††thanks: These authors contribute equally.††thanks: These authors contribute
equally.

# Delocalization of topological edge states

Weiwei Zhu  Department of Physics, National University of Singapore, Singapore
117542, Singapore  Wei Xin Teo  Department of Physics, National University of
Singapore, Singapore 117542, Singapore  Linhu Li<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing $\&$
School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus),
Zhuhai 519082, China  Jiangbin Gong<EMAIL_ADDRESS>Department of Physics,
National University of Singapore, Singapore 117542, Singapore

###### Abstract

The non-Hermitian skin effect (NHSE) in non-Hermitian lattice systems depicts
the exponential localization of eigenstates at system’s boundaries. It has led
to a number of counter-intuitive phenomena and challenged our understanding of
bulk-boundary correspondence in topological systems. This work aims to
investigate how the NHSE localization and topological localization of in-gap
edge states compete with each other, with several representative static and
periodically driven 1D models, whose topological properties are protected by
different symmetries. The emerging insight is that at critical system
parameters, even topologically protected edge states can be perfectly
delocalized. In particular, it is discovered that this intriguing
delocalization occurs if the real spectrum of the system’s edge states falls
on the same system’s complex spectral loop obtained under the periodic
boundary condition. We have also performed sample numerical simulation to show
that such delocalized topological edge states can be safely reconstructed from
time-evolving states. Possible applications of delocalized topological edge
states are also briefly discussed.

##  I Introduction

The so-called non-Hermitian skin effect (NHSE) has been recognized as one of
the seminal findings in non-Hermitian lattice systems Yao and Wang (2018); Yao
_et al._ (2018); Martinez Alvarez  _et al._ (2018); Kunst  _et al._ (2018);
Lee  _et al._ (2019); Yokomizo and Murakami (2019); Lee and Thomale (2019);
Longhi (2019); Herviou  _et al._ (2019); Zhang and Gong (2020); Li  _et al._
(2020a, b); Liu  _et al._ (2020); Weidemann  _et al._ (2020); Scheibner  _et
al._ (2020); Hofmann  _et al._ (2019); Brandenbourger  _et al._ (2019); Song
_et al._ (2019a). Due to NHSE, eigenstates of a non-Hermitian system tend to
exponentially pile up at its boundaries. This challenges our conventional
thinking of bulk-boundary correspondence (BBC) Gong  _et al._ (2018); Takata
and Notomi (2018); Foa Torres (2019); Ghatak and Das (2019); Brzezicki and
Hyart (2019); Deng and Yi (2019); Edvardsson  _et al._ (2019); Ezawa (2019);
Kunst and Dwivedi (2019); Wang  _et al._ (2019); Song  _et al._ (2019b);
Longhi (2020); Yi and Yang (2020); Kawabata  _et al._ (2020); Yang  _et al._
(2020), which often connects eigenstate localization with band topology alone.
For example,  normally the generalized Brillouin zone approach Yao and Wang
(2018); Yokomizo and Murakami (2019); Lee and Thomale (2019) has to be
introduced to remove the complications of NHSE so that we can correctly
describe the topological properties of non-Hermitian systems by non-Bloch
topological invariants Kawabata  _et al._ (2020); Yang  _et al._ (2020); Li
_et al._ (2019); Lee  _et al._ (2020); Imura and Takane (2019); Jin and Song
(2019); Ghatak  _et al._ (2019); Helbig  _et al._ (2020); Xiao  _et al._
(2020). On the other hand, NHSE itself has a topological origin arising from
point-gapped spectrum with spectral winding on the complex plane Kawabata  _et
al._ (2019); Okuma  _et al._ (2020); Zhang  _et al._ (2020). With NHSE
demonstrated in several experiments Weidemann  _et al._ (2020); Scheibner  _et
al._ (2020); Hofmann  _et al._ (2019); Brandenbourger  _et al._ (2019); Ghatak
_et al._ (2019); Helbig  _et al._ (2020); Xiao  _et al._ (2020), it is of even
more interest to explore new faces of NHSE in topological non-Hermitian
systems.

In this work, our focus is to investigate the localization/delocalization of
topological edge states in several representative non-Hermitian systems.
Intuitively, the behavior of topological edge states depends on both
topological localization and NHSE as two competing factors. Topological
localization can be largely connected with the width of a topological band
gap, whereas NHSE is of a different origin. How these two factors compete then
presents an interesting subject to study. The main job of this work is to
identify details of such a competition. This can become highly nontrivial in
systems with more than one characteristic length scales of NHSE, e.g., in a
periodic driven system where the NHSE can be challenging to analyze Zhang and
Gong (2020) or in a three-band system with a complicated spectrum on the
complex plane. As a key finding, we are able to identify a key condition to
observe completely delocalized topological edge states. For one model, we have
also performed explicit simulations to show that the profile of such
delocalized topological edge states can be reconstructed from time-evolving
states.

The main message of this work arises from our computational observations
rather than a formal theory. What we shall uncover here is one overlooked and
yet very useful aspect of the spectrum of non-Hermitian systems under the
periodic boundary condition (PBC). Indeed, in previous studies, the PBC
spectrum of a non-Hermitian system is largely used, via the generalized
Brillouin zone approach Yao and Wang (2018); Lee and Thomale (2019); Yokomizo
and Murakami (2019); Kawabata  _et al._ (2020); Yang  _et al._ (2020); Li  _et
al._ (2019); Lee  _et al._ (2020), to predict and understand the bulk spectrum
of the system under the open boundary condition (OBC). In this type of routine
by which one unravels the non-Hermitian pumping Lee  _et al._ (2020), the
properties of the possible topological edge states are not in the picture. It
is hence remarkable that this work shows that the PBC spectrum is actually
essential to understand under what conditions the topological edge states may
become completely delocalized.

This work is organized as follows. Section II starts our analysis with a non-
Hermitian Su-Schrieffer-Heeger (SSH) model, in connection with the emergence
of exceptional points and half-integer Bloch topological invariants. This is
followed by Sec.III, where a periodically driven (Floquet) SSH model is
studied. Such a Floquet two-band model is richer, because it has now two
quasi-energy band gaps. Results there enhance our analysis in the static case,
insofar as the delocalization of topological edge states associated with both
gaps are observed and it occurs at the exceptional point of the two-band
system. Besides, some numerical simulations presented there also show that the
profile of such delocalized edge states can be reconstructed from time-
evolving states. With these results, one might infer that the delocalization
of topological edge states is a property of exceptional band topology.
However, in Sec. IV we are able to identify a more general condition using a
three-band system without the kind of chiral symmetry in the previous two
examples. The magic condition for delocalization of topological edge states
turns out to be the same in the all examples studied, namely, it occurs when
the spectrum of the system under the periodic boundary condition touches that
of the topological edge states. Sec.V concludes this paper, with a brief
discussion on possible applications of delocalized edge states.

##  II Delocalized topological edge state in non-Hermitian SSH model

Figure 1: (a) Non-Hermitian SSH model. $t_{1}$ is the intracellular couplings
and asymmetric couplings are introduced into the intercellular couplings
described by $t_{2}$ and $g$. (b)(c) Non-Hermitian Floquet SSH model. (b) The
non-Hermitian Floquet SSH model is composed of four steps. In odd steps, the
lattice is fully dimerized with the site only coupled with one of its nearest
sites. In even steps, the lattice has alternative gain and loss. (c) The
equivalent two-step model, with asymmetric couplings in even steps. The
intracellular couplings is $\theta$ and the intercellular couplings is
$\gamma$. $g$ is the gain strength. (d) Non-Hermitian three-band extended SSH
model. $t_{1,2}$ are the intracellular couplings and asymmetric couplings are
introduced to the intercellular couplings described by $t_{3}$ and $g$. This
model is conjugated-inversion symmetric when $t_{1}=t_{2}$.

We first study the non-Hermitian SSH model shown in Fig. 1(a) which is
composed of two sublattices. The intracellular couplings are $t_{1}$ and the
intercellular couplings are asymmetric, i.e. $t_{2}e^{g}$ for right couplings
and $t_{2}e^{-g}$ for left couplings. The Hamiltonian in momentum space is

$H(k)=\left(\begin{array}[]{cc}0&t_{1}+t_{2}e^{i(k-ig)}\\\
t_{1}+t_{2}e^{-i(k-ig)}&0\end{array}\right),$ (1)

satisfying a chiral symmetry $\sigma_{z}H(k)\sigma_{z}=-H(k)$ that protects a
$Z$-type topology in 1D. In complex plane, $e^{ik}$ gives the Brillouin zone
(BZ) as a unit circle centered at origin, and $e^{i\tilde{k}}$ with
$\tilde{k}=k+ig$ gives the generalized Brillouin zone (GBZ) as another circle
with the same center but a different radius of $e^{-g}$. According to the non-
Bloch bulk-boundary correspondence, the topological properties under open
boundary conditions (OBCs) are described by $H(\tilde{k})$ with
$\tilde{k}=k+ig$ corresponding to the GBZ, meaning that they are determined
only by $t_{1}$ and $t_{2}$, but not by the value of $g$. Namely, the system
is topologically trivial when $t_{1}>t_{2}$, and nontrivial with topological
edge states under OBCs when $t_{1}<t_{2}$. The absence and presence of these
edge states are characterized by a winding number ($0$ and $1$ respectively)
defined in the GBZ, which can be obtained from $2\pi
i\tilde{\nu}=\oint_{\tilde{k}}d\tilde{z}/\tilde{z}$, with
$\tilde{z}=t_{1}+t_{2}e^{i\tilde{k}}$.

We show an example in Fig. 2(a) with $t_{1}=1$ and $t_{2}=e$. Without loss of
generality, we only consider $t_{1}$, $t_{2}$ and $g$ with positive values.
The nonzero $g$ introduces NHSE into the system and all bulk states are
localized at right edge.

Due to the divergence of BZ and GBZ, another Bloch topological invariant can
also be defined in the BZ, which can be determined by the average of two
winding numbers Yin  _et al._ (2018)

$\nu=\frac{\nu_{+}+\nu_{-}}{2}$ (2)

where $2\pi i\nu_{\pm}=\oint_{k}dz_{\pm}/z_{\pm}$, $z_{\pm}=t_{1}+t_{2}e^{\pm
g}e^{ik}$. Unlike the GBZ winding number which is always quantized, $\nu$ can
be half-quantized when $\nu_{+}\neq\nu_{-}$. Specifically in our model, we
have $\nu=-1/2$ when $g>g_{c}\equiv|\ln(t_{1}/t_{2})|$. Such a half integer
means that the two bands connect to each other head-to-tail in the BZ, forming
one big loop encircling an EP Dembowski  _et al._ (2001); Gao  _et al._
(2015); Shen  _et al._ (2018); Li  _et al._ (2019). Fig. 3(a)-(c) show the
results for topological trivial states with $t_{1}=e$ and $t_{2}=1$. And Fig.
3(d)-(f) show the results for topological nontrivial states with $t_{1}=1$ and
$t_{2}=e$. Their critical points are all $g_{c}=1$. We notice the spectrum is
gapped in Fig. 3(a)(d) with $g<1$ and gapless in Fig. 3(c)(f) with $g>1$. At
the critical point $g_{c}$, the system supports EP shown in Fig. 3(b)(e).

Figure 2: NHSE induced delocalized topological edge state in non-Hermitian SSH
model. (a) Spectrum in open boundary condition, $e-1$ and $e+1$ indicated on
the panel are boundary values of the bulk spectrum. (b1)-(d1) Eigenstate
profile of topological edge state most occupying $A$ sublattice. (c1) The
delocalized topological edge state. (b2)-(d2) Eigenstate profile of
topological edge state most occupying $B$ sublattice. $t_{1}$ and $t_{2}$ are
set as $1$ and $e$, respectively. (b) for $g=0.5$, (c) for $g=1$ and (d) for
$g=1.5$. Figure 3: Spectrum of non-Hermitian SSH model in periodic boundary
condition and EPs. (a)-(c) for topological trivial state with $t_{1}=e$ and
$t_{2}=1$. (d)-(f) for topological nontrivial state with $t_{1}=1$ and
$t_{2}=e$. For both cases the critical point is at
$g_{c}\equiv|\ln(t_{1}/t_{2})|=1$. (c)(f) are gapless states with a half-
integer topological invariant $\nu=-1/2$.

As the winding number $\nu$ depends on the value of $g$, it is not expected to
be directly related to the existence of topological edge states under OBCs.
Indeed, both the cases with and without such edge states can accompany a half-
integer $\nu=-1/2$, as shown in Fig. 3(c)(f). On the other hand, We notice
that $g$ describes the localization strength of the NHSE, and
$g_{c}\equiv|\ln(t_{1}/t_{2})|$ corresponds to the topological localization
strength of the edge states. Thus the half-integer topological invariant is
actually the results of strong NHSE whose strength is larger than the
topological localization from band gap. Therefore we can expect the change of
localizing direction of the edge states with $g$ tuned across $g_{c}$,
provided the two types of localizations are toward opposite directions. In
other words, a transition of localizing direction of topological edge states
shall occur when $\nu$ changes between an integer and a half-integer.

To give a concrete example, we illustrate the distribution of the topological
edge states in Fig. 2(b) to (d). For $g<g_{c}$, the two edge states localized
at different ends of the 1D chain, as shown in Figs. 2(b1) and (b2). Note that
they exhibit different localization strengths as the NHSE contributes to the
same localizing direction for all eigenstates. With increasing $g$, the
$A$-sublattice edge state is delocalized at $g=g_{c}$ [Fig. 2(c1)] and
localized at the other end when $g>g_{c}$ [Fig. 2(d1)]. The localization
strength of $B$-sublattice edge state is also getting stronger as shown in
Figs. 2(c2) and (d2). The localization of two edge states can be described by
$-g_{c}+g$ and $g_{c}+g$, negative value means localized at left edge. Such a
transition of localizing direction of topological edge states also coincides
with the geometric transition of the PBC spectrum, where the two bands touch
at the origin and give rise to an EP [Figs. 3(b) and (e)].

Figure 4: NHSE induced delocalized topological edge state in non-Hermitian
Floquet SSH model and EPs. (a) Phase diagram. There are four phases: trivial
phase, $0$ phase which supports $0$ mode, $\pi$ phase which supports $\pi$
mode and $0\pi$ phase which supports both $0$ mode and $\pi$ mode. (b)
Spectrum in periodic boundary condition with different values of $g$. (c)
Spectrum in open boundary condition as function of $g$. (d) The non-Bloch
winding number for $0$ gap and $\pi$ gap. (e)(f) Two edge states with
quasienergy $0$. (g)(h) Two edge states with quasienergy $\pi$. (e2)(g4) are
delocalized topological edge states. In (b)-(h), we choose $\theta=0.6\pi$ and
$\gamma=0.75\pi$.

##  III Delocalized topological edge state in non-Hermitian Floquet SSH model

The non-Hermitian SSH model in Fig. 1(a) requires asymmetric couplings which
is hard to realize in experiment. Here we propose a non-Hermitian Floquet SSH
model shown in Fig. 1(b) which are composed of four steps. In odd steps, the
lattice is fully dimerized with site only coupled with one of its nearest
site. In even steps, the lattice has alternative gain and loss on potential.
Such four-steps model is equivalent to a two-step model with asymmetric
couplings shown in Fig. 1(c). The time dependent Bloch Hamiltonian is

$\displaystyle H(k,t)=\left\\{\begin{array}[]{c}H_{a}(k),\,\,0<t\leq T/2\\\
H_{b}(k),\,\,T/2<t\leq T\end{array}\right.$ (5)

where $H_{a}(k)=2\theta\sigma_{x}$ and

$H_{b}(k)=2\gamma\left(\begin{array}[]{cc}0&e^{i(k-ig)}\\\
e^{-i(k-ig)}&0\end{array}\right)$ (6)

The Floquet theory then leads us to the Floquet states
$U_{t_{0}}(k,T)|\Psi\rangle=e^{i\varepsilon T}|\Psi\rangle$, with
$U_{t_{0}}(k,T)\equiv\mathcal{T}\mathrm{exp}\big{[}i\int_{t_{0}}^{t_{0}+T}H(\tau)\,d\tau\big{]}$
being time evolution operator in one period, $\mathcal{T}$ is the time-
ordering operator and $t_{0}$ is a reference time. In our case, we set the
period $T=1$.

Similar to the non-Hermitian SSH model, the generalized Brillouin zone of such
system is a circle with center at origin and radius $e^{-g}$. Its topological
properties are determined by $\theta$ and $\gamma$, and independent with $g$.
Topological phase transition can happen at $0$ or $\pi$ quasi-energy gap and
splits the parameter space into four phases shown in Fig. 4(a). Those phases
can be described by two non-Bloch winding numbers corresponding to $0$ gap and
$\pi$ gap Asbóth and Obuse (2013); Zhou and Gong (2018), which are obtained
from $2\pi
i\tilde{\nu}^{\varepsilon}=\oint_{\tilde{k}}d\tilde{z}^{\varepsilon}/\tilde{z}^{\varepsilon}$,
where $\varepsilon=\\{0,\pi\\}$,
$\tilde{z}^{0}=i\cos\frac{\gamma}{2}\sin\frac{\theta}{2}+i\sin\frac{\gamma}{2}\cos\frac{\theta}{2}e^{i\tilde{k}}$
and
$\tilde{z}^{\pi}=\cos\frac{\gamma}{2}\cos\frac{\theta}{2}-\sin\frac{\gamma}{2}\sin\frac{\theta}{2}e^{-i\tilde{k}}$.

Similar to the static SSH model, while the non-Bloch winding number only takes
integer, the Bloch winding number can take a half integer. The half-integer
topological invariant comes from the competition between NHSE strength and
band gap width. Differently, the Floquet model has two gaps at $0$ and $\pi$.
So one needs two Bloch topological invariants to describe the system. They are
determined by

$\displaystyle\nu^{0}$ $\displaystyle=$
$\displaystyle\frac{\nu_{+}^{0}+\nu_{-}^{0}}{2}$ (7) $\displaystyle\nu^{\pi}$
$\displaystyle=$ $\displaystyle\frac{\nu_{+}^{\pi}+\nu_{-}^{\pi}}{2}$ (8)

where $2\pi
i\nu_{\pm}^{\varepsilon}=\oint_{k}dz_{\pm}^{\varepsilon}/z_{\pm}^{\varepsilon}$,
$\varepsilon=0,\pi$

$\displaystyle z_{\pm}^{0}$ $\displaystyle=$ $\displaystyle
i\cos\frac{\gamma}{2}\sin\frac{\theta}{2}+i\sin\frac{\gamma}{2}\cos\frac{\theta}{2}e^{\pm
g}e^{ik}$ (9) $\displaystyle z_{\pm}^{\pi}$ $\displaystyle=$
$\displaystyle\cos\frac{\gamma}{2}\cos\frac{\theta}{2}-\sin\frac{\gamma}{2}\sin\frac{\theta}{2}e^{\mp
g}e^{-ik}$ (10)

There are two critical points obtained from $|z_{+}^{0}||z_{-}^{0}|=0$ and
$|z_{+}^{\pi}||z_{-}^{\pi}|=0$,

$\displaystyle g_{c}^{0}$ $\displaystyle\equiv$
$\displaystyle\left|\ln\left(\tan\frac{\theta}{2}\cot\frac{\gamma}{2}\right)\right|$
(11) $\displaystyle g_{c}^{\pi}$ $\displaystyle\equiv$
$\displaystyle\left|\ln\left(\tan\frac{\theta}{2}\tan\frac{\gamma}{2}\right)\right|,$
(12)

corresponding to the EPs at $0$ and $\pi$ quasienergies, and determining the
transition of localizing direction of topological zero modes and $\pi$ modes
respectively. For trivial phase and $\pi$ phase, $\nu^{0}=0$ with
$g<g_{c}^{0}$ and $\nu^{0}=-1/2$ with $g>g_{c}^{0}$. For $0$ phase and $0\pi$
phase, $\nu^{0}=-1$ with $g<g_{c}^{0}$ and $\nu^{0}=-1/2$ with $g>g_{c}^{0}$.
For trivial phase and $0$ phase, $\nu^{\pi}=0$ with $g<g_{c}^{\pi}$ and
$\nu^{0}=1/2$ with $g>g_{c}^{\pi}$. For $\pi$ phase and $0\pi$ phase,
$\nu^{\pi}=1$ with $g<g_{c}^{\pi}$ and $\nu^{\pi}=1/2$ with $g>g_{c}^{\pi}$.

Next we focus on special case with $\theta=0.6\pi$ and $\gamma=0.75\pi$, which
belongs to the $0\pi$ phase. The two critical points approximately equal to
$g_{c}^{0}\approx 0.56$ and $g_{c}^{\pi}\approx 1.2$. At those critical
points, the spectrum in periodic boundary condition support EP at $0$ gap and
$\pi$ gap as shown in Figs. 4(b2) and (b4). Within $g<g_{c}^{0}$, the spectrum
forms two separated loops in Fig. 4(b1). Within $g_{c}^{0}<g<g_{c}^{\pi}$,
$\nu^{0}$ is half-integer and the spectrum is gapless at quasienergy $0$ shown
in Fig. 4(b3). Within $g>g_{c}^{\pi}$, both $\nu^{0}$ and $\nu^{\pi}$ are
half-integer, and the spectrum is gapless at quasienergy $0$ and $\pi$ in Fig.
4(b5).

Under OBC, the spectrum is independent of $g$, as shown in Fig. 4(c). The
system supports $0$ mode and $\pi$ mode, of which the presence can be digested
through the nonzero non-Bloch winding number (winding number defined in
generalized Brillouin zone). The generalized Brillouin zone is used to remove
the NHSE. Indeed, one can obtain the non-Bloch winding number by setting $g=0$
and calculating the Bloch winding number $\nu^{0}$ and $\nu^{\pi}$. Fig. 5(d)
show the nonzero non-Bloch winding numbers for $0$ gap and $\pi$ gap.

The localization of topological edge state is determined by band gap width and
NHSE. The profile of two $0$ modes are shown in Figs. 4(e)(f). With
$g<g_{c}^{0}$, two $0$ modes are localized at opposite edges in Figs.
4(e1)(f1). With $g>g_{c}^{0}$, two $0$ modes are localized at same sides in
Figs. 4(e3)-(e5),(f3)-(f5). At critical point $g_{c}^{0}$, one $0$ mode is
delocalized in Fig. 4(e2). The profile of two $\pi$ modes are shown in Figs.
4(g)(h). With $g<g_{c}^{\pi}$, two $\pi$ modes are localized at opposite edges
in Figs. 4(g1)-(g3),(h1)-(h3). With $g>g_{c}^{\pi}$, two $\pi$ modes are
localized at same sides in Figs. 4(g5)(h5). At critical point $g_{c}^{\pi}$,
one $\pi$ mode is delocalized in Fig. 4(g4).

Figure 5: Reconstruction of the profile of delocalized topological edge states
by a simple time averaging approach. (a) Extracted topological edge states
with quasienergy $0$. (b) Extracted topological edge states with quasienergy
$\pi$. $\theta=0.6\pi$ and $\gamma=0.75\pi$. The slope of dashed lines
connecting the respective dots in each panel indicate the localization
exponents of one of the two topological edge states, with the zero slop
corresponding to delocalization of the concerned edge state.

To end this section, we proceed to show that the delocalized topological edge
states identified above can be directly relevant in experiments. For example,
in Ref. Xiao  _et al._ (2020) it was shown that the topological edge state in
a non-Hermitian Floquet system can be extracted from the time-evolving state
profile $|\Phi(t)\rangle$ by the following simple operation,

$|\Phi^{\varepsilon}(t)\rangle=\sum_{t^{\prime}=0}^{t}\frac{e^{-i\varepsilon
t^{\prime}}}{t+1}|\Phi(t^{\prime})\rangle\,\,\,(\varepsilon=0,\pi).$ (13)

The underlying insight motivating this approach is the following: For a
sufficiently long time $t$ used for the averaging above, the contribution to
the above expression from bulk state cancels out and only topological edge
state with the specified quasienergy $\varepsilon$ retains. This being the
case, one can next extract the profile of a topological edge state by assuming
that the initial state has a finite overlap with the corresponding topological
edge state. We perform precisely such type of simulation, with the profile of
the extracted topological edge states at quasienergy $0$ shown in Fig. 5(a).
It is seen that both edge states (left and right) are excited in Fig. 5(a1)
with $g=0.2$. At $g_{c}^{0}\approx 0.56$, the left part is delocalized in Fig.
5(a2). With $g=0.9>g_{c}^{0}$, the left part is also localized to right in
Fig. 5(a3). The extracted edge states with quasienergy $\pi$ is shown in
parallel in Fig. 5(b). The left part is delocalized at $g_{c}^{\pi}\approx
1.2$ in Fig. 5(b2).

##  IV A three-band extended SSH model

In previous sections, we have studied both static and Floquet SSH model, where
a chiral symmetry ensures the absence of the third Pauli matrices, and thus
allows the topological winding numbers to be well-defined for the systems.
Nonetheless, there are different symmetries which can protect the 1D topology,
such as the inversion symmetry in a generalized multi-band system Hughes  _et
al._ (2011); Guo and Chen (2015); Yang  _et al._ (2015); Zhu  _et al._ (2018).
Therefore, to further explore the emergence of the delocalization of the
topological edge states, we consider a three-band extended SSH model as shown
in Fig. 1(d). It is described by the following Hamiltonian:

$H(k)=\begin{pmatrix}0&t_{1}&t_{3}e^{i(k-ig)}\\\ t_{1}&0&t_{2}\\\
t_{3}e^{-i(k+ig)}&t_{2}&0\end{pmatrix},$ (14)

where $t_{1,2}$ are the intracellular couplings, and the intercellular
couplings are asymmetric, with $t_{3}e^{g}$ for right couplings and
$t_{3}e^{-g}$ for left couplings.

In the Hermitian case with $g=0$, the three-band extended SSH model is
inversion symmetric when $t_{1}=t_{2}$. The non-Hermitian coupling strength
$g$ has to break the inversion symmetry as it takes different signs toward
different spacial direction. Nevertheless, the overall Hamiltonian still
satisfies a conjugated-inversion symmetry, defined as
$IH(k)I^{-1}=H^{\dagger}(-k)$ with the symmetry operator
$I_{ij}=\delta_{i,4-j}$. This symmetry also coincides with the inversion
symmetry when $g=0$, where the Hamiltonian becomes Hermitian, $H=H^{\dagger}$.
Later we shall see that the degeneracy of a pair of in-gap edge states is
lifted when the symmetry is broken, suggesting a corresponding symmetry
protection of the topological properties in this model.

We first consider the conjugated-inversion symmetric case where $t_{1}=t_{2}$.
Analogous to the SSH model, the topological properties in this model is also
only determined by the intercellular and intracellular couplings. That is, the
system is topologically nontrivial if $t_{3}>t_{1,2}$, and trivial otherwise.
In Fig. 6, we illustrate an Hermitian example with $t_{1}=t_{2}=0.8$,
$t_{3}=1.7$ and $g=0$, where 2 pairs of degenerate topological edge states
emerge in this system, each lies in a different band gap. The two topological
edge states in each pair also localize at different ends respectively.

Figure 6: Energy spectra of a three-band SSH system without NHSE. The blue
dots are the energy spectrum under open boundary condition. The eigenstate
profile of two pairs of degenerate topological edge states are shown in the
insets. Parameters: $t_{1}=0.8$, $t_{2}=0.8$, $t_{3}=1.7$ and $g=0$.

By turning on the non-reciprocal coupling $g$, the PBC spectrum will form loop
and the NHSE occurs. Increasing $g$ will enlarge the loop formed by the PBC
spectrum as shown in Fig. 7. The loop will eventually touch the topological
edge states, which is the critical point where the topological edge states
delocalized, as shown in Fig. 7(b2). In contrast to the results in previous
sections, exceptional point does not present in this case.

Figure 7: Conjugated-inversion symmetry case. (a1)-(c1) Energy spectra in both
periodic boundary conditions (pink, green, red lines) and open boundary
conditions (blue dots). The loop formed by the PBC spectrum is enlarging when
$g$ increases. (b1) The PBC spectrum touches the topological edge states.
(a2)-(c2) Eigenstate profile of the topological edge state which originally
localized at left. (b2) The delocalized topological edge state. (a3)-(c3)
Eigenstate profile of the right topological edge state which originally
localized at right. The negative gradient is slightly increasing when $g$
increases. Parameters: $t_{1}=0.8$, $t_{2}=0.8$, $t_{3}=1.7$. (a) $g=0.5$ (b)
$g=0.755$ and (c) $g=0.9$. We only consider the topological edge state with
positive eigen-energies.

With these results obtained, it finally becomes clear that the delocalization
of topological edge states does not necessarily occur at the exceptional
point. As such, the reason why previously there is a connection between
delocalization and EP lies in the presence of the chiral symmetry. That is, in
the static and Floquet SSH models, the chiral symmetry ensures that the
topological edge states are located at the center of band gap, which is also
where the PBC spectral loop touches the real axis at the EP point. Therefore,
in previous cases, the delocalization occurs when the topological edge states
touches the EP, and hence touches the PBC spectrum. This hence implies the
following: the general magic condition for the delocalization of the
topological edge states is that the PBC spectrum touches the topological edge
states.

Interestingly, this condition identified above applies not only to the
topological edge states, but also non-topological in-gap edge states that are
not protected by any symmetry. For instance, if we choose $t_{1}\neq t_{2}$,
the conjugated-inversion symmetry is broken. A pair of in-gap edge states is
observed in each band gap, but is no longer degenerate, as shown in Fig.
8(a1)-(c1). Thus they can now move and merge into different bands without
closing the band gap and are no longer topologically protected. On the other
hand, the loop formed by the PBC spectrum also enlarges with the increasing
$g$, and the critical point where PBC spectrum touches the edge state occurs
at $g=1.2$ for the parameters we choose. At the critical point, the edge state
which is originally localized left will be delocalized, as shown in Fig.
8(b2).

Figure 8: Conjugated-inversion symmetry broken case. (a1)-(c1) Energy spectra
in both periodic boundary conditions (pink, green, red lines) and open
boundary conditions (blue dots). The loop formed by the PBC spectrum is
enlarging when $g$ increases. (b1) The PBC spectrum touches the in-gap edge
states. (a2)-(c2) Eigenstate profile of the in-gap edge state which originally
localized at left with eigen energy around $E\approx 0.8$. (b2) The
delocalized in-gap edge state. (a3)-(c3) Eigenstate profile of the right in-
gap edge state which originally localized at right with eigen energy around
$E\approx 0.5$. The negative gradient is slightly increasing when $g$
increases. Parameters: $t_{1}=0.8$, $t_{2}=0.5$, $t_{3}=1.7$. (a) $g=0.5$ (b)
$g=1.2$ and (c) $g=1.35$.

The above seen critical point represents a critical NHSE strength, for which
the localization associated with NHSE is exactly canceled by the localization
of in-gap states. Interestingly, this transition can be regarded as a
topological transition as well, because the spectral winding number with
respect to the energy of the concerned edge state also jumps at this point. In
particular, the spectral winding number with respect to a reference energy
$E_{r}$ is defined as follow Gong  _et al._ (2018); Okuma  _et al._ (2020);
Zhang  _et al._ (2020) :

$\nu(E_{r})=\frac{1}{2\pi}\oint_{BZ}\frac{d}{dk}\arg\det[H(k)-E_{r}]dk.$ (15)

This spectral winding number has been used to characterize the presence of the
NHSE from the topological feature of the energetics on the complex plane. If
there exists a reference energy $E_{r}$ such that the winding number with
respect to $E_{b}$ is nonzero, then the NHSE presents in the system under open
boundary condition and causes localization of the eigenstates at two different
ends according to the sign of the winding number, i.e. positive sign indicates
the localization to the left and negative sign indicates localization to the
right. By taking the reference energy to be the energy of the edge state of
which the localization direction is opposite to that of the NHSE, the spectral
winding number, by definition, changes where the PBC spectrum passes through
the edge state, which is also the critical point for the delocalization of the
same edge state. Furthermore, the non-zero spectral winding number with
respect to the energy of the edge state also represents the regime where NHSE
strength has overpowered the topological localization. Therefore, the spectral
winding number has a richer physics meaning now as it can be used to indicate
the competition between the topological localization and the non-Hermitian
pumping.

##  V  Conclusion and discussion

In this work, we have studied the localization properties of non-Hermitian
topological edge states in three representative 1D models, including the non-
Hermitian SSH model, non-Hermitian Floquet SSH model, and three-band extended
SSH model. At a critical NHSE strength, the delocalization of topological edge
states is observed in all three models. In the first two models with chiral
symmetry, it is described by that the energy (quasienergy) of the edge state
is located at the EP of the PBC spectrum. Numerical simulations are also
presented in the Floquet SSH model, showing that delocalized topological edges
should be within the reach of experimental detection. Furthermore, as shown in
the third model with only conjugated-inversion symmetry, the delocalization of
topological edge states does not occur at the EP, but only requires that the
real energy of the topological edge falls exactly on the PBC spectral loop.
Such a requirement turns out to be the general condition for edge-state
delocalization in all the models studied in this work. Notably, the spectral
winding number with respect to the energy of the edge state can serve as a
tool to reflect the inherent competition between the topological localization
and the NHSE. With this understanding the critical NHSE strength at which
topological edge states are delocalized becomes the transition point at which
the spectral winding number makes a jump.

Delocalized topological edge states are fascinating because of the following
two simultaneous features: they are gapped from the bulk and yet they are
extended states. Because such edge states are real valued and remain isolated
from the bulk spectrum, it should be experimentally convenient to
adiabatically manipulate them, without much nonadiabatic contamination from
the bulk. For example, we envision that with the adiabatic following of the
edge states, an edge state initially localized at the left end can be
adiabatically switched to an extended state, and then further to a localized
state at the right end. This kind of scenario may be highly useful for
realizing edge state pumping or state transfer. Since via this process an
extended state is also adiabatically generated from a localized state, one may
synthesize a giant Schrödinger cat state starting from a localized edge state,
if the non-Hermitian lattice indeed accommodates a quantum particle. We plan
to explore these interesting possibilities in our future work.

###### Acknowledgements.

J.G. acknowledges funding support by the Singapore Ministry of Education
Academic Research Fund Tier-3 (Grant No. MOE2017-T3-1-001 and WBS No.
R-144-000-425-592) and by the Singapore NRF Grant No. NRF-NRFI2017-04 (WBS No.
R-144-000-378- 281).

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# Outlier-Insensitive Kalman Filtering using NUV Priors

###### Abstract

The kf (kf) is a widely-used algorithm for tracking the latent state of a
dynamical system from noisy observations. For systems that are well-described
by lg ss models, the kf minimizes the mse (mse). However, in practice,
observations are corrupted by outliers, severely impairing the kf’s
performance. In this work, an outlier-insensitive kf is proposed, where
robustness is achieved by modeling each potential outlier as a nuv (nuv). The
nuv variances are estimated online, using both em (em) and am (am). The former
was previously proposed for the task of smoothing with outliers and was
adapted here to filtering, while both em and am obtained the same performance
and outperformed the other algorithms, the AM approach is less complex and
thus requires $40\%$ less runtime. Our empirical study demonstrates that the
mse of our proposed outlier-insensitive kf outperforms previously proposed
algorithms, and that for data clean of outliers, it reverts to the classic kf,
i.e., mse optimality is preserved.

Index Terms—  kf, outliers, am

##  1 Introduction

Se of dynamical systems from noisy observations plays a key role in various
scientific and technological fields such as radar target tracking, complex
image processing, navigation, and positioning [1]. The celebrated kf [2] is an
efficient recursive state estimation algorithm that is mse optimal for
dynamical systems obeying a lg ss (ss) model. However, the quadratic form of
its objective, i.e., mse, makes it sensitive to deviations from nominal noise.
Thus, the kf is severely impaired when outliers are present in the
measurements [3, 4]. As sensory data is often populated with outliers,
robustness to outliers is essential [5, 6, 7]. A common approach for dealing
with outliers is to detect and then disregard influential observations. Such
detection can be achieved using appropriate statistical diagnostics [8] on the
posterior distribution, e.g., $Z\textrm{-test}$ [9] and
$\chi^{2}\textrm{-test}$ [10, 11]. The main drawbacks of these approaches are
that they need to be carefully tuned for a required false alarm, and that
potentially useful outlier information is not accounted for in the estimation
process. Alternatively, one can limit the effect of outliers by reweighting
the covariance of the observation noise at each data sample when estimating
the current state [5, 6]. These techniques require careful tuning of multiple
hyperparameters to operate reliably as well. A different approach formulates
the kf as a linear regression problem, detecting outliers via a sparsifying
$\ell_{1}$-penalty [4, 12], tackled via optimization techniques, which may be
computationally complex.

In this work, a new approach for oikf (oikf) is proposed that leverages ideas
from _sparse_ Bayesian learning [13]. Here, each potential outlier is modeled
as a nuv [14, 15, 7], i.e., an additive component on top of the observation
noise. nuv incorporation effectively yields a modified overall sparsity-aware
objective [14], which is shown to yield a robust outlier detection statistical
test with a relatively low false alarm. When an outlier is reliably detected,
we incorporate its variance into the overall covariance matrix of the
observation noise, thereby balancing its contribution to the information
fusion (i.e., update) step in the kf, and the outlier information is used in
the se process.

To estimate the nuv online, we first adapt the em algorithm [16, 17, 18],
which was previously proposed for offline smoothing [7], to the online
filtering task. em is based on computing second-order moments, i.e., the full
state observation posterior covariance. We then present an implementation,
which has not been considered to date, using a _simpler_ am algorithm [19, 20,
21]. Unlike em, am uses only first-order moments as an empirical surrogate for
outlier detection, without sacrificing performance and with improved
robustness. We evaluate the oikf for tracking based on the wna (wna) motion
model with outliers [22]. We empirically demonstrate the superiority of our
proposed algorithm compared to previous robust variants of the kf, achieving
improved performances with low complexity operations.

The rest of this paper is organized as follows: Section 2 formulates the
system and the kf with outliers. Section 3 presents the oikf with its
estimation for the outlier’s variance, and Section 4 empirically evaluates our
methods.

##  2 System Model and the Kalman Filter

ss models in dt are a common characterization of dynamical systems [22]. Such
representations capture the relationship between an unknown latent state
vector $\mathbf{x}_{t}$ and an observed vector $\mathbf{y}_{t}$, where
$t\in\mathbb{Z}$ is the time index. Here, a Gaussian and continuous ss model
is considered, namely

$\displaystyle\mathbf{x}_{t}$
$\displaystyle=\mathbf{F}\cdot{\mathbf{x}_{t-1}}+\mathbf{e}_{t},\quad\,\,\,\,\,\,\mathbf{e}_{t}\sim\mathcal{N}\left(\mathbf{0},\mathbf{Q}\right),\quad\mathbf{x}_{t}\in\mathbb{R}^{m}$
(1a) $\displaystyle\mathbf{y}_{t}$
$\displaystyle=\mathbf{H}\cdot\mathbf{x}_{t}+\mathbf{z}_{t}+\mathbf{u}_{t},\,\,\,\,\mathbf{z}_{t}\sim\mathcal{N}\left(\mathbf{0},\mathbf{R}\right),\quad\mathbf{y}_{t}\in\mathbb{R}^{n}.$
(1b)

In (1a), the state evolves by an evolution matrix $\mathbf{F}$ and by awgn
(awgn) $\mathbf{e}_{t}$ with covariance matrix $\mathbf{Q}$. In (1b), the
state observations are generated by the linear mapping $\mathbf{H}$ corrupted
by an awgn $\mathbf{z}_{t}$ with a diagonal covariance matrix
$\mathbf{R}=\textrm{diag}\left(\mathbf{r}^{2}\right)$, and by additive outlier
impulsive noise $\mathbf{u}_{t}$ with an _unknown_ distribution. The kf is an
efficient online recursive filter that estimates the state $\mathbf{x}_{t}$
from the observations $\\{\mathbf{y}_{\tau}\\}_{\tau\leq t}$. It is mse
optimal for the ss model (1) _without_ the outliers $\mathbf{u}_{t}$. It can
be conceptualized as a two-step procedure in each time step $t$, predict and
update, in which the joint probability distribution over the variables is
computed, using the first- and second-order moments of the Gaussian
distribution. In the _predict_ step, the _prior_ distribution is computed,
namely

$\displaystyle\hat{\mathbf{x}}_{t\left|t-1\right.}$
$\displaystyle=\mathbf{F}\cdot{\hat{\mathbf{x}}_{t-1}},\hskip
12.80365pt\bm{\Sigma}_{t\left|t-1\right.}\\!=\\!\mathbf{F}\cdot\bm{\Sigma}_{t-1}\cdot\mathbf{F}^{\top}\\!+\\!\mathbf{Q},$
(2a) $\displaystyle\hat{\mathbf{y}}_{t\left|t-1\right.}$
$\displaystyle=\mathbf{H}\cdot{\hat{\mathbf{x}}_{t\left|t-1\right.}},\hskip
5.69046pt\mathbf{S}_{t\left|t-1\right.}\\!=\\!\mathbf{H}\cdot\bm{\Sigma}_{t\left|t-1\right.}\cdot\mathbf{H}^{\top}\\!+\\!\mathbf{R}.$
(2b)

In the _update_ step the _posterior_ distribution is computed by fusing the
new observation $\mathbf{y}_{t}$ with the previously predicted prior
$\hat{\mathbf{x}}_{t\left|t-1\right.}$, where the kg (kg) matrix
$\mathbfcal{K}_{t}$ is used to _balance_ the contributions of both parts,
namely

$\hat{\mathbf{x}}_{t}=\hat{\mathbf{x}}_{t\left|t-1\right.}+\mathbfcal{K}_{t}\cdot\Delta\mathbf{y}_{t},\quad{\bm{\Sigma}}_{t}\\!=\\!{\bm{\Sigma}}_{t\left|t-1\right.}\\!-\\!\mathbfcal{K}_{t}\cdot{\mathbf{S}}_{t\left|t-1\right.}\cdot\mathbfcal{K}^{\top}_{t},$
(3)
$\mathbfcal{K}_{t}={\bm{\Sigma}}_{t\left|t-1\right.}\cdot{\mathbf{H}^{\top}}\cdot{\mathbf{S}}^{-1}_{t\left|t-1\right.},\quad\Delta\mathbf{y}_{t}=\mathbf{y}_{t}-\hat{\mathbf{y}}_{t\left|t-1\right.}.$
(4)

##  3 OUTLIER-INSENSITIVE KALMAN filtering

Next we propose our _outlier-robust_ online filter. In Subsection 3.1 we
present the nuv modeling. Then we present the oikf algorithm in Subsection
3.2, after which we derive two considered methods for nuv estimation based on
em (Subsection 3.3) and am (Subsection 3.4).

Fig. 1: Factor graph of the system model at time step $t$

###  3.1 NUV Modeling

Given an observation sample $\mathbf{y}_{t}$ (1b), we define $\mathbf{v}_{t}$
to be the error vector as the sum of two independent sources: the observation
noise $\mathbf{z}_{t}$ and the outlier-causing impulsive noise
$\mathbf{u}_{t}$, modeled as nuv with unknown variance vector
${\boldsymbol{\gamma}}^{2}_{t}$. Thus, the covariance of $\mathbf{v}_{t}$,
$\mathbf{\Gamma}_{t}$, is diagonal and comprises the sum of variances of the
two noise sources, namely

$\displaystyle\mathbf{v}_{t}$
$\displaystyle\triangleq\mathbf{y}_{t}-\mathbf{H}\cdot\mathbf{x}_{t}=\mathbf{z}_{t}+\mathbf{u}_{t},\quad\mathbf{u}_{t}\sim\mathcal{N}\left(\mathbf{0},{\boldsymbol{\gamma}}^{2}_{t}\right),$
(5a) $\displaystyle\mathbf{v}_{t}$
$\displaystyle\sim\mathcal{N}\left(\mathbf{0},\mathbf{\Gamma}_{t}\right),\quad\mathbf{\Gamma}_{t}=\textrm{diag}\left({\bm{\nu}}^{2}_{t}\right),\quad{\bm{\nu}}^{2}_{t}\triangleq\mathbf{r}^{2}+{\boldsymbol{\gamma}}^{2}_{t}.$
(5b)

The motivation for utilizing the nuv framework stems from its ability to
systematically incorporate interference of a bursty nature [14, 15], and is
thus useful for handling outliers with sparse ls (quadratic) models [7]. The
incorporation of the nuv representation to the overall ss model (1), is
illustrated as a factor graph [23] in Figure 1.

###  3.2 OIKF Algorithm

The proposed oikf uses map (map) estimation to instantaneously estimate the
unknown variance ${\boldsymbol{\gamma}}^{2}_{t}$. By either applying em or am
in each time step $t$, we obtain an estimate for the variance. In the em
version, the second-order moment $\mathbf{\nu}_{t}$ (5b) is directly
estimated, while in am version it is obtained from estimating $\mathbf{v}_{t}$
(5a), i.e., from the first-order moment. The oikf thus uses

${\hat{\boldsymbol{\gamma}}}^{2}_{t}=\max\left\\{\bm{\nu}_{t}^{2}-\mathbf{r}^{2},0\right\\},\hskip
4.97931pt\bm{\nu}_{t}^{2}=\left\\{\textrm{EM}:\hat{\bm{\nu}}_{t}^{2},\textrm{AM}:\mathbf{\hat{v}}^{2}_{t}\right\\}.$
(6)

A key property of the map estimator (6) is that it tends to be _sparse_ [14],
thus providing a _robust_ statistical test to detect the presence of outliers.
Due to the sparsity property, in most of the time steps, outliers are not
detected, and filtering coincides with the standard kf, thus preserving its
optimality for data without outliers. When outlier is detected, namely, when
$\hat{\boldsymbol{\gamma}}^{2}_{t}\neq\mathbf{0}$, its contribution to the
update step is balanced via its estimated variance; it is integrated into the
overall error covariance $\mathbf{\Gamma}_{t}$, which in turn becomes the
observation noise and therefore affects the kg in the update equation.

The above procedure is repeated iteratively, and the pseudo-code for the
proposed oikf is summarized in Algorithm 1. It can be done for fixed $N$
iterations, or until convergence. In [7], $\hat{\boldsymbol{\gamma}}^{2}_{t}$
was initialized to $\mathbf{0}$. Here, empirical experience suggests to
initialize it to the value of the prior moments. While retaining performance,
it converges faster.

Algorithm 1 oikf at time instance $t$

Number of iterations $N$

Predict: Estimate _a priori_ for
$\mathbf{\hat{x}}_{t\left|t-1\right.}^{i=0},\mathbf{\bm{\Sigma}}_{t\left|t-1\right.}^{i=0}$
via (2a)

for $i=0,...,N-1$ do

AM: Compute
$\mathbf{\hat{v}}_{t}^{i}=\mathbf{y}_{t}-\mathbf{H}\cdot\mathbf{\hat{x}}_{t}^{i}$

Estimate $\left(\hat{\boldsymbol{\gamma}}^{i}_{t}\right)^{2}$ via (15)

EM: Estimate $\left(\hat{\boldsymbol{\gamma}}^{i}_{t}\right)^{2}$ via (10)
with the $2$nd-order

moment ${\hat{\mathbf{X}}_{t}^{i}}$ as in (8)

Compute
$\mathbf{\Gamma}_{t}^{i}=\textnormal{diag}\left(\mathbf{r}^{2}+\left(\hat{\boldsymbol{\gamma}}^{i}_{t}\right)^{2}\right)$

Compute
$\mathbf{\hat{y}}_{t\left|t-1\right.}^{i},\mathbf{S}_{t\left|t-1\right.}^{i}$
via (2b) with $\mathbf{R}=\mathbf{\Gamma}_{t}^{i}$.

Update: Estimate _a posteriori_ for
$\mathbf{\hat{x}}_{t}^{i},\bm{\Sigma}_{t}^{i}$ via (3)

end for

###  3.3 Expectation Maximization

For an observation sample $\mathbf{y}_{t}$, the map estimate for
$\hat{\boldsymbol{\gamma}}^{2}_{t}$ is

$\displaystyle{\hat{\boldsymbol{\gamma}}_{t}^{2}}\left(\mathbf{y}_{t}\right)={\arg}\max_{\boldsymbol{\gamma}^{2}_{t}}p\left(\mathbf{y}_{t}\left|\boldsymbol{\gamma}^{2}_{t}\right.\right)\cdot
p\left(\boldsymbol{\gamma}^{2}_{t}\right).$ (7)

To compute the map estimate using EM, we assume a _plain_ nuv, i.e., a uniform
prior $p\left(\boldsymbol{\gamma}^{2}_{t}\right)\propto 1$ [14]. We can now
evaluate the standard em by alternating between the E-step, i.e., the
conditional expectation, and the M-step, i.e., maximizing this expression with
respect to $\boldsymbol{\gamma}^{2}_{t}$. Here, the expectation step
corresponds to the kf, from which we get first- and second-order _posterior_
moments, namely

$\hat{\mathbf{x}}_{t}^{i},\qquad{\hat{\mathbf{X}}_{t}^{i}}\triangleq\bm{\Sigma}^{i}_{t}+\hat{\mathbf{x}}_{t}^{i}\cdot{\hat{\mathbf{x}}^{i}_{t}}^{\top}.\vspace{-0.9mm}$
(8)

The _maximization_ step corresponds to recovering the covariance (5b) using
the following estimate [7, 18]:

${\mathbf{\hat{\Gamma}}_{t}^{i}}=\mathbf{y}_{t}\cdot\mathbf{y}^{\top}_{t}-\mathbf{H}\cdot\hat{\mathbf{x}}_{t}^{i}\cdot{\mathbf{y}^{\top}_{t}}-{\mathbf{y}}_{t}\cdot{{\mathbf{\hat{x}}^{i}_{t}}}^{\top}\cdot\mathbf{H}^{\top}-\mathbf{H}\cdot{\hat{\mathbf{X}}_{t}^{i}}\cdot\mathbf{H}^{\top}.$
(9)

We exploit the fact that $\mathbf{{\Gamma}}_{t}$ in (5b) is diagonal, to
estimate the variance $\left(\mathrm{\hat{\nu}}^{i}_{t,k}\right)^{2}$ for each
dimension $k\in\\{1,\dots,n\\}$ in a scalar manner. Since ${\gamma}_{t,k}^{2}$
is non-negative, we get the following expression for its estimate:

$\left({\hat{\gamma}}^{i}_{t,k}\right)^{2}=\max\left\\{\left({\hat{\nu}}^{i}_{t,k}\right)^{2}-\mathrm{r}_{k}^{2},0\right\\}.$
(10)

This demonstrates the sparsifying property of the nuv modeling, maybe more
specifically the map estimate for $\mathrm{u}_{t,k}$ from
$\hat{\gamma}^{2}_{t,k}$ [14] is

$\displaystyle\hat{\mathrm{u}}_{t,k}$
$\displaystyle=\mathrm{v}_{t,k}\cdot\frac{\hat{\gamma}^{2}_{t,k}}{\hat{\gamma}^{2}_{t,k}+{\mathrm{r}^{2}_{k}}}=\max\left\\{\mathrm{v}_{t,k}\cdot\frac{\hat{\nu}^{2}_{t,k}-{\mathrm{r}^{2}_{k}}}{\hat{\nu}^{2}_{t,k}},0\right\\}.$
(11)

Namely, $\hat{\gamma}^{2}_{t,k}$ leads to $\hat{\mathrm{u}}_{t,k}=0$. When the
outlier is identified as zero, oikf coincides with the kf.

###  3.4 Alternating Maximization

We next describe an iterative am method to compute the joint map estimate
based on the nuv representation [15]

$[\hat{\mathbf{v}}_{t},\hat{\mathrm{\boldsymbol{\gamma}}}^{2}_{t}](\mathbf{y}_{t})\\!=\\!\underset{{\mathbf{v}_{t},\boldsymbol{\gamma}^{2}_{t}}}{\arg\max}\,p\left({\mathbf{y}_{t}\left|\mathbf{v}_{t}\right.}\right)\cdot
p\big{(}\mathbf{v}_{t}\big{|}{\boldsymbol{\gamma}^{2}_{t}}\big{)}\cdot
p\left(\boldsymbol{\gamma}^{2}_{t}\right).$ (12)

am iterates between a maximization step over the error state
${\mathbf{v}}_{t}$ with fixed variance ${\boldsymbol{\gamma}^{2}_{t}}$:

$\hat{\mathbf{v}}_{t}\\!=\\!\underset{{\mathbf{v}_{t}}}{\arg\max}\,p\left({\mathbf{y}_{t}\left|\mathbf{v}_{t}\right.}\right)\cdot
p\left(\mathbf{v}_{t}\left|\boldsymbol{\gamma}\right.^{2}_{t}\right)\cdot
p\left(\boldsymbol{\gamma}^{2}_{t}\right),$ (13)

and a maximization step over the unknown variance
${\boldsymbol{\gamma}^{2}_{t}}$ based on $\mathbf{v}_{t}$, i.e., finding

$\hat{\boldsymbol{\gamma}}^{2}_{t}=\underset{{\boldsymbol{\gamma}^{2}_{t}}\geq
0}{\arg\max}\,p\left(\mathbf{v}_{t}\left|\boldsymbol{\gamma}\right.^{2}_{t}\right)\cdot
p\left(\boldsymbol{\gamma}^{2}_{t}\right).\vspace{-3mm}$ (14)

Similarly to our derivation of em in Subsection 3.3, we assume a uniform prior
on ${\boldsymbol{\gamma}}_{t}^{2}$. However, while em utilizes estimates of
both the first- and second-order moments of the states, namely,
$\hat{\mathbf{x}}_{t}^{i}$ and ${\hat{\mathbf{X}}_{t}^{i}}$ (8), am uses only
$\hat{\mathbf{x}}_{t}^{i}$.

For convenience, we formulate the second step in a scalar manner, which
extends to multivariate observations by assuming that the observation noise
$\mathbf{z}_{t}$ and the outlier $\mathbf{u}_{t}$ in each dimension are
independent. In particular, by replacing $\mathbf{v}_{t}$ in (14) with its
instantaneous estimate
$\mathbf{\hat{v}}_{t}^{i}=\mathbf{y}_{t}-\mathbf{H}\cdot\mathbf{\hat{x}}_{t}^{i}$
computed from the kf, we obtain the following update rule for the $k$th entry
of the unknown variance

$\left({\hat{\gamma}}^{i}_{t,k}\right)^{2}\\!=\\!\max\left\\{{({\mathrm{\hat{v}}^{i}_{t,k}})^{2}\\!-\\!{\mathrm{r}^{2}_{k}},0}\right\\}.$
(15)

We obtain an analytic expression of $\hat{\gamma}^{2}_{t,k}$ for the update
step, which is parameter-free. Furthermore, combining am with the nuv-prior
results in an equivalent cost function
$\mathcal{L}\left(\mathbf{v}_{t}\right)\triangleq-\log\,p\left(\mathbf{v}_{t}\right)$
that is non-convex [15]. This is equivalent loss function useful for sparse ls
models, e.g., kf with outliers [15], and $\hat{\boldsymbol{\gamma}}_{k}^{2}$
is likely to be sparse, following the same arguments as in Subsection 3.3.

(a) Noisy data clean of outliers

(b) Noisy data with outliers

(c) Vehicle estimated trajectory

Fig. 2: Sub-Fig. 2(a) and 2(b) present the obtained MSE for different
algorithms and Sub-Fig. 2(c) presents the measured vehicle position obtained
from the noisy GPS readings (orange line), and the trajectory estimated by our
oikf-am (green dashed line).

###  3.5 Discussion

The proposed oikf is designed to track in the presence of outliers by
iteratively refining its update step with map estimation of the outlier
variance. Both em and am algorithms can be used to tackle this map estimation,
which arises from the nuv modelling. The main difference between the
algorithms is that em uses the second-order moment of the state vector, while
am uses the first-order moment only, hence it does not require the posterior
covariance matrix $\bm{\Sigma}_{t}$ in each iteration. Consequently, am avoids
possibly unnecessary heuristics in the model noise, making it more robust
compared to em, as empirically demonstrated in Section 4. The fact that am
does not explicitly rely on the second-order moments is expected to facilitate
its augmentation with trainable data-driven variants of the Kalman filter,
e.g., [24, 25, 26]. Such a fusion of oikf with data-aided computations, which
we intend to explore in future work, bears the potential of facilitating
robust filtering in partially known ss models.

oikf provides a degree of freedom in choosing the prior of
$\boldsymbol{\gamma}_{k}^{2}$. We chose it to be uniform, which is parameter
free, and can be shown to effectively modify the overall loss function to
account for sparse outliers [15]. Alternative settings of this prior would
result in different effective loss functions such as the convex Huber cost
function [27, 15]. We leave the investigation of oikf with different priors to
future work.

##  4 Empirical Evaluation

To evaluate the proposed oikf-am and oikf-em, we tested these algorithms on a
standard localization task111The source code and additional information on the
empirical study can be found at https://github.com/KalmanNet/OIKF_ICASSP23..
We compare our proposed algorithms performance in terms of position error to
the following alternative algorithms: $1)$ the standard kf; $2)$ the well-
known $\chi^{2}\textrm{-test}$ [10] (with a $95\%$ confidence level); $3)$ the
weighted covariance methods, i.e., the WRKF from [5]; and $4)$ the ORKF from
[6]. We evaluate these algorithms in two settings: A synthetic dataset
generated from a known SS model, and real-world dynamics data based on the
Michigan NCLT dataset [28]. In both cases, the state vector is given by
$\mathbf{x}=\left(\mathrm{p,v}\right)^{\top}$, where $\mathrm{p}$ and
$\mathrm{v}$ are the position and velocity, respectively.

For the synthetic data we define the dynamic wna model [22], defined by a
linear ss model. The system is fully observable, thus the observation matrix
is the identity matrix $\mathbf{H}=\mathbf{I}_{n}$ and the observation noise
covariance matrix is defined as
$\mathbf{R}=\mathrm{r}^{2}\cdot\mathbf{I}_{n}$. The state-evolution noise
variance $\textnormal{q}^{2}$ is set to a constant value of
$-10\left[\textrm{dB}\right]$. Uncertainty in the dynamics and measurements is
accounted for by Gaussian i.i.d input, state, and measurements noise, whereas
the outliers are modeled with intensity sampled from a Rayleigh distribution
$\mathcal{RAY}\left(\beta\right)$ with scale parameter $\beta=30$. The
outliers’ time steps are drawn from a Bernoulli distribution, namely
$\mathcal{B}\left(p\right)$, where ${p}$ is set to 0.2. For the real-world
data we use the NCLT dataset from session with date 2013-04-05, which contains
noisy GPS (GPS) readings and the corresponding ground truth location of a
moving Segway robot. For the filter process, we define the dynamic wna
model[22]. The only observable output is the GPS position, thus the
observation matrix is $\mathbf{H}=\left(\begin{matrix}1&0\end{matrix}\right)$
and the observation noise covariance matrix is $\mathbf{R}={\mathrm{r}^{2}}$.

The results reported in Fig. 2(a) examine the performance for synthetic data
clean of outliers. In this case, it is shown that the oikf-am achieves the
optimal minimal mse bound, as most $\hat{\gamma}^{2}_{t}$ values are estimated
to be zero, which means the model turns back to be the kf. When the synthetic
data is populated with outliers, the oikf-am has the best performance in terms
of mse compared to other algorithms for different values of observation noise
variance $\mathrm{r}^{2}$, as presented in Fig 2(b). More specifically, it
coincides with the oikf-em, and even outperforms it for low observation noise,
without using a second-order moment as in em.

We proceed to the NCLT data set and demonstrate the tracking of a single
trajectory in Fig. 2(c). We observe in Fig. 2(c) the robustness of oikf-am in
estimating the position from real-world data while reliably smoothing the
outliers. Table 1 presents the obtained position rmse (rmse) and mse for each
of the algorithms when the observation noise variance is set to
$\mathrm{r}^{2}=3^{2}$, which is a GPS ”textbook” error, while
$\mathrm{r}^{2}$ is selected for each algorithm separately by grid search to
yield the lowest mse. For both scenarios $\mathrm{q}^{2}$ was optimized by
grid search. We can see in Table 1 that our proposed oikf-em and oikf-am
produce the lowest estimation errors for both settings, when the latter
coincides with the oikf-em, even without using the second-order moment.
Furthermore, the OIKF-AM exhibits the shortest runtime in the domain of
algorithms for outlier detection and weighting (except the
$\chi^{2}\textrm{-test}$ that only detects and then rejects the outliers). For
instance, in comparison to our other suggested method the oikf-em, the oikf-am
showcases an almost $40\%$ reduced runtime compared to it.

Table 1: Position error for different values of $\mathrm{r}^{2}$ | ${\mathrm{r}^{2}=3^{2}}$ | optimal ${\mathrm{r}^{2}}$ |  Runtime
---|---|---|---
|  rmse[m] |  mse$\left[\textrm{dB}\right]$ |  rmse[m] |  mse$\left[\textrm{dB}\right]$ |  [ms]
Noisy GPS |  349.31 |  50.86 |  349.31 |  50.86 |  -
kf |  94.35 |  39.49 |  92.28 |  39.3 |  0.05
ORKF |  75.79 |  37.59 |  27.74 |  28.86 |  2.83
$\chi^{2}\textrm{-test}$ |  45.3 |  33.12 |  14 |  22.92 |  0.06
oikf-am |  10.87 |  20.72 |  10.38 |  20.33 |  0.28
oikf-em |  10.73 |  20.61 |  10.35 |  20.29 |  0.44

##  5 Conclusions

In this work we derived a novel approach for outlier insensitive Kalman-
filtering oikf. Based on sparse Bayesian learning concepts, we modeled
outliers as NUV with am or em approaches, resulting in a sparse outlier
detection. Both algorithms are parameter-free and amount essentially to a
short iterative process during the _update step_ of the kf. The presented
empirical evaluations demonstrate that oikf-am and oikf-em present better
performance compared to the other algorithms in terms of mse and rmse,
highlighting the robustness and accuracy of oikf for systems that rely on
high-quality sensory data.

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  *[kf]:
  *[mse]:
  *[nuv]:
  *[em]:
  *[am]:
  *[ss]:
  *[oikf]:
  *[wna]:
  *[awgn]:
  *[kg]:
  *[map]:
  *[GPS]:
  *[rmse]: