$\displaystyle|\boldsymbol{w}_{r,(j)}(t)-\boldsymbol{w}_{r,(j)}(0)|\leq\int_{0}^{t}|\dot{\boldsymbol{w}}_{r,(j)}(s)|\mathrm{d}s\leq\frac{4\sqrt{3}B\|\boldsymbol{y}\|_{2}}{\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))\sqrt{n}}\sqrt{\frac{\log
m}{m}}\leq R^{\prime},$
$\displaystyle|b_{r}(t)-b_{r}(0)|\leq\int_{0}^{t}|\dot{b}_{r}(s)|\mathrm{d}s\leq\frac{4\sqrt{3}\|\boldsymbol{y}\|_{2}}{\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))\sqrt{n}}\sqrt{\frac{\log
m}{m}}\leq R^{\prime},$

hold for all $r\in[2m],j\in[d]$. ∎

###### Proof of Proposition C.12.

For every $\omega\in\mathcal{B}\cap\mathcal{R}\cap\mathcal{C}$, let
$\tau=\min\\{\tau_{\lambda},\tau_{\boldsymbol{\theta}},\tau_{\boldsymbol{u}}\\}$,
where

$\displaystyle\tau_{\lambda}=\inf\left\\{t~{}\middle|~{}\lambda_{\min}(K_{\boldsymbol{\theta}(t)}^{m}(\boldsymbol{X},\boldsymbol{X}))<\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))/2\right\\},$
$\displaystyle\tau_{\boldsymbol{\theta}}=\inf\left\\{t~{}\middle|~{}\boldsymbol{\theta}(t)\notin\boldsymbol{\Theta}(\boldsymbol{\theta}(0),R^{\prime})\right\\},$
$\displaystyle\tau_{\boldsymbol{u}}=\inf\left\\{t~{}\middle|~{}\|\boldsymbol{u}^{m}(t)\|_{2}^{2}>e^{-\frac{\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))}{n}t}\|\boldsymbol{u}(0)\|_{2}^{2}\right\\}.$

We will show that $\tau=\infty$ by contradiction, so that
$\omega\in\mathcal{A}$. Notice that $\tau_{\lambda}<\tau_{\boldsymbol{u}}$ and
$\tau_{\lambda}<\tau_{\boldsymbol{\theta}}$ according to Lemma C.13 and Lemma
C.15 respectively. However if $\tau=\tau_{\lambda}<\infty$, which suggests
that
$\boldsymbol{\theta}(\tau)\notin\boldsymbol{\Theta}(\boldsymbol{\theta}(0),R)$
by Lemma C.14. Then there must exists some $t_{0}$ such that $0\leq
t_{0}<\tau$ and
$\boldsymbol{\theta}(t_{0})\notin\boldsymbol{\Theta}(\boldsymbol{\theta}(0),R^{\prime})$
since $R^{\prime}<R$, which however violates the assumption that
$\tau=\tau_{\lambda}$. ∎

###  C.2 Proof of Proposition 3.2

Since we have
$f_{\boldsymbol{\theta}(0)}^{m}(\boldsymbol{x})=f_{0}^{\mathtt{NTK}}(\boldsymbol{x})=0$
for every $\boldsymbol{x}\in\mathcal{X}$ under our initialization setting, we
can bound the difference between
$f_{\boldsymbol{\theta}(t)}^{m}(\boldsymbol{x})$ and
$f_{t}^{\mathtt{NTK}}(\boldsymbol{x})$ by bounding the difference between
their derivative, i.e.,

$|f_{\boldsymbol{\theta}(t)}^{m}(\boldsymbol{x})-f_{t}^{\mathtt{NTK}}(\boldsymbol{x})|=\left|\int_{0}^{t}\dot{f}_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{x})-\dot{f}_{s}^{\mathtt{NTK}}(\boldsymbol{x})\mathrm{d}s\right|\leq\int_{0}^{t}|\dot{f}_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{x})-\dot{f}_{s}^{\mathtt{NTK}}(\boldsymbol{x})|\mathrm{d}s.$

Recall that

$\displaystyle\dot{f}_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{x})=-\frac{1}{n}K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{x},\boldsymbol{X})\boldsymbol{u}^{m}(s),$
$\displaystyle\dot{f}_{s}^{\mathtt{NTK}}(\boldsymbol{x})=-\frac{1}{n}K_{d}(\boldsymbol{x},\boldsymbol{X})\boldsymbol{u}^{\mathtt{NTK}}(s),$

where
$\boldsymbol{u}^{m}(s)=f_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X})-\boldsymbol{y}$
and
$\boldsymbol{u}^{\mathtt{NTK}}(s)=f_{s}^{\mathtt{NTK}}(\boldsymbol{X})-\boldsymbol{y}$.
Let

$\Delta=\sup_{t\geq
0}\sup_{\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathcal{X}}|K_{\boldsymbol{\theta}(t)}^{m}(\boldsymbol{x},\boldsymbol{x}^{\prime})-K_{d}(\boldsymbol{x},\boldsymbol{x}^{\prime})|.$

Then

$\displaystyle\quad|f_{\boldsymbol{\theta}(t)}^{m}(\boldsymbol{x})-f_{t}^{\mathtt{NTK}}(\boldsymbol{x})|$
$\displaystyle\leq\frac{1}{n}\int_{0}^{t}|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{x},\boldsymbol{X})\boldsymbol{u}^{m}(s)-K_{d}(\boldsymbol{x},\boldsymbol{X})\boldsymbol{u}^{\mathtt{NTK}}(s)|\mathrm{d}s$
$\displaystyle\leq\frac{1}{n}\int_{0}^{t}\left(\|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{x},\boldsymbol{X})^{\top}-K_{d}(\boldsymbol{x},\boldsymbol{X})^{\top}\|_{2}+\|K_{d}(\boldsymbol{x},\boldsymbol{X})^{\top}\|_{2}\right)\|\boldsymbol{u}^{m}(s)-\boldsymbol{u}^{\mathtt{NTK}}(s)\|_{2}\mathrm{d}s$
$\displaystyle\quad+\frac{1}{n}\int_{0}^{t}\|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{x},\boldsymbol{X})^{\top}-K_{d}(\boldsymbol{x},\boldsymbol{X})^{\top}\|_{2}\|\boldsymbol{u}^{\mathtt{NTK}}(s)\|_{2}\mathrm{d}s$
$\displaystyle\leq\frac{1}{n}\cdot(\sqrt{n}\Delta+\sqrt{n}C)\cdot\int_{0}^{t}\|\boldsymbol{y}\|_{2}\Delta
se^{-\frac{1}{n}(\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))-n\Delta)s}\mathrm{d}s$
$\displaystyle\quad+\frac{1}{n}\cdot\sqrt{n}\Delta\cdot\int_{0}^{t}e^{-\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))s}\|\boldsymbol{y}\|_{2}\mathrm{d}s$
$\displaystyle\leq\frac{1}{n}\cdot(\sqrt{n}\Delta+\sqrt{n}C)\cdot\|\boldsymbol{y}\|_{2}\Delta(\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))/2)^{-2}$
$\displaystyle\quad+\frac{1}{n}\cdot\sqrt{n}\Delta\cdot\left(\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))\right)^{-1}\|\boldsymbol{y}\|_{2}$
$\displaystyle\leq\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon,$

where $C>0$ is a constant depending only on $B$ and we apply Lemma C.16 and
Lemma C.17 in the third inequality and the last line follows from Proposition
3.3 that for sufficiently large $m$, we have

$\Delta\leq\min\left\\{C,\frac{\epsilon(\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X})))^{2}\sqrt{n}}{16C\|\boldsymbol{y}\|_{2}},\frac{\epsilon\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))}{2\sqrt{n}\|\boldsymbol{y}\|_{2}}\right\\},$

with probability at least $1-\delta$ over initialization.

######  Lemma C.16.

For all $t\geq 0$, we have $\|\boldsymbol{u}^{\mathtt{NTK}}(t)\|_{2}\leq
e^{-\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))t}\|\boldsymbol{y}\|_{2}$.

###### Proof.

Recall that
$\dot{\boldsymbol{u}}^{\mathtt{NTK}}(t)=-\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})\boldsymbol{u}^{\mathtt{NTK}}(t)$.
Notice that $K_{d}(\boldsymbol{X},\boldsymbol{X})$ is fixed, so we can write
the explicit form
$\boldsymbol{u}^{\mathtt{NTK}}(t)=e^{-\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})t}\boldsymbol{u}^{\mathtt{NTK}}(0)$,
where $\boldsymbol{u}^{\mathtt{NTK}}(0)=-\boldsymbol{y}$. Then
$\|\boldsymbol{u}^{\mathtt{NTK}}(t)\|_{2}\leq
e^{-\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))t}\|\boldsymbol{y}\|_{2}$,
since
$\|e^{-\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})t}\|_{2}=e^{-\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))t}$.
∎

######  Lemma C.17.

For all $t\geq 0$, we have

$\|\boldsymbol{u}^{m}(t)-\boldsymbol{u}^{\mathtt{NTK}}(t)\|_{2}\leq\|\boldsymbol{y}\|_{2}\Delta
te^{-\frac{1}{n}(\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))-n\Delta)t}$

###### Proof.

Recall that we can express explicitly for all $s\geq 0$,

$\displaystyle\dot{\boldsymbol{u}}^{m}(s)=-\frac{1}{n}K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X})\boldsymbol{u}^{m}(s)$
$\displaystyle\text{~{}and~{}}\dot{\boldsymbol{u}}^{\mathtt{NTK}}(s)=-\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})\boldsymbol{u}^{\mathtt{NTK}}(s).$

Notice that

$\frac{\mathrm{d}}{\mathrm{d}s}e^{\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})s}\left(\boldsymbol{u}^{m}(s)-\boldsymbol{u}^{\mathtt{NTK}}(s)\right)=\frac{1}{n}e^{\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})s}(K_{d}(\boldsymbol{X},\boldsymbol{X})-K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X}))\boldsymbol{u}^{m}(s).$

Since $\boldsymbol{u}^{m}(0)=\boldsymbol{u}^{\mathtt{NTK}}(0)$, integrating
gives

$e^{\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})t}\left(\boldsymbol{u}^{m}(t)-\boldsymbol{u}^{\mathtt{NTK}}(t)\right)=\frac{1}{n}\int_{0}^{t}e^{\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})s}(K_{d}(\boldsymbol{X},\boldsymbol{X})-K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X}))\boldsymbol{u}^{m}(s)\mathrm{d}s,$

then

$\boldsymbol{u}^{m}(t)-\boldsymbol{u}^{\mathtt{NTK}}(t)=\frac{1}{n}\int_{0}^{t}e^{\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})(s-t)}(K_{d}(\boldsymbol{X},\boldsymbol{X})-K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X}))\boldsymbol{u}^{m}(s)\mathrm{d}s.$

Bounding the norm implies

$\displaystyle\|\boldsymbol{u}^{m}(t)-\boldsymbol{u}^{\mathtt{NTK}}(t)\|_{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\int_{0}^{t}\|e^{\frac{1}{n}K_{d}(\boldsymbol{X},\boldsymbol{X})(s-t)}(K_{d}(\boldsymbol{X},\boldsymbol{X})-K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X}))\boldsymbol{u}^{m}(s)\|_{2}\mathrm{d}s$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\int_{0}^{t}e^{\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))(s-t)}\|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X})-K_{d}(\boldsymbol{X},\boldsymbol{X})\|_{2}\|\boldsymbol{u}^{\mathtt{NTK}}(s)\|_{2}\mathrm{d}s$
$\displaystyle+\frac{1}{n}\int_{0}^{t}e^{\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))(s-t)}\|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X})-K_{d}(\boldsymbol{X},\boldsymbol{X})\|_{2}\|\boldsymbol{u}^{m}(s)-\boldsymbol{u}^{\mathtt{NTK}}(s)\|_{2}\mathrm{d}s.$

For all $s\geq 0$, let

$\displaystyle
u(s)=e^{\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))s}\|\boldsymbol{u}^{m}(s)-\boldsymbol{u}^{\mathtt{NTK}}(s)\|_{2},$
$\displaystyle\alpha(s)=\frac{1}{n}\int_{0}^{s}e^{\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))s^{\prime}}\|K_{\boldsymbol{\theta}(s^{\prime})}^{m}(\boldsymbol{X},\boldsymbol{X})-K_{d}(\boldsymbol{X},\boldsymbol{X})\|_{2}\|\boldsymbol{u}^{\mathtt{NTK}}(s^{\prime})\|_{2}\mathrm{d}s^{\prime},$
$\displaystyle\text{~{}and~{}}\beta(s)=\frac{1}{n}\|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X})-K_{d}(\boldsymbol{X},\boldsymbol{X})\|_{2}.$

Notice that $\beta(\cdot)$ is non-negative, $\alpha(\cdot)$ is non-decreasing
and $u(\cdot)$ satisfies that
$u(s)\leq\alpha(s)+\int_{0}^{s}\beta(s^{\prime})u(s^{\prime})\mathrm{d}s^{\prime}$
for all $s\geq 0$. Applying the Grönwall’s inequality [64] yields
$u(s)\leq\alpha(s)e^{\int_{0}^{s}\beta(s^{\prime})\mathrm{d}s^{\prime}}$ for
all $s\geq 0$. Then we have

$\displaystyle\quad\|\boldsymbol{u}^{m}(t)-\boldsymbol{u}^{\mathtt{NTK}}(t)\|_{2}$
$\displaystyle\leq\frac{1}{n}\int_{0}^{t}e^{\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))(s-t)}\|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X})-K_{d}(\boldsymbol{X},\boldsymbol{X})\|_{2}\|\boldsymbol{u}^{\mathtt{NTK}}(s)\|_{2}\mathrm{d}s$
$\displaystyle\quad\cdot
e^{\frac{1}{n}\int_{0}^{t}\|K_{\boldsymbol{\theta}(s)}^{m}(\boldsymbol{X},\boldsymbol{X})-K_{d}(\boldsymbol{X},\boldsymbol{X})\|_{2}\mathrm{d}s}$
$\displaystyle\leq\frac{1}{n}\cdot n\Delta\cdot
e^{-\frac{1}{n}\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))t}\|\boldsymbol{y}\|_{2}\cdot
t\cdot e^{\frac{1}{n}\cdot n\Delta\cdot t}=\|\boldsymbol{y}\|_{2}\Delta
te^{-\frac{1}{n}(\lambda_{\min}(K_{d}(\boldsymbol{X},\boldsymbol{X}))-n\Delta)t},$

where Lemma C.16 is applied in the second inequality. ∎

##  Appendix D Proof of Theorem 4.1

We recollect some essentials of spectral algorithms for the convenience of the
readers here. For a thorough introduction on spectral algorithms, we refer the
interested readers to [44] and references therein. For simplicity, let
$\mathcal{X}\subseteq\mathbb{R}^{d}$ be a compact set and
$K:\mathcal{\mathcal{X}}\times\mathcal{X}$ a kernel function which is
continuous and measurable. Denote the RKHS of $K$ by $\mathcal{H}$. Assume
that the kernel $K$ satisfies that
$\sup_{\boldsymbol{x}\in\mathcal{X}}K(\boldsymbol{x},\boldsymbol{x})\leq\kappa^{2}$
for some $\kappa<\infty$. Notice $K_{d}$ satisfies all the assumptions above.
Let $K_{\boldsymbol{x}}:\mathbb{R}\to\mathcal{H}$ be the mapping $x\mapsto
xK(\cdot,\boldsymbol{x})$ and the adjoint operator
$K_{\boldsymbol{x}}^{*}:\mathcal{H}\to\mathbb{R}$ such that
$K_{\boldsymbol{x}}^{*}f\mapsto f(\boldsymbol{x})$. We further introduce
$T_{\boldsymbol{x}}=K_{\boldsymbol{x}}K_{\boldsymbol{x}}^{*}:\mathcal{H}\to\mathcal{H}$
and $T_{\boldsymbol{X}}=\frac{1}{n}\sum_{i=1}^{n}T_{\boldsymbol{x}_{i}}$.

######  Definition 3 (Filter functions).

Let $\Lambda$ be a subset of $[0,\infty)\cup\\{\infty\\}$ and define
$\infty^{-1}=0$. The functions
$\\{\mathcal{G}_{\lambda}:[0,\kappa^{2}]\to[0,\infty),\lambda\in\Lambda\\}$
are the filter functions with qualification $\tau\geq 1$ if there exists
absolute constant $E$ and constant $F_{\tau}$ depending on $\tau$ such that

$\sup_{\lambda\in\Lambda}\sup_{\alpha\in[0,1]}\sup_{u\in[0,\kappa^{2}]}u^{\alpha}\mathcal{G}_{\lambda}(u)\lambda^{1-\alpha}\leq
E$

and

$\sup_{\lambda\in\Lambda}\sup_{\alpha\in[0,\tau]}\sup_{u\in[0,\kappa^{2}]}u^{\alpha}|1-u\mathcal{G}_{\lambda}(u)|\lambda^{-\alpha}\leq
F_{\tau}.$

######  Definition 4 (Spectral algorithms).

Given the filter functions $\mathcal{G}_{\lambda}(u)$, define

$\mathcal{G}_{\lambda}(T_{\boldsymbol{X}})=\sum_{j=1}^{\infty}\mathcal{G}_{\lambda}(\hat{\lambda}_{j})\langle\cdot,\hat{e}_{j}\rangle_{\mathcal{H}}\hat{e}_{j},$

where $\\{\hat{\lambda}_{j},j\geq 1\\}$ and $\\{\hat{e}_{j},j\geq 1\\}$ are
eigenvalues and eigenfunctions of $T_{\boldsymbol{X}}$. The estimator
$\hat{f}_{\lambda}$ reads as follows,

$\hat{f}_{\lambda}=\mathcal{G}_{\lambda}(T_{\boldsymbol{X}})\frac{1}{n}\sum_{i=1}^{n}y_{i}K(\cdot,\boldsymbol{x}_{i}).$

######  Lemma D.1.

The filter function corresponding to gradient flow is

$\mathcal{G}_{\lambda}(u)=(1-e^{-u/\lambda})/u,$

where $\Lambda=(0,\infty)\cup\\{\infty\\}$, $\tau$ could be any real number
which is greater than or equal to $1$, $E=1$, $F_{\tau}=(\tau/e)^{\tau}$.

###### Proof.

By the definition of gradient flow,

$\dot{f}=-\frac{\partial\hat{\mathcal{L}}_{n}}{\partial
f}=-\frac{1}{n}\sum_{i=1}^{n}-K_{\boldsymbol{x}_{i}}(y_{i}-f(\boldsymbol{x}_{i}))=-(T_{\boldsymbol{X}}f-\frac{1}{n}\sum_{i=1}^{n}y_{i}K(\cdot,\boldsymbol{x}_{i})),$

where
$\hat{\mathcal{L}}_{n}(f)=\frac{1}{2n}\sum_{i=1}^{n}(y_{i}-f(\boldsymbol{x}_{i}))^{2}$
and $f_{0}=0\in\mathcal{H}$. Hence

$f_{t}=T_{\boldsymbol{X}}^{-1}(I-e^{-tT_{\boldsymbol{X}}})\frac{1}{n}\sum_{i=1}^{n}y_{i}K(\cdot,\boldsymbol{x}_{i}),$

where $T_{\boldsymbol{X}}^{-1}$ is the Moore-Penrose inverse of
$T_{\boldsymbol{X}}$. So $\mathcal{G}_{\lambda}(u)=(1-e^{-u/\lambda})/u$,
where we parameterize $t=1/\lambda$. It could be verified that
$\mathcal{G}_{\lambda}(u)=\sum_{j=1}^{\infty}\frac{(-1)^{j-1}(1/\lambda)^{j}}{j!}u^{j-1}$
is continuous for all $u\geq 0$ so $\mathcal{G}_{\lambda}(\cdot)$ can be
applied to $T_{\boldsymbol{X}}$ by Theorem 5.1.11 from [55]. It could also be
checked that $u^{\alpha}\mathcal{G}_{\lambda}(u)\lambda^{1-\alpha}\leq 1$ by
the fact that $1-e^{-x}\leq x$ for all $x\geq 0$ and
$u^{\alpha}|1-u\mathcal{G}_{\lambda}(u)|\lambda^{-\alpha}\leq(\alpha/e)^{\alpha}$.
∎

######  Proposition D.2 (Corollary 4.4 in [44]).

Suppose Assumption 1 holds and we observed $n$ i.i.d. samples
$\\{(\boldsymbol{x}_{i},y_{i}),i\in[n]\\}$ from the model (1). For any given
$\delta\in(0,1)$, if the training process is stopped at $t_{\star}\propto
n^{2/3}$ for the NTK regression, then for sufficiently large $n$, there exists
a constant $C$ independent of $\delta$ and $n$, such that

$\mathcal{E}(f_{t_{\star}}^{\mathtt{NTK}})=Cn^{-\frac{2}{3}}\log^{2}\frac{6}{\delta}$

holds with probability at least $1-\delta$ over the training data.

Setting $\epsilon=Cn^{-\frac{2}{3}}\log^{2}\frac{6}{\delta}$ in Theorem 3.1
yields

$\displaystyle\mathcal{E}(f_{\boldsymbol{\theta}(t_{\star})}^{m})$
$\displaystyle\leq|\mathcal{E}(f_{\boldsymbol{\theta}(t_{\star})}^{m})-\mathcal{E}(f_{t_{\star}}^{\mathtt{NTK}})|+\mathcal{E}(f_{t_{\star}}^{\mathtt{NTK}})$
$\displaystyle\leq 2Cn^{-\frac{2}{3}}\log^{2}\frac{6}{\delta}.$

##  Appendix E Proof of Theorem 4.2

###### Proof of Theorem 4.2 $i)$.

Theorem 4.2 $i)$ is a direct corollary of the following lemmas and Proposition
3.2.

######  Lemma E.1 (Overfitted NTK model can be approximated by the linear
interpolation).

Suppose that we have observed $n$ data $\\{(x_{i},y_{i}),i\in[n]\\}$ from the
model (1) and $x_{i}=\frac{i-1}{n-1}$, $i\in[n]$. With the probability at
least $1-C_{1}/n$, the overfitted NTK model with zero initialization
$f^{\mathtt{NTK}}_{\infty}(x)=K_{1}(x,\boldsymbol{X})K_{1}^{-1}(\boldsymbol{X},\boldsymbol{X})\boldsymbol{y}$
can be approximated by the linear interpolation, i.e.,

$\displaystyle\sup_{x\in[0,1]}|f_{\infty}^{\mathtt{NTK}}(x)-f_{\mathtt{LI}}(x)|\leq
C_{2}\sqrt{\log n}/(n-1)^{2}$ (59)

for some absolute constants $C_{1}$ $C_{2}$.

######  Lemma E.2.

Suppose that we have observed $n$ data $\\{(x_{i},y_{i}),i\in[n]\\}$ from the
model (1) and $x_{i}=\frac{i-1}{n-1}$, $i\in[n]$. If $t\geq C_{1}n^{2}\log n$,
we have

$\sup_{x\in[0,1]}|f_{t}^{\mathtt{NTK}}(x)-f_{\infty}^{\mathtt{NTK}}(x)|\leq\frac{C_{2}}{(n-1)^{3}}$
(60)

for some absolute constants $C_{1}$, $C_{2}$.

∎

###### Proof of lemma E.1.

Since

$y_{i}=K(x_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}$

and

$y_{i+1}=K(x_{i+1},\boldsymbol{X})K^{-1}\boldsymbol{y},$

the Taylor Expansion and intermediate theorem imply that for $\forall
x\in(x_{i},x_{i+1})$, there are $\xi_{i}$ and
$\hat{\xi}_{i}\in(x_{i},x_{i+1})$ such that

$\displaystyle K(x,\boldsymbol{X})K^{-1}\boldsymbol{y}-y_{i}$
$\displaystyle=(x-x_{i})K_{+}^{\prime}(x_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}+\frac{(x-x_{i})^{2}}{2}K^{\prime\prime}(\xi_{i},\boldsymbol{X})K^{-1}\boldsymbol{y},$
(61) $\displaystyle y_{i+1}-y_{i}$
$\displaystyle=(x_{i+1}-x_{i})K_{+}^{\prime}(x_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}+\frac{(x_{i+1}-x_{i})^{2}}{2}K^{\prime\prime}(\hat{\xi}_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}$
(62)

where $K^{\prime}_{+}(x_{i},\boldsymbol{X})=\lim_{x\rightarrow
x_{i}^{+}}K^{\prime}_{+}(x,\boldsymbol{X})=\lim_{x\rightarrow
x_{i}^{+}}\frac{\partial K(x,\boldsymbol{X})}{\partial x}$. Thus,

$\displaystyle K(x,\boldsymbol{X})K^{-1}\boldsymbol{y}$ $\displaystyle-
y_{i}-\frac{(x-x_{i})}{x_{i+1}-x_{i}}(y_{i+1}-y_{i})$ (63)
$\displaystyle=-\frac{(x-x_{i})}{x_{i+1}-x_{i}}\frac{(x_{i+1}-x_{i})^{2}}{2}K^{\prime\prime}(\hat{\xi}_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}$
$\displaystyle+\frac{(x-x_{i})^{2}}{2}K^{\prime\prime}(\xi_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}.$

The second-order derivative can be bounded by the following lemma:

######  Lemma E.3 (Bounded second order derivative of overfitted NTK
regression).

Suppose that we have observed $n$ data $\\{(x_{i},y_{i}),i\in[n]\\}$ from the
model (1) and $x_{i}=\frac{i-1}{n-1}$, $i\in[n]$. With the probability at
least $1-\frac{2}{n}$, we have

$\displaystyle\sup_{x\in(x_{i},x_{i+1})}|K_{1}^{\prime\prime}(x,\boldsymbol{X})K_{1}^{-1}(\boldsymbol{X},\boldsymbol{X})\boldsymbol{y}|\leq
C\sqrt{\log n}$ (64)

for $\forall i\in[n]$ and for some absolute constant $C$.

By Lemma E.3,
$|K^{\prime\prime}(\xi_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}|<C\log n$ for
some constant $C$ and the RHS of (63) is bounded by $\frac{C\sqrt{\log
n}}{(n-1)^{2}}$ for some constant $C$. ∎

###### Proof of Lemma E.2.

$K(x,x^{\prime})$ is bounded by a constant By Lemma E.9, $|y_{i}|$ are also
bounded by $C\sqrt{\log n}$ with the probability at least $1-2/n$. By Theorem
2.2, $\lambda_{\min}=C_{1}/n$ for some constant $C_{1}$. If $t\geq
C_{2}n^{2}\log(n^{6})=C_{3}n^{2}\log n$ for some constants $C_{2}$, $C_{3}$,
we have

$\begin{split}|f_{t}^{\mathtt{NTK}}(x)-f_{\infty}^{\mathtt{NTK}}(x)|&=e^{-\frac{tK_{1}(\boldsymbol{X},\boldsymbol{X})}{n}}|K_{1}(x,\boldsymbol{X})K_{1}^{-1}(\boldsymbol{X},\boldsymbol{X})\boldsymbol{y}|\\\
&\leq e^{-\frac{t\lambda_{\min}}{n}}\lambda_{\min}^{-1}n\sqrt{\log n}\\\
&\leq\frac{C_{4}}{(n-1)^{3}}\end{split}$

for some constant $C_{4}$. ∎

###### Proof of Theorem 4.2 $ii)$.

Theorem 4.2 $ii)$ is a direct corollary of the following lemma and Theorem 4.2
$i)$.

######  Lemma E.4 (Linear Interpolation cannot Generalize Well).

Suppose that we have observed $n$ data $\\{(x_{i},y_{i}),i\in[n]\\}$ from the
model (1) and $x_{i}=\frac{i-1}{n-1}$, $i\in[n]$. Let $f_{\mathtt{LI}}$ be a
linear interpolation estimator. Then there exists a positive constant $C$ such
that

$\mathcal{E}(f_{\mathtt{LI}}(x))\geq\frac{1}{3}\sigma^{2}.$ (65)

holds with probability at least $1-\frac{C}{n}$.

∎

###### Proof of Lemma E.4.

For $x\in[x_{i},x_{i+1}]$, the linear interpolation takes the form

$f_{\mathtt{LI}}(x)=\lambda_{i}(x)y_{i}+(1-\lambda_{i}(x))y_{i+1},$

where $\lambda_{i}(x)=\frac{x_{i+1}-x}{x_{i+1}-x_{i}}$.

Denote
$b^{2}(x)=(\mathbf{E}_{\boldsymbol{\varepsilon}}f_{\mathtt{LI}}(x)-f_{\star}(x))^{2}$
and
$\sigma^{2}(x)=(f_{\mathtt{LI}}(x)-\mathbf{E}_{\boldsymbol{\varepsilon}}f_{\mathtt{LI}}(x))^{2}$
to be the bias and variance term respectively, where we denote by
$\mathbf{E}_{\boldsymbol{\varepsilon}}$ taking expectations with respect to
the noise $\varepsilon_{1},\dots,\varepsilon_{n}$. Thus the excess risk of the
linear interpolation $\mathcal{E}(f_{\mathtt{LI}})$ can be formulated as

$\displaystyle\mathcal{E}(f_{\mathtt{LI}})$
$\displaystyle=\int_{0}^{1}(b(x)+\sigma(x))^{2}\mathrm{d}x$ (66)
$\displaystyle=c_{1}+\sum_{i=1}^{n-1}\frac{c_{2,i}}{n-1}(\varepsilon_{i}+\varepsilon_{i+1})+\frac{1}{3(n-1)}\sum_{i=1}^{n-1}(\varepsilon_{i}^{2}+\varepsilon_{i+1}^{2}+\varepsilon_{i}\varepsilon_{i+1}).$
(67)

for some positive constant $c_{1}$ and a uniformly bounded sequence
$\\{c_{2,i}\\}_{i=1}^{n-1}$. The last equation is the result of Lemma E.5. The
expectation of $\mathcal{E}(f_{\mathtt{LI}})$

$\displaystyle\mathbf{E}_{\boldsymbol{\varepsilon}}\mathcal{E}(f_{\mathtt{LI}})=c_{1}+\frac{2}{3}\sigma^{2}\geq\frac{2}{3}\sigma^{2}$
(68)

and the variance of $\mathcal{E}(f_{\mathtt{LI}})$

$\displaystyle\operatorname{Var}_{\boldsymbol{\varepsilon}}(\mathcal{E}(f_{\mathtt{LI}}))\leq\frac{c_{3}}{n}$
(69)

for some constant $c_{3}$. By Chebyshev’s inequality, we have

$\displaystyle\mathbf{P}(|\mathcal{E}(f_{\mathtt{LI}})-\mathbf{E}_{\boldsymbol{\varepsilon}}(\mathcal{E}(f_{\mathtt{LI}}))|\geq\frac{1}{3}\sigma^{2})\leq\frac{\operatorname{Var}_{\boldsymbol{\varepsilon}}(\mathcal{E}(f_{\mathtt{LI}}))}{\frac{1}{9}\sigma^{4}}.$
(70)

Thus, we conclude that with probability at least $1-\frac{c_{4}}{n}$,

$\displaystyle\mathcal{E}(f_{\mathtt{LI}})\geq\frac{1}{3}\sigma^{2}.$ (71)

for some constant $c_{4}$. ∎

######  Lemma E.5.

Denote
$b^{2}(x)=(\mathbf{E}_{\boldsymbol{\varepsilon}}f_{\mathtt{LI}}(x)-f_{\star}(x))^{2}$
and
$\sigma^{2}(x)=(f_{\mathtt{LI}}(x)-\mathbf{E}_{\boldsymbol{\varepsilon}}f_{\mathtt{LI}}(x))^{2}$.
$\mathcal{E}(f_{\mathtt{LI}})=\int_{0}^{1}(b(x)+\sigma(x))^{2}dx$ can be
reformulated as

$\displaystyle\mathcal{E}(f_{\mathtt{LI}})=c_{1}+\sum_{i=1}^{n-1}\frac{c_{2,i}}{n-1}(\varepsilon_{i}+\varepsilon_{i+1})+\frac{1}{3(n-1)}\sum_{i=1}^{n-1}(\varepsilon_{i}^{2}+\varepsilon_{i+1}^{2}+\varepsilon_{i}\varepsilon_{i+1})$
(72)

for some positive constant $c_{1}$ and a uniformly bounded sequence
$\\{c_{2,i}\\}_{i=1}^{n}$.

###### Proof of Lemma E.5.

Denote

$\displaystyle b_{i}(x)$
$\displaystyle=(1-\frac{x-x_{i}}{x_{i+1}-x_{i}})(f_{\star}(x_{i})-f_{\star}(x))+\frac{x-x_{i}}{x_{i+1}-x_{i}}(f_{\star}(x_{i+1})-f_{\star}(x))$
(73) $\displaystyle\sigma_{i}(x)$
$\displaystyle=(1-\frac{x-x_{i}}{x_{i+1}-x_{i}})\varepsilon_{i}+\frac{x-x_{i}}{x_{i+1}-x_{i}}\varepsilon_{i+1}.$
(74)

$\displaystyle\mathcal{E}(f_{\mathtt{LI}})$
$\displaystyle=\int_{0}^{1}b^{2}(x)+2b(x)\sigma(x)+\sigma^{2}(x)\mathrm{d}x$
(75)
$\displaystyle=\sum_{i=1}^{n-1}\int_{x_{i}}^{x_{i+1}}b^{2}_{i}(x)+2b_{i}(x)\sigma_{i}(x)+\sigma^{2}_{i}(x)\mathrm{d}x$
(76)

Since $f_{\star}$ is bounded, $b_{i}(x)\in(C_{1},C_{2})$ where $C_{1}$ and
$C_{2}$ depend on $f_{\star}$. Thus, by the mean value theorems, there exists
a positive constant $c_{1}$ and a uniformly bounded sequence
$\\{c_{2,i}\\}_{i=1}^{n}$ such that

$\displaystyle\mathcal{E}(f_{\mathtt{LI}})=c_{1}+\sum_{i=1}^{n-1}\frac{c_{2,i}}{n-1}(\varepsilon_{i}+\varepsilon_{i+1})+\frac{1}{3(n-1)}\sum_{i=1}^{n-1}(\varepsilon_{i}^{2}+\varepsilon_{i+1}^{2}+\varepsilon_{i}\varepsilon_{i+1}).$
(77)

∎

###  E.1 Technical Lemmas

In the following content, to simplify the notations, denote
$K=K(\boldsymbol{X},\boldsymbol{X})$, $G=G_{1}(\boldsymbol{X},\boldsymbol{X})$
and $\Pi=2\Pi_{1}(\boldsymbol{X},\boldsymbol{X})$.

###### Proof of lemma E.3.

Let $\xi\in(x_{i},x_{i+1}).$We only present the proof for $2\leq i\leq n-2$.
When $i\in\\{1,n-1\\}$, one can prove the statement in a similar way.

Let $e_{k}$ be the $k$-th vector in the standard basis of $\mathbb{R}^{n}$.
Denote $I_{k}=e_{k}e_{k}^{\top}$. Let us consider the rank one decomposition
of $\Pi$:

$\displaystyle\Pi$
$\displaystyle=\sum_{k\not\in\\{1,i,i+1,n\\}}I_{k}\Pi+I_{i}\Pi+I_{i+1}\Pi+I_{1}\Pi+I_{n}\Pi$
(78)
$\displaystyle=\underbrace{I_{2}\Pi}_{\triangleq\Pi_{1}}+\cdots+\underbrace{I_{i-1}\Pi}_{\triangleq\Pi_{i-2}}+\underbrace{I_{i+2}\Pi}_{\triangleq\Pi_{i-1}}+\cdots+\underbrace{I_{n-1}\Pi}_{\triangleq\Pi_{n-4}}+\underbrace{I_{i}\Pi}_{\triangleq\Pi_{n-3}}+\underbrace{I_{i+1}\Pi}_{\triangleq\Pi_{n-2}}+\underbrace{I_{1}\Pi}_{\triangleq\Pi_{n-1}}+\underbrace{I_{n}\Pi}_{\triangleq\Pi_{n}}.$

We denote by $S$ (resp. $S^{-1}$ ) the transformation (resp. inverse
transform) between the indices appeared in (78), i.e., $S(1)=2$, $S(2)=3$, …,
$S(i-2)=i-1$, $S(i-1)=i+2$,…, $S(n-4)=n-1$, $S(n-3)=i$, $S(n-2)=i+1$,
$S(n-1)=1$, and $S(n)=n$. It is clear that $\Pi_{k}=I_{S(k)}\Pi$.

Let $D_{k}=G+\Pi_{1}+...+\Pi_{k-1}$, $k=1,2,\ldots,n$. It is clear that
$D_{1}=G$ and $D_{n+1}=G+\Pi=K$. To proceed with the proof, we need the
following lemma:

######  Lemma E.6.

Suppose that $n>22$. Let
$\Gamma=\operatorname{diag}\\{1,-(n-1),\cdots,-(n-1),1\\}$ be an $n\times n$
diagonal matrix. There exists a constant $C$ such that the following
statements hold.

  1. 1.

For any $p\in[n]$, $D_{p}$ is an invertible matrix and
$g_{p}=(1+\operatorname{tr}(\Pi_{k}D_{p}^{-1}))^{-1}\in(0,C]$.

  2. 2.

Let $H^{(p)}=\Pi D_{p}^{-1}$ and $C^{(p)}=\Gamma H^{(p)}$. Then for any
$a\in[n]$, we have $|C^{(p)}_{i,j}|\leq C$ for any $i,j\in[n]$.

We remind that $\Pi$ is an invertible matrix (please see Lemma B.2). Thus, if
$D_{k}$’s are invertible, then $H^{(k)}$’s are invertible.

$\bullet$ $\underline{p=1}$; Since
$K^{\prime\prime}(\xi_{i},\boldsymbol{X})=\frac{4}{\pi((1+\xi_{i}^{2})^{2})}(|\xi_{i}-x_{1}|,\dots,|\xi_{i}-\boldsymbol{X}|)$,
we can easily verify that there exists a constant $C$ such that

$\left|\left(K^{\prime\prime}(\xi_{i},\boldsymbol{X})D_{1}^{-1}\right)_{j}\right|=\left|\left(K^{\prime\prime}(\xi,\boldsymbol{X})G^{-1}\right)_{j}\right|\leq\begin{cases}\quad
C&j\in\\{1,i,i+1,n\\},\\\\[6.0pt] \quad 0&j\not\in\\{1,i,i+1,n\\}.\end{cases}$
(79)

In other words, $K^{\prime\prime}G^{-1}=K^{\prime\prime}D_{1}^{-1}$ is a row
vector with at most 4 non-zero entries located in $\\{1,i,i+1,n\\}$.

$\bullet$ $\underline{p\in\\{2,\ldots,n-4\\}}$; Thanks to the Lemma E.6,
$D_{k}$’s are invertible matrices. Thus, the Sherman–Morrison formula gives us
that for any $p=1,2,\ldots,n$,

$\displaystyle K^{\prime\prime}D_{p+1}^{-1}$
$\displaystyle=K^{\prime\prime}(D_{p}+\Pi_{p})^{-1}=K^{\prime\prime}D_{p}^{-1}-g_{p}K^{\prime\prime}D_{p}^{-1}\Pi_{p}D_{p}^{-1}.$
(80)

Because that for any $p\in\\{1,2,\dots,n-4\\}$, $S(p)\not\in\\{1,i,i+1,n\\}$,
we know from the definition of $\Pi_{p}$ that for $p\in\\{1,2,\ldots,n-4\\}$,
$K^{\prime\prime}D_{p}^{-1}\Pi_{p}=0$ and
$K^{\prime\prime}D_{p+1}^{-1}=K^{\prime\prime}D_{p}^{-1}$. In particular, we
know that

$\displaystyle
K^{\prime\prime}D_{n-3}^{-1}=K^{\prime\prime}D_{n-4}^{-1}=\cdots=K^{\prime\prime}D_{1}^{-1}=K^{\prime\prime}G^{-1}.$
(81)

In other words, $K^{\prime\prime}D_{p}^{-1},p=2,3,\dots,n-4$ are row vectors
with at most 4 non-zero entries located in $\\{1,i,i+1,n\\}$.

$\bullet$ $p\in\\{n-3,n-2,n-1,n\\}$; Since
$g_{p}K^{\prime\prime}D_{p}^{-1}\Pi_{p}D_{p}^{-1}$ is no longer 0 for $p\geq
n-3$, we do not have
$K^{\prime\prime}D_{n-2}^{-1}=K^{\prime\prime}D^{-1}_{n-3}$ anymore. We need
to treat them separately. Again, the Sherman–Morrison formula gives us that

$\displaystyle K^{\prime\prime}D_{n-2}^{-1}$
$\displaystyle=K^{\prime\prime}(D_{n-3}+\Pi_{n-3})^{-1}=K^{\prime\prime}D_{n-3}^{-1}-g_{n-3}K^{\prime\prime}D_{n-3}^{-1}\Pi_{n-3}D_{n-3}^{-1}.$
(82)

Thus, there exists an absolute constant $C$, such that

$\displaystyle|(K^{\prime\prime}$
$\displaystyle(\xi_{i},\boldsymbol{X})D^{-1}_{n-2})_{j}|\leq|(K^{\prime\prime}(\xi_{i},\boldsymbol{X})D^{-1}_{n-3})_{j}|+g_{n-3}|(K^{\prime\prime}(\xi_{i},\boldsymbol{X})D^{-1}_{n-3}\Pi_{n-3}D^{-1}_{n-3})_{j}|$
$\displaystyle=|(K^{\prime\prime}(\xi_{i},\boldsymbol{X})D^{-1}_{n-3})_{j}|+g_{n-3}|(K^{\prime\prime}(\xi_{i},\boldsymbol{X})D^{-1}_{1}\Pi_{n-3}D^{-1}_{n-3})_{j}|\leq\begin{cases}C,&j\in\\{1,i,i+1,n\\}\\\\[6.0pt]
C\frac{1}{n-1}&j\neq\\{1,i,i+1,n\\}\end{cases}$

where the last inequality follows from the Lemma E.6 and
$\Pi_{n-1}D^{-1}_{n-3}=\frac{1}{n-1}I_{S(n-1)}C^{(n-3)}$. We can prove the
results for $p=n-2,n-1,n$ in a similar way.

In other words, we have shown that there exists an absolute constant $C$ such
that

$\displaystyle|(K^{\prime\prime}(\xi_{i},\boldsymbol{X})K^{-1})_{j}|=|(K^{\prime\prime}(\xi_{i},\boldsymbol{X})D^{-1}_{n+1})_{j}|\leq\begin{cases}C,&j=1,i,i+1,n\\\\[6.0pt]
C\frac{1}{n-1}&j\neq 1,i,i+1,n\end{cases}.$ (83)

Denote $y_{\max}=\max_{i\in[n]}y_{i}$. Then
$|K^{\prime\prime}(\xi_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}|\leq
C|y_{\max}|$.

By Lemma E.9, we have $|y_{\max}|\leq C\sqrt{\log n}$ with probability
$1-\frac{2}{n}$. Thus,

$|K^{\prime\prime}(\xi_{i},\boldsymbol{X})K^{-1}\boldsymbol{y}|\leq
C\sqrt{\log n}$ (84)

with probability $1-\frac{2}{n}$ and for some constant $C$. ∎

###### Proof of Lemma E.6.

We prove the statements through induction on $k$.

$\bullet$ $k=1$; It is clear that $D_{1}=G$ is invertible. The second
statement follows the following lemma:

######  Lemma E.7.

There exists an absolute constant $C$ such that

$\displaystyle(\Pi
G^{-1})_{i,j}\in\begin{cases}\quad(-0.54-\frac{1}{n-1},C]&\quad
i\in[n],j=1\\\\[4.0pt] \quad(0,C]&\quad i\in[n],j=n\\\\[4.0pt]
\quad[-2\frac{|i-j|+1}{(n-1)^{2}},0],&\quad i\in[n],j\neq 1,n.\end{cases}$

Specifically, $(\Pi G^{-1})_{1,1}>1.2-\frac{1}{n-1}$.

Moreover, the above lemma also shows that
$g_{1}=(1+\operatorname{tr}(\Pi_{1}D_{1}^{-1}))^{-1}>0$ is bounded. Thus, we
proved the Lemma E.6 for $k=1$.

$\bullet$ $k>1$; Suppose that the inductive hypotheses hold for any $1\leq
k^{\prime}\leq k-1$.

Since $g_{k-1}=(1+\operatorname{tr}(\Pi_{k-1}D_{k-1}^{-1}))^{-1}\in(0,C]$,
Sherman–Morrison formula implies that $D_{k}=D_{k-1}+\Pi_{k-1}$ is invertible.
Thus, we have

$\displaystyle C^{(k)}$ $\displaystyle=\Gamma\Pi
D_{k}^{-1}=C^{(k-1)}-g_{k-1}C^{(k-1)}\Pi_{k-1}D_{k-1}^{-1}$ (85)
$\displaystyle=C^{(k-1)}-g_{k-1}C^{(k-1)}I_{S(k-1)}\Gamma^{-1}C^{(k-1)}.$ (86)

Since both $\Pi$, $\Gamma$ and $D_{k}$ are invertible matrices, we know that
$C^{(k)}$ is invertible and the Sherman-Morrison formula gives us

$\displaystyle(C^{(k)})^{-1}=(C^{(k-1)})^{-1}-\frac{I_{S(k-1)}}{n-1}.$ (87)

The desired bound about $C^{(k)}$ is provided by the following lemma:

######  Lemma E.8.

Assume that $k\leq n-1$ and $C^{(j)},j=1,2,\dots,k$ are invertible matrices.
There exists an absolute constant $C$ such that,

$\displaystyle|C^{(k)}_{i,j}|\leq 21,\mbox{~{}if~{}}2\leq i,j\leq
n-1\mbox{~{}and~{}}|C^{(k)}_{i,j}|\leq C,\mbox{ ~{}if~{} }i\mbox{ or
}j\in\\{1,n\\}.$ (88)

First, Lemma E.8 implies that the second statement in Lemma E.6 hold for $k$.
Second, since $n\geq 23$, Lemma E.8 implies that $C^{(k)}_{S(k),S(k)}\leq
21<n-1$. Thus, for any constant $C>22$, we have
$g_{k}=\left(1-\frac{C^{(k)}_{S(k),S(k)}}{n-1}\right)^{-1}\in(0,C]$.

∎

Lemma E.7 There exists an absolute constant $C$ such that

$\displaystyle(\Pi
G^{-1})_{i,j}\in\begin{cases}\quad(-0.54-\frac{1}{n-1},C]&\quad
i\in[n],j=1\\\\[4.0pt] \quad(0,C]&\quad i\in[n],j=n\\\\[4.0pt]
\quad[-2\frac{|i-j|+1}{(n-1)^{2}},0],&\quad i\in[n],j\neq 1,n.\end{cases}$

Moreover, we can prove that $(\Pi G^{-1})_{1,1}>1.2-\frac{1}{n-1}$.

###### Proof.

Since $\Pi(x,y)$ are continuous differentiable of 2nd order, the Taylor
expansion gives us that for any $i$, there exist $\xi_{i}$ and
$\xi_{i}^{\prime}\in[x_{i},x_{i+1}]$ such that

$\displaystyle\Pi(x.x_{i+1})-\Pi(x,x_{i})=\Pi^{\prime}(x,\xi_{i})(x_{i+1}-x_{i}),$
(89)
$\displaystyle\Pi(x.x_{i+1})-\Pi(x,x_{i})=\Pi^{\prime}(x,x_{i})(x_{i+1}-x_{i})+\frac{1}{2}\Pi^{\prime\prime}(x,\xi_{i}^{\prime})(x_{i+1}-x_{i})^{2}.$
(90)

$\bullet$ $j=1$; For any $i\in[n]$,

$\begin{split}(\Pi
G^{-1})_{i,1}&=\frac{\pi}{2}(n-1)(\Pi(x_{i},x_{1})-\Pi(x_{i},x_{2}))+\frac{\pi}{2}\frac{\Pi(x_{i},x_{1})+\Pi(x_{i},x_{n})}{2\pi-1}\\\
&=\frac{\pi}{2}\Pi^{{}^{\prime}}(x_{i},\xi_{1})+\frac{\pi}{2}\frac{\Pi(x_{i},x_{1})+\Pi(x_{i},x_{n})}{2\pi-1}\\\
&=-\xi_{1}\frac{|x_{i}-\xi_{1}|}{1+\xi_{1}^{2}}-x_{i}(\pi-\psi(\xi_{1},x_{i}))+\frac{\pi}{2}\frac{\Pi(x_{i},x_{1})+\Pi(x_{i},x_{n})}{2\pi-1}\end{split}$
(91)

Since $x_{i}\in[0,1]$, It is clear that there exists a constant $C$ such that
$(\Pi G^{-1})_{i,1}\leq C$. On the other hand,

$\displaystyle(\Pi G^{-1})_{i,1}$ $\displaystyle\geq-
x_{2}-x_{i}(\pi-\psi(x_{1},x_{i}))+\frac{\pi}{2}\frac{\Pi(x_{i},x_{1})+\Pi(x_{i},x_{n})}{2\pi-1}$
$\displaystyle=-\frac{1}{n-1}-\frac{i-1}{n-1}(\pi-\psi(x_{1},x_{i}))+\frac{\pi-\psi(x_{1},x_{i})+(1+x_{i})(\pi-\psi_{(}x_{n},x_{i}))+1}{2\pi-1}$
$\displaystyle\geq-0.54-\frac{1}{n-1}$

Finally, we have $(\Pi
G^{-1})_{1,1}\geq-\frac{1}{n-1}+\frac{2\pi-\frac{\pi}{4}+1}{2\pi-1}>1.2-\frac{1}{n-1}$.

$\bullet$ $j=n$;

$\begin{split}(\Pi
G^{-1})_{i,n}&=\frac{\pi}{2}(n-1)(\Pi(x_{i},x_{n})-\Pi(x_{i},x_{n-1}))+\frac{\pi}{2}\frac{\Pi(x_{i},x_{1})+\Pi(x_{i},x_{n})}{2\pi-1}\\\
&=\frac{\pi}{2}\Pi^{{}^{\prime}}(x_{i},\xi_{n-1})+\frac{\pi}{2}\frac{\Pi(x_{i},x_{1})+\Pi(x_{i},x_{n})}{2\pi-1}.\end{split}$
(92)

Since $x_{i}\in[0,1]$, it is clear there exists a constant $C$ such that $(\Pi
G^{-1})_{i,n}\leq C$. On the other hand, we have

$\displaystyle(\Pi G^{-1})_{i,n}$
$\displaystyle=\xi_{1}\frac{|x_{i}-\xi_{n-1}|}{1+\xi_{i}^{2}}+x_{i}(\pi-\psi(\xi_{n-1},x_{i}))+\frac{\pi}{2}\frac{\Pi(x_{i},x_{1})+\Pi(x_{i},x_{n})}{2\pi-1}>0.$
(93)

$\bullet$ $2\leq j\leq n-1$;

$\begin{split}(\Pi
G^{-1})_{i,j}&=\frac{\pi}{2}(n-1)\left(2\Pi(x_{i},x_{j})-\Pi(x_{i},x_{j+1})-\Pi(x_{i},x_{j-1})\right)\\\
&=-\frac{\pi}{2}(n-1)(\Pi^{{}^{\prime\prime}}(x,\xi_{j})\frac{(x_{j}-x_{j+1})^{2}}{2}+\Pi^{{}^{\prime\prime}}(x,\xi_{j-1})\frac{(x_{j}-x_{j-1})^{2}}{2})\\\
&=-\frac{1}{(n-1)}\left(\frac{|x_{i}-\xi_{j}|}{(1+\xi_{j}^{2})^{2}}+\frac{|x_{i}-\xi_{j-1}|}{(1+\xi_{j-1}^{2})^{2}}\right)\end{split}$
(94)

It is clear that $(\Pi G^{-1})_{i,j}\leq 0$. One the other hand, since
$\xi_{j}^{\prime}\in(x_{j},x_{j+1})$ and
$\xi_{j-1}^{\prime}\in(x_{j-1},x_{j})$, we have

$\displaystyle(\Pi G^{-1})_{i,j}$
$\displaystyle\geq-\frac{1}{(n-1)}\frac{|x_{i}-\xi_{j}|+|x_{i}-\xi_{j-1}|}{(1+x_{j-1}^{2})^{2}}\geq-2\frac{|x_{i}-x_{j}|+\frac{1}{n-1}}{(n-1)}=-2\frac{|i-j|+1}{(n-1)^{2}}.$
(95)

∎

Lemma E.8 Assume that $k\leq n-1$ and $C^{(j)},j=1,2,\dots,k$ are invertible
matrices. There exists an absolute constant $C$ such that,

$\displaystyle|C^{(k)}_{i,j}|\leq 21,\mbox{~{}if~{}}2\leq i,j\leq
n-1\mbox{~{}and~{}}|C^{(k)}_{i,j}|\leq C,\mbox{ ~{}if~{} }i\mbox{ or
}j\in\\{1,n\\}.$ (96)

###### Proof.

We prove this lemma by induction on $k$.

$\bullet$ $k=1$; Recall that Lemma E.7 implies that

$\displaystyle C^{(1)}_{i,j}=\begin{cases}\quad-(n-1)(\Pi G^{-1})_{i,j}\leq
2\frac{|i-j|+1}{(n-1)}\leq 2&i\in[n],j\neq 1,n\\\\[8.0pt]
\quad\quad\quad\quad\quad(\Pi G^{-1})_{i,j}\leq C&i\in[n],j=1,n\\\
\end{cases}.$ (97)

Thus the statements hold for $k=1$.

$\bullet$ $k>1$; Suppose that the inductive hypotheses hold for $k$. Then

$\displaystyle(C^{(k+1)})^{-1}=(C^{(k)})^{-1}-\frac{1}{n-1}I_{S_{k}}=(C^{(1)})^{-1}-\frac{1}{n-1}\sum_{j=1}^{k}I_{S(j)}.$
(98)

Denote $\frac{1}{n-1}\sum_{j=1}^{k}I_{S(j)}$ by $T_{k}$. Then, we have

$\displaystyle
C^{(k+1)}=C^{(1)}+C^{(1)}T_{k}C^{(1)}+C^{(1)}\left(T_{k}C^{(1)}\right)^{2}+\cdots=\mathcal{Q}+\mathcal{Q}\left(T_{k}C^{(1)}\right)^{3}+\mathcal{Q}\left(T_{k}C^{(1)}\right)^{6}+\cdots$

where
$\mathcal{Q}=C^{(1)}\left(1+T_{k}C^{(1)}+\left(T_{k}C^{(1)}\right)^{2}\right)$.
Simple calculations show that (please see Lemma E.11 ), for any $2\leq i,j\leq
n-1$ and $q\in\mathbb{N}$, we have

$\displaystyle|C^{(1)}_{i,j}|\leq
2,~{}\left|\left(C^{(1)}T_{k}C^{(1)}\right)_{i,j}\right|\leq\frac{4}{3},~{}\left|\left(C^{(1)}\left(T_{k}C^{(1)}\right)^{2}\right)_{i,j}\right|\leq\frac{4}{5},~{}\left|\left(\left(T_{k}C^{(1)}\right)^{3q}\right)_{i,j}\right|\leq\frac{\left(\frac{4}{5}\right)^{q}}{n-1}.$
(99)

Note that the first row and last row of $T_{k}$ are zero vectors. Thus, for
any $2\leq i,j\leq n-1$, we get

$\displaystyle|\mathcal{Q}_{i,j}|=\left|\left(C^{(1)}\left(1+T_{k}C^{(1)}+\left(T_{k}C^{(1)}\right)^{2}\right)\right)_{i,j}\right|\leq
2+\frac{4}{3}+\frac{4}{5}=\frac{62}{15}$ (100)

and

$\displaystyle|C^{(k+1)}_{i,j}|$ $\displaystyle\leq\sum_{q\geq
0}\left|\left(\mathcal{Q}\left(T_{k}C^{(1)}\right)^{3q}\right)_{i,j}\right|\leq\sum_{q\geq
0}\sum_{p=1}^{n}|\mathcal{Q}|_{i,p}\left|\left(T_{k}C^{(1)}\right)^{3q}_{p,j}\right|\leq\sum_{q\geq
0}\sum_{p=2}^{n-1}\frac{62}{15}\left(\frac{4}{5}\right)^{q}\frac{1}{(n-1)}<21$
(101)

If $i$ or $j\in\\{1,n\\}$, we can prove $|C^{(k+1)}_{i,j}|\leq C$ similarly.

Finally, we can show that
$\left(C^{(1)}\left(T_{n-2}C^{(1)}\right)^{p}\right)_{1,1}\geq 0$ for
$p=0,1,...$ and

$C^{(n-1)}_{1,1}=C^{(1)}_{1,1}+\left(C^{(1)}T_{n-2}C^{(1)}\right)_{1,1}+\cdots\geq
C^{(1)}_{1,1}\geq 1.2-\frac{1}{n-1}>0.$ (102)

∎

Under the bounded input $x_{i}$, we can get the bounded $f^{*}(x_{i})$. Let
$y_{max}=\max_{i\in[n]}y_{i}$. we can have the upper bound of $y_{max}$
through the following lemma:

######  Lemma E.9.

With the definition of Equation (1), with the probability at least
$1-\frac{2}{n}$, we have

$|y_{\max}|\leq C\sqrt{\log n}$ (103)

and for some constant $C$.

###### Proof of Lemma E.9.

Denote $\epsilon_{\max}=\max_{i\in[n]}\epsilon_{i}$. For $\forall t>0$, we
have

$\exp(t\mathbf{E}(\epsilon_{\max}))\leq\mathbf{E}(\exp(t\epsilon_{\max}))\leq\sum_{i=1}^{n}\mathbf{E}(\exp(t\epsilon_{i}))=n\exp(t^{2}\sigma^{2}/2).$
(104)

The first inequality is Jensen’s inequality, the second is the union bound,
and the final equality follows from the definition of the moment-generating
function. Taking the logarithm of both sides of this inequality, we have

$\mathbf{E}(\epsilon_{\max})\leq\frac{\log n}{t}+\frac{\sigma^{2}t}{2}.$ (105)

Let $t=\frac{\sqrt{2\log n}}{\sigma}$, we have

$\mathbf{E}(\epsilon_{\max})\leq\sigma\sqrt{2\log n}.$ (106)

By Borell-ITS inequality, since
$\sigma_{max}^{2}=\max_{i\in[n]}E(\epsilon_{i}^{2})=\sigma^{2}$, for $t\geq
0$, we have

$P(|\epsilon_{\max}-\mathbf{E}(\epsilon_{\max})|\geq
t)<2exp(-\frac{t^{2}}{2\sigma_{max}^{2}}).$ (107)

Let $t=\sigma\sqrt{2\log n}$, we have

$P(|\epsilon_{\max}|\leq 2\sigma\sqrt{2\log n})>1-\frac{2}{n}.$ (108)

Since $f_{\star}(x)$ is bounded, we have $\max_{x\in[0,1]}|f_{\star}(x)|\leq
C\sqrt{\log n}$ for some constant C and $n>2$. Thus, with the probability at
least $1-\frac{2}{n}$, we have

$|y_{\max}|\leq C\sqrt{\log n}$ (109)

for some constant $C$. ∎

######  Lemma E.10.

For any $x,x^{\prime}\in[0,1]$, the function
$\Pi(x^{\prime},x)=2\left(\frac{\pi-\psi(x^{\prime},x)}{\pi}(1+x^{\prime}x)+\frac{\lvert
x^{\prime}-x\rvert}{\pi}\right)$ is a twice continuously differentiable
function where
$\psi(x^{\prime},x)=\arccos\frac{1+x^{\prime}x}{\sqrt{(1+x^{{}^{\prime}2})(1+x^{2})}}$.
Moreover,

$\frac{\partial\Pi(x^{\prime},x)}{\partial
x}=\frac{2x}{\pi}\frac{|x-x^{\prime}|}{1+x^{2}}+2x^{\prime}-\frac{2}{\pi}x^{\prime}\psi\mbox{~{}~{}
and~{}~{} }\frac{\partial^{2}\Pi(x^{\prime},x)}{\partial
x^{2}}=\frac{4}{\pi}\frac{|x-x^{\prime}|}{(1+x^{2})^{2}}.$ (110)

###### Proof of Lemma E.10.

Simple calculations give us that

$\begin{split}\frac{\partial\psi(x,x^{\prime})}{\partial
x}&=\frac{\operatorname{sgn}(x-x^{\prime})}{(1+x^{2})}\mbox{~{} and
~{}}\frac{\partial^{2}\psi(x,x^{\prime})}{\partial
x^{2}}=-\frac{2x\operatorname{sgn}(x-x^{\prime})}{(1+x^{2})^{2}}\end{split}$
(111)

and the desired results. ∎

######  Lemma E.11.

Detailed calculations in equation (99).

###### Proof.

For $k\leq n-2$, we have

$C^{(1)}=\begin{pmatrix}C^{(1)}_{1,1}&u&C^{(1)}_{1,n}\\\ l&M&r\\\
C^{(1)}_{n,1}&d&C^{(1)}_{n,n}\end{pmatrix},\quad
T_{k}C^{(1)}=\frac{1}{n-1}\begin{pmatrix}0&0&0\\\ l&M&r\\\ 0&0&0\end{pmatrix}$
(112)

where $u$ and $d$ are $1\times(n-2)$ vectors, $l$ and $r$ are $(n-2)\times 1$
vectors, and $M$ is a $(n-2)\times(n-2)$ matrix. Simple calculations imply
that

$\left(T_{k}C^{(1)}\right)^{p}=\left(\frac{1}{n-1}\right)^{p}\begin{pmatrix}0&0&0\\\
M^{p-1}l&M^{p}&M^{p-1}r\\\ 0&0&0\end{pmatrix}.$ (113)

For any $2\leq i,j\leq n-1$, we have

$\displaystyle\left(C^{(1)}\right)_{i,j}$ $\displaystyle=M_{i,j}=-(n-1)(\Pi
G^{-1})_{i,j}\leq 2\frac{|i-j|+1}{n-1},$ (114)
$\displaystyle\left(C^{(1)}T_{k}C^{(1)}\right)_{i,j}$
$\displaystyle=\left(\frac{M^{2}}{n-1}\right)_{i-1,j-1}\leq\frac{1}{n-1}\sum_{k=1}^{n-2}\frac{2(|i-1-k|+1)}{n-1}\frac{2(|k-j+1|+1)}{n-1}$
$\displaystyle\leq\frac{4}{3}$
$\displaystyle\left(C^{(1)}\left(T_{k}C^{(1)}\right)^{2}\right)_{i,j}$
$\displaystyle=\left(\frac{M^{3}}{(n-1)^{2}}\right)_{i-1,j-1}$
$\displaystyle\leq\frac{1}{(n-1)^{5}}\sum_{l=1}^{n-2}\sum_{k=1}^{n-2}8(|i-1-k|+1)(|k-l|+1)(|l-j+1|+1)$
$\displaystyle\leq\frac{1}{(n-1)^{5}}\sum_{l=1}^{n-2}\sum_{k=1}^{n-2}8(n-1-k)(|k-l|+1)l$
$\displaystyle=\frac{2}{15}\frac{(n-2)n(2n-1)(3n-7)}{(n-1)^{5}}\leq\frac{4}{5}.$

Finally, we have

$\left(\left(T_{k}C^{(1)}\right)^{3q}\right)_{i,j}\leq\left(\frac{4}{5}\right)^{q}\frac{1}{(n-1)}.$
(115)

∎


February 2024

Time Crystals

and

Phase-Space Noncommutative Quantum Mechanics

Orfeu Bertolami1,2

1 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do
Porto,

Rua do Campo Alegre s/n, 4169-007 Porto, Portugal

2 Centro de Física das Universidades do Minho e do Porto, Rua do Campo Alegre
s/n, 4169-007 Porto, Portugal

and

A. E. Bernardini

Departamento de Física, Universidade Federal de São Carlos,

PO Box 676, 13565-905, São Carlos, SP, Brasil.

###### Abstract

We argue that time crystal properties naturally arise from phase-space
noncommutative quantum mechanics. In order to exemplify our point we consider
the 2-dimensional noncommutative quantum harmonic oscillator and show that it
exibihits periodic oscillations that can be identified as time crystals.

E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
Based on talk presented by one of us (O.B.) at the Third Minkowsky Meeting on
the Foundations of the Spacetime Physics at Albena, Bulgaria, 11-14 September
2023.

##  1 Introduction

In this contribution we review the arguments presented in Ref. [1] where it
was shown that time crystal features arise in the context of the phase-space
noncommutative 2-dimensional quantum harmonic oscillator. As discussed in the
following this is yet another new property emerging from phase-space
noncommutative quantum mechanics (PSNCQM).

Time crystals are time-periodic self-organized structures that presumably
arise due to the spontaneous breaking of time translation symmetry [2, 3].
They are analogous to spatial crystal lattices that form when the spontaneous
breaking of space translation symmetry takes place [4]. Time crystal features
were claimed to appear in ultra-cold atoms [5, 6] and spin-based solid state
systems [7, 8, 9, 10, 11, 12], through which it has been argued that
periodically driven systems exhibit a discrete time symmetry [13, 14]. In
fact, these experiments suggest that novel phases of matter do exist [10, 11,
12] which exhibit a discrete time translation symmetry hinting the breakdown
of the continuous time translation symmetry, $\hat{\mathcal{T}}_{H}\equiv
e^{-i\hat{H}t}$. For a contextualization of time crystals with respect to the
research on the physics of time, see, for instance, Ref. [15].

As is well known, if a time-independent system driven by a time-independent
Hamiltonian, $H$, is prepared in an eigenstate $|\psi_{n}\rangle$, such that
$H|\psi_{n}\rangle=E_{n}|\psi_{n}\rangle$, for the energy eigenvalue $E_{n}$,
in the context of quantum mechanics (QM) the probability density at a fixed
position in the configuration space is also time-independent. Nevertheless,
the mentioned experiments suggest that time crystals exist and thus,
$[\hat{H},\,\rho_{n}]\equiv[\hat{\mathcal{T}}_{H},\,\rho_{n}]\neq 0$ for
$\rho_{n}=|\psi_{n}\rangle\langle\psi_{n}|$.

As originally argued [2], this would correspond to a spontaneous breakdown of
time translation symmetry followed by a non-stationary behaviour of the
eigensystem solutions. However, a no-go theorem [16], based on the time-
dependent correlation functions of the order parameter, rules out the
possibility of time crystals defined in this way for the ground state and for
a canonical ensemble of a general Hamiltonian. We argue that the emergence of
a non-stationary behaviour, and its connection with time crystal properties
can be explained in terms of both position and/or momentum noncommutativity in
the phase-space [1], in opposition to the ab initio breaking symmetry
assumptions proposed in Refs. [2, 3].

##  2 Phase-Space Noncommutative Quantum Mechnics

We present now some of the main features of PSNCQM. Noncommutativity was
firstly considered in the space coordinate domain as a way to regularize
quantum field theories [17] and subsequently in string theory [18, 19, 20,
21]. The PSNCQM extension [22, 23, 24, 25, 26, 27, 28], considered here, can
be formulated in terms of the Weyl-Wigner-Groenewold-Moyal (WWGM) framework
[29, 30, 31], supported by a $2n$-dimensional phase-space deformed Heisenberg-
Weyl algebra, where position and momentum operators, $\hat{q}_{i}$ and
$\hat{p}_{j}$, obey the commutation relations,

$[\hat{q}_{i},\hat{q}_{j}]=i\theta_{ij},\hskip
14.22636pt[\hat{q}_{i},\hat{p}_{j}]=i\hbar\delta_{ij},\hskip
14.22636pt[\hat{p}_{i},\hat{p}_{j}]=i\eta_{ij},$ (1)

where $i,j=1,...,d$, and $\eta_{ij}$ and $\theta_{ij}$ are the entries of
invertible antisymmetric real constant ($d\times d$) matrices, ${\bf\Theta}$
and ${\bf N}$, such that an equally invertible matrix, ${\bf\Sigma}$, with
$\Sigma_{ij}\equiv\delta_{ij}+\hbar^{-2}\theta_{ik}\eta_{kj}$, exists, which
demands that $\theta_{ik}\eta_{kj}\neq-\hbar^{2}\delta_{ij}$. Of course, given
that $\eta_{ij}\neq 0,\,\theta_{ij}\neq 0$, the relations from Eq. (1) can
affect the symmetries related to conserved quantities associated to quantum
operators, $\hat{\mathcal{O}}$, for which
$d\langle\hat{\mathcal{O}}\rangle/dt=i\hbar^{-1}\,\langle[\hat{H},\,\hat{\mathcal{O}}]\rangle=0$.

In this context, the key issue is if quantum operators identified by
$\hat{\mathcal{O}}\rightarrow\hat{\mathcal{O}}(\\{\hat{q}_{i},\hat{p}_{i}\\})$
do present a time crystal behaviour arising from the breakdown of time
translational symmetries,
$\langle[\hat{H},\,\hat{\mathcal{O}}(\\{\hat{q}_{i},\hat{p}_{i}\\})]\rangle\neq
0$, in opposition to usual QM. In order to investigate this point, the NC
algebra, Eq. (1) can be mapped into the Heisenberg-Weyl algebra through the
linear Seiberg-Witten (SW) transformation [20],

$\hat{q}_{i}=A_{ij}\mathit{\hat{Q}}_{j}+B_{ij}\hat{\Pi}_{j},\hskip
28.45274pt\hat{p}_{i}=C_{ij}\mathit{\hat{Q}}_{j}+D_{ij}\hat{\Pi}_{j},$ (2)

where $A_{ij},B_{ij},C_{ij}$ and $D_{ij}$ are real entries of constant
matrices, ${\bf A},{\bf B},{\bf C}$ and ${\bf D}$. In this case, one recovers
the algebra of ordinary QM,

$[\hat{Q}_{i},\hat{Q}_{j}]=0,\quad[\hat{Q}_{i},\hat{P}_{j}]=i\hbar\delta_{ij},\hskip
14.22636pt[\hat{P}_{i},\hat{P}_{j}]=0,$ (3)

through the following matrix equation constraints [24], ${\bf A}{\bf
D}^{T}-{\bf B}{\bf C}^{T}={\bf I}_{d\times d}$, ${\bf A}{\bf B}^{T}-{\bf
B}{\bf A}^{T}=\hbar^{-1}{\bf\Theta}$, and ${\bf C}{\bf D}^{T}-{\bf D}{\bf
C}^{T}={\hbar^{-1}}{\bf N}$, where the superscript $T$ denotes matrix
transposition.

From the WWGM framework [24, 26] for the algebra, Eq. (1), it is possible to
show that the resulting quantum mechanical extensions have some striking
features which include putative violations of the Robertson-Schrödinger
uncertainty relation [32, 33], quantum correlations and information collapse
in gaussian quantum systems [34, 35, 36, 37, 38], new regularizing features in
minisuperspace quantum cosmology models [32, 39] and in black-hole physics
[40, 41, 42], putative violations of the Equivalence Principle [43, 44] and,
likewise ordinary QM, non-locality properties that can be captured by the Bell
operator [45]. In fact, the generalized WWGM star-product, the extended Moyal
bracket and the noncommutative (NC) Wigner function framework ensure that
observables are independent of any particular choice of the SW map [26].

##  3 The Noncommutative 2-dimensional Quantum Harmonic Oscillator and the
Emergence of Time Crystal Behaviour

Aiming to exemplify the emerging time crystal behaviour we consider the
$2$-dimensional harmonic oscillator in the PSNCQM [46] with Hamiltonian,

$\hat{H}_{HO}(\hat{\mathbf{q}},\hat{\mathbf{p}})={\hat{\mathbf{p}}^{2}\over
2m}+{\textstyle{{1}\over{2}}}m\omega^{2}\hat{\mathbf{q}}^{2},$ (4)

on the NC “$x-y$” plane, with position and momentum satisfying the NC algebra,
Eq. (1), now with $i,j=1,2$, $\theta_{ij}=\theta\epsilon_{ij}$ and
$\eta_{ij}=\eta\epsilon_{ij}$, where $\epsilon_{ij}$ is the $2$-dimensional
Levi-Civita tensor. The map to commutative operators is given by

$\mathit{\hat{Q}}_{i}=\mu\left(1-{\theta\eta\over\hbar^{2}}\right)^{-1/2}\left(\hat{q}_{i}+{\theta\over
2\lambda\mu\hbar}\epsilon_{ij}\hat{p}_{j}\right),\quad\hat{\Pi}_{i}=\lambda\left(1-{\theta\eta\over\hbar^{2}}\right)^{-1/2}\left(\hat{p}_{i}-{\eta\over
2\lambda\mu\hbar}\epsilon_{ij}\hat{q}_{j}\right),$ (5)

in terms of the SW map,

$\hat{q}_{i}=\lambda\mathit{\hat{Q}}_{i}-{\theta\over
2\lambda\hbar}\epsilon_{ij}\hat{\Pi}_{j},\hskip 14.22636pt\hskip
14.22636pt\hat{p}_{i}=\mu\hat{\Pi}_{i}+{\eta\over
2\mu\hbar}\epsilon_{ij}\mathit{\hat{Q}}_{j}~{},$ (6)

which is invertible for $\theta\eta\neq\hbar^{2}$, and the parameters
$\lambda$ and $\mu$ satisfying the condition

${\theta\eta\over 4\hbar^{2}}=\lambda\mu(1-\lambda\mu).$ (7)

The Hamiltonian in terms of the commutative variables, $\mathit{\hat{Q}}_{i}$
and $\hat{\Pi}_{i}$, reads [46]

$\hat{H}_{HO}(\hat{\mbox{\bf\em
Q}},\hat{\mathbf{\Pi}})=\alpha^{2}\hat{\mbox{\bf\em
Q}}^{2}+\beta^{2}\hat{\mathbf{\Pi}}^{2}+\gamma\sum_{i,j=1}^{2}{\epsilon_{ij}\hat{\Pi}_{i}\mathit{\hat{Q}}_{j}},$
(8)

where ${\alpha}^{2}\equiv
m\omega^{2}\lambda^{2}/2+\eta^{2}/(8m\hbar^{2}\mu^{2})$,
${\beta}^{2}\equiv{\mu^{2}/(2m)}+{m\omega^{2}\theta^{2}/(8\hbar^{2}\lambda^{2})}$,
and ${\gamma}\equiv m\omega^{2}{\theta}/({2\hbar})+{\eta}/({2m\hbar})$, from
which one obtains the following set of coupled equations of motion,

$\displaystyle\dot{\Pi}_{i}$ $\displaystyle=$
$\displaystyle-{\textstyle{{i}\over{\hbar}}}\langle\left[\hat{\Pi}_{i},\,\hat{H}_{HO}\right]\rangle=-2\alpha^{2}\,\mathit{Q}_{i}-\gamma\,\varepsilon_{ji}\Pi_{j},$
$\displaystyle\dot{\mathit{Q}}_{i}$ $\displaystyle=$
$\displaystyle-{\textstyle{{i}\over{\hbar}}}\langle\left[\hat{\mathit{Q}}_{i},\,\hat{H}_{HO}\right]\rangle=~{}~{}2\beta^{2}\,\Pi_{i}-\gamma\,\varepsilon_{ji}\mathit{Q}_{j},$
(9)

with $\mathit{Q}_{i}\equiv\langle\hat{\mathit{Q}}_{i}\rangle$ and
${\Pi}_{i}\equiv\langle\hat{\Pi}_{i}\rangle$. In this case, $\mbox{\bf\em
Q}=(\mathit{Q}_{1},\,\mathit{Q}_{2})$ and
$\mathbf{\Pi}=({\Pi}_{1},\,{\Pi}_{2})$ may be interpreted as the dynamical
variables within the WWGM formalism for which the solutions are given by [46]

$\displaystyle\mathit{Q}_{1}(t)$ $\displaystyle=$ $\displaystyle
x\,\cos(\Omega t)\cos(\gamma t)+y\,\cos(\Omega t)\sin(\gamma
t)+{\textstyle{{\beta}\over{\alpha}}}\left[\pi_{y}\,\sin(\Omega t)\sin(\gamma
t)+\pi_{x}\,\sin(\Omega t)\cos(\gamma t)\right],$
$\displaystyle\mathit{Q}_{2}(t)$ $\displaystyle=$ $\displaystyle
y\,\cos(\Omega t)\cos(\gamma t)-x\,\cos(\Omega t)\sin(\gamma
t)-{\textstyle{{\beta}\over{\alpha}}}\left[\pi_{x}\,\sin(\Omega t)\sin(\gamma
t)-\pi_{y}\,\sin(\Omega t)\cos(\gamma t)\right],$ $\displaystyle\Pi_{1}(t)$
$\displaystyle=$ $\displaystyle\pi_{x}\,\cos(\Omega t)\cos(\gamma
t)+\pi_{y}\,\cos(\Omega t)\sin(\gamma
t)-{\textstyle{{\alpha}\over{\beta}}}\left[y\,\sin(\Omega t)\sin(\gamma
t)+x\,\sin(\Omega t)\cos(\gamma t)\right],~{}~{}$ $\displaystyle\Pi_{2}(t)$
$\displaystyle=$ $\displaystyle\pi_{y}\,\cos(\Omega t)\cos(\gamma
t)-\pi_{x}\,\cos(\Omega t)\sin(\gamma
t)+{\textstyle{{\alpha}\over{\beta}}}\left[x\,\sin(\Omega t)\sin(\gamma
t)-y\,\sin(\Omega t)\cos(\gamma t)\right],~{}~{}$ (10)

where $x,\,y,\,\pi_{x},$ and $\pi_{y}$ are arbitrary parameters, and

$\Omega=2\alpha\beta=\sqrt{(2\lambda\mu-1)^{2}\omega^{2}+\gamma^{2}}=\sqrt{\omega^{2}+\gamma^{2}-{\theta\eta\over\hbar^{2}}},$
(11)

with $\lambda$ and $\mu$ being eliminated by the constraint Eq. (7). Of
course, if one sets $\theta=\eta=0$, and therefore $\gamma=0$, one recovers
the solutions for the $2$-dimensional harmonic oscillator with uncoupled $x-y$
coordinates and $\Omega=\omega$. For $\theta,\,\eta\neq 0$, the above results
lead to two decoupled time-invariant quantities,

$\displaystyle\sum_{i=1}^{2}{\left({\textstyle{{\alpha}\over{\beta}}}\mathit{Q}_{i}(t)^{2}+{\textstyle{{\beta}\over{\alpha}}}\Pi_{i}(t)^{2}\right)}$
$\displaystyle=$
$\displaystyle{\textstyle{{\alpha}\over{\beta}}}(x^{2}+y^{2})+{\textstyle{{\beta}\over{\alpha}}}(\pi_{x}^{2}+\pi_{y}^{2}),$
$\displaystyle\sum_{i,j=1}^{2}{\left(\epsilon_{ij}\mathit{Q}_{i}(t)\,\Pi_{j}(t)\right)}$
$\displaystyle=$ $\displaystyle x\,\pi_{y}-y\,\pi_{x}.$ (12)

The changes introduced by the NC variables can be evinced by setting
$\pi_{x}=\pi_{y}=\sqrt{\alpha\hbar/2\beta}$, and
$x=y=\sqrt{\beta\hbar/2\alpha}$, so that the associated $x$ and $y$
translational energy contributions evolve as

$\small
E_{i}=\alpha\beta{\left({\textstyle{{\alpha}\over{\beta}}}\mathit{Q}_{i}(t)^{2}+{\textstyle{{\beta}\over{\alpha}}}\Pi_{i}(t)^{2}\right)}={\hbar\Omega\over
2}\left(1-(-1)^{i}\,\sin(2\gamma t)\right),$ (13)

with $i=1,\,2$, from which a typical low frequency $\gamma$-dependent beating
behaviour is encountered [46]. Such a time-dependent periodic modification is
a new feature of the NC harmonic oscillator ground state. The stargenfunctions
for the Hamiltonian, Eq. (8), are obtained from the stargen value equation,

$H^{W}_{HO}\star\rho_{n_{\tiny 1},n_{\tiny 2}}^{W}(\mbox{\bf\em
Q},\mathbf{\Pi})=E_{n_{\tiny{1}},n_{\tiny{2}}}\,\rho^{W}_{n_{\tiny{1}},n_{\tiny{2}}}(\mbox{\bf\em
Q},\mathbf{\Pi}),$ (14)

where $W(\mbox{\bf\em Q},\mathbf{\Pi})$ is the eigenstate associated Wigner
function, from which one has [24],

$\rho_{n_{\tiny{1}},n_{\tiny{2}}}^{W}(\mbox{\bf\em
Q},\mathbf{\Pi})={(-1)^{n_{1}+n_{2}}\over\pi^{2}\hbar^{2}}\exp\left[{-{\textstyle{{1}\over{\hbar}}}\left({\textstyle{{\alpha}\over{\beta}}}\mbox{\bf\em
Q}^{2}+{\textstyle{{\beta}\over{\alpha}}}{\mathbf{\Pi}}^{2}\right)}\right]\,L^{0}_{n_{1}}\left(\Omega_{+}/\hbar\right)\,L^{0}_{n_{2}}\left(\Omega_{-}/\hbar\right),$
(15)

where $L^{0}_{n}$ are the associated Laguerre polynomials, $n_{1}$ and $n_{2}$
are non-negative integers, and

${\Omega}_{\pm}={\alpha\over\beta}\mbox{\bf\em
Q}^{2}+{\beta\over\alpha}\mathbf{\Pi}^{2}\mp
2\sum_{i,j=1}^{2}{\left(\epsilon_{ij}\Pi_{i}\mathit{Q}_{j}\right)},$ (16)

such that the energy spectrum is given by $E_{n_{\tiny 1},n_{\tiny
2}}=\hbar\left[2\alpha\beta(n_{1}+n_{2}+1)+\gamma(n_{1}-n_{2})\right]$.

It has been shown that the $2$-dimensional harmonic oscillator on the NC plane
approaches the classical limit, and exhibits well-marked quantum effects such
as state swapping, quantum beating, and some extent of loss of quantum
coherence. These properties are not due to extrinsic or artificial time-
dependent effects, but due to the entanglement [34] induced by NC “$x$” and
“$y$” Hilbert spaces mapped by the time-independent Hamiltonian. From Eq. (8)
one sees that quantum states associated to $x$ and $y$ degrees of freedom are
no longer independent. This differs from the standard quantum mechanical
configuration, for which $x$ and $y$ modes are each of them associated to
decoupled stationary behaviour [46]. In order to clarify the relation with the
time crystal behaviour, one should get back to the Hamiltonian Eq. (4) and
examine the contributions of $\\{\hat{q}_{1},\,\hat{p}_{1}\\}$ and
$\\{\hat{q}_{2},\,\hat{p}_{2}\\}$ to the energy and to the eigenstates.
Indeed, identifying the associated energy of each $i$-sector ($i=1,\,2$) as

$\hat{\xi}_{i}={\hat{p}_{i}^{2}\over
2m}+{\textstyle{{1}\over{2}}}m\omega^{2}\hat{q}_{i}^{2},$ (17)

from standard QM, one would have
$\dot{\xi}_{i}=i\hbar^{-1}\langle[H,\,\hat{\xi}_{i}]\rangle=0$ (with
$\langle\hat{\xi}_{i}\rangle\equiv{\xi}_{i}$). However, after recasting
$\\{\hat{q}_{i},\,\hat{p}_{i}\\}$ in terms of the SW map, Eqs. (5)-(6), with
$\Omega$, $\omega$ and $\gamma$ constrained by Eq. (11), one finds an
unexpected non-stationary behaviour for each of the energy contributions,

$\displaystyle\xi_{i}(t)$ $\displaystyle=$ $\displaystyle{\hbar\Omega\over
2}\left\\{1-(-1)^{i}\left[\sqrt{1-{\textstyle{{\omega^{2}}\over{\Omega^{2}}}}}\left(\cos(2\gamma
t)\,\cos(2\Omega t)-{\textstyle{{\gamma}\over{\Omega}}}\sin(2\gamma
t)\,\sin(2\Omega t)\right)\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.+{\textstyle{{\omega}\over{\Omega}}}\sqrt{1-{\textstyle{{\gamma^{2}}\over{\Omega^{2}}}}}\sin(2\gamma
t)\right]\right\\},$

from which arise the time crystal non-stationary behaviour driven by $\Omega$
and a beating behaviour driven by $\gamma$. This is depicted in Fig. 1; if
either $\theta$ or $\eta$ vanishes, one has $\Omega^{2}=\omega^{2}+\gamma^{2}$
and

$\xi_{i}(t)={\hbar\Omega\over
2}\left\\{1-(-1)^{i}\left[{\textstyle{{\gamma}\over{\Omega}}}\left(\cos(2\gamma
t)\,\cos(2\Omega t)-{\textstyle{{\gamma}\over{\Omega}}}\sin(2\gamma
t)\,\sin(2\Omega
t)\right)+\left(1-{\textstyle{{\gamma^{2}}\over{\Omega^{2}}}}\right)\sin(2\gamma
t)\right]\right\\}.$ (19)

For the arbitrary choice of $\gamma/\Omega=0.002$, it is shown in the smaller
window of Fig. 1, the energy decoupled $\gamma$-frequency NC quantum beating
(dashed lines) and the externally driven $\Omega$-frequency time crystal
behaviour (dotted lines) for $\gamma t\gtrsim 0$. In Fig. 2 the time
derivative of the energy is depicted, from which the magnitude of the time
crystal oscillating behaviour can be quantified. From Eq. (3), the externally
driven oscillation amplitude, $\hbar\Omega/2$, is modulated by a factor
$\gamma/\Omega$.

Figure 1: (Colour on line) Dimensionless NC associated energies ,
$(\hbar\Omega)^{-1}{\xi}_{1(2)}$ (black (blue) line) as function of $\Omega
t$, for $\gamma/\Omega=0.002$. Decoupled $\gamma$-frequency NC quantum beating
(dashed lines) and $\Omega$-frequency time crystal behaviour (dotted lines)
for $\gamma t\gtrsim 0$ are identified in the zoom in window. Figure from Ref.
[1] Figure 2: (Colour on line) Dimensionless time derivative,
$(\hbar\Omega^{2})^{-1}\dot{\xi}_{1}$ as function of $\Omega t$, for
$\gamma/\Omega=0.002$. The NC beating behaviour is depicted by the first plot
and the time crystal periodic behaviour, driven by the amplitude modulation,
$\hbar\gamma\Omega(\equiv\gamma/\Omega\times\hbar\Omega^{2})$ (red line) is
depicted in zoom in plot. Figure from Ref. [1]

Given that the corrections due to the $\gamma$ parameter are small, the
beating oscillations are presumably difficult to measure. On the other hand,
an effect is acessible for $\gamma\ll\Omega$ and $\gamma t\gtrsim 0$ implying
that

$\xi_{i}(t)\approx{\hbar\Omega\over
2}\left[1-(-1)^{i}{\textstyle{{\gamma}\over{\Omega}}}\left(2\,\Omega
t+\cos(2\Omega t)\right)\right],$ (20)

at first order in $\gamma$. In this case,

$\dot{\xi}_{i}(t)\approx(-1)^{i+1}\hbar\gamma\Omega\left[1-\sin(2\Omega
t)\right],$ (21)

a time crystal periodic behaviour arise with a measurable energy time
derivative oscillation amplitude, $\hbar\gamma\Omega$, driven by both the NC
parameter, $\gamma$, and the external oscillation frequency
$\Omega\sim\omega$.

It should be added that the states of the $2$-dimensional NC quantum harmonic
oscillator here examined satisfy the no-cloning and no-deleting theorems
without additional constraints [47], given that these theorems depend only on
the unitarity of QM, which is shared by PSNCQM. From these results it follows
that some specific features of the system studied here cannot be ruled out on
the basis of the same no-go theorems that render the Wilczek’s hypothesis
untenable [16].

Let us close this section pointing out that, as discussed in Ref. [1] ,the
effects presented above can also be inferred from the behaviour of the time
derivative of the Wigner eigenfunctions for each $i$-sector of the
$2$-dimensional harmonic oscillator.

##  4 Discussion and Conclusions

Let us briefly discuss our results. First of all, it is natural to expect that
for too small values of $\gamma/\Omega$ (say $\gamma/\Omega\ll 0.002$), time
crystal and NC beating patterns are hard to measure. Nevertheless, it is
interesting that 2-dimensional (or 3-dimensional) Bose-Einstein condensates
with time crystal like behaviour have been detected through a resonance
between two (or three) oscillating mirror atoms [48, 49]. Our results suggest
that the observed behaviour is a natural explanation for the quasi-periodic
eigenstates driven by $\xi_{1}$, $\xi_{2}$, and $\gamma$. In fact, for
extended time intervals, in which the NC quantum beating takes place, the
short time scale $\Omega$-frequency periodic behaviour turns into a quasi-
periodic one, due to the periodic corrections from $\gamma$-frequency. This
suggests that a connection of our results with the spontaneous formation of
time quasi-crystals from atoms bouncing between a pair of orthogonal mirror
atoms [48, 50] is possible.

On general grounds, our results show that the non-stationary behaviour
associated to time crystals, arises entirely from either position ($q$) or
momentum ($p$) noncommutativity, i.e., from $[\hat{q}_{i},\,\hat{q}_{j}]\neq
i\theta\epsilon_{ij}$ and/or $[\hat{p}_{i},\,\hat{p}_{j}]\neq
i\eta\epsilon_{ij}$, with no need of an ab initio hypothesis of spontaneous
breaking of time translation symmetry. Thus, we can conclude that the NC
parameters naturally give origin to periodic oscillations that resemble time
crystals. Conversely, besides accounting for the emergence of such unexpected
properties, the measurable oscillation amplitude $\propto\hbar\gamma\Omega$
($\gamma/\Omega$ driven by the NC parameters, $\gamma$, and the external
oscillation frequency $\Omega\sim\omega$, can themselves be tested in order to
set bounds to the NC parameters. Thus, we hope that our discussion in the
context of the 2-dimensional noncommutative quantum harmonic oscillator might
stimulate further attempts to experimentally test such a fascinating phenomena
as time crystals. In fact, recent claims on the observation of continuous time
crystal behaviour in quantum processors [51, 52], atom-cavity system excited
by photon oscillations [53] and in an electron-nuclear spin system [54] show
that there is a vivid interest in searchimg for concrete experimental evidence
of time crystals in Nature.

As a final remark, we point out that some discrete or Floquet time crystals
have been considered, for instance, in finite dimensional Hilbert spaces [55,
56], however these might not be related to the ones discussed here, which
emerge from the continuous deformed algebra, Eq. (1).

Acknowledgments

The work of one of us (O.B.) is partially supported by FCT (Fundação para a
Ciência e Tecnologia, Portugal) through the project CERN/FIS-PAR/0027/2021,
with DOI identifier 10.54499/CERN/FI.

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