_{{j_{i}}}}\left(t\right),{\beta_{{j_{i}}}}\left(t\right)}\right)$,
where ${\alpha_{{j_{i}}}}\left(t\right)$ and ${\beta_{{j_{i}}}}\left(t\right)$
are the azimuth and elevation angles of the $j$th scatterer, respectively. The
delay of the echo path is expressed as
${{\tau}_{{{j}_{i}}}}\left(t\right)=2{{d}_{{{j}_{i}}}}\left(t\right)/c$, and
the Doppler shift is
${{f}_{D,{{j}_{i}}}}\left(t\right)=2{{v}_{{{j}_{i}}}}\left(t\right)/\lambda\
$, where $c$ and ${{v}_{{{j}_{i}}}}\left(t\right)$ are the velocities of light
and the $j$th scatterer, respectively.

####  II-A2  Communication Channel Model

The communication channel refers to the channel between ISAC BS and user. At
the same time, this channel can also be regarded as a bi-static channel, in
which ISAC BS and users are located at different locations. The channel model
consists of LoS $h_{qp}^{LoS}\left(t,\tau\right)$ and non-LoS (NLoS)
$h_{qp,{{m}_{n}}}^{NLoS}\left(t,\tau\right)$ components. The communication CIR
can be represented as

$\displaystyle h_{qp}^{c}\left({t,\tau}\right)=$
$\displaystyle\sqrt{\frac{{K\left(t\right)}}{{K\left(t\right)+1}}}h_{qp}^{LoS}\left({t,\tau}\right)+$
(2)
$\displaystyle\sqrt{\frac{1}{{K\left(t\right)+1}}}\sum\limits_{n=1}^{{N_{c}}\left(t\right)}{\sum\limits_{m=1}^{{M_{n}}\left(t\right)}{h_{qp,{m_{n}}}^{NLoS}\left({t,\tau}\right)}}$

where $K\left(t\right)$ denotes the Rician factor, ${{N}_{c}}\left(t\right)$
denotes the number of clusters at time $t$, and ${{M}_{n}}\left(t\right)$
denotes the number of scatterers in the $n$th cluster.

The LoS component $h_{qp}^{LoS}\left(t,\tau\right)$ can be calculated as

$\displaystyle
h_{qp}^{LoS}\left({t,\tau}\right)={\left[\begin{array}[]{l}{F_{q,V}}\left({\beta_{L}^{R}\left(t\right),\alpha_{L}^{R}\left(t\right)}\right)\\\
{F_{q,H}}\left({\beta_{L}^{R}\left(t\right),\alpha_{L}^{R}\left(t\right)}\right)\end{array}\right]^{T}}\left[\begin{array}[]{ccc}{e^{j\xi_{L}^{VV}}}&0\\\
0&-{e^{j\xi_{L}^{HH}}}\end{array}\right]$ (3)
$\displaystyle\left[\begin{array}[]{l}{F_{p,V}}\left({\beta_{L}^{T}\left(t\right),\alpha_{L}^{T}\left(t\right)}\right)\\\
{F_{p,H}}\left({\beta_{L}^{T}\left(t\right),\alpha_{L}^{T}\left(t\right)}\right)\end{array}\right]{e^{j2\pi{f_{c}}\tau_{qp}^{LoS}\left(t\right)}}\delta\left({\tau-\tau_{qp}^{LoS}\left(t\right)}\right)$

where ${\left\\{\cdot\right\\}^{T}}$ means transposition,
$\delta\left(\cdot\right)$ means Dirac delta function,
${F_{p\left(q\right),V}}$ and ${F_{p\left(q\right),H}}$ are vertical and
horizontal polarizations of the antenna patterns for ISAC BS (Rx),
respectively. The variables $\alpha_{L}^{T}\left(t\right)$
($\alpha_{L}^{R}\left(t\right)$) and $\beta_{L}^{T}\left(t\right)$
($\beta_{L}^{R}\left(t\right)$) are the AAoD (AAoA) and EAoD (EAoA) of LoS
path, respectively. The initial phases $\xi_{L}^{HH}$ and $\xi_{L}^{VV}$
follow the uniform distribution of $[0,2\pi]$. The variable
$\tau_{qp}^{LoS}\left(t\right)$ represents the delay of LoS path between the
$p$th transmit antenna and the $q$th receive antenna.

The NLoS component is defined as

$\displaystyle h_{qp,{m_{n}}}^{NLoS}\left({t,\tau}\right)=$
$\displaystyle{\left[\begin{array}[]{l}{F_{q,V}}\left({\beta_{{m_{n}}}^{R}\left(t\right),\alpha_{{m_{n}}}^{R}\left(t\right)}\right)\\\
{F_{q,H}}\left({\beta_{{m_{n}}}^{R}\left(t\right),\alpha_{{m_{n}}}^{R}\left(t\right)}\right)\end{array}\right]^{T}}$
(4)
$\displaystyle\left[\begin{array}[]{cc}{e^{j\xi_{{m_{n}}}^{VV}}}&\sqrt{\mu\kappa_{{m_{n}}}^{-1}\left(t\right)}{e^{j\xi_{{m_{n}}}^{VH}}}\\\
\sqrt{\kappa_{{m_{n}}}^{-1}\left(t\right)}{e^{j\xi_{{m_{n}}}^{HV}}}&\sqrt{\mu}{e^{j\xi_{{m_{n}}}^{HH}}}\end{array}\right]$
$\displaystyle\left[\begin{array}[]{l}{F_{p,V}}\left({\beta_{{m_{n}}}^{T}\left(t\right),\alpha_{{m_{n}}}^{T}\left(t\right)}\right)\\\
{F_{p,H}}\left({\beta_{{m_{n}}}^{T}\left(t\right),\alpha_{{m_{n}}}^{T}\left(t\right)}\right)\end{array}\right]$
$\displaystyle\sqrt{{P_{qp,{m_{n}}}}\left(t\right)}{e^{j2\pi{f_{c}}\tau_{qp,{m_{n}}}^{NLoS}\left(t\right)}}\delta\left({\tau-\tau_{qp,{m_{n}}}^{NLoS}\left(t\right)}\right)$

where, $\mu$ is the co-polarization imbalance, ${\kappa_{{m_{n}}}}$ is the
cross-polarization power ratio,
$\alpha_{{m_{n}}}^{T\left(R\right)}\left(t\right)$ and
$\beta_{{m_{n}}}^{T\left(R\right)}\left(t\right)$ are AAoD (AAoA) and EAoD
(EAoA) of the $m$th ray in the $n$th cluster at time $t$,
${{P_{qp,{m_{n}}}}\left(t\right)}$ is the power of path, and
$\tau_{qp,{{m}_{n}}}^{NLoS}\left(t\right)$ is the delay of NLoS path between
the $p$th transmit antenna and the $q$th receive antenna via the $m$th
scatterer of the $n$th cluster. The delay is given by
$\tau_{qp,{m_{n}}}^{NLoS}\left(t\right)={{{D_{qp,{m_{n}}}}\left(t\right)}\mathord{\left/{\vphantom{{{D_{qp,{m_{n}}}}\left(t\right)}c}}\right.\kern-1.2pt}c}+\tilde{\tau}\left(t\right)$,
where ${{D_{qp,{m_{n}}}}\left(t\right)}$ is the distance of NLoS path between
the $p$th transmit antenna and the $q$th receive antenna, and
$\tilde{\tau}\left(t\right)$ is the virtual delay. The distance
${{D_{qp,{m_{n}}}}\left(t\right)}$ can be computed according to

$\displaystyle{D_{qp,{m_{n}}}}\left(t\right)=\left|{{{\bf{D}}_{p,{m_{n}}}}\left(t\right)}\right|+\left|{{{\bf{D}}_{q,{m_{n}}}}\left(t\right)}\right|$
(5)

where ${{{\bf{D}}_{p,{m_{n}}}}\left(t\right)}$ and
${{{\bf{D}}_{q,{m_{n}}}}\left(t\right)}$ are the distance vectors from $p$th
transmit antenna to the $m$th scatterer and from the $q$th receive antenna to
the $m$th scatterer, respectively. These distance vectors are given as

$\displaystyle{{\bf{D}}_{p,{m_{n}}}}\left(t\right)={\bf{D}}_{{m_{n}}}^{T}\left(t\right)-{\bf{A}}_{p}^{T}$
(6)
$\displaystyle{{\bf{D}}_{q,{m_{n}}}}\left(t\right)={\bf{D}}_{{m_{n}}}^{R}\left(t\right)-{\bf{A}}_{q}^{R}$
(7)

where ${\bf{D}}_{{m_{n}}}^{T(R)}\left(t\right)$ is the distance vector from
the first antenna at the Tx (Rx) to the $m$th scatterer at time $t$, and
${\bf{A}}_{p}^{T}({\bf{A}}_{q}^{R})$ is the vector from the first antenna of
the transmitter (receiver) to the $p$th ($q$th) antenna. The vectors
${\bf{D}}_{{m_{n}}}^{T}\left(t\right)$ and
${\bf{D}}_{{m_{n}}}^{R}\left(t\right)$ can be calculated as

$\displaystyle{\bf{D}}_{{m_{n}}}^{T}\left(t\right)=D_{{m_{n}}}^{T}\left(t\right){\left[\begin{array}[]{c}\cos\left({\beta_{{m_{n}}}^{T}\left(t\right)}\right)\cos\left({\alpha_{{m_{n}}}^{T}\left(t\right)}\right)\\\
\cos\left({\beta_{{m_{n}}}^{T}\left(t\right)}\right)\sin\left({\alpha_{{m_{n}}}^{T}\left(t\right)}\right)\\\
\sin\left({\beta_{{m_{n}}}^{T}\left(t\right)}\right)\end{array}\right]}$ (8)
$\displaystyle{\bf{D}}_{{m_{n}}}^{R}\left(t\right)=D_{{m_{n}}}^{R}\left(t\right)\left[\begin{array}[]{c}\cos\left({\beta_{{m_{n}}}^{R}\left(t\right)}\right)\cos\left({\alpha_{{m_{n}}}^{R}\left(t\right)}\right)\\\
\cos\left({\beta_{{m_{n}}}^{R}\left(t\right)}\right)\sin\left({\alpha_{{m_{n}}}^{R}\left(t\right)}\right)\\\
\sin\left({\beta_{{m_{n}}}^{R}\left(t\right)}\right)\end{array}\right]$ (9)

where ${{D}}_{{m_{n}}}^{T}\left(t\right)$ and
${{D}}_{{m_{n}}}^{R}\left(t\right)$ are the amplitudes of the distance
vectors. The locations of multi-bounce scatterers in the channel can be
estimated by particle filter. The delay and angle parameters of scatterers
mentioned above can also be obtained, rather than by random generation.

###  II-B Particle Filter-based Localization

The first-bounce scatterers of communication channel model, which is
distributed around the Tx, can be sensed by ISAC BS. With the assistance of
sensing, the last-bounce scatterers in the communication channel can be
located and tracked by particle filter. The state vector of particle filter at
time ${{t}_{k}}$ is given as

$\displaystyle\mathbf{s}\left({{t}_{k}}\right)={{\left[{{\mathbf{s}}_{u}}\left({{t}_{k}}\right),{{\mathbf{s}}_{CA}}\left({{t}_{k}}\right),{{\mathbf{s}}_{CZ}}\left({{t}_{k}}\right)\right]}^{T}}$
(10)

where ${{\mathbf{s}}_{u}}\left({{t}_{k}}\right)$ is the state vector of user,
${{\bf{s}}_{CA}}\left({{t_{k}}}\right)={\left[{...,{{\bf{s}}_{CA,{m_{n}}}}\left({{t_{k}}}\right),...}\right]^{T}}$
and
${{\bf{s}}_{CZ}}\left({{t_{k}}}\right)={\left[{...,{{\bf{s}}_{CZ,{m_{n}}}}\left({{t_{k}}}\right),...}\right]^{T}}$,
(${m=1,\ldots,{M_{n}}\left({{t_{k}}}\right),n=1,\ldots,{N_{c}}\left({{t_{k}}}\right)}$)
are the state vectors of the first-bounce scatterer and the last-bounce
scatterer, respectively. The vectors
${{\mathbf{s}}_{u}}\left({{t}_{k}}\right)$,
${{\mathbf{s}}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right)$, and
${{\mathbf{s}}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right)$ can be represented as

$\displaystyle{{\mathbf{s}}_{u}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[{{\mathbf{p}}_{u}}\left({{t}_{k}}\right),{{\mathbf{v}}_{u}}\left({{t}_{k}}\right)\right]}^{T}}$
(11a) $\displaystyle{{\mathbf{s}}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[{{\mathbf{p}}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right),{{\mathbf{v}}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right)\right]}^{T}}$
(11b) $\displaystyle{{\mathbf{s}}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[{{\mathbf{p}}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right),{{\mathbf{v}}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right)\right]}^{T}}.$
(11c)

The location vectors of user, the first-bounce scatterer, and the last-bounce
scatterer are represented as

$\displaystyle{{\mathbf{p}}_{u}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[{{x}_{u}}\left({{t}_{k}}\right),{{y}_{u}}\left({{t}_{k}}\right),{{z}_{u}}\left({{t}_{k}}\right)\right]}^{T}}$
(12a) $\displaystyle{{\mathbf{p}}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[{{x}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right),{{y}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right),{{z}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right)\right]}^{T}}$
(12b) $\displaystyle{{\mathbf{p}}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[{{x}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right),{{y}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right),{{z}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right)\right]}^{T}}$
(12c)

where ${{x}_{u}}\left({{t}_{k}}\right)$, ${{y}_{u}}\left({{t}_{k}}\right)$,
and ${{z}_{u}}\left({{t}_{k}}\right)$ are the coordinates of user in
rectangular coordinate system, respectively. Similarly,
${{x}_{CA\left(CZ\right),{{m}_{n}}}}\left({{t}_{k}}\right)$,
${{y}_{CA\left(CZ\right),{{m}_{n}}}}\left({{t}_{k}}\right)$, and
${{z}_{CA\left(CZ\right),{{m}_{n}}}}\left({{t}_{k}}\right)$ are the
coordinates of the first-bounce (last-bounce) scatterer, respectively. The
velocity vectors of user, the first-bounce scatterer, and the last-bounce
scatterer are represented as

$\displaystyle{{\mathbf{v}}_{u}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[v_{u}^{x}\left({{t}_{k}}\right),v_{u}^{y}\left({{t}_{k}}\right),v_{u}^{z}\left({{t}_{k}}\right)\right]}^{T}}$
(13a) $\displaystyle{{\mathbf{v}}_{CA,{{m}_{n}}}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[v_{CA,{{m}_{n}}}^{x}\left({{t}_{k}}\right),v_{CA,{{m}_{n}}}^{y}\left({{t}_{k}}\right),v_{CA,{{m}_{n}}}^{z}\left({{t}_{k}}\right)\right]}^{T}}$
(13b) $\displaystyle{{\mathbf{v}}_{CZ,{{m}_{n}}}}\left({{t}_{k}}\right)=$
$\displaystyle{{\left[v_{CZ,{{m}_{n}}}^{x}\left({{t}_{k}}\right),v_{CZ,{{m}_{n}}}^{y}\left({{t}_{k}}\right),v_{CZ,{{m}_{n}}}^{z}\left({{t}_{k}}\right)\right]}^{T}}$
(13c)

where $v_{u}^{x}\left({{t}_{k}}\right)$, $v_{u}^{y}\left({{t}_{k}}\right)$,
and $v_{u}^{z}\left({{t}_{k}}\right)$ are the magnitudes of the user velocity
vector along the $x$, $y$, and $z$ axis, respectively. Symbols
$v_{CA\left(CZ\right),{{m}_{n}}}^{x}\left({{t}_{k}}\right)$,
$v_{CA\left(CZ\right),{{m}_{n}}}^{y}\left({{t}_{k}}\right)$, and
$v_{CA\left(CZ\right),{{m}_{n}}}^{z}\left({{t}_{k}}\right)$ are the speeds of
the first-bounce (last-bounce) scatterer along the $x$, $y$, and $z$ axis,
respectively. Utilizing the state values, the distance parameters of channel
model can be calculated as

$\displaystyle\hat{D}_{{m_{n}}}^{T}\left({{t_{k}}}\right)=\left[{{{{x^{2}_{CA,{m_{n}}}}\left({{t_{k}}}\right)}}}+{{{{y^{2}_{CA,{m_{n}}}}\left({{t_{k}}}\right)}}}+{{{{z^{2}_{CA,{m_{n}}}}\left({{t_{k}}}\right)}}}\right]^{\frac{1}{2}}$
(14) $\displaystyle\hat{D}_{{m_{n}}}^{R}\left({{t_{k}}}\right)=$
$\displaystyle\left[{{\left({{x_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{x_{u}}\left({{t_{k}}}\right)}\right)}^{2}}+{{\left({{y_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-}\right.}}\right.$
(15)
$\displaystyle\left.{{\left.{{y_{u}}\left({{t_{k}}}\right)}\right)}^{2}}+{{\left({{z_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{z_{u}}\left({{t_{k}}}\right)}\right)}^{2}}\right]^{\frac{1}{2}}$

where $\left({\hat{\cdot}}\right)$ means the estimated value. We assume
${\Upsilon_{1}}=\arctan\left({\frac{{{y_{CA,{m_{n}}}}\left({{t_{k}}}\right)}}{{{x_{CA,{m_{n}}}}\left({{t_{k}}}\right)}}}\right)$,
the estimated AAoD is calculated as

$\displaystyle\begin{split}\hat{\alpha}_{{m_{n}}}^{T}\left({{t_{k}}}\right)=\left\\{\begin{array}[]{ll}{\Upsilon_{1}},&{{x_{CA,{m_{n}}}}\left({{t_{k}}}\right)>0,{y_{CA,{m_{n}}}}\left({{t_{k}}}\right)\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle<}\vphantom{{}_{x}}}$}}0}\\\
{\Upsilon_{1}}+\pi,&{{x_{CA,{m_{n}}}}\left({{t_{k}}}\right)<0,{y_{CA,{m_{n}}}}\left({{t_{k}}}\right)>0}\\\
{\Upsilon_{1}}-\pi,&{{x_{CA,{m_{n}}}}\left({{t_{k}}}\right)<0,{y_{CA,{m_{n}}}}\left({{t_{k}}}\right)<0}.\end{array}\right.\end{split}$
(16)

Similarly, we assume the
${\Upsilon_{2}}=\arctan\left({\frac{{{y_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{y_{u}}\left({{t_{k}}}\right)}}{{{x_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{x_{u}}\left({{t_{k}}}\right)}}}\right)$,
the estimated AAoA can be calculated as

$\displaystyle\begin{split}\hat{\alpha}_{{m_{n}}}^{R}\left({{t_{k}}}\right)=\left\\{\begin{array}[]{ll}{\Upsilon_{2}},&\begin{array}[]{l}{x_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{x_{u}}\left({{t_{k}}}\right)>0,\\\
{y_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{y_{u}}\left({{t_{k}}}\right)\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle<}\vphantom{{}_{x}}}$}}0\end{array}\\\
{\Upsilon_{2}}+\pi,&\begin{array}[]{l}{x_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{x_{u}}\left({{t_{k}}}\right)<0,\\\
{y_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{y_{u}}\left({{t_{k}}}\right)>0\end{array}\\\
{\Upsilon_{2}}-\pi,&\begin{array}[]{l}{x_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{x_{u}}\left({{t_{k}}}\right)<0,\\\
{y_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{y_{u}}\left({{t_{k}}}\right)<0.\end{array}\end{array}\right.\end{split}$
(17)

The EAoD and EAoA can be estimated as

$\displaystyle\hat{\beta}_{{m_{n}}}^{T}\left({{t_{k}}}\right)=\arccos\left({\frac{{{z_{CA,{m_{n}}}}\left({{t_{k}}}\right)}}{{\hat{D}_{{m_{n}}}^{T}\left({{t_{k}}}\right)}}}\right)$
(18)

$\displaystyle\hat{\beta}_{{m_{n}}}^{R}\left({{t_{k}}}\right)=\arccos\left({\frac{{{z_{CZ,{m_{n}}}}\left({{t_{k}}}\right)-{z_{u}}\left({{t_{k}}}\right)}}{{\hat{D}_{{m_{n}}}^{R}\left({{t_{k}}}\right)}}}\right).$
(19)

Utilizing the particle filter, the state vector is recursive in the time
domain, and the location and velocity vectors of user and scatterers at
different times can be obtained. For the reason that the mono-static channel
model and communication channel model has an intersection, the initial state
vector can be determined jointly by both of them. It is assumed that the
locations of Tx and Rx, and the velocity of Rx at initial time are known. To
acquire the communication channel information, the ISAC BS transmits the pilot
signal to the Rx at the initial time. At the same time, the receiving array of
the ISAC BS will receive the echo signals backscattered from the scatterers
around the ISAC BS. The received signals of Rx and receiving array of the ISAC
BS are processed separately to obtain the communication channel information
and mono-static sensing channel information. The initial state of the first-
bounce scatterer ${{\mathbf{s}}_{CA}}\left({{t}_{1}}\right)$ can be determined
by the intersection of mono-static sensing channel and communication channel.
The state vector of the last-bounce scatterer at initial time
${{\mathbf{s}}_{CZ}}\left({{t}_{1}}\right)$ is determined by the communication
channel individually. After the state vector is initialized, it needs to
recurse in the time domain. The state transition equation from time $t_{k-1}$
to $t_{k}$ can be represented as

$\displaystyle\mathbf{s}\left({{t}_{k}}\right)=\mathbf{As}\left({{t}_{k-1}}\right)+{{\mathbf{n}}_{s}}\left({{t}_{k}}\right)$
(20)

where $\mathbf{A}$ is state transition matrix,
${{\mathbf{n}}_{s}}\left({{t}_{k}}\right)$ is system noise, which follows the
Gaussian random distribution with zero mean value. The state transition matrix
$\mathbf{A}$ can be computed as

$\displaystyle{\bf{A}}=\left[{\begin{array}[]{*{20}{c}}{\bf{Q}}&{\bf{0}}&{...}&{\bf{0}}&{\bf{0}}\\\
{\bf{0}}&{\bf{Q}}&{...}&{\bf{0}}&{\bf{0}}\\\ {...}&{...}&{...}&{...}&{...}\\\
{\bf{0}}&{\bf{0}}&{...}&{\bf{Q}}&{\bf{0}}\\\
{\bf{0}}&{\bf{0}}&{...}&{\bf{0}}&{\bf{Q}}\end{array}}\right]$ (21)

where

$\displaystyle{\bf{Q}}=\left[\begin{array}[]{cccccc}1&0&0&{T_{\rm{s}}}&0&0\\\
0&1&0&0&{T_{s}}&0\\\ 0&0&1&0&0&{T_{s}}\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\end{array}\right]$ (22)

${{T}_{s}}$ is the sampling interval. Each scatterer is approximated by $C$
particles, so that for the
${{N}_{c}}\left({{t}_{k}}\right){{M}_{n}}\left({{t}_{k}}\right)$ scatterers of
channel model, a total of ${{N}_{c}}\left(t\right){{M}_{n}}\left(t\right)C$
particles are used for approximation. After that, resampling is needed to
extract the particle with the largest weight among the $C$ particles for each
scatterer. The weight can be calculated as the Euclidean distance between the
$c$th particle state vector and the observation vector
$\mathbf{z}\left({{t}_{k}}\right)$, and it is denoted as

$\displaystyle
w^{\left(c\right)}\left({{t}_{k}}\right)=p\left(\left.{{\mathbf{z}}}\left({{t}_{k}}\right)\right|\mathbf{s}^{\left(c\right)}\left({{t}_{k}}\right)\right)=\left|{{\mathbf{z}}}\left({{t}_{k}}\right)-\mathbf{s}^{\left(c\right)}\left({{t}_{k}}\right)\right|.$
(23)

The observation vector is denoted as
${\bf{z}}\left({{t_{k}}}\right)=\left[{{\bf{\tau}}\left({{t_{k}}}\right),{{\bf{\alpha}}^{T}}\left({{t_{k}}}\right),{{\bf{\beta}}^{T}}\left({{t_{k}}}\right),{{\bf{\alpha}}^{R}}\left({{t_{k}}}\right),{{\bf{\beta}}^{R}}\left({{t_{k}}}\right)}\right]$,
which contains the time-varying channel parameters of the communication
channel, including delay, AAoD, EAoD, AAoA, and EAoA.

##  III Statistical properties

###  III-A Delay Spread

The delay spread is caused by the time differences of multipath arrival at Rx.
It is an intuitive and important statistical property in describing the
multipath time dispersion of the channel. The delay spread is expressed as

$\displaystyle{\sigma_{\tau}}\left(t\right)=\sqrt{{{\bar{\tau}}^{2}}\left(t\right)-{{\left({{\mu_{\tau}}\left(t\right)}\right)}^{2}}}$
(24)

where

$\displaystyle{\bar{\tau}^{2}}\left(t\right)=\frac{{\sum\limits_{n=1}^{{N_{c}}\left(t\right)}{\sum\limits_{m=1}^{{M_{n}}\left(t\right)}{{P_{qp,{m_{n}}}}\left(t\right){{\left({\tau_{qp,{m_{n}}}^{NLoS}\left(t\right)}\right)}^{2}}}}}}{{\sum\limits_{n=1}^{{N_{c}}\left(t\right)}{\sum\limits_{m=1}^{{M_{n}}\left(t\right)}{{P_{qp,{m_{n}}}}\left(t\right)}}}}$
(25)
$\displaystyle{\mu_{\tau}}\left(t\right)=\frac{{\sum\limits_{n=1}^{{N_{c}}\left(t\right)}{\sum\limits_{m=1}^{{M_{n}}\left(t\right)}{{P_{qp,{m_{n}}}}\left(t\right)\tau_{qp,{m_{n}}}^{NLoS}\left(t\right)}}}}{{\sum\limits_{n=1}^{{N_{c}}\left(t\right)}{\sum\limits_{m=1}^{{M_{n}}\left(t\right)}{{P_{qp,{m_{n}}}}\left(t\right)}}}}.$
(26)

###  III-B Angular Spread

The angular spread describes the degree of dispersion of multipath in space
domain. The AAoD (AAoA) and EAoD (EAoA) spreads are defined as

$\displaystyle{\sigma_{{\alpha^{T\left(R\right)}}}}\left(t\right)=\sqrt{{{\bar{\alpha}}^{T\left(R\right)}}{{\left(t\right)}^{2}}-{{\left({{\mu_{{\alpha^{T\left(R\right)}}}}\left(t\right)}\right)}^{2}}}$
(27)
$\displaystyle{\sigma_{{\beta^{T\left(R\right)}}}}\left(t\right)=\sqrt{{{\bar{\beta}}^{T\left(R\right)}}{{\left(t\right)}^{2}}-{{\left({{\mu_{{\beta^{T\left(R\right)}}}}\left(t\right)}\right)}^{2}}}.$
(28)

The calculation of ${{\bar{\alpha}}^{T\left(R\right)}}$
(${{\bar{\beta}}^{T\left(R\right)}}$) and
${{\mu_{{\alpha^{T\left(R\right)}}}}\left(t\right)}$
(${{\mu_{{\beta^{T\left(R\right)}}}}\left(t\right)}$) is similar to (25) and
(26), replacing ${\tau_{qp,{m_{n}}}^{NLoS}\left(t\right)}$ with
$\alpha_{{m_{n}}}^{T\left(R\right)}\left(t\right)$ and
$\beta_{{m_{n}}}^{T\left(R\right)}\left(t\right)$, respectively.

##  IV Results and analysis

In this section, the statistical properties of the proposed localization-
assisted ISAC channel model are simulated and analyzed. Because we are not
able to conduct the corresponding channel measurements, RT is used to verify
the proposed channel model. The RT simulation is conducted in outdoor scenario
at 28 GHz. The Tx equips a $32\times 4$ planar array antenna, and the Rx
equips a $2\times 2$ planar array antenna. The map of the RT simulation
environment is shown in Fig. 2. Tx is located at the coordinate origin and its
height is 10 m. Rx moves at a speed of 1 m/s from route1 to route2. The
coordinates of the starting point and the ending point at route1 are (-164, 0)
and (0, 20), respectively. In addition, route2 is from the starting point (0,
20) to the ending point (0, 231). The height of Rx is 1.2 m. The simulation
results of RT are taken as the observation state of particle filter. When Rx
is located at (-104, 7.32, 1.2) of route1 and (0, 120, 1.2) of route2, it is
selected as the initial state of the particle filter. The estimated locations
of user, the first- and last-scatterers 20 s after the initial state are shown
in Fig. 3. It can be found that the scatterers are located around the
buildings in the physical environment. The channel state, including delay and
angular parameters, within 20 s after the initial time is estimated by the
particle filter. The corresponding statistical properties are compared with
the RT simulation results.

Figure 2: RT simulation environment map at outdoor scenario.

Figure 3: Estimated locations of user, the first- and last-scatterers 20 s
after the initial state, (a) route1 and (b) route2.

###  IV-A Delay Spread

Fig. 4 compares the delay spreads of localization-assisted ISAC channel model
simulation results and RT results. The delay spreads when Rx at (-84.2, 9.76,
1.2) of route1 and (0, 140, 1.2) of route2 are shown in Fig. 4 (a) and (b),
respectively. These two points are 20 s after the initial state points. It
shows that the delay spread simulation results of the proposed channel model
are almost identical to the RT simulation results for both of these two
points. This indicates that the proposed channel model can estimate and track
the multipath delay parameter of the channel.

Figure 4: Cumulative distribution functions (CDFs) of delay spread comparison
between ISAC channel model simulation and RT results, (a) route1 and (b)
route2.

###  IV-B Angular Spread

####  IV-B1 Azimuth Angle Spread

Fig. 5 (a) and (b) compare the AAoD spread and AAoA spread simulation results
of the proposed channel model with that of RT results, respectively. It is
found that the azimuth angular spreads of the two points in route1 and route2
both provide a good approximation to the RT results. It shows that the
proposed channel model can track and simulate the azimuth parameters well.

Figure 5: CDFs of azimuth angle spread comparison between ISAC channel model
simulation and RT results, (a) AAoD spread and (b) AAoA spread.

####  IV-B2 Elevation Angle Spread

Fig. 6 (a) compares the EAoD spread of the point at route1 with the RT
results. Fig. 6 (b) compares the EAoA spread of the point at route2 with the
RT results. The EAoD and EAoA spreads can both match well with the RT results.
However, it can be seen that the proposed channel model simulation results
have a long tail at both ends of the curve. It may be caused by fewer antenna
elements in the elevation direction and insufficient angular resolution.

Figure 6: CDFs of elevation angle spread comparison between ISAC channel model
simulation and RT results, (a) route1 and (b) route2.

##  V Conclusions

In this paper, a novel 3D non-stationary localization-assisted ISAC channel
model has been proposed. The particle filter algorithm has been utilized to
localize and track the first- and last-bounce scatterers in the channel model.
The locations of the scatterers have been used to further assist the
communication channel modeling. The locations of scatterers estimated by
particle filter have shown a difference at different time instances, which
explains the non-stationarity of the channel model. The channel properties
have been simulated and compared with RT results, including delay spread, AAoD
(AAoA) spread, and EAoD (EAoA) spread. The simulation results of the channel
model can accurately fit with RT results, which verifies the correctness of
the proposed channel model.

## Acknowledgment

This work was supported by the National Key R$\&$D Program of China under
Grant 2018YFB1801101, the National Natural Science Foundation of China (NSFC)
under Grants 61960206006 and 62271147, the Key Technologies R$\&$D Program of
Jiangsu (Prospective and Key Technologies for Industry) under Grants BE2022067
and BE2022067-1, the Frontiers Science Center for Mobile Information
Communication and Security, the EU H2020 RISE TESTBED2 project under Grant
872172, and the High Level Innovation and Entrepreneurial Doctor Introduction
Program in Jiangsu under Grant JSSCBS20210082.

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Paper-and-pencil cosmological calculator

S.V. Pilipenko
Astrospace Center of the Lebedev Physical Institute, 117997 Moscow,
Profsojuznaja 84/32
Moscow Institute of Physics and Technology, 141700 Dolgoprudnyj, Russia,
Institutskij per. 9

A cosmological calculator is usually a program which computes relations
between redshift, distance, physical and angular sizes, luminosity and
apparent magnitude for some cosmological model characterized by a convenient
set of parameters such as the Hubble constant at present, $H_{0}$, the
dimensionless matter and dark energy densities $\Omega_{m}$ and
$\Omega_{\Lambda}$. Several on-line calculators are available at NASA NED web
site (http://ned.ipac.caltech.edu/help/cosmology_calc.html). Although these
calculators are ready to use, there are situations when there is no the
Internet or a PC at hand. Online calculators also may be not enough vivid to
use them in education since they usually calculate the answer only for a
single input redshift. One solution to these two problems is a calculator
based on the nomogram method.

The proposed paper-and-pencil calculator is designed for the $\Lambda$CDM
cosmological model with recent cosmological parameters from the Planck
mission: $H_{0}=67.15$ km/s/Mpc, $\Omega_{\Lambda}=0.683$ and
$\Omega_{m}=0.317$ [1]. The calculator contains the following quantities:

  * •

z — redshift;

  * •

H — current value of the Hubble constant, km/s/Mpc;

  * •

r_comov — comoving distance, Mpc;

  * •

dm — distance modulus;

  * •

age — age of the Universe, Gyr;

  * •

time — lookback time, Gyr;

  * •

size 1” — physical size of an object which is seen as an 1” arc on the sky,
kpc;

  * •

angle 1kpc — angular size of a rod with physical size 1 kpc, arcsec.

In order to use the calculator, one needs to find a known value on a
respective vertical scale. All the other values are situated on the same
horizontal level. For the comfort of using a ruler, the redshift scale is
repeated twice. The calculators are available for three redshift intervals:
$z<20$, $z<1$, $z<0.1$. The space between major (labelled) tics on each
vertical scale is always divided into ten equally-spaced intervals of the
denoted value.

The code used to produce these calculators is public available and can be
found at http://code.google.com/p/cosmonom/

## References

  * [1] Planck collaboration XVI, 2013, submitted to A&A, arXiv:astro-ph/1303.5076