# Fundamental constraints on the observability
of non-Hermitian effects in passive systems

Henning Schomerus  Department of Physics, Lancaster University, Lancaster, LA1
4YB, United Kingdom

###### Abstract

Utilizing scattering theory, we quantify the consequences of physical
constraints that limit the visibility of non-Hermitian effects in passive
devices. The constraints arise from the fundamental requirement that the
system obeys causality, and can be captured concisely in terms of an internal
time-delay operator, which furthermore provides a direct quantitative measure
of the visibility of specific non-Hermitian phenomena in the density of
states. We illustrate the implications by contrasting different symmetry
classes and non-Hermitian effects, including exceptional points and the non-
Hermitian skin effect, whose underlying extreme mode nonorthogonality turns
out to be effectively disguised.

Hermitian systems support orthogonal stationary modes with identical,
infinite, life times. If the Hermiticity is of a fundamental nature, as in
quantum mechanics, this plays an important role in guaranteeing the dynamical
stability of the system. The situation is more complex in effectively non-
Hermitian systems [1, 2], for instance photonic systems with gain or loss [3,
4, 5, 6, 7], where nonorthogonal modes of different life times appear. These
systems attract considerable attention because they can display a wide range
of phenomena that can be exploited for unique applications, such as power
oscillations [8, 9, 10], unidirectional transport [11] and invisibility [12],
coherent absorption [13, 14], mode selection [15] and lasing [16, 17, 18, 19,
20, 21]. Many of these applications purposefully utilize the life-time
differences to enhance desired modes relative to undesired modes—hence,
achieve a clear visibility of specific modes incorporating a particular
functionality by their long life time. Furthermore, many of these applications
make explicit use of the mode nonorthogonality. Prominent examples are
enhanced sensors [22, 23, 24] operating near exceptional points [25, 26, 27],
which are non-Hermitian degeneracies where eigenmodes align, and directed
amplifiers and sensors [28, 29, 30, 31, 32] facilitated by the non-Hermitian
skin effect [33, 34, 35, 36, 37], a prominent feature of non-reciprocal
systems where all bulk modes become systematically distorted towards one side
of the system.

The design of such systems has received further impetus by the realization
that in non-Hermitian systems, modes with predetermined frequencies, life
times, and mode profiles can be protected by generalised symmetries [38, 39,
40, 41]. This leads to rich scenarios that transcend Hermitian topological
physics [42, 43, 44], both practically [45] as well as in their mathematical
complexity [46, 47, 48]. However, many of these symmetries can only be
realized in active systems, which require a sustained supply of energy and
introduce noise. For instance, parity-time (PT) symmetric systems with
balanced gain and loss [4, 5] can provide a spectrum of infinite life-time
modes, but these are intrinsically destabilised by the quantum noise in the
active regions [49, 50, 51]. Furthermore, present experimental realizations of
the non-Hermitian skin effect [52, 53, 54, 55, 56] all invoke active elements.
Analogously, noise limits the precision of exceptional-point sensors [57], and
non-adiabatic transients from life-time differences limit the observability of
the half-integer Berry phase of these points [58, 59]. Even in the passive
setting, many of the most coveted non-Hermitian effects can be made visible
only with specially tailored excitations. In view of these challenges, it is
highly desirable to base non-Hermitian functionality on the generic scattering
response of passive stationary devices in which specific modes display long
life times, both absolutely as well as relatively when compared to other
modes.

In this work, we establish and evaluate a stringent fundamental constraint on
this objective, which arises from the requirement that the underlying
microscopic physics obeys causality [60]. This requirement is stronger than
just insisting on the dynamical stability of the modes in the system (hence,
on non-negative lifetimes), and takes care of the fact that the system is not
isolated, but couples to the components supplying the loss. However, the
constraints can be readily formulated in general terms, and evaluated
quantitatively in given systems. From this we can determine general limits of
the visibility of specific non-Hermitian effects.

To establish these links systematically, we first formulate the constraints
compactly by phrasing them in terms of the language of scattering theory.
Causality is then encoded into the internal time-delay operator, a central
object from scattering theory that can be obtained from the microscopic model.
From this, one can obtain the critical threshold value of overall losses that
a model necessarily needs to include to be realizable in a passive device.
These conditions can be classified by symmetries inherited from the effective
Hamiltonian, which establishes a systematic link between non-Hermitian and
Hermitian symmetry classes. Furthermore, the time-delay operator determines a
spectral function (the scattering density of states) that serves as a
quantitative measure of the visibility of specific non-Hermitian effects. We
apply this to scenarios of particular theoretical and experimental interest,
comprising systems displaying exceptional points, the non-Hermitian skin
effect, and edge states. Remarkably, in the passive setting, the extreme mode
nonorthogonality underlying exceptional points and the skin effect turns out
to be effectively disguised.

_General formulation of causality constraints._ For concreteness, we base the
considerations on coupled-mode theory, which enjoys a wide range of
applications from photonic to mechanical systems and mimics the language of
quantum mechanics. In this theory, one employs a basis corresponding to a
suitable set of bare modes, often identified with individual components such
as resonators or waveguides, and collects their intrinsic properties and
couplings in a matrix $H$ that serves as an effective Hamiltonian. We assume
that the elements of $H$ are frequency-independent, but note that effects of
additional frequency dispersion can be accounted for by including auxiliary
components [61]. This effective Hamiltonian serves as the input to model and
design experiments, and also is the ubiquitous starting points for theoretical
considerations, such as about the role of symmetries in these systems, as
specified further below. For the moment, the key feature of the effective
Hamiltonian is that it can be non-Hermitian, $H\neq H^{\dagger}$. On the basis
of the microscopic model, we capture this quantitatively by separating out the
Hermitian and nontrivial anti-Hermitian parts as well as an overall level of
uniform scalar background losses, $H=H_{0}+iF-i\gamma$, where
$H_{0}=H_{0}^{\dagger}$, $F=F^{\dagger}$, and $\gamma$ set so that
$\mathrm{tr}\,F=0$. As we will see, the model becomes realizable in a passive
device when $\gamma$ exceeds a certain threshold value $\gamma_{c}$, which we
will call the _causality threshold_ , and constitutes a central quantity in
our considerations.

To identify this threshold and establish the ensuing limits on the visibility
of non-Hermitian effects, we probe the system from the outside and adopt a
scattering perspective. Let us assume that the system is uniformly coupled to
the outside at a coupling strength $\Gamma$, and state as our aim to resolve
the desired modes in the scattered signal. In the wide-band limit, where all
spectral features in the scattered signal are due to the system, the standard
scattering matrix is given by

$\displaystyle S(\omega)$
$\displaystyle=(1-i\Gamma\mathcal{G}(\omega))(1+i\Gamma\mathcal{G}(\omega))^{-1},$
(1)

where

$\displaystyle\mathcal{G}(\omega)$ $\displaystyle=(\omega-
H_{0}-iF+i\gamma)^{-1}$ (2)

is the Greens function of the system. From the scattering perspective, the
system is passively realizable if it does not amplify any incoming signal
$\mathbf{a}$, i.e., if $||S\mathbf{a}||\leq||\mathbf{a}||$, irrespective of
whether the incoming signal is designed to couple into a specific mode or not.
This means that the operator combination $S^{\dagger}S$ has no eigenvalues
exceeding 1, or equivalently, that the expression $1-S^{\dagger}S$ is positive
semidefinite. With Eq. (1), this condition can be reformulated instructively
by writing

$1-S^{\dagger}S=2\Gamma Q_{\Gamma},$ (3)

where

$Q_{\Gamma}=2\mathcal{G}(\omega+i\Gamma)^{\dagger}(\gamma-F)\mathcal{G}(\omega+i\Gamma)$
(4)

is the time-delay operator in an open system [62]. Via Eq. (3), its elements
can be directly determined in experiments from the scattering strength. For a
passively realizable system we then have to demand that $Q_{\Gamma}$ is
positive semidefinite.

This formulation is useful because it provides a unifying perspective on
different aspects of the system.

Firstly, it confirms that on the simplest level, a microscopic model can be
physically realized in a passive system when the combination $\gamma-F$ itself
is positive semidefinite [63], hence, if $\gamma$ is larger than the largest
eigenvalue of $F$. We denote the eigenvalues of $F$ as $f_{k}$, so that
$\gamma_{c}=\mathrm{max}\,f_{k}$. This constraint is both _simpler_ and
_stronger_ than requiring positive life times $\tau_{k}$ of all modes, which
are encoded in the eigenvalues $\Omega_{k}=\omega_{k}-i/2\tau_{k}$ of the
effective Hamiltonian $H$ itself, and related mathematical constraints such as
the Lee-Wolfenstein and Bell-Steinberger relations [63, 64, 65].

Secondly, the positive semidefiniteness of the delay times guarantees
causality of the system, which clarifies the fundamental physical nature of
the constraint, and justifies to call $\gamma_{c}$ the causality threshold.

Thirdly, the time-delay operator delivers a direct measure of the generic
visibility of physical effects, the scattering density of states
$\rho(\omega)=(2\pi)^{-1}\,\mathrm{tr}\,Q_{\Gamma}(\omega)$, which accounts
for the mode broadening by the intrinsic and radiative losses. We read off
that for fixed $H_{0}$ and $F$, the passive visibility of individual modes in
this measure is maximized for $\Gamma=0$ and $\gamma=\gamma_{c}$, hence at the
causality threshold of a weakly probed system. Below, we will use this measure
to quantify the properties of specific systems from different symmetry
classes, whose general features we describe next.

_Evaluation in specific symmetry classes._ Effectively non-Hermitian
Hamiltonians appear in many guises, and the physical symmetries of a system
translate into different mathematical forms that reflect the context. E.g.,
the effective Hamiltonian may generate the time evolution of a wavefunction in
a slowly-varying envelope coupled-mode description, which is analogous to the
Schrödinger equation, it may feature in the propagation of a density matrix,
or it may appear in a stability analysis or Bogoliubov theory that includes
complex-conjugated fields [39, 40]. In the identification of the causality
constraints above, we adopted the language of scattering theory, and we
therefore apply the corresponding notions when identifying the symmetries of
the system [66, 67, 44]. Specifically, conventional time-reversal (T) symmetry
then entails $H=H^{*}$, and in a reciprocal system we have $H=H^{T}$ (we will
refer to this as the T′ symmetry). Furthermore, PT symmetry dictates
$\mathcal{X}H\mathcal{X}=H^{*}$, and a non-Hermitian charge-conjugation (C)
symmetry entails $\mathcal{X}H\mathcal{X}=-H^{*}$, involving in each case a
suitable unitary operator fulfilling $\mathcal{X}^{2}=1$. We will also
consider variants of these systems with $\mathcal{X}H\mathcal{X}=H^{\dagger}$
(PTT′) and $\mathcal{X}H\mathcal{X}=-H^{\dagger}$ (CT′), which constitute
separate symmetry classes when the system is non-reciprocal.

It should be noted that several of these symmetries can only be realized for
$\gamma=0$. To extend them to the passive setting, we hence assume that they
hold for $H=H_{0}+iF$, and then consider the same model at finite $\gamma$,
following the established example of passive PT symmetry [9]. The symmetry can
then be realized passively for $\gamma\geq\gamma_{c}$. These symmetries can be
combined in different ways, leading to an extensive classification [39, 40],
where the cases above are of particular theoretical and experimental interest.
For instance, non-reciprocal systems with $H=H^{*}\neq H^{T}$ are the simplest
setting in which the non-Hermitian skin effect appears.

Figure 1: Disguised exceptional points in passive non-Hermitian systems.
Complex eigenvalue spectrum $\Omega_{k}$ (top) and density of states
$\rho(\omega)$ (bottom) for the effective Hamiltonian (5). The couplings are
fixed to $b=c^{*}=0.4-0.3i$, while (a) $a=\sqrt{-bc}-0.2i$ (PT symmetric
case), (b) $a=\sqrt{-bc}$ (EP), (c) $a=\sqrt{-bc}+0.1i$ (dynamically broken
phase), and (d) $a=\sqrt{-bc}+0.2$ (symmetry explicitly broken). The thick
lines are for the limit of weak couplings $\Gamma=0$, where the life times are
maximized. At the exceptional point, $\rho(\omega)$ is then a simple
Lorentzian, which disguises the extreme mode nonorthogonality of the system.
The thin lines are for finite $\Gamma=0.1$, where a squared-Lorentzian
background of limited contrast appears at the EP.

Before we address the visibility of such effects, let us examine the general
structure of the causality constraints in these symmetry classes. We express
this compactly in terms of the symmetries inherited by the operator $F$
capturing the nontrivial non-Hermitian content of the model, and refer to 111
See the Appendix for concrete constructions of the operator $F$ in different
symmetry classes as well as additional analytical and numerical results. for a
detailed description in terms of the block structure of this operator.

We start with the case of a PT-symmetric system. From the definition of the
symmetry in terms of the effective Hamiltonian, we see that the operator $F$
then obeys a Hermitian charge-conjugation symmetry,
$\mathcal{X}F\mathcal{X}=-F^{*}$, as encountered in a superconductor [44].
This enforces a symmetry of its spectrum, with eigenvalues appearing in pairs
$\pm|f_{k}|$. For a PTT′ symmetry, the operator $F$ displays a chiral
symmetry, $\mathcal{X}F\mathcal{X}=-F$, again leading to a spectral symmetry
of its eigenvalues. Both variants coincide if the system is reciprocal,
$H=H^{T}$, where $F$ is real and obeys time-reversal symmetry.

For systems with a C symmetry, $F=\mathcal{X}F^{*}\mathcal{X}$ displays a
generalized time-reversal symmetry, while CT′ entails that
$F=\mathcal{X}F\mathcal{X}$ obeys a unitary symmetry, and hence can be block-
diagonalized into the symmetry sectors of $\mathcal{X}$. Finally, in a non-
reciprocal system with passive $T$ symmetry, $F=-F^{T}$, as is typical for a
topologically nontrivial superconductor in the so-called Majorana basis [44].
In all cases, we see that there is a systematic link from a non-Hermitian
symmetry class to a Hermitian symmetry class.

This concludes or first main objective of formulating a convenient general
framework. We now apply it to quantify the visibility of non-Hermitian effects
in concrete settings.

_Observability of specific non-Hermitian effects._ As mentioned in the
introduction, exceptional points (from here on EP’s), where eigenvalues
collide in the complex plane and eigenmodes align, are one of the most
prominent features of non-Hermitian systems. Their effects are most easily
seen for modes of long life times, such as in active photonic systems near the
laser threshold [16, 17, 18, 19], which then are highly sensitive to
perturbations [22, 23, 24]. The enhanced sensitivity also applies to the noise
from spontaneous emission, which at the EP results in an unconventional
_squared-Lorentzian_ lineshape [50, 69]. The squared Lorentzian also shows up
as a background when a passive system is forced into a stationary state by
external driving [70, 71]. This behaviour is linked to the drastic violation
of mode orthogonality at the EP.

Figure 2: Disguised non-Hermitian skin effect, illustrated by model (8) in
finite systems of $20$ sites, $v_{1}=-0.8i$, $v_{2}=-0.2i$,
$u_{1}^{\pm}=\bar{u}\mp 0.4$, $u_{2}^{\pm}=1$, $w=0$, and (a) $\bar{u}=0.8$,
(b) $\bar{u}=2$, with open (top) or periodic (bottom) boundary conditions.
Here $\rho_{n}(\omega)$ and $\rho(\omega)$ are the spatially resolved and
overall density of states, for which we contrast the case $\Gamma=0$ (limit of
vanishing coupling, left supanels/solid curves) and $\Gamma=0.1$ (strong
coupling, right supanels/dashed curves). This physical data is compared to the
summed eigenvector profile $I_{n}$ displaying the skin effect, and the complex
eigenvalues $\Omega$ of $H$.

To assess the observability of these features in the scattering density of
states, we adopt the standard reduced $2\times 2$ Hamiltonian

$H=\begin{pmatrix}a-i\gamma&b\\\ c&-a-i\gamma\end{pmatrix},$ (5)

where we initially allow for the most general case with complex parameters
$a$, $b$, and $c$. The causality constraint then follows from the matrix

$F=(-i/2)\begin{pmatrix}a-a^{*}&b-c^{*}\\\ c-b^{*}&-a+a^{*}\end{pmatrix},$ (6)

giving the threshold value
$\gamma_{c}=\sqrt{(\mathrm{Im}\,a)^{2}+|b-c^{*}|^{2}/4}$.

In Fig. 1 we show the threshold scattering density of states $\rho(\omega)$
for different scenarios at and away from the EP, which occurs for
$a^{2}+bc=0$. In the figure, the EP signals a spontaneous symmetry-breaking
transition, where the eigenvalue pair moves away from the spectral symmetry
line $\mathrm{Im}\,\omega=-\gamma$. At the EP, this density of states is given
by

$\rho^{(\mathrm{EP})}(\omega)=\frac{1}{\pi}\frac{|b|+|c|}{\omega^{2}+(|b|+|c|)^{2}/4}.$
(7)

This is a _simple_ Lorentzian normalized to 2, hence accounts for both states
but fully disguises the extreme mode nonorthogonality at the degeneracy. A
squared-Lorentzian background of limited contrast only appears at finite
coupling $\Gamma$ (dashed lines). However, this background comes with a
negative spectral weight that never exceeds 1/4 of the Lorentz contribution
[68], with the optimal contrast attained at $\Gamma\sim\gamma_{c}$ where the
density of states is already strongly suppressed due to the much reduced
lifetime to the states.

In other symmetry classes, functionality can arise collectively from the bulk,
or individually from particular states. To determine their visibility we
consider a flexible one-dimensional model encompassing a wide range of
paradigms [72, 73, 38, 74, 75, 76, 34, 77, 78, 79], based on an effective
Hamiltonian

$\displaystyle H_{nm}$
$\displaystyle=\delta_{n,m}v_{n}+\delta_{n,m-1}u_{n}^{-}+\delta_{n,m+1}u_{m}^{+}$
$\displaystyle{}+\delta_{n,m-2}w_{n}+\delta_{n-2,m}w_{m}.$ (8)

The imaginary parts of the complex onsite-potentials $v_{n}$ induce
distributed losses, while the real parts allow to define a corrugated
potential. The couplings $u_{n}^{\pm}$ can be arranged to induce zero modes,
and when they are non-reciprocal they induce the non-Hermitian skin effect.
For the moment, we also include reciprocal real next-nearest neighbour
couplings $w_{n}$ as they allow to obtain the skin effect from scalar losses
and real magnetic fields [74], and feature in some experiments [52].

All these effects appear in a periodic dimer arrangement with a two-site unit
cell, which induces a non-trivial band structure governed by the Bloch
Hamiltonian

$H(k)=\begin{pmatrix}v_{1}+2w_{1}\cos k&u_{1}^{-}+u_{2}^{+}e^{-ik}\\\
u_{1}^{+}+u_{2}^{-}e^{ik}&v_{2}+2w_{2}\cos k\end{pmatrix}.$ (9)

With conventional periodic boundary conditions, $k$ is real, but to fulfill
the open boundary conditions of a finite system one generically needs to
combine spectrally degenerate non-reciprocal modes with complex $k$ [80, 81,
82]. Hence, the modes display an exponential spatial profile distorted toward
the edge, which is the essence of the non-Hermitian skin effect. In the
Hermitian limit, these modes become conventional extended Bloch states, with
the possible exception of a finite number of edge states that maintain an
exponentially decaying profile. This complex interplay of effects explains why
the non-Hermitian skin effect has received considerable attention.

In contrast, the causality constraints on the bulk and edge modes are governed
much more simply by the Bloch version of the operator $F$,

$F(k)=\begin{pmatrix}\Delta v&\Delta u_{1}^{*}+\Delta u_{2}e^{-ik}\\\ \Delta
u_{1}+\Delta u_{2}^{*}e^{ik}&-\Delta v\end{pmatrix},$ (10)

where $\Delta v=\mathrm{Im}\,(v_{1}-v_{2})/2$ and $\Delta
u_{n}=-i(u_{n}^{+}-{u_{n}^{-}}^{*})/2$. This is a standard Hermitian dimer
chain with two symmetric bands, encompassing the Su-Schrieffer Heeger [83] and
Rice-Mele [84] models as special cases. These models can support edge states,
but they occur in the middle of the spectrum, while we are interested in the
upper edge of the spectrum. Furthermore, being Hermitian, $F$ does not suffer
from any complications of the skin effect, so that this upper edge can be
found from the Bloch version. Therefore, in large systems the causality
threshold

$\gamma_{c}=\sqrt{(\Delta v)^{2}+(|\Delta u_{1}|+|\Delta u_{2}|)^{2}}$ (11)

coincides for both types of boundary conditions, and hence is also insensitive
to the skin effect.

To assess how visible these effects are in the passive setting, we use the
spatially resolved density of states $\rho_{n}=Q_{nn}/2\pi$, again evaluated
to the causality threshold. Figure 2 illustrates this in different scenarios
[68]. The top row shows results for a finite system with open boundary
conditions, where each panel further contrasts weak and strong coupling
$\Gamma$. In panel (a), the system supports an edge state, which for
$\Gamma=0$ is clearly visible in the local and total density of states, while
no such edge state exists in panel (b). In both panels, the eigenstates
display the non-Hermitian skin effect, as quantified by the summed profile
$I_{n}\equiv(UU^{\dagger})_{nn}$ of normalized eigenvectors, collected in the
diagonalizing matrix $U$. However, this effect is completely disguised in the
density of states, to the extent that it does not even appear as a background
effect at finite $\Gamma$. Indeed, the spatial profile of the bulk density of
states is similar to that in a system with periodic boundary conditions
(bottom row), even though the eigenfunctions and eigenvalue spectra of both
cases dramatically differ.

_Conclusions._ In summary, we have established general constraints on the
observability of non-Hermitian effects in passive devices, and evaluated the
implications for prominent paradigms, in particular exceptional points, the
non-Hermitian skin effect, and symmetry-protected edge states. Some of the
most widely studied features are effectively disguised in the density of
states, in particular to the signatures of drastic mode nonorthogonality.
These findings highlight the essential role of active elements in devices that
aim to exploit these signatures.

Being formulated in a unifying scattering approach, the general results apply
to various platforms, such as photonic, mechanical and acoustic systems,
electronic circuits, or microwave networks, and can be readily used to
quantify the visibility of effects in a wide range of models. This also
provides guidance for the design of passive devices in which the visibility is
maximized, and allows to discard designs relying on effects whose
observability is severely limited.

###### Acknowledgements.

I gratefully thank Jan Wiersig for very useful discussions. All data produced
in this work has been directly processed into the figures.

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##  Appendix A Concrete forms of the symmetries of the nontrivial non-
Hermitian content $F$

In the main text, we identified the symmetries inherited by the operator $F$
in general terms. Here, we express this symmetries concretely in terms of the
block structure of this operator. We again start with the case of a PT-
symmetric system. Taking $\mathcal{X}$ of the form of a Pauli-$x$ block
matrix, as commonly done to reflect the physical design with symmetrically
placed balanced gain and loss components, the effective Hamiltonian is of the
form

$H=\begin{pmatrix}A-i\gamma&B\\\ B^{*}&A^{*}-i\gamma\end{pmatrix}$ (S.1)

with general subblocks $A$ and $B$. The nontrivial non-Hermitian content is
captured by the operator

$F=\frac{-i}{2}\begin{pmatrix}A-A^{\dagger}&B-B^{\dagger}\\\
B^{*}-B^{T}&A^{*}-A^{T}\end{pmatrix}.$ (S.2)

This displays the symmetries of a superconductor with a Hermitian charge-
conjugation symmetry, $\mathcal{X}F\mathcal{X}=-F^{*}$, and enforces the
symmetry of its spectrum, with eigenvalues paired as $\pm|f_{k}|$.

For a PTT′ symmetry,

$H=\begin{pmatrix}A-i\gamma&B\\\ C&A^{\dagger}-i\gamma\end{pmatrix}$ (S.3)

where $B=B^{\dagger}$ and $C=C^{\dagger}$. Therefore,

$F=\frac{-i}{2}\begin{pmatrix}A-A^{\dagger}&B-C\\\
C-B&A^{\dagger}-A\end{pmatrix}$ (S.4)

now displays a chiral symmetry, $\mathcal{X}F\mathcal{X}=-F$, again leading to
a spectral symmetry of its eigenvalues. The block structures of the PT and PT′
variants coincide if the system is reciprocal, $H=H^{T}$, where $F$ is real
and obeys time-reversal symmetry.

For systems with a C symmetry, one conventionally chooses $\mathcal{X}$ of the
form of a Pauli-$z$ block matrix, as this allows for cases with a nontrivial
topological index $\nu=\mathrm{tr}\,\mathcal{X}$. The effective Hamiltonian
then has the structure

$H=\begin{pmatrix}iA-i\gamma&B\\\ C&iD-i\gamma\end{pmatrix}$ (S.5)

with real matrices $A$, $B$, $C$, $D$, and $\gamma$ chosen such that
$\mathrm{tr}(A+D)=0$. This entails that

$F=\frac{-i}{2}\begin{pmatrix}i(A+A^{T})&B-C^{T}\\\
C-B^{T}&i(D+D^{T})\end{pmatrix}=\mathcal{X}F^{*}\mathcal{X}$ (S.6)

displays a generalized time-reversal symmetry.

Analogously, for a system with CT′ symmetry,

$H=\begin{pmatrix}iA-i\gamma&B\\\ B^{\dagger}&iD-i\gamma\end{pmatrix}$ (S.7)

with general $B$ and Hermitian $A$ and $D$, once more obeying
$\mathrm{tr}(A+D)=0$. This entails that

$F=\begin{pmatrix}A&0\\\ 0&D\end{pmatrix}=\mathcal{X}F\mathcal{X}$ (S.8)

is indeed block-diagonalized into the symmetry sectors of $\mathcal{X}$.

Finally, in a non-reciprocal system with passive $T$ symmetry,

$H=\begin{pmatrix}A-i\gamma&B\\\ C&D-i\gamma\end{pmatrix}$ (S.9)

with real blocks $A$, $B$, $C$, $D$, and

$F=(-i/2)\begin{pmatrix}A-A^{T}&B-C^{T}\\\
C-B^{T}&D-D^{T}\end{pmatrix}=-F^{T},$ (S.10)

which indeed coincides with the block structure of the Hamiltonian for a
topologically nontrivial superconductor in the Majorana basis [44].

##  Appendix B Analytical discussion of the visibility of exceptional points

Here, we provide further analytical details for the signatures of mode
nonorthogonality in the density of states near an exceptional point, as
obtained from the model Hamiltonian Eq. (5). From the definitions, this
density of states can be written analytically as

$\displaystyle\rho(\omega)$
$\displaystyle=\frac{2\mathrm{Re}\,[(a^{2}+bc)(\gamma+\Gamma+i\omega)]}{\pi|a^{2}+bc+(\gamma+\Gamma+i\omega)^{2}|^{2}}$
$\displaystyle+\frac{2\gamma|\gamma+\Gamma+i\omega|^{2}}{\pi|a^{2}+bc+(\gamma+\Gamma+i\omega)^{2}|^{2}}$
$\displaystyle-\frac{\Gamma(2|a|^{2}+|b|^{2}+|c|^{2})}{\pi|a^{2}+bc+(\gamma+\Gamma+i\omega)^{2}|^{2}}.$
(S.11)

At the exceptional point $a^{2}+bc=0$, this reduces to

$\displaystyle\rho^{(\mathrm{EP})}(\omega)$
$\displaystyle=\rho^{(1)}(\omega)+\rho^{(2)}(\omega),$ (S.12)
$\displaystyle\rho^{(1)}(\omega)$
$\displaystyle=\frac{2\gamma}{\pi|\gamma+\Gamma+i\omega|^{2}},$ (S.13)
$\displaystyle\rho^{(2)}(\omega)$
$\displaystyle=-\frac{\Gamma(|b|+|c|)^{2}}{\pi|\gamma+\Gamma+i\omega|^{4}},$
(S.14)

hence, the sum of a simple Lorentzian and a squared Lorentzian, where the
latter one only appears for finite $\Gamma$. Equation (7) is obtained for
$\Gamma=0$, $\gamma=\gamma_{c}=(|b|+|c|)/2$, where the width $\gamma+\Gamma$
of the simple Lorentzian is minimized in a passive system, and its weight

$S_{1}(\Gamma)=\int_{-\infty}^{\infty}\rho^{(1)}(\omega)d\omega=\frac{2\gamma}{\gamma+\Gamma}$
(S.15)

is maximized.

The squared Lorentzian carries a negative weight

$S_{2}(\Gamma)=\int_{-\infty}^{\infty}\rho^{(2)}(\omega)d\omega=-\frac{\Gamma(|b|+|c|)^{2}}{2(\gamma+\Gamma)^{3}}.$
(S.16)

Therefore, using again $\gamma\geq\gamma_{c}=(|b|+|c|)/2$, in a passive system
the relative weight

$\frac{|S_{2}(\Gamma)|}{S_{1}(\Gamma)}=\frac{\Gamma(|b|+|c|)^{2}}{4\gamma(\gamma+\Gamma)^{2}}\leq\frac{\Gamma\gamma}{(\gamma+\Gamma)^{2}}\leq\frac{1}{4},$
(S.17)

where the maximum is attained at $\gamma=\Gamma=\gamma_{c}$.

Following the same steps, we can also compare the relative peak heights of
these two contributions,

$\frac{|\rho^{(2)}(0)|}{\rho^{(1)}(0)}=\frac{\Gamma(|b|+|c|)^{2}}{2\gamma(\gamma+\Gamma)^{2}}\leq\frac{1}{2},$
(S.18)

which again is maximized at $\gamma=\Gamma=\gamma_{c}$. We note that the
density of states formally turns negative for $\gamma<\gamma_{c}/\sqrt{2}$,
which is less stringent than the causality constraint.

##  Appendix C Additional numerical results

Figure S1: Analogous to Fig. 1 in the main text, but with complex non-
reciprocal coupling parameters $b=0.3-0.4i$, $c=0.4+0.3i$ (top) $b=0.6$,
$c=0.4-0.2i$, while again (a) $a=\sqrt{-bc}-0.2i$, (b) $a=\sqrt{-bc}$ (EP),
(c) $a=\sqrt{-bc}+0.1i$, and (d) $a=\sqrt{-bc}+0.2$. At the exceptional point,
$\rho(\omega)$ is again a simple Lorentzian.  Figure S2: (a) Analogous to Fig.
2 in the main text, but for $\bar{u}=0.25$, where the bulk modes in the system
with open boundary conditions have moved away from the symmetry line in the
complex frequency plane. $(b)$ Evolution of the eigenvalues with the parameter
$\bar{u}$.  Figure S3: Analogous to Figs. 2 and S2, but for $\Gamma=0$ only,
and the other parameters set to $v_{1}=-0.3i-i\gamma_{c}$,
$v_{2}=-0.2+0.3i-i\gamma_{c}$, $u_{1}^{+}=\bar{u}-0.4+0.4i$,
$u_{1}^{-}=\bar{u}+0.4-0.1i$, $u_{2}^{+}=1-0.1i$, $u_{2}^{-}=0.9$,
$w_{1}=w_{2}=0$, and (a) $\bar{u}=0.25$, (b) $\bar{u}=0.8$, (c) $\bar{u}=2$.
Panel (d) shows the evolution of the eigenvalues with the parameter $\bar{u}$.
Figure S4: Analogous to Fig. S3, but including next-nearest neighbour
couplings $w_{1}=0.2$, $w_{2}=0.4$, while here (a) $\bar{u}=0$, (b)
$\bar{u}=1$, (c) $\bar{u}=2$.

In the main text we showed numerical results for particularly interesting
scenarios in which the underlying non-Hermitian effects are realized cleanly.
To illustrate the general nature of our findings, we show in Fig. S1
additional results for the EP model (5), evaluated at parameters where PT and
PTT′ symmetries are manifestly broken also by the couplings. Analogously, we
show additional results for the non-Hermitian skin effect, covering the case
where some bulk states in the open system have moved away from the symmetry
line in the complex plane (Fig. S2), as well as parameter configurations in
which the all spectral symmetries are explicitly broken (Fig. S3), including
by next-nearest-neighbour couplings (Fig. S4).


# Anomaly detection for the identification of volcanic unrest in satellite
imagery

###### Abstract

Satellite images have the potential to detect volcanic deformation prior to
eruptions, but while a vast number of images are routinely acquired, only a
small percentage contain volcanic deformation events. Manual inspection could
miss these anomalies, and an automatic system modelled with supervised
learning requires suitably labelled datasets. To tackle these issues, this
paper explores the use of unsupervised deep learning on satellite data for the
purpose of identifying volcanic deformation as anomalies. Our detector is
based on Patch Distribution Modeling (PaDiM), and the detection performance is
enhanced with a weighted distance, assigning greater importance to features
from deeper layers. Additionally, we propose a preprocessing approach to
handle noisy and incomplete data points. The final framework was tested with
five volcanoes, which have different deformation characteristics and its
performance was compared against the supervised learning method for volcanic
deformation detection.

Index Terms—  Unsupervised learning, Anomaly detection, Deep learning, InSAR.

##  1 Introduction

Volcano monitoring is an important process as more than 500 million people
live within 100 km of an active volcano worldwide [1]. Satellite imagery,
acquired periodically, offers insights into volcanic behaviour through
Interferometric Synthetic Aperture Radar (InSAR) techniques, where deformation
signals exhibit a substantial statistical correlation with eruption [2].
However, the rapid growth of satellite technology results in a data volume
beyond the capacity for manual inspection, necessitating an automated system
to flag interferograms indicating potential ground deformation and combine
them with other monitoring data to forecast hazards.

InSAR methods leverage the phase discrepancy between two radar images to
deduce changes in the radar signal’s path length between the satellite and the
Earth’s surface. These radar image pairs are commonly referred to as
interferograms. The wrapped interferograms, having values ranging from -$\pi$
to $\pi$, show fringes where ground deformation occurs, as seen in Figure 1c.
These fringes are unwrapped to obtain absolute magnitudes of deformation.
However, a challenge arises as volcanoes are often surrounded by vegetation or
water bodies, which cause loss of signal coherence, and water vapour in the
atmosphere causes phase delays that produce artefacts as seen in Figure 1b.

Fig. 1: Interferograms at Nevados Casiri, Peru ($17.47^{\circ}S$,
$69.813^{\circ}W$). (a) Normal sample (no deformation) with minor atmospheric
effects. (b) Normal sample with significant atmospheric effects. (c) Abnormal
sample (deformation). Top row is wrapped interferograms. Bottom row is
unwrapped interferograms. Each interferogram is 50 km across.

The existing automated deformation detection frameworks using interferograms
are based on supervised learning [3], relying heavily on labelled datasets.
This presents a challenge in detecting deformations with unknown
characteristics. Additionally, supervised learning requires a balanced
training dataset, creating issues for volcanic deformation detection, where
positive signals are scarce in InSAR images. Attempts to address this through
synthetic interferograms have resulted in poor generalization to real InSAR
data, as only simple geometric sources could be formulated[4]. A computational
approach like independent component analysis (ICA) has been utilised to
distinguish deformation signals from atmospheric interference [5]. ICA
operates under the assumption that source signals are mutually independent.
However, the deformation signal and atmosphere exhibit a significant
correlation, especially concerning the volcano’s topography.

To address those problems, here we apply unsupervised machine learning
techniques to identify anomalous behaviour in the deformation patterns of
volcanoes in interferograms. These methods rely on training the model only on
the normal data. This strategy is well-suited for volcanic deformation
detection due to the extensive satellite data available for stable volcanic
ground, providing abundant spatial and temporal information. This unsupervised
learning approach enables precise estimation of the characteristics and
statistical attributes of normal images.

Our proposed unsupervised learning framework identifies volcanic deformation
using Patch Distribution Modeling (PaDiM) [6], originally designed for
detecting anomalies in natural images. We enhance PaDiM’s detection
performance by introducing a weighted Mahalanobis distance [7], where deeper
layers are assigned greater importance. This approach is more suitable for
distinguishing characteristics of volcanic deformation and atmospheric effects
observed in InSAR data. This is because atmospheric noise is likely to be
captured in the low-level layers, while the deformation carries semantic
meaning captured in the higher-level layers. Detecting anomalies in volcanic
deformation is challenging due to data uncertainties, noise, and the dynamic
nature of volcanic systems. Therefore, the proposed framework integrates a
preprocessing module enabling the utilisation of noisy and sparse InSAR data
and enhancing the performances of deep neural networks. We evaluate the
performance of our proposed framework through various case studies
representing diverse characteristics of volcanic events. Additionally, we
evaluate its effectiveness against the supervised-learning method [8] for
volcanic deformation detection.

##  2 Related work

Anomaly detection: Various deep learning techniques, including CNNs, GANs, and
Variational Autoencoder (VAEs), have been explored for image anomaly
detection. Typically, these models are trained on healthy (non-anomalous)
images, learning to generate normal samples. During testing, anomalies are
identified by comparing the generated image with the original one in pixel
space [9, 10, 11]. Anomalies are expected to appear significantly different
from normal samples. Alternatively, anomalies can be detected if their data
lie outside the manifold of the learned representation [6, 12]. For a
comprehensive review, refer to [13].

CNNs serve as effective feature extractors, capturing patterns and statistics
from training data. Pre-trained models like PaDiM [6] and CCD [14] are
commonly utilized for feature extraction across different levels. These
methods then employ distance metrics such as Mahalanobis distance [6] and
contrastive loss [14] to identify anomalies. CCD [14] learns fine-grained
feature representations by predicting augmented data distributions and image
contexts simultaneously using contrastive learning. CutPaste [15] classifies
images into three categories: normal, two types of generated defects, and
abnormal using t-SNE. Gaussian density estimation is employed for
localization, and data preprocessing involves rotation and colour shifts.

Unsupervised Deep Learning in InSAR applications: Only a few unsupervised deep
learning methods have been proposed for detecting deformation in InSAR data to
date. Bountos et al. [16] base their model on self-supervised contrastive
learning by empling SimCLR for extracting visual representations from
interferograms, showing success in detecting unrest episodes before the
Fagradalsfjall volcanic eruption in 2020-2021. However, it is limited to a
single case study. Shakeel et al. [17] utilize a VAE trained on InSAR time
series, achieving over 91% accuracy on synthetic deformations and successfully
identifying a real earthquake (magnitude 5.7). This method is tailored for the
unique structure of InSAR time series, with 26 interferograms as input.

Fig. 2: Diagram of the proposed framework

##  3 Methodology

The proposed framework comprises i) data preprocessing, ii) patch-based
feature extraction, and iii) anomaly identification. We exploit unwrapped
interferograms in this paper as offering smooth spatial gradients and
providing an easier interpretation of displacement.

###  3.1 Data preprocessing

The preprocessing aims to ensure the data is suitable for deep learning-based
frameworks. This includes data interpolation, normalisation and atmospheric
correction.

Interpolation: Interferograms often have missing values, due to poor coherence
caused by water bodies or vegetation. However, the anomaly detection methods
in this study utilise convolutional layers, which depend on the spatial or
sequential attributes of dense data for effective learning. Thus, it is
essential to preprocess the data by interpolating it to resemble a dense
image.

For interpolation, we first mask the regions with missing values. This
involves applying morphological dilation to the masked image using a disk of
size $2$, followed by image closing with a disk of size $5$ to further refine
the masked areas. Subsequently, to interpolate over the masked regions, we use
MATLAB’s regionfill function, which calculates the discrete Laplacian over the
regions and solves the Dirichlet boundary value problem.

Normalisation: Deep neural networks typically accept inputs within the range
of $[-1,1]$. Normalizing unwrapped interferograms is complex due to varying
values. We analyze the global dataset, subtract each interferogram by its
mean, and observe that values outside the -30 to 30 radians range likely
indicate deformation signals. Therefore, we use -30 and 30 as the min and max
for data normalization. If an interferogram has values beyond this interval,
they are clamped to the nearest limit.

Atmospheric correction Interferograms are susceptible to atmospheric
turbulence, which can introduce false signals into the data. To address this
issue, we employ atmospheric corrections using the Generic Atmospheric
Correction Online Service for InSAR (GACOS) [18, 19, 20]. GACOS utilizes an
Iterative Tropospheric Decomposition (ITD) model to differentiate between
stratified and turbulent signals within tropospheric total delays. It
generates high-resolution zenith total delay maps to correct InSAR
measurements.

###  3.2 Patch-based feature extraction

Given that the interferograms of volcanoes obtained through LiCSAR have
dimensions of 500 by 500 pixels, we crop the images into smaller patches that
match the input size of the neural networks. We opt for the overlapping
patches approach, as splitting the image into non-overlapping patches can
adversely affect performance by posing a risk that the model may not correctly
identify features in either patch.

Following PaDiM [6], we employ a ResNet18 model [21] pre-trained on ImageNet
[22] to extract embedded features from images, which are concatenated to form
embedding vectors. We randomly remove embeddings in order to decrease the
dimensionality of the embedding vectors and minimize redundant information, as
suggested in the original paper. The vectors are then used to generate
multivariate Gaussian distributions. When testing, the distance between the
embeddings of an image and the learned distributions is calculated, thus an
abnormal image would have features that sit outside those distributions. This
framework is shown in Fig. 2.

###  3.3 Anomaly identification

Anomaly score $S$: To determine if the test sample is anomalous, the
statistics of its feature maps, extracted using the above process, are
compared with those of the training data. The distance between the
distributions of the testing and training features is then computed to
generate an anomaly score, denoted as $S$. A high value of $S$ indicates a
high probability of being an anomaly, as an abnormal image is likely to have
features that deviate from the learned distribution.

Vanilla PaDiM employs the Mahalanobis distance, which measures how many
standard deviations away the point $\mathbf{x}$ is from the mean $\bm{\mu}$,
taking into account the correlation between different variables as captured by
the covariance matrix $\mathbf{C}$. It can be expressed as:

$D_{M}(\mathbf{x},\bm{\mu},\mathbf{C})=\sqrt{(\mathbf{x}-\bm{\mu})^{T}\mathbf{C}^{-1}(\mathbf{x}-\bm{\mu})},$
(1)

A limitation of the Mahalanobis distance is that the covariates influence the
distance only through their covariance structure, not their importance [7]. To
enhance the accuracy of our $S$, we employ the weighted Mahalanobis distance.
The updated distance incorporates user-defined weights in the form of a
diagonal matrix $\mathbf{W}$. The distance can be expressed as:

$D_{M}(\mathbf{x},\bm{\mu},\mathbf{C},\mathbf{W})=\sqrt{(\mathbf{x}-\bm{\mu})^{T}\mathbf{W}\mathbf{C}^{-1}\mathbf{W}(\mathbf{x}-\bm{\mu})},$
(2)

In this paper, we empirically assigned weights of 0, 1, and 5 to the three
layers. The initial layer, characterized by significant noise, contributes
insignificantly to the distance measurement. Conversely, the final layer,
capturing high-level features, assumes greater importance in determining the
anomaly score of a sample, thus justifying its higher weight.

Additionally, we test the usage of the negative logarithm of the matching
likelihood (NLML) as $S$ to measure the distance between the testing sample
and the normal data. This is because the negative logarithm of the matching
likelihood has been shown to be capable of better capturing the distance
between a distribution and new data points[23]. By evaluating a probability
density function $\mathcal{N}(\bm{\mu},\mathbf{C})$ at different values of
$\mathbf{x}$, you can analyze the likelihood of observing different outcomes
under the given distribution. This is called the matching likelihood and can
be written as:

$f(\mathbf{x})=\frac{1}{\sqrt{(2\pi)^{k}\det(\mathbf{C})}}\exp\left(-\frac{1}{2}(\mathbf{x}-\bm{\mu})^{\top}\mathbf{C}^{-1}(\mathbf{x}-\bm{\mu})\right)$
(3)

where $k$ is the dimensionality of the observation vector. Thus, the matching
likelihood describes the probability distribution of a continuous random
variable. It can be seen that the Mahalanobis distance is the specific form of
the previous formula. When applying different weights, it can be verified by
expanding Equation 3 as a negative log-likelihood as follows:

$-\log f(\mathbf{x})=\frac{1}{2}\left(k\log
2\pi+D_{M}^{2}(\mathbf{x},\bm{\mu},\mathbf{C},\mathbf{W})+\log\mathbf{C}\right).$
(4)

This shows that NLML captures more information than the Mahalanobis distance.

Final probability map: The patch scores ($S$) are combined using a Gaussian
distribution $\mathcal{N}(\frac{s}{2},\frac{s}{6})$, where $s$ is the patch
size, producing a score map. The image’s score is the average value of this
map, using the mean to avoid errors from interferogram artefacts. The score
map indicates the anomaly signal location.

Anomalies in newly arriving interferograms are flagged based on a threshold.
In our case, with unlabeled volcanoes, we set the threshold to classify 95% of
the training dataset as normal. This prevents outliers from influencing the
threshold, and a unique threshold is calculated for each volcano due to
differing characteristics.

The final probability map ($P$) is generated by the score values, flagging
anomalies when probabilities exceed 0.5. Using Eq. 5, we divide the $S$ by
double the threshold ($T_{h}$). We do this in order to scale the loss values
to the same interval used by the supervised learning model, which uses 0.5 as
the threshold. Normal samples have a loss below 0.5, abnormal losses are
above, and values greater than double the threshold are set to 1.

$P=\text{min}\left(1,\frac{S}{2T_{h}}\right).$ (5)

##  4 Experimental results and discussion

###  4.1 Data and case studies

The InSAR data were obtained from the Sentinel-1 satellites and processed
using LiCSAR [24], an automated InSAR processing system developed by the
Centre for Observation and Modelling of Earthquakes, Volcanoes, and Tectonics
(COMET). Based on the global dataset reported in [8], we selected five
volcanoes to examine the methods outlined in the previous section and to
compare them with the supervised learning approach proposed in [8]. The
training period was chosen to extend up to 6 months before a deformation event
happened at the volcano. If the volcano had no deformation events, then half
the images were used for training. The time series showing the cumulative
displacement of these volcanoes can be found in the supplementary material.

i) Taal is located in the Philippines and had a major eruption in January
2020, leading to significant lateral magma movement [25]. This case is
relatively straightforward, with large and distinct deformation signals
observed. 589 interferograms were used for training and 335 interferograms for
testing, including 17 positive samples.

ii) Agung, a volcano situated in Indonesia, exhibited deformation accompanied
by an earthquake swarm in September 2017 that preceded the eruption on 21
November 2017 [26]. The signal is less distinct than observed at Taal and was
initially obscured by atmospheric artefacts[26]. The displacement persists
after the eruption, posing challenges for anomaly detection. 202
interferograms were used for training and 961 interferograms for testing,
including 139 positive samples.

iii) Casiri, located in Peru, exhibited a deformation signal in July 2020 [8],
evident in a sudden uplift signal. Despite this, the spatial signal is small
(see Fig. 4c), and the interferograms are impacted by atmospheric noise,
rendering this case challenging

iv) Lamongan, located in Indonesia and surrounded by vegetation, experiences
atmospheric effects in interferograms, making it challenging to discern
deformation. Deformation was detected in November 2019 but has not previously
been reported.

v) Lawu situated in Indonesia and surrounded by vegetation, is prone to noise
and atmospheric artefacts in interferograms, leading to patterns that resemble
real signals. Deformation signals were claimed to have been observed at Lawu
[27], but at least some of those signals can be attributed to atmospheric
artefacts [28].

Table 1: Performance comparison reported in Area Under the Receiver Operating Characteristic (AUROC). Denote that Maha. is Mahalanobis distance, wNLML is weighted NLML distance, and wMaha. is weighted Mahalanobis distance. Methods | Taal | Agung | Casiri | Lamongan | Lawu
---|---|---|---|---|---
PaDiM NLML | 0.97 | 0.92 | 0.36 | 0.95 | 7 FP
PaDiM Maha. | 0.96 | 0.90 | 0.32 | 0.94 | 9 FP
PaDiM wNLML | 0.97 | 0.93 | 0.65 | 0.97 | 8 FP
PaDiM wMaha. | 0.95 | 0.91 | 0.69 | 0.96 | 9 FP
Ganomaly | 0.93 | 0.73 | 0.51 | 0.85 | 11 FP
Diffusion | 0.90 | 0.84 | 0.33 | 0.91 | 12 FP
Supervised learning | 0.94 | 0.52 | 0.47 | 0.88 | 7 FP

###  4.2 Detection results

We compared the performance of our proposed method with two state-of-the-art
unsupervised anomaly detection methods, namely Ganomaly [29] and Diffusion
model [30], as well as the supervised learning employed in [8]. The outcomes
derived from the examination of five distinct study volcanoes are presented in
Table 1, showing that PaDiM achieves the best AUROC score. Note that, due to
the absence of positive signals at Lawu, AUROC for validation could not be
calculated. Consequently, we reported the number of false positives instead.
The examples of detection results overlaid on the interferograms are shown in
Fig. 3

Different distances were considered for PaDiM, and the results in Table 1
demonstrate that integrating the weighted Mahalanobis distance improves
PaDiM’s performance, whether using the original version or NLML as the
distance metric. This enhancement is attributed to assigning greater weights
to the deeper layers. The initial layer, characterized by significant noise,
has minimal impact on the distance measurement, while the final layer,
capturing high-level features, plays a more crucial role in determining the
anomaly score of a sample.

Amongst the five volcanoes, the anomaly detection methods performed the worst
at Casiri because the deformation signal was small. However, both our method
and the supervised learning model flag 5 interferograms correctly.
Additionally, our model can detect processing artefacts (see Fig.4). This is
useful for identifying errors that happened during the automated process of
generating the interferograms.

In Lamongan’s case, the supervised learning model fails to flag any image that
shows deformation. In comparison, the unsupervised model flags 22
interferograms as abnormal, out of which 3 have real deformation and 7 have
processing artefacts, the rest being false positives.

Fig. 3: Results at Lamongan, Indonesia ($7.981^{\circ}S$, $113.341^{\circ}E$),
using (a) PaDiM (original), (b) PaDiM with weighted Mahalanobis (proposed),
(c) Ganomaly, (d) Diffusion, (e) Supervised learning. Top row is real
deformation (anomaly). Bottom row is no deformation (normal). The brighter
yellow means higher probability. Areas inside dark and bright green contours
are where $P>0.5$ and $P>0.8$, respectively. Each image is 50 km across. Fig.
4: Example results from Nevados Casiri, Peru ($17.47^{\circ}S$,
$69.813^{\circ}W$), using (left) our model and (right) supervised learning
model, showing (A) deformation (anomaly), (B) artefacts (anomaly), and (C) no
deformation (normal). The brighter yellow means higher probability. Areas
inside dark and bright green contours are where $P>0.5$ and $P>0.8$,
respectively. Each image is 50 km across.

###  4.3 Study of input formats

Here, we present the performance of our method when trained and tested with
wrapped interferograms, contrasting with the proposed input – unwrapped
interferograms. Columns two and three in Table 2 demonstrate that the use of
unwrapped interferograms yields significantly better results compared to
wrapped interferograms, with an 8% improvement in our method and 80%
improvement in the GANomaly case. This shows that the discontinuity resulting
from a phase change from -$\pi$ to $\pi$ degrades anomaly detection.

The last column of Fig. 2 indicates that the interpolation process
significantly improved the performance of all three detectors. This confirms
the importance of spatial correlation for convolutional neural networks.

Table 2: Performance comparison (AUROC) when using different input formats at Taal methods | wrapped | unwrapped | interpolated unwrapped
---|---|---|---
PaDiM | 0.86 | 0.93 | 0.98
GANomaly | 0.44 | 0.79 | 0.96
DDPM | 0.59 | 0.69 | 0.93

##  5 Conclusion

We propose a novel unsupervised learning framework for anomaly detection based
on Patch Distribution Modeling (PaDiM). The primary contribution is the
utilization of weighted NLML distance measurement for detection, with higher
weights assigned to deeper layers due to their increased significance.
Additionally, we introduce a preprocessing approach to handle noisy and
incomplete data points, enhancing the model’s performance. Our model
demonstrates the capability to identify volcanic deformation and processing
artefacts that go unnoticed by the supervised learning model. Future work will
concentrate on integrating additional volcano characteristics with
interferograms to enhance training.

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# An extension of Birkhoff’s representation theorem to infinite distributive
lattices

Dale R. Worley<EMAIL_ADDRESS>
(Date: 13 March 2023)

###### Abstract.

Birkhoff’s representation theorem for finite distributive lattices states that
any finite distributive lattice is isomorphic to the lattice of order ideals
(lower sets) of the partial order of the join-irreducible elements of the
lattice. This theorem can be extended as follows: A non-finite distributive
lattice that is locally finite and has a $\hat{0}$ is isomorphic to the
lattice of finite order ideals of the partial order of the join-irreducible
elements of the lattice. In addition, certain “well ordering” properties are
shown to be equivalent to the premises of the extended theorem.

Note added to the 2nd version: Since the publication of the 1st version, I
have been informed that the main result of this paper (2) was previously
published as [Stan2012]*Prop. 3.4.3.

Birkhoff’s representation theorem for finite distributive lattices has long
been known:

######  Theorem 1 (Birkhoff [Birk1937][Birk1967]*ch. III sec. 3 p. 58).

Any finite distributive lattice $L$ is isomorphic to the lattice of order
ideals of the partial order of the join-irreducible elements of $L$.

###### Proof sketch.

The proof is to demonstrate that there is a one-to-one order-preserving
correspondence between elements of $L$ and order ideals of the partial order.
The order ideal corresponding to an element $x\in L$ is simply the set of
join-irreducible elements of $L$ that are $\leq x$, and the element of $L$
corresponding to an order ideal $S$ of join-irreducible elements is $\bigvee
S$. ∎

(Note that in all of this discussion, it is required that $\hat{0}$ in $L$
corresponds to the empty order ideal of the partial order. If $\hat{0}$ is
seen as join-irreducible, then both $\varnothing$ and $\\{\hat{0}\\}$ are
order ideals, which does not match the structure of $L$. So we define that an
element $x$ is “join-irreducible” if it cannot be expressed as the join of a
finite set of smaller elements; thus $\hat{0}$ is the join of zero elements
and not join-irreducible.)

The motivation for this extension is that in the combinatorics of tableaux,
the distributive lattice of tableaux is formally defined as the lattice of
finite order ideals of an infinite base poset. The archetype example is that
the Young lattice of Ferrers diagrams equiv.111Two equivalent terms or
conditions are conjoined with “equiv.” (unshifted) shapes is the finite order
ideals of the quadrant of the plane, $\mathbb{P}^{2}$.222I use $\mathbb{P}$
for $\\{1,2,3,\ldots\\}$ and $\mathbb{N}$ for $\\{0,1,2,\ldots\\}$. Another
example is that the lattice of shifted shapes is the finite order ideals of
the octant of the plane, $\\{(x,y):x,y\in\mathbb{P}\text{ and }x\geq y\\}$. In
these cases, the poset generating the lattice is already known, of course, but
it’s clear that the construction of a lattice from a poset is a general
pattern. (See e.g. [Fom1994]*sec. 2 and the $J(\cdot)$ operator.)

The extension theorem shows that this construction is universal.

######  Theorem 2.

If a distributive lattice $L$ is locally finite and has a $\hat{0}$, then $L$
is isomorphic to the lattice of finite order ideals of the partial order of
the join-irreducible elements of $L$.

###### Proof.

The proof is to demonstrate that there is a one-to-one order-preserving
correspondence between elements of $L$ and finite order ideals of the partial
order. Composing these two maps in either order gives the identity map on the
appropriate domain, showing that they are the desired correspondence and its
inverse.

The order ideal corresponding to an element $x\in L$ is the set of join-
irreducible elements of $L$ that are $\leq x$, and the element of $L$
corresponding to a finite order ideal $S$ of join-irreducible elements is
$\bigvee S$.

For any element $x\in L$, let $S$ be the join-irreducible elements in $L$ that
are $\leq x$, which is an order ideal of the join-irreducible elements.
Clearly the mapping from $x$ to $S$ is order-preserving. $S$ is finite because
it is a subset of the interval $[\hat{0},x]$.

Now let $y$ be the join of $S$. We need to show $y=x$. As a join of elements
$\leq x$, $y\leq x$. Since $[\hat{0},x]$ is finite, any decomposition of $x$
into joins must terminate, so $x$ is the join of some finite set $I$ of join-
irreducibles in $[\hat{0},x]$. $I\subset S$, so $x=\bigvee I\leq\bigvee S=y$.

For any finite order ideal $S$ of join-irreducible elements of $L$, let $x$ be
$\bigvee S$. Clearly this mapping is order-preserving. Let $T$ be the set of
join-irreducible elements $\leq x$. We need to show $S=T$.

Every element of $S$ clearly belongs to $T$. Conversely, any element of $T$ is
a join-irreducible element $\leq x$ and must (by the join-primality of the
elements of the finite set $S$, which is implied by join-irreducibility in
distributive lattices) be $\leq$ one of the members of $S$, and therefore must
itself (by the assumption that $S$ is an order ideal) belong to $S$.

∎

While determining the antecedents of Theorem 2, I found that certain “well
ordering” properties are alternative antecedents:

######  Theorem 3.

For distributive lattice $L$, the conditions

  1. (1)

$L$ is well-partially-ordered equiv. $L$ contains no infinite descending
chains equiv. there does not exist $a_{1},a_{2},a_{3},...\in L$ such that
$a_{1}>a_{2}>a_{3}>...$, and

  2. (2)

$L$ contains no bounded infinite ascending chains equiv. there does not exist
$a_{1},a_{2},a_{3},...,b\in L$ such that $a_{1}<a_{2}<a_{3}<...<b$,

are equivalent to the conditions that $L$ is locally finite and has a
$\hat{0}$.

###### Proof.

Suppose $L$ satisfies the well-ordering conditions (1) and (2). Then it must
have a minimal element, as otherwise an infinite descending chain could be
constructed. Since $L$ is a lattice, there can be only one minimal element,
hence that element is $\hat{0}$.

Let $I$ be the set of elements of $S=[x,y]$ that are join-irreducible within
$S$. Every element of $S$ can be decomposed into a finite join of elements of
$I$. If $S$ is infinite, then $I$ must be infinite to allow constructing an
infinite number of different joins of elements of $I$. Then by the infinite
Ramsey theorem, $I$ would contain either an infinite chain or an infinite
antichain.

An infinite chain must be well-ordered (because $L$ contains no infinite
descending chains), and any infinite well-ordered set contains an initial
subset that is order-isomorphic to $\mathbb{N}$. This subset would be an
infinite ascending chain, all of whose elements are $<y$, which contradicts
(2).

An infinite antichain would contain a countably infinite subset
$i_{1},i_{2},i_{3},...$ of incomparable elements of $I$. Construct
$i_{1},i_{1}\vee i_{2},i_{1}\vee i_{2}\vee i_{3},...$ . Each of these is
strictly greater than the previous, because if

$\bigvee_{j=1}^{n}i_{j}=\bigvee_{j=1}^{n-1}i_{j}$

then

$i_{n}\vee\bigvee_{j=1}^{n-1}i_{j}=\bigvee_{j=1}^{n-1}i_{j}$
$i_{n}\leq\bigvee_{j=1}^{n-1}i_{j}$ $i_{n}\leq i_{j}\text{ for some
}j<n\text{, because }\bigvee_{j=1}^{n-1}i_{j}\text{ is a finite join of join-
irreducibles}$

contradicting that the $i_{j}$ form an antichain. Thus the
$\bigvee_{j=1}^{n}i_{j}$ form an infinite ascending chain, all of which are
$<x$, which is assumed not to exist. Thus, $S$ is finite.

Conversely, suppose $L$ is locally finite and has a $\hat{0}$. There can be no
infinite descending chain $a_{1}>a_{2}>a_{3}>...\in L$, because it would be
contained in $[\hat{0},a_{1}]$, which is finite. Similarly, there can be no
bounded infinite ascending chain $a_{1}<a_{2}<a_{3}<...<b\in L$, because it
would be contained in $[a_{1},b]$, which is finite. ∎

  * []*labels=alphabetic
Fom1994FominSergeyDuality of graded graphsJournal of Algebraic
Combinatorics31994357–404Documenthttps://link.springer.com/content/pdf/10.1023/A:1022412010826.pdfhttps://scholar.google.com/scholar?cluster=3401296478290474488@article{Fom1994,
label = {Fom1994}, author = {Fomin, Sergey}, title = {Duality of Graded
Graphs}, journal = {Journal of Algebraic Combinatorics}, volume = {3}, date =
{1994}, pages = {357–404}, doi = {doi:10.1023/A:1022412010826}, eprint =
{https://link.springer.com/content/pdf/10.1023/A:1022412010826.pdf}, note =
{https://scholar.google.com/scholar?cluster=3401296478290474488}}
Birk1937BirkhoffGarrettRings of setsDuke Mathematical
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mathematical-journal/volume-3/issue-3/Rings-of-
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label = {Birk1937}, author = {Birkhoff, Garrett}, title = {Rings of sets},
journal = {Duke Mathematical Journal}, volume = {3}, date = {1937}, pages =
{443–454}, doi = {doi:10.1215/S0012-7094-37-00334-X}, eprint =
{https://projecteuclid.org/journals/duke-mathematical-
journal/volume-3/issue-3/Rings-of-sets/10.1215/S0012-7094-37-00334-X.short},
note = {https://scholar.google.com/scholar?cluster=10180976689018188837}}
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https://scholar.google.com/scholar?cluster=10180976689018188837@book{Birk1967,
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https://archive.org/details/latticetheory0000birk
https://scholar.google.com/scholar?cluster=10180976689018188837}}
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