# Topological Nanophononic Interface States Using High-order Bandgaps in the
One-Dimensional Su-Schrieffer-Heeger Model

A. Rodriguez  Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120 Palaiseau, France  K. Papatryfonos  Université Paris-
Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau,
France  E.R. Cardozo de Oliveira  Université Paris-Saclay, CNRS, Centre de
Nanosciences et de Nanotechnologies, 91120 Palaiseau, France  N. D.
<EMAIL_ADDRESS>Université Paris-Saclay,
CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

###### Abstract

Topological interface states in periodic lattices have emerged as valuable
assets in the fields of electronics, photonics, and phononics, owing to their
inherent robustness against disorder. Unlike electronics and photonics, the
linear dispersion relation of hypersound offers an ideal framework for
investigating higher-order bandgaps. In this work, we propose a design
strategy for the generation and manipulation of topological nanophononic
interface states within high-order bandgaps of GaAs/AlAs multilayered
structures. These states arise from the band inversion of two concatenated
superlattices that exhibit inverted spatial mode symmetries around the
bandgap. By adjusting the thickness ratio of the unit cells in these
superlattices, we are able to engineer interface states in different bandgaps,
enabling the development of versatile topological devices spanning a wide
frequency range. Moreover, we demonstrate that such interface states can also
be generated in hybrid structures that combine two superlattices with bandgaps
of different orders centered around the same frequency. These structures open
up new avenues for exploring topological confinement in high-order bandgaps,
providing a unique platform for unveiling and better understanding complex
topological systems.

##  1 Introduction

A periodic lattice containing two elements per unit cell can be described by
the one-dimensional Su-Schrieffer-Heeger (SSH) model in the tight-binding
approximation. This description has been a significant breakthrough for
developing materials with topological properties [1, 2]. Topological states
have since been demonstrated for a wide variety of excitations, including
photons [3, 4, 5, 6, 7, 8], phonons [9, 10, 11, 12, 13], vibrations [14, 15,
16, 17, 18, 19, 20, 21], polaritons [22, 23, 24], plasmons [25], and magnons
[26]. In the context of periodic lattices, multilayered structures, such as
distributed Bragg reflectors (DBRs), have high-reflectivity regions associated
with bandgaps. The states at the edge of these bandgaps present different
spatial symmetries [27]. When concatenating two DBRs with inverted spatial
mode symmetries around a gap, a topological interface state emerges [6, 28,
29].

Nanophononics [30, 31, 32], i.e., the engineering of acoustic nanowaves,
appears as a versatile simulation platform [9, 33]. Unlike in optics or
electronics [34, 35, 36], the linear dispersion relation of acoustic phonons
[37] allows for the study of topological interface states in a broad frequency
range. In particular, nanoacoustic topological states have been evidenced in
superlattices working at acoustic frequencies in the tens to hundreds of GHz
range, and demonstrated exceptional agreement between theory and experiments.
[38, 39] In these reported cases, the unit cell was formed by two materials
whose thickness ratio was optimized to reverse the mode symmetries around a
specific bandgap while keeping the acoustic thickness constant. These studies
demonstrated the robustness of the topologically protected states against
thickness perturbations that do not affect the Zak phase ($\theta^{Zak}$) [38,
39], which is a key parameter to characterize topological phases in the SSH
model [40]. Formally, the Zak phase is defined as the integral of the
displacement across the Brillouin zone, and it can be associated with the sign
of the reflection phase of a finite-size DBR [6, 16]. Each individual band of
the acoustic dispersion relation has an associated $\theta^{Zak}$, which can
take only two values, 0 or $\pi$. In this work, we theoretically investigate
topological interface states by concatenating DBRs with inverted spatial mode
symmetries at high-order bandgaps. Based on the different Zak phase
configurations for the different bandgaps at specific unit cell thickness
ratios, we engineer interface modes at higher bandgap orders. Furthermore, we
benefit from the linear dispersion relation to generate hybrid topological
resonators. We establish the interface states of these resonators by
concatenating two superlattices with different bandgap orders that are
overlapping in frequency. We thus show that the presence of the topological
states does not depend only on the spatial mode symmetry but also on the
relative bandgap order of the concatenated superlattices. This platform allows
to experimentally map textbook cases, and test innovative and counterintuitive
physical situations. Our findings unlock a new degree of freedom to design
topological nanoacoustic resonators.

The paper is organized as follows. Section 2 describes the principle of band
inversion in the context of nanoacoustics. In Section 3, we present the method
for generating interface states at high-order bandgaps and discuss their
robustness compared to Fabry-Perot resonators. Sections 4 and 5 introduce our
new designs of topological acoustic resonators.

##  2 Principle of Band Inversion

In topological superlattices, band inversion refers to the situation where two
modes with opposite symmetries at the edges of a bandgap exchange their
ordering in energy. This inversion can be achieved by adjusting the relative
thickness of the two layers in the unit cell of the superlattice. By
concatenating two superlattices with different layer thicknesses, i.e.,
superlattices presenting inverted bands, topologically protected interface
states can be created.

In the case of multilayered GaAs/AlAs structures composed of two concatenated
distributed Bragg reflectors (DBRs), we introduce a parameter
$\delta\in[-1,:1]$ to represent the relative thickness of AlAs and GaAs. [28]
In DBRs with a centro-symmetric unit cell centered around the AlAs layer, GaAs
is distributed equally on both sides of AlAs as follows:
$\frac{\lambda_{GaAs}}{8}(1+\delta)$, $\frac{\lambda_{AlAs}}{4}(1-\delta)$,
$\frac{\lambda_{GaAs}}{8}(1+\delta)$. Figure 1 illustrates the evolution of
the first four bandgaps as a function of $\delta$, while maintaining the unit
cell acoustic thickness constant. The orange lines indicate modes symmetric
with respect to the center of the unit cell, while the blue lines indicate
anti-symmetric modes. It can be observed that the modes exhibit a sinusoidal
dependence on the parameter $\delta$. Starting from the second bandgap, there
is a consecutive opening and closing of the bandgap as $\delta$ increases,
accompanied by an inversion of symmetry. The number of nodes in the modes is
directly related to the order of the bandgap. The first bandgap (Fig.1(a))
opens and closes only once, with a maximum amplitude at $\delta=0$. The second
bandgap (Fig.1(b)) opens twice and closes at $\delta=0$, with a symmetry
inversion of the edge modes around this point. The third bandgap exhibits two
symmetry inversions (Fig.1(c)), while the fourth gap undergoes three symmetry
changes across the full range of $\delta$ (Fig.1(d)). Generally, the $n^{th}$
bandgap experiences $(n-1)$ symmetry inversions.

Figure 1: Left: Frequency of the band-edges bounding the bandgap as a function
of the parameter $\delta$. (a)-(d) Band inversion of the acoustic bandgap
around 9.3 GHz (a), 18.6 GHz (b), 28 GHz (c), and 37.3 GHz (d). The mode
symmetries are indicated by orange (symmetric) and blue (anti-symmetric)
lines. The thickness ratio of GaAs/AlAs is indicated by the vertical dashed
line in each case. Right: The unit cell is displayed with light and dark gray
colors, representing AlAs and GaAs, respectively. The band-edge modes are
shown in the unit cells for each bandgap.

The topological properties of a multilayered acoustic device can be
characterized using the Zak phase, which is the 1D equivalent of the Berry
phase [41]. For a periodic phononic 1D system with a periodicity $a$, the Zak
phase of the $n^{th}$ band is calculated by integrating across the Brillouin
zone:

$\theta_{n}^{Zak}=\int_{-\pi/a}^{\pi/a}\left[i\int_{\text{unit
cell}}\frac{1}{2\rho(z)v^{2}(z)}dzu^{*}{n,k}(z)\partial_{k}u{n,k}(z)\right]dk,$
(1)

where $u_{n,k}(z)$ is the acoustic displacement of the $n^{th}$ band and wave-
vector $k$ at position $z$, and $\rho(z)$ and $v(z)$ correspond to the mass
density and speed of sound in the materials. [16, 40, 28]

In a periodic system with inversion symmetry, where the unit cell is centro-
symmetric around AlAs, the Zak phase can only take on two discrete values: 0
or $\pi$ [6, 16]. It is associated with the symmetries of the Bloch modes at
the band edge [6]. When the modes at both ends of the same $n^{th}$ band
(i.e., at the edge and center of the Brillouin zone) have the same symmetries,
the Zak phase $\theta_{n}^{Zak}$ for that band is 0. Conversely, if the band
has edge modes with opposite symmetries, $\theta_{n}^{Zak}=\pi$ [6]. This
parameter is crucial for predicting interface states and characterizing one-
dimensional topological systems.

##  3 Interface states at high-order bandgaps

Fundamentally, an interface state is formed whenever two DBRs with opposite
reflection phase signs are concatenated [28, 6]. The reflection phase can be
either positive or negative depending on the structure of the DBR. For the
$n^{th}$ bandgap, one can determine the sign of the reflection phase by
evaluating the relation [6]:

$\displaystyle
sgn(\phi)=(-1)^{n}(-1)^{l}\times\exp{\left(i\sum_{m=0}^{n-1}\theta_{m}^{Zak}\right)},$
(2)

where $l$ is the number of closed bandgaps below the $n^{th}$ bandgap.
Therefore, for the $n^{th}$ bandgap, there is an interface state at the
condition that $\sum_{m=0}^{n-1}\theta_{m}^{Zak}=0+2p\pi,p\in\mathbb{N}$ for
one DBR and $\sum_{m=0}^{n-1}\theta_{m}^{Zak}=\pi+2p\pi,p\in\mathbb{N}$ for
the other. The creation of an interface state at the $n^{th}$ bandgap does not
necessarily imply the generation of an interface state in other bands, as the
sum of $\theta^{Zak}$ for each band might differ.

Figures 2 (a)-(c) display reflectivity spectra for the third bandgap obtained
by combining two DBRs with different values of $\delta$, indicated by the
dashed lines on the corresponding top panels. In each case, $\delta$ is chosen
such that the amplitude of the bandgap of the corresponding superlattice is
maximized, which corresponds to the values $\delta=0$ and $\pm 0.66$. In
panels (a) and (c), the two DBRs have inverted symmetry around the bandgap. In
both cases, the acoustic reflectivity contains a dip centered in the high
reflectivity region featuring the interface state. In contrast, in panel (b),
the two DBRs have the same symmetry around the bandgap. Thus, this structure
acts as a standard DBR, with a high reflectivity region. Generally, for the
third bandgap, by concatenating one DBR with $\delta\in[-0.33,0.33]$ and
another with $\delta<0.33$ (Fig.2(a)) or $\delta>0.33$ (Fig.2(c)), the band
inversion is preserved, and an interface state is generated.

Figure 2: Top panels: Band inversion of the (a)-(c) third acoustic bandgap
around 28 GHz, and (d)-(i) fourth bandgap around 37.3 GHz. Bottom panels:
calculated acoustic reflectivity spectra for the concatenated DBRs with the
corresponding GaAs/AlAs thickness ratio $\delta$ marked as dashed lines on the
top panels.

Regarding the fourth bandgap, Figs. 2 (d)-(i) show six possible combinations
between the two DBRs. In all shown cases, the values of $\delta$ are chosen
such that the amplitude of the bandgap is maximized, resulting in high
reflectivity regions centered at $\sim$37.3 GHz. Depending on the mode
symmetries around the fourth gap, an interface state is either present or
absent. In the cases shown in Figs. 2(e) and (h), the modes of the two
concatenated DBRs have the same symmetry (both modes at the bottom band are
symmetric in panel (e), while they are anti-symmetric in panel (h)). Thus, the
acoustic reflectivity spectra present high reflectivity regions like a
standard DBR. On the contrary, in Figs. 2 (d), (f), (g), and (i), the two DBRs
that are concatenated have inverted mode symmetries. As a result, an interface
state between the two concatenated DBRs is generated. Generally, the approach
presented for the third and fourth bandgaps can be extended to higher bandgap
orders. More specifically, an interface state is generated when concatenating
any two DBRs corresponding to superlattices with an even and odd bandgap
opening.

These interface states can be accessed experimentally through Brillouin
scattering measurements. [10, 38] Nevertheless, the scattering cross-section
($\sigma$), which represents the magnitude of the scattered signal, relies on
the relative thickness of GaAs and AlAs within the two unit cells constituting
each superlattice. Consequently, not all theoretically predicted interface
states can be accessed experimentally. To identify the superlattice
combinations that yield experimentally accessible interface states, we
employed the transfer matrix method and a photoelastic model to simulate the
Brillouin cross-section of the interface states. [42, 43] The Brillouin cross-
section is defined by the overlap integral between the incident laser electric
field $E(z)$, the strain, which is given by the derivative of the displacement
$\frac{\partial u(\omega,z)}{\partial z}$, and the photoelastic constant
$p(z)$ over the whole structure in the form:

$\sigma(\omega)=\int|E(z)|^{2}p(z)\frac{\partial u(\omega,z)}{\partial z}dz,$
(3)

where $p(z)$ is material dependent, being $p=1$ ($p=0$) in Ga-rich (Al-rich)
layers [38]. We considered here the whole acoustic structure as an optical
$\lambda$-cavity embedded in vacuum, where $\lambda_{\textrm{opt}}\sim 1600$
nm. The cross-section depends on the overlap between the electric and acoustic
fields $|E(z)|^{2}(\partial u(\omega,z)/\partial z)$ in the regions where the
photoelastic constant distribution $p(z)$ is non-zero [38].

The integrand of Eq. 3 is displayed in Fig. 3 for structures supporting
interface states in the 3rd and 4th bandgaps. We can analyze the integrand by
splitting it into quadrants: left/right superlattice and positive/negative
amplitude contributions. The integrand features signals composed of either
double peaks (thick lines) or single peaks (thin lines). Figs. 3 (a) and (b),
display the integrand of the modes in the third bandgap, associated with the
topological structures presented in Figs. 2(a) and (c), respectively. The
peaks with the maximum amplitude at the interface between the two DBRs are the
main contributors to the overall cross-section, resulting in a high Brillouin
cross-section in panel (a). Conversely, the positive and negative
contributions of the integrand displayed in panel (b) cancel each other,
leading to a low Brillouin cross-section. The calculated Brillouin cross-
sections for the mode in the third bandgap on both cases are, respectively,
$\sigma=8291$ and $\sigma=4$.

Figure 3: Integrand of the Brillouin cross-section for interface states in the
third bandgap (a),(b), associated to the structures shown in Figs. 2(a) and
(c), respectively; and the fourth bandgap (c)-(f), associated to the
structures of Figs. 2(d), (f), (g) and (i), respectively. The vertical red
line indicates the interface between the two DBRs.

Figs. 3(c)-(f) display the integrand of the modes in the fourth bandgap,
associated to the cases of Figs. 2(d), (f), (g), and (i), respectively. The
integrand in Figs. 3(c) and (f) exhibit positive contribution across the
entire structure, while negative amplitude contribution is asymmetric between
left and right quadrants. This results in an overall positive signal with high
Brillouin cross-sections of $\sigma=278720$ and $\sigma=259010$, respectively.
In contrast, the Brillouin cross-section in Figs. 3(d) and (e) is smaller;
$\sigma=328$ and $\sigma=6$, respectively. A zoomed-in version of the graphs
depicting the integrands in more detail is shown in the Supplementary
Information. We note that although the apparent overall contribution is
positive, the positive right quadrant contribution in panel (d) is not
substantial, resulting in a fairly small cross-section. Similarly, in panel
(e), even though the positive left quadrant displays double peaks, the
negative peaks are thicker than the positive contributions, resulting in a
small Brillouin cross-section as well.

Topological interface states have been demonstrated to exhibit exceptional
robustness against disorder, which is useful for transport and error-free data
communication. [44, 45, 46]. The band inversion principle exploited here to
build topological resonators preserves the center of the bandgap when varying
$\delta$. We have numerically demonstrated that the robustness of the
interface mode applies to all bandgap orders when introducing fluctuations in
the layer thickness ratio. In practice, such fluctuations might emerge, for
example, by material intermixing or composition fluctuations [47, 48] during
the epitaxial growth of the layers that form the superlattice. The
fluctuations considered here, concern changes in the GaAs/AlAs ratio while
maintaining a constant unit-cell acoustic thickness. In the model, we
implement this by using a flat distribution of random numbers with an
amplitude $\Delta\delta/\delta$ ranging from zero (unperturbed system) to 0.5.
We note that a noise amplitude set to 0.999 effectively closes the targeted
bandgap for each designed superlattice.

In Figs. 4(a) and (b), we compare the topological interface state generated in
the third bandgap (corresponding to the structure shown in Fig. 2(c)) to the
confined mode in a Fabry-Perot (FP) cavity. The FP cavity considered here is
formed by non-centrosymmetric unit cells of GaAs/AlAs with $\delta=0.66$,
chosen such that it preserves the maximum bandgap opening. The two DBRs
surround a spacer of thickness $\lambda$, and are embedded in a GaAs
background.

Figure 4: (a) Resonant frequency in the third bandgap under random
perturbations, with a uniform distribution of width $\Delta\delta/\delta$. The
acoustic resonance frequency stays trapped at the bandgap center for the
topological mode (blue) but undergoes variations in the Fabry–Perot (orange).
(b) Acoustic quality factor under random perturbations. For both types of
resonators, the acoustic quality factor drops by a factor of ten. Similarly,
(c) and (d) display the same comparisons for the fourth bandgap.

Figure 4(a) displays the acoustic frequencies of the topological (blue) and
the Fabry-Perot (orange) resonators as a function of $\delta$-fluctuations. In
the FP resonator, the fluctuations are introduced in the thickness ratio of
GaAs and AlAs constituting the DBR unit cell, while the spacer has no
perturbation. In this case, the topological resonator maintains its resonance
at the bandgap center. In contrast, the resonance frequency of the FP
resonator undergoes large fluctuations, spanning up to 0.5 GHz away from the
center frequency. Figure 4(b) shows the influence of thickness fluctuations on
the acoustic quality factors of the two structures. As seen, both quality
factors decrease by a factor of 10 at high perturbation amplitudes. This
effect can be explained by the effective reduction in the width of the
bandgap, which results in an increase of the evanescent decay length of the
confined mode, and so an enhancement of the leakage through the DBRs towards
the background [38]. Despite the decrease in quality factors for both
structures with increasing perturbation amplitudes, the topological structure
shows more consistent Q-factors compared to the FP resonator. This is related
to the fact that the fluctuations in the topological mode frequency are less
prominent, and so its associated Q-factor decreases at a slower pace.

Likewise, in Figs. 4(c) and (d), we compare the robustness of the topological
interface state generated in the fourth bandgap to an FP cavity
($\delta=0.75$), with a spacer of thickness $\lambda$/2. Figure 4(c) shows the
resonant frequencies of both resonators as a function of the fluctuations. The
observed behavior is similar to the case of the third bandgap: the frequency
is clamped at the bandgap center for the topological structure and fluctuates
for the Fabry-Perot resonator. These results show that the robustness
characteristic of topological devices, protecting the acoustic resonance
against disorder, is also preserved at high bandgap orders. However, as we see
by comparing Figs. 4(b) and (d), the acoustic quality factor of the fourth
bandgap (Fig. 4(d)) in both structures (FP and topological) is more sensitive
to fluctuations in comparison to the third bandgap (Fig. 4(b)). This can be
understood intuitively by comparing the opening and closing of the bandgaps as
a function of unit cell composition, as shown, for example, in Fig. 1. When
transitioning to higher-order bandgaps, the opening/closing of each bandgap
necessitates gradually smaller adjustments in material thicknesses.
Consequently, interface states at higher-order bandgaps become progressively
more susceptible to inaccuracies in material thicknesses.

##  4 Multimode engineering

In the previous section, we showed that we could engineer the interface states
at the $n^{th}$ bandgap in topological acoustic resonators by carefully tuning
the unit-cell material thickness ratios in both juxtaposed DBRs. In this
section, we will show that we can also generate interface states at multiple
bandgap orders simultaneously. By varying the unit-cell relative thickness
ratio, $\delta$, the bandgap amplitudes at all the orders are, in fact,
simultaneously altered. However, the closing and reopening of the bandgaps are
not coincident for every bandgap order. As a result, by changing $\delta$, we
can reach different combinations of bandgap symmetries, and so we can engineer
the formation of interface states at different bands.

Figure 5: Engineering of topological interface states. (a) Band inversion of
the acoustic bandgaps. The dots show the bandgaps opening and symmetries at
the given $\delta$. Inset: Symmetry of the two DBRs concatenated. There is
inversion only for the fourth bandgap. (b) Schematic of the unit cell
configurations at the interface between the two DBRs, with the dark blue and
green colors representing GaAs and the corresponding light colors representing
AlAs. (c) Simulated acoustic reflectivity spectra for two topological
resonators formed by two DBRs concatenated embedded in GaAs with
$\delta_{top}=-0.33$ and $\delta_{bottom}=+0.33$. Likewise in (d)-(f)
$\delta_{top}=-0.15$ and $\delta_{bottom}=-0.85$; (g)-(i) $\delta_{top}=-0.8$
and $\delta_{bottom}=+0.2$; and (j)-(l) $\delta_{top}=+0.4$ and
$\delta_{bottom}=+0.6$.

Figure 5 presents different conditions to generate interface states at
different gaps. In the first case (Figs. 5(a)-(c)), we optimize the GaAs/AlAs
thickness ratio to generate interface states at the second and fourth
bandgaps, with $\delta=\pm 0.33$ in each DBR. Fig. 5(a) shows the dependence
of the bandgaps on $\delta$, with the dashed vertical lines indicating the
$\delta$ values of each DBR ($\delta=\pm 0.33$), and the orange (blue) dots
corresponding to the symmetric (anti-symmetric) modes. Fig. 5(b) shows the
unit cell configurations at the interface for the chosen $\delta$, in which
the dark blue and green colors represent GaAs whereas light blue and green
represent AlAs. The inset at the right side of Fig. 5(a) shows the mode
symmetry of the two DBRs in each band. As shown, at the particular value of
$\delta$ chosen here, there is an inversion of symmetry at the second and the
fourth bandgap, whereas the third bandgap is closed (see black dots). In the
calculated acoustic reflectivity spectrum shown in Fig. 5(c), three bandgaps
can be seen. In these bandgaps, the band-inversion interface states are
present for the 2nd and 4th gaps, as indicated by the dips centered in the
high reflectivity regions at $\sim$18 GHz and $\sim$37 GHz, respectively. As
we already saw in Fig. 5(a), the third gap is closed for this combination of
$\delta$s in the DBRs, and so that gap is absent in the reflectivity spectrum.

Figures 5(d)-(f) present the conditions that generate interface states in the
third and fourth bandgap simultaneously. The structure is designed with
$\delta_{top/bottom}=-0.15/-0.85$ for the top and bottom DBRs. In comparison
to the previous case, the four bandgaps are open for both superlattices (see
panel (d)), resulting in four high reflectivity regions in the reflectivity
spectrum, as shown in panel (f). For this configuration of unit cells,
interface states are present in the third and fourth bandgaps. They are
induced by the inversion of symmetry of the modes around the respective
bandgaps, as shown in panel (d). On the contrary, there is no interface mode
in the first and second bandgaps, as they have the same band-edge symmetries
for both superlattices.

We note that the mode is not centered in the third bandgap, as shown in Fig.
5(f). To generate an interface that is centered in the bandgap, there are two
necessary conditions. First, the two DBRs should have a bandgap with equal
central frequency and, second, they should have the same bandwidth. The first
condition is required for topological robustness [38]. The second condition
results in similar evanescent decay lengths into both DBRs. In case the values
of $\delta$ for each superlattice are not equidistant from a band inversion
point, the interface state generated is not centered in the bandgap, as we can
see in Figs. 5(f) and (i) at the third and second bandgaps, respectively.
Figures 5(g)-(i) present the conditions to generate interface states in the
second and third bandgaps, associated to $\delta_{top/bottom}=-0.8/+0.2$. In
Fig. 5(g) the two DBRs have inverted symmetry at the targeted bandgaps,
whereas the mode symmetries at the fourth bandgap on both DBRs are the same,
even though they fall into different bandgap openings. This results in two
interface states at bandgap orders 2 and 3, as seen in Fig. 5(i). Figures
5(j)-(l) present the conditions to generate interface states only in the
fourth bandgap, with $\delta_{top/bottom}=+0.4/+0.6$. Despite the slight
disparity between the unit cells of the two superlattices in this arrangement,
the interface state is generated (Fig. 5(k)). Only the modes at the fourth
bandgap have inverted symmetries, whereas all the other gaps exhibit the same
mode symmetry. As a result, the acoustic reflectivity spectrum, displayed in
Fig. 5(l), presents three high reflectivity regions and one interface mode in
the fourth bandgap.

The first bandgap does not undergo any symmetry inversion of the topological
phase over the entire $\delta$ range, which makes it impossible to create an
interface state at this band. It is important to point out that this does not
mean that interface states cannot be conceived in the first bandgap. For
instance, one can tune the impedance of the materials to switch the symmetry
of the modes. [6, 29]

##  5 Hybrid topological resonators

The formation of interface states is not limited to the same bandgap order of
the constituting DBRs. In fact, the general rule for engineering topological
states is associated with the sign of the reflection phase as well as the
overlap of the high reflectivity regions from both reflectors. So far, we have
fulfilled these conditions and created interface modes at higher-order
bandgaps by concatenating two DBRs designed at the same acoustic frequency. In
this section, we extend this concept to generate topological states between
two DBRs designed at different fundamental frequencies, resulting in bandgaps
of different orders sharing the same frequency range.

Figures 6(a) and (b) display the band inversion diagrams of two superlattices,
S1 and S2, intended to have the first bandgap at different frequencies. The
first superlattice (S1) is designed to have a fundamental bandgap centered
$\sim$9.3 GHz (Fig. 6(a)), while the first bandgap of S2 is centered at
$\sim$14 GHz (Fig. 6(b)). These designs result in the third bandgap of S1
centered at the same frequency as the second bandgap of S2, which is
represented by the alignment of both bands in panels (a) and (b). Fig. 6(c)
shows the acoustic reflectivity spectra of the two DBRs associated with the
band structures in panels (a) and (b). The reflectivities are calculated for
values of $\delta$ in which all the bandgaps are open. We see that there is a
complete overlap of the high reflectivity regions around 28 GHz. By
concatenating two such superlattices with overlapping bandgaps of different
order, we can generate an interface state at the frequency in which they
overlap.

Figure 6: Hybrid topological acoustic resonator. (a),(b) Band inversion of the
acoustic bandgaps associated to the two concatenated superlattices, named (a)
S1 (green frame), and (b) S2 (blue frame). The third bandgap of S1 and the
second bandgap of S2 share the same central frequency, around 28 GHz. (c)
Simulated acoustic reflectivity of the two DBRs. The green (blue) line
corresponds to the band inversion diagram displayed in panel (a) (panel (b)).
(d),(e) Simulated acoustic reflectivity spectra for different concatenated
DBRs embedded in GaAs. The relative thickness ratio $\delta$ of the
corresponding DBRs is marked by dashed lines on panels (a) and (b). (f),(g)
Acoustic displacement $|u(z)|^{2}$ of the modes at the edge of the bandgaps
plotted on top of the unit cell for the two superlattices. The green (blue)
unit cell corresponds to the band inversion diagram displayed in panel (a)
(panel (b)), and the dark (light) colors represent GaAs and AlAs,
respectively.

Figs. 6(d) and (e) depict the calculated acoustic reflectivity spectra for two
different combinations of such DBRs. The chosen values of $\delta$ are marked
on the band inversion figures (Figs. 6(a) and (b)) with the vertical dashed
lines. The matching labels between panels (a) and (b) indicate the
corresponding acoustic reflectivity spectrum. The first superlattice on both
structures is the same, where $\delta=0.66$ corresponds to the thickness ratio
for which the third bandgap opening is maximized. On the other hand, the
thickness ratio of the second superlattice is chosen to maximize the second
bandgap, where $\delta=0.5$ ($\delta=-0.5$) correspond to the same (inverted)
band edge mode symmetries compared to the third order bandgap of S1. In both
cases (panels (d) and (e)), there are five high-reflectivity regions. The
regions below 25 GHz and above 35 GHz correspond to the individual bandgaps of
S1 and S2 that have no overlap. At 28 GHz, different bandgaps of the two DBRs
do overlap, but an interface state is either present (panel (d)) or absent
(panel (e)) in the two shown examples. Contrary to what one might have
expected based on the results of the previous sections, the combination of
superlattices supporting an interface state (panel (d)) have the same band
edge mode symmetries. Likewise, when the symmetry of the modes between S1 and
S2 is inverted, no interface state is generated. Therefore, the rule of
creating an interface state by band inversion cannot be blindly applied for
bandgaps of different orders.

To understand this, we investigate the relation between the generation of an
interface state and the acoustic displacement field in the unit cell of each
superlattice. In Fig. 6(f) and (g), we can see a schematic of the unit cells
of superlattices S1 and S2, associated with the acoustic reflectivity spectra
shown in Figs. 6(d) and (e). As before, the dark- and light-colored regions
correspond to GaAs and AlAs, respectively. On top of these superlattices, we
show the corresponding relevant acoustic displacements. More specifically, in
the top (bottom) panels of this schematic, we show the acoustic displacement
of the modes at the higher (lower) frequency band edges of the bandgap
centered at 28 GHz in each superlattice. As we can see in Fig. 6(f), there is
a discontinuity of the displacement field at the interface between these
superlattices. This discontinuity leads to the generation of an interface
state, even though the band edge modes have the same symmetry. On the other
hand, in Fig. 6(g), the displacement field at the interface between S1 and S2
is continuous, and so prevents the formation of an interface state. In
general, when the order of one bandgap is even, and the other one is odd, an
interface state is generated if both have the same symmetries. On the
contrary, if two odd or even bandgaps are concatenated, they must have
inverted symmetries to generate an interface state. The difference in phases
of the reflection coefficient does not only depend on the sum of the Zak
phases anymore but also on the order of each bandgap. The case of two DBRs
with different lattice parameters was briefly considered in Ref. [6].

##  6 Conclusions

We theoretically presented a method to generate acoustic interface states in
topological nanoacoustic resonators based on the band inversion principle. We
simulated a series of topological optophononic resonators with different
combinations of concatenated superlattices. By changing the thickness ratio of
GaAs and AlAs in the unit cell, we were able to control the symmetries of the
modes around each bandgap. In general, an interface state can be generated
when two superlattices with inverted symmetries are concatenated. Here, we
extended this principle to create interface states in high-order bandgaps. The
modes that we presented can be accessed experimentally in a Brillouin or pump-
probe experiment. We numerically discussed the Brillouin efficiency of
different combinations of superlattices forming topological states at the
third and fourth bandgap orders. The accessibility of these states in
Brillouin scattering experiments is directly associated with the unit cell
thickness ratio between GaAs and AlAs of both concatenated superlattices. In
addition, we studied the robustness of our structures against disorder and
compared them with Fabry-Perot resonators. The use of GaAs and AlAs enables
the study of electronic and optical resonances effects, [49, 50] and is
compatible with the integration with quantum dots and quantum wells. However,
the concepts presented in this work can be easily extended to other materials
platforms.

Moreover, we demonstrated that multiple topological acoustic interface states
can be generated simultaneously at different bandgap orders. Importantly, the
generated interface states are robust against disorder in the unit cell
thickness ratio across a broad frequency range that does not affect the
associated Zak phases. A promising extension of this system might thus arise
by introducing spatial periodicity to the system. Then, the platform developed
here could potentially enable the development and study of synthetic
dimensions, where high-order interface states would serve as an additional
lattice dimension for the topological system [51, 52]. Furthermore, we
demonstrated the presence of interface states in hybrid structures, i.e.,
structures combining two superlattices with bandgaps of different orders
centered at the same frequency. The interface states in high-order bandgaps
presented here can potentially be a useful tool to explore a full class of
hybrid topological resonators. One could, for instance, exploit the
versatility of hybrid topological resonators to generate multiple interface
states at higher-order frequency-matched bandgaps, that are difficult to
access by electronics or optics due to their respective dispersion relations.
Overall, our results constitute an important step in the development of
nanophononics for robust noise-insensitive communication, data processing, and
quantum technologies.

##  7 Acknowledgments

The authors gratefully acknowledge M. Esmann for fruitful discussions and
support at an early stage of the project. The authors acknowledge funding from
European Research Council Consolidator Grant No.101045089 (T-Recs). This work
was supported by the European Commission in the form of the H2020 FET
Proactive project No. 824140 (TOCHA).

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&+O\Big{(}\frac{L\beta_{1}M\sqrt{d}(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{n}}\Big{)}+O\Big{(}\frac{L(\beta_{1}M\sqrt{d}+\Delta_{\zeta}^{\eta})(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{\pi^{2}(1-\pi)m}}\Big{)}.\end{split}$

###### Proof of Lemma 13.

Using Lemma 13 with $\eta=8/9$, we obtain that

$\begin{split}&R_{\mathcal{S}_{T}}^{-}(\mathbf{g}_{\bm{\theta}})-R_{\mathcal{S}_{\text{wild}}^{\text{out}}}^{-}(\mathbf{g}_{\bm{\theta}})\\\
\leq&\frac{1.2L\delta(T)}{1-\delta(T)}+\frac{9L\beta_{1}(1-\pi)}{T\pi(1-\delta(T))}R_{\text{in}}^{*}+\frac{L\epsilon(n)}{\Delta(T)}+\frac{L\bar{\epsilon}(n)}{\pi\Delta(T)}+\frac{L\epsilon(m)}{\Delta(T)}\\\
+&\frac{8L\beta_{1}R^{*}_{\text{in}}}{\pi^{2}\Delta(T)T}O\big{(}\sqrt{\frac{1}{m}}\big{)}+\frac{L\bar{\epsilon}(m)}{{\pi}^{2}\Delta(T)}O\big{(}\sqrt{\frac{1}{m}}\big{)}+\frac{L\bar{\epsilon}(m)}{{\pi}\Delta(T)}+\frac{L\bar{\epsilon}(n)}{{\pi}^{2}\Delta(T)}O\big{(}\sqrt{\frac{1}{m}}\big{)},\end{split}$

where

$\epsilon(n)=O\Big{(}\frac{\beta_{1}M\sqrt{d}}{1-T/M^{\prime}}\sqrt{\frac{1}{{\pi}^{2}n}}\Big{)},~{}~{}~{}~{}\epsilon(m)=O\Big{(}\frac{{\beta_{1}M\sqrt{d}}+\sqrt{1-\pi}\Delta_{\zeta}^{\eta}/\pi}{1-T/M^{\prime}}\sqrt{\frac{1}{{\pi}^{2}(1-\pi)m}}\Big{)}.$
$\bar{\epsilon}(n)=O\Big{(}\frac{\beta_{1}M\sqrt{d}}{T}\sqrt{\frac{1}{n}}\Big{)},~{}~{}~{}\bar{\epsilon}(m)=O\Big{(}\frac{\beta_{1}M\sqrt{d}}{T}\sqrt{\frac{1}{(1-\pi)m}}\Big{)}.$

Using the condition that
$0<T<0.9M^{\prime}\min\\{1,\Delta_{\zeta}^{\eta}/\pi\\}$, we have

$\frac{1}{\Delta(T)}\big{[}\frac{1}{T}+\frac{1}{1-T/{M^{\prime}}}\big{]}\leq
O\big{(}\frac{T+1}{\min\\{1,\Delta_{\zeta}^{\eta}/\pi\\}T}\big{)}.$

Then, we obtain that

$\begin{split}R_{\mathcal{S}_{T}}^{-}(\mathbf{g}_{\bm{\theta}})-R_{\mathcal{S}_{\text{wild}}^{\text{out}}}^{-}&(\mathbf{g}_{\bm{\theta}})\leq\frac{1.2L}{1-\delta(T)}\delta(T)+\frac{9L\beta_{1}(1-\pi)}{T\pi(1-\delta(T))}R_{\text{in}}^{*}\\\
&+O\Big{(}\frac{L\beta_{1}M\sqrt{d}(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{n}}\Big{)}+O\Big{(}\frac{L(\beta_{1}M\sqrt{d}+\Delta_{\zeta}^{\eta})(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{\pi^{2}(1-\pi)m}}\Big{)}.\end{split}$

Using the similar strategy, we can obtain that

$\begin{split}R_{\mathcal{S}_{\text{wild}}^{\text{out}}}^{-}(\mathbf{g}_{\bm{\theta}})-R_{\mathcal{S}_{T}}^{-}&(\mathbf{g}_{\bm{\theta}})\leq\frac{1.2L}{1-\delta(T)}\delta(T)+L\delta(T)\\\
&+O\Big{(}\frac{L\beta_{1}M\sqrt{d}(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{n}}\Big{)}+O\Big{(}\frac{L(\beta_{1}M\sqrt{d}+\Delta_{\zeta}^{\eta})(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{\pi^{2}(1-\pi)m}}\Big{)}.\end{split}$

∎

######  Lemma 14.

Under the conditions of Theorem 1, when

$n=\Omega\Big{(}\frac{\tilde{M}^{2}d}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}^{2}}\Big{)},~{}~{}~{}m=\Omega\Big{(}\frac{\tilde{M}^{2}d+\Delta_{\zeta}^{\eta}}{\pi^{2}(1-\pi)\min\\{\pi,\Delta_{\zeta}^{\eta}\\}^{2}}\Big{)},$

with the probability at least $0.895$, for any
$0<T<0.9M^{\prime}\min\\{1,\Delta_{\zeta}^{\eta}/\pi\\}$,

$\begin{split}R_{\mathcal{S}_{T}}^{-}(\mathbf{g}_{\bm{\theta}})&-R_{\mathbb{P}_{\text{out}}}^{-}(\mathbf{g}_{\bm{\theta}})\leq\frac{1.2L}{1-\delta(T)}\delta(T)+\frac{9L\beta_{1}(1-\pi)}{T\pi(1-\delta(T))}R_{\text{in}}^{*}\\\
&+O\Big{(}\frac{L\beta_{1}M\sqrt{d}(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{n}}\Big{)}+O\Big{(}\frac{L\max\\{\beta_{1}M\sqrt{d},\sqrt{d^{\prime}},\Delta_{\zeta}^{\eta}\\}(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{\pi^{2}(1-\pi)m}}\Big{)},\end{split}$
$\begin{split}R_{\mathbb{P}_{\text{out}}}^{-}(\mathbf{g}_{\bm{\theta}})&-R_{\mathcal{S}_{T}}^{-}(\mathbf{g}_{\bm{\theta}})\leq\frac{1.2L}{1-\delta(T)}\delta(T)+L\delta(T)\\\
&+O\Big{(}\frac{L\beta_{1}M\sqrt{d}(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{n}}\Big{)}+O\Big{(}\frac{L\max\\{\beta_{1}M\sqrt{d},\sqrt{d^{\prime}},\Delta_{\zeta}^{\eta}\\}(1+T)}{\min\\{\pi,\Delta_{\zeta}^{\eta}\\}T}\sqrt{\frac{1}{\pi^{2}(1-\pi)m}}\Big{)}.\end{split}$

###### Proof.

By Lemmas 1 and 3, under the condition of this lemma, we can obtain that with
the high probability,

$\big{|}R^{-}_{\mathbb{P}_{\text{out}}}(\mathbf{g}_{\bm{\theta}})-R^{-}_{\mathcal{S}_{\text{wild}}^{\text{out}}}(\mathbf{g}_{\bm{\theta}})\big{|}\leq
O\big{(}L\sqrt{\frac{d^{\prime}}{\pi m}}\big{)}.$

Then by Lemma 13, we can prove this lemma. ∎

##  Appendix F Empirical Verification on the Main Theorems

Verification on the regulatory conditions. In Table 5, we provide empirical
verification on whether the distribution discrepancy $\zeta$ satisfies the
necessary regulatory condition in Theorem 2, i.e., $\zeta\geq
2.011\sqrt{8\beta_{1}R_{\rm in}^{*}}+1.011\sqrt{\pi}$. We use Cifar-100 as ID
and Textures as the wild OOD data.

Since $R^{*}_{{\rm in}}$ is the optimal ID risk, i.e., $R^{*}_{{\rm
in}}=\min_{\mathbf{w}\in\mathcal{W}}\mathbb{E}_{(\mathbf{x},y)\sim\mathbb{P}_{\mathcal{X}\mathcal{Y}}}\ell(\mathbf{h}_{\mathbf{w}}(\mathbf{x}),y)$,
it can be a small value close to 0 in over-parametrized neural networks (Frei
et al., 2022; Bartlett et al., 2020). Therefore, we can omit the value of
$2.011\sqrt{8\beta_{1}R_{\rm in}^{*}}$. The empirical result shows that
$\zeta$ can easily satisfy the regulatory condition in Theorem 2, which means
our bound is useful in practice.

Table 5: Discrepancy value $\zeta$ with different ratios $\pi$. $\pi$ | 0.05 | 0.1 | 0.2 | 0.5 | 0.7 | 0.9 | 1.0
---|---|---|---|---|---|---|---
$\zeta$ | 0.91 | 1.09 | 1.43 | 2.49 | 3.16 | 3.86 | 4.18
$1.011\sqrt{\pi}$ | 0.23 | 0.32 | 0.45 | 0.71 | 0.84 | 0.96 | 1.01

Verification on the filtering errors and OOD detection results with varying
$\pi$. In Table 6, we empirically verify the value of ${\rm ERR}_{\rm out}$
and ${\rm ERR}_{\rm in}$ in Theorem 1 and the corresponding OOD detection
results with various mixing ratios $\pi$. We use Cifar-100 as ID and Textures
as the wild OOD data. The result aligns well with our observation of the
bounds presented in Section 4.1 of the main paper.

Table 6: The values of ${\rm ERR}_{\rm in}$, ${\rm ERR}_{\rm out}$ and the OOD detection results with various mixing ratios $\pi$. $\pi$ | 0.05 | 0.1 | 0.2 | 0.5 | 0.7 | 0.9 | 1.0
---|---|---|---|---|---|---|---
${\rm ERR}_{\rm out}$ | 0.37 | 0.30 | 0.22 | 0.20 | 0.23 | 0.26 | 0.29
${\rm ERR}_{\rm in}$ | 0.031 | 0.037 | 0.045 | 0.047 | 0.047 | 0.048 | 0.048
FPR95 | 5.77 | 5.73 | 5.71 | 5.64 | 5.79 | 5.88 | 5.92

##  Appendix G Additional Experimental Details

Dataset details. For Table 1, following WOODS (Katz-Samuels et al., 2022), we
split the data as follows: We use 70% of the OOD datasets (including Textures,
Places365, Lsun-Resize and Lsun-C) for the OOD data in the wild. We use the
remaining samples for testing-time OOD detection. For Svhn, we use the
training set for the OOD data in the wild and use the test set for evaluation.

Training details. Following WOODS (Katz-Samuels et al., 2022), we use Wide
ResNet (Zagoruyko & Komodakis, 2016) with 40 layers and widen factor of 2 for
the classification model $\mathbf{h}_{\mathbf{w}}$. We train the ID classifier
$\mathbf{h}_{\mathbf{w}}$ using stochastic gradient descent with a momentum of
0.9, weight decay of 0.0005, and an initial learning rate of 0.1. We train for
100 epochs using cosine learning rate decay, a batch size of 128, and a
dropout rate of 0.3. For the OOD classifier $\mathbf{g}_{\bm{\theta}}$, we
load the pre-trained ID classifier of $\mathbf{h}_{\mathbf{w}}$ and add an
additional linear layer which takes in the penultimate-layer features for
binary classification. We set the initial learning rate to 0.001 and fine-tune
for 100 epochs by Eq. 8. We add the binary classification loss to the ID
classification loss and set the loss weight for binary classification to 10.
The other details are kept the same as training $\mathbf{h}_{\mathbf{w}}$.

##  Appendix H Additional Results on Cifar-10

In Table 7, we compare our SAL with baselines with the ID data to be Cifar-10,
where the strong performance of SAL still holds.

Table 7: OOD detection performance on Cifar-10 as ID. All methods are trained
on Wide ResNet-40-2 for 100 epochs with $\pi=0.1$. For each dataset, we create
corresponding wild mixture distribution
$\mathbb{P}_{\text{wild}}:=(1-\pi)\mathbb{P}_{\text{in}}+\pi\mathbb{P}_{\text{out}}$
for training and test on the corresponding OOD dataset. Values are percentages
averaged over 10 runs. Bold numbers highlight the best results. Table format
credit to (Katz-Samuels et al., 2022).

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
With $\mathbb{P}_{\text{in}}$ only
MSP | 48.49 | 91.89 | 59.48 | 88.20 | 30.80 | 95.65 | 52.15 | 91.37 | 59.28 | 88.50 | 50.04 | 91.12 | 94.84
ODIN | 33.35 | 91.96 | 57.40 | 84.49 | 15.52 | 97.04 | 26.62 | 94.57 | 49.12 | 84.97 | 36.40 | 90.61 | 94.84
Mahalanobis | 12.89 | 97.62 | 68.57 | 84.61 | 39.22 | 94.15 | 42.62 | 93.23 | 15.00 | 97.33 | 35.66 | 93.34 | 94.84
Energy | 35.59 | 90.96 | 40.14 | 89.89 | 8.26 | 98.35 | 27.58 | 94.24 | 52.79 | 85.22 | 32.87 | 91.73 | 94.84
KNN | 24.53 | 95.96 | 25.29 | 95.69 | 25.55 | 95.26 | 27.57 | 94.71 | 50.90 | 89.14 | 30.77 | 94.15 | 94.84
ReAct | 40.76 | 89.57 | 41.44 | 90.44 | 14.38 | 97.21 | 33.63 | 93.58 | 53.63 | 86.59 | 36.77 | 91.48 | 94.84
DICE | 35.44 | 89.65 | 46.83 | 86.69 | 6.32 | 98.68 | 28.93 | 93.56 | 53.62 | 82.20 | 34.23 | 90.16 | 94.84
ASH | 6.51 | 98.65 | 48.45 | 88.34 | 0.90 | 99.73 | 4.96 | 98.92 | 24.34 | 95.09 | 17.03 | 96.15 | 94.84
CSI | 17.30 | 97.40 | 34.95 | 93.64 | 1.95 | 99.55 | 12.15 | 98.01 | 20.45 | 95.93 | 17.36 | 96.91 | 94.17
KNN+ | 2.99 | 99.41 | 24.69 | 94.84 | 2.95 | 99.39 | 11.22 | 97.98 | 9.65 | 98.37 | 10.30 | 97.99 | 93.19
With $\mathbb{P}_{\text{in}}$ and $\mathbb{P}_{\text{wild}}$
OE | 0.85 | 99.82 | 23.47 | 94.62 | 1.84 | 99.65 | 0.33 | 99.93 | 10.42 | 98.01 | 7.38 | 98.41 | 94.07
Energy (w/ OE) | 4.95 | 98.92 | 17.26 | 95.84 | 1.93 | 99.49 | 5.04 | 98.83 | 13.43 | 96.69 | 8.52 | 97.95 | 94.81
WOODS | 0.15 | 99.97 | 12.49 | 97.00 | 0.22 | 99.94 | 0.03 | 99.99 | 5.95 | 98.79 | 3.77 | 99.14 | 94.84
SAL | 0.02 | 99.98 | 2.57 | 99.24 | 0.07 | 99.99 | 0.01 | 99.99 | 0.90 | 99.74 | 0.71 | 99.78 | 93.65
(Ours) |  ±0.00 |  ±0.00 |  ±0.03 |  ±0.00 |  ±0.01 |  ±0.00 |  ±0.00 |  ±0.00 |  ±0.02 |  ±0.01 |  ±0.01 |  ±0.00 |  ±0.57

##  Appendix I Additional Results on Unseen OOD Datasets

In Table 8, we evaluate SAL on unseen OOD datasets, which are different from
the OOD data we use in the wild. Here we consistently use 300K Random Images
as the unlabeled wild dataset and Cifar-10 as labeled in-distribution data. We
use the 5 different OOD datasets (Textures, Places365, Lsun-Resize, Svhn and
Lsun-C) for evaluation. When evaluating on 300K Random Images, we use 99% of
the 300K Random Images dataset (Hendrycks et al., 2019) as the wild OOD data
and the remaining 1% of the dataset for evaluation. $\pi$ is set to 0.1. We
observe that SAL can perform competitively on unseen datasets as well,
compared to the most relevant baseline WOODS.

Table 8: Evaluation on unseen OOD datasets. We use Cifar-10 as ID and 300K
Random Images as the wild data. All methods are trained on Wide ResNet-40-2
for 50 epochs. Bold numbers highlight the best results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | 300K Rand. Img.
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
OE | 13.18 | 97.34 | 30.54 | 93.31 | 5.87 | 98.86 | 14.32 | 97.44 | 25.69 | 94.35 | 30.69 | 92.80 | 94.21
Energy (w/ OE) | 8.52 | 98.13 | 23.74 | 94.26 | 2.78 | 99.38 | 9.05 | 98.13 | 22.32 | 94.72 | 24.59 | 93.99 | 94.54
WOODS | 5.70 | 98.54 | 19.14 | 95.74 | 1.31 | 99.66 | 4.13 | 99.01 | 17.92 | 96.43 | 19.82 | 95.52 | 94.74
SAL | 4.94 | 97.53 | 14.76 | 96.25 | 2.73 | 98.23 | 3.46 | 98.15 | 11.60 | 97.21 | 10.20 | 97.23 | 93.48

Following (He et al., 2023), we use the Cifar-100 as ID, Tiny ImageNet-crop
(TINc)/Tiny ImageNet-resize (TINr) dataset as the OOD in the wild dataset and
TINr/TINc as the test OOD. The comparison with baselines is shown below, where
the strong performance of SAL still holds.

Table 9: Additional results on unseen OOD datasets with Cifar-100 as ID. Bold
numbers are superior results.

Methods | OOD Datasets
---|---
TINr | TINc
FPR95 | AUROC | FPR95 | AUROC
STEP | 72.31 | 74.59 | 48.68 | 91.14
TSL | 57.52 | 82.29 | 29.48 | 94.62
SAL (Ours) | 43.11 | 89.17 | 19.30 | 96.29

##  Appendix J Additional Results on Near OOD Detection

In this section, we investigate the performance of SAL on near OOD detection,
which is a more challenging OOD detection scenario where the OOD data has a
closer distribution to the in-distribution. Specifically, we use the Cifar-10
as the in-distribution data and Cifar-100 training set as the OOD data in the
wild. During test time, we use the test set of Cifar-100 as the OOD for
evaluation. With a mixing ratio $\pi$ of 0.1, our SAL achieves an FPR95 of
24.51% and AUROC of 95.55% compared to 38.92% (FPR95) and 93.27% (AUROC) of
WOODS.

In addition, we study near OOD detection in a different data setting, i.e.,
the first 50 classes of Cifar-100 as ID and the last 50 classes as OOD. The
comparison with the most competitive baseline WOODS is reported as follows.

Table 10:  Near OOD detection with the first 50 classes of Cifar-100 as ID and the last 50 classes as OOD. Bold numbers are superior results.  Methods | OOD dataset
---|---
Cifar-50
FPR95 | AUROC | ID ACC
WOODS | 41.28 | 89.74 | 74.17
SAL | 29.71 | 93.13 | 73.86

##  Appendix K Additional Results on Using Multiple Singular Vectors

In this section, we ablate on the effect of using $c$ singular vectors to
calculate the filtering score (Eq. 6). Specifically, we calculate the scores
by projecting the gradient
$\nabla\ell(\mathbf{h}_{\mathbf{w}_{\mathcal{S}^{\text{in}}}}\big{(}\tilde{\mathbf{x}}_{i}),\widehat{y}_{\tilde{\mathbf{x}}_{i}})-\bar{\nabla}$
for the wild data $\tilde{\mathbf{x}}_{i}$ to each of the singular vectors.
The final filtering score is the average over the $c$ scores. The result is
summarized in Table 11. We observe that using the top 1 singular vector for
projection achieves the best performance. As revealed in Eq. 7, the top 1
singular vector $\mathbf{v}$ maximizes the total distance from the projected
gradients (onto the direction of $\mathbf{v}$) to the origin (sum over all
points in $\mathcal{S}_{\rm wild}$), where outliers lie approximately close to
and thus leads to a better separability between the ID and OOD in the wild.

Table 11: The effect of the number of singular vectors used for the filtering score. Models are trained on Wide ResNet-40-2 for 100 epochs with $\pi=0.1$. We use Textures as the wild OOD data and Cifar-100 as the ID. Number of singular vectors $c$ | FPR95 | AUROC
---|---|---
1 | 5.73 | 98.65
2 | 6.28 | 98.42
3 | 6.93 | 98.43
4 | 7.07 | 98.37
5 | 7.43 | 98.27
6 | 7.78 | 98.22

##  Appendix L Additional Results on Class-agnostic SVD

In this section, we evaluate our SAL by using class-agnostic SVD as opposed to
class-conditional SVD as described in Section 3.1 of the main paper.
Specifically, we maintain a class-conditional reference gradient
$\bar{\nabla}_{k}$, one for each class $k\in[1,K]$, estimated on ID samples
belonging to class $k$. Different from calculating the singular vectors based
on gradient matrix with $\mathbf{G}_{k}$ (containing gradient vectors of wild
samples being predicted as class $k$), we formulate a single gradient matrix
$\mathbf{G}$ where each row is the vector
$\nabla\ell(\mathbf{h}_{\mathbf{w}_{\mathcal{S}^{\text{in}}}}\big{(}\tilde{\mathbf{x}}_{i}),\widehat{y}_{\tilde{\mathbf{x}}_{i}})-\bar{\nabla}_{\widehat{y}_{\tilde{\mathbf{x}}_{i}}}$,
for $\tilde{\mathbf{x}}_{i}\in\mathcal{S}_{\rm wild}$. The result is shown in
Table 12, which shows a similar performance compared with using class-
conditional SVD.

Table 12: The effect of using class-agnostic SVD. Models are trained on Wide
ResNet-40-2 for 100 epochs with $\pi=0.1$. Cifar-100 is the in-distribution
data. Bold numbers are superior results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
SAL (Class-agnostic SVD) | 0.12 | 99.43 | 3.27 | 99.21 | 0.04 | 99.92 | 0.03 | 99.27 | 5.18 | 98.77 | 1.73 | 99.32 | 73.31
SAL | 0.07 | 99.95 | 3.53 | 99.06 | 0.06 | 99.94 | 0.02 | 99.95 | 5.73 | 98.65 | 1.88 | 99.51 | 73.71

##  Appendix M Additional Results on Post-hoc Filtering Score

We investigate the importance of training the binary classifier with the
filtered candidate outliers for OOD detection in Tables 13 and 14.
Specifically, we calculate our filtering score directly for the test ID and
OOD data on the model trained on the labeled ID set $\mathcal{S}^{\text{in}}$
only. The results are shown in the row ”SAL (Post-hoc)” in Tables 13 and 14.
Without explicit knowledge of the OOD data, the OOD detection performance
degrades significantly compared to training an additional binary classifier (a
15.73% drop on FPR95 for Svhn with Cifar-10 as ID). However, the post-hoc
filtering score can still outperform most of the baselines that use
$\mathbb{P}_{\text{in}}$ only (_c.f._ Table 1), showcasing its effectiveness.

Table 13: OOD detection results of using post-hoc filtering score on Cifar-10
as ID. SAL is trained on Wide ResNet-40-2 for 100 epochs with $\pi=0.1$. Bold
numbers are superior results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
SAL (Post-hoc) | 15.75 | 93.09 | 23.18 | 86.35 | 6.28 | 96.72 | 15.59 | 89.83 | 23.63 | 87.72 | 16.89 | 90.74 | 94.84
SAL | 0.02 | 99.98 | 2.57 | 99.24 | 0.07 | 99.99 | 0.01 | 99.99 | 0.90 | 99.74 | 0.71 | 99.78 | 93.65

Table 14: OOD detection results of using post-hoc filtering score on Cifar-100
as ID. SAL is trained on Wide ResNet-40-2 for 100 epochs with $\pi=0.1$. Bold
numbers are superior results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
SAL (post-hoc) | 39.75 | 81.47 | 35.94 | 84.53 | 23.22 | 90.90 | 32.59 | 87.12 | 36.38 | 83.25 | 33.58 | 85.45 | 75.96
SAL | 0.07 | 99.95 | 3.53 | 99.06 | 0.06 | 99.94 | 0.02 | 99.95 | 5.73 | 98.65 | 1.88 | 99.51 | 73.71

##  Appendix N Additional Results on Leveraging the Candidate ID data

In this section, we investigate the effect of incorporating the filtered wild
data which has a score smaller than the threshold $T$ (candidate ID data) for
training the binary classifier $\mathbf{g}_{\bm{\theta}}$. Specifically, the
candidate ID data and the labeled ID data are used jointly to train the binary
classifier. The comparison with SAL on CIFAR-100 is shown as follows:

Table 15: OOD detection results of selecting candidate ID data for training on
Cifar-100 as ID. SAL is trained on Wide ResNet-40-2 for 100 epochs with
$\pi=0.1$. Bold numbers are superior results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
Candidate ID data | 1.23 | 99.87 | 2.62 | 99.18 | 0.04 | 99.95 | 0.02 | 99.91 | 4.71 | 98.97 | 1.72 | 99.58 | 73.83
SAL (Ours) | 0.07 | 99.95 | 3.53 | 99.06 | 0.06 | 99.94 | 0.02 | 99.95 | 5.73 | 98.65 | 1.88 | 99.51 | 73.71

The result of selecting candidate ID data (and combine with labeled ID data)
shows slightly better performance, which echoes our theory that the
generalization bound of the OOD detector will be better if we have more ID
training data (Theorem 3).

##  Appendix O Analysis on Using Random Labels

We present the OOD detection result of replacing the predicted labels with the
random labels for the wild data as follows. The other experimental details are
kept the same as SAL.

Table 16: OOD detection results of using random labels for the wild data on
Cifar-100 as ID. SAL is trained on Wide ResNet-40-2 for 100 epochs with
$\pi=0.1$. Bold numbers are superior results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
w/ Random labels | 39.36 | 89.31 | 77.98 | 78.31 | 47.46 | 88.90 | 67.28 | 80.23 | 54.86 | 86.92 | 57.39 | 84.73 | 73.68
SAL (Ours) | 0.07 | 99.95 | 3.53 | 99.06 | 0.06 | 99.94 | 0.02 | 99.95 | 5.73 | 98.65 | 1.88 | 99.51 | 73.71

As we can observe, using the random labels leads to worse OOD detection
performance because the gradient of the wild data can be wrong. In our
theoretical analysis (Theorem 5), we have proved that using the predicted
label can lead to a good separation of the wild ID and OOD data. However, the
analysis using random labels might hold since it violates the assumption
(Definitions 2 and 3) that the expected gradient of ID data should be
different from that of wild data.

##  Appendix P Details of the Illustrative Experiments on the Impact of
Predicted Labels

For calculating the filtering accuracy, SAL is trained on Wide ResNet-40-2 for
100 epochs with $\pi=0.1$ on two separate ID datasets. The other training
details are kept the same as Section 5.1 and Appendix G.

##  Appendix Q Details of Figure 2

For Figure 2 in the main paper, we generate the in-distribution data from
three multivariate Gaussian distributions, forming three classes. The mean
vectors are set to $[-2,0],[2,0]$ and $[0,2\sqrt{3}]$, respectively. The
covariance matrix for all three classes is set to
$\left[\begin{array}[]{ll}0.25&0\\\ 0&0.25\end{array}\right]$. For each class,
we generate $1,000$ samples.

For wild scenario 1, we generate the outlier data in the wild by sampling
$100,000$ data points from a multivariate Gaussian
$\mathcal{N}([0,\frac{2}{\sqrt{3}}],7\cdot\mathbf{I})$ where $\mathbf{I}$ is
$2\times 2$ identity matrix, and only keep the $1,000$ data points that have
the largest distance to the mean vector $[0,\frac{2}{\sqrt{3}}]$. For wild
scenario 2, we generate the outlier data in the wild by sampling $1,000$ data
points from a multivariate Gaussian
$\mathcal{N}([10,\frac{2}{\sqrt{3}}],0.25\cdot\mathbf{I})$. For the in-
distribution data in the wild, we sample $3,000$ data points per class from
the same three multivariate Gaussian distributions as mentioned before.

##  Appendix R Software and Hardware

We run all experiments with Python 3.8.5 and PyTorch 1.13.1, using NVIDIA
GeForce RTX 2080Ti GPUs.

##  Appendix S Results with Varying Mixing Ratios

We provide additional results of SAL with varying $\pi$, i.e., 0.05, 0.2, 0.5,
0.9, and contrast with the baselines, which are shown below (Cifar-100 as the
in-distribution dataset). We found that the advantage of SAL still holds.

Table 17: OOD detection results with multiple mixing ratios $\pi$ with
Cifar-100 as ID. SAL is trained on Wide ResNet-40-2 for 100 epochs. Bold
numbers are superior results.

Methods | OOD Datasets | ID ACC |
---|---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures |
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC |
| $\pi=0.05$ |  |
OE | 2.78 | 98.84 | 63.63 | 80.22 | 6.73 | 98.37 | 2.06 | 99.19 | 32.86 | 90.88 | 71.98 |
Energy w/ OE | 2.02 | 99.17 | 56.18 | 83.33 | 4.32 | 98.42 | 3.96 | 99.29 | 40.41 | 89.80 | 73.45 |
WOODS | 0.26 | 99.89 | 32.71 | 90.01 | 0.64 | 99.77 | 0.79 | 99.10 | 12.26 | 94.48 | 74.15 |
SAL (Ours) | 0.17 | 99.90 | 6.21 | 96.87 | 0.94 | 99.79 | 0.84 | 99.37 | 5.77 | 97.12 | 73.99 |
| $\pi=0.2$ |  |
OE | 2.59 | 98.90 | 55.68 | 84.36 | 4.91 | 99.02 | 1.97 | 99.37 | 25.62 | 93.65 | 73.72 |
Energy w/ OE | 1.79 | 99.25 | 47.28 | 86.78 | 4.18 | 99.00 | 3.15 | 99.35 | 36.80 | 91.48 | 73.91 |
WOODS | 0.22 | 99.82 | 29.78 | 91.28 | 0.52 | 99.79 | 0.89 | 99.56 | 10.06 | 95.23 | 73.49 |
SAL (Ours) | 0.08 | 99.92 | 2.80 | 99.31 | 0.05 | 99.94 | 0.02 | 99.97 | 5.71 | 98.71 | 73.86 |
| $\pi=0.5$ |  |
OE | 2.86 | 99.05 | 40.21 | 88.75 | 4.13 | 99.05 | 1.25 | 99.38 | 22.86 | 94.63 | 73.38 |
Energy w/ OE | 2.71 | 99.34 | 34.82 | 90.05 | 3.27 | 99.18 | 2.54 | 99.23 | 30.16 | 94.76 | 72.76 |
WOODS | 0.17 | 99.80 | 21.87 | 93.73 | 0.48 | 99.61 | 1.24 | 99.54 | 9.95 | 95.97 | 73.91 |
SAL (Ours) | 0.02 | 99.98 | 1.27 | 99.62 | 0.04 | 99.96 | 0.01 | 99.99 | 5.64 | 99.16 | 73.77 |
| $\pi=0.9$ |  |
OE | 0.84 | 99.36 | 19.78 | 96.29 | 1.64 | 99.57 | 0.51 | 99.75 | 12.74 | 94.95 | 72.02 |
Energy w/ OE | 0.97 | 99.64 | 17.52 | 96.53 | 1.36 | 99.73 | 0.94 | 99.59 | 14.01 | 95.73 | 73.62 |
WOODS | 0.05 | 99.98 | 11.34 | 95.83 | 0.07 | 99.99 | 0.03 | 99.99 | 6.72 | 98.73 | 73.86 |
SAL (Ours) | 0.03 | 99.99 | 2.79 | 99.89 | 0.05 | 99.99 | 0.01 | 99.99 | 5.88 | 99.53 | 74.01 |

##  Appendix T Comparison with Weakly Supervised OOD Detection Baselines

We have additionally compared with the two related works (TSL (He et al.,
2023) and STEP (Zhou et al., 2021)). To ensure a fair comparison, we strictly
follow the experimental setting in TSL, and rerun SAL under the identical
setup. The comparison on Cifar-100 is shown as follows.

Table 18: Comparison with relevant baselines on Cifar-100. Bold numbers are
superior results.

Methods | OOD Datasets
---|---
Lsun-C | Lsun-Resize
FPR95 | AUROC | FPR95 | AUROC
STEP | 0.00 | 99.99 | 9.81 | 97.87
TSL | 0.00 | 100.00 | 1.76 | 99.57
SAL (Ours) | 0.00 | 99.99 | 0.58 | 99.95

##  Appendix U Additional Results on Different Backbones

We have additionally tried ResNet-18 and ResNet-34 as the network
architectures—which are among the most used in OOD detection literature. The
comparison with the baselines on Cifar-100 is shown in the following tables,
where SAL outperforms all the baselines across different architectures. These
additional results support the effectiveness of our approach.

Table 19: OOD detection performance on Cifar-100 as ID. All methods are
trained on ResNet-18 for 100 epochs. For each dataset, we create corresponding
wild mixture distribution
$\mathbb{P}_{\text{wild}}=(1-\pi)\mathbb{P}_{\text{in}}+\pi\mathbb{P}_{\text{out}}$
for training and test on the corresponding OOD dataset. Bold numbers highlight
the best results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
With $\mathbb{P}_{\text{in}}$ only
MSP | 81.32 | 77.74 | 83.06 | 74.47 | 70.11 | 83.51 | 82.46 | 75.73 | 85.11 | 73.36 | 80.41 | 76.96 | 78.67
ODIN | 40.94 | 93.29 | 87.71 | 71.46 | 28.72 | 94.51 | 79.61 | 82.13 | 83.63 | 72.37 | 64.12 | 82.75 | 78.67
Mahalanobis | 22.44 | 95.67 | 92.66 | 61.39 | 68.90 | 86.30 | 23.07 | 94.20 | 62.39 | 79.39 | 53.89 | 83.39 | 78.67
Energy | 81.74 | 84.56 | 82.23 | 76.68 | 34.78 | 93.93 | 73.57 | 82.99 | 85.87 | 74.94 | 71.64 | 82.62 | 78.67
KNN | 83.62 | 72.76 | 82.09 | 80.03 | 65.96 | 84.82 | 71.05 | 81.24 | 76.88 | 77.90 | 75.92 | 79.35 | 78.67
ReAct | 70.81 | 88.24 | 81.33 | 76.49 | 39.99 | 92.51 | 54.47 | 89.56 | 59.15 | 87.96 | 61.15 | 86.95 | 78.67
DICE | 54.65 | 88.84 | 79.58 | 77.26 | 0.93 | 99.74 | 49.40 | 91.04 | 65.04 | 76.42 | 49.92 | 86.66 | 78.67
CSI | 49.98 | 89.57 | 82.87 | 75.64 | 76.39 | 80.38 | 74.21 | 83.34 | 58.23 | 81.04 | 68.33 | 81.99 | 74.23
KNN+ | 43.21 | 90.21 | 84.62 | 74.21 | 50.12 | 82.48 | 76.92 | 80.81 | 63.21 | 84.91 | 63.61 | 82.52 | 77.03
With $\mathbb{P}_{\text{in}}$ and $\mathbb{P}_{\text{wild}}$
OE | 3.29 | 97.93 | 62.90 | 80.23 | 7.07 | 95.93 | 4.06 | 97.98 | 33.27 | 90.03 | 22.12 | 92.42 | 74.89
Energy (w/ OE) | 3.12 | 94.27 | 59.38 | 82.19 | 9.12 | 91.23 | 7.28 | 95.39 | 43.92 | 90.11 | 24.56 | 90.64 | 77.92
WOODS | 3.92 | 96.92 | 33.92 | 86.29 | 5.19 | 94.23 | 2.95 | 96.23 | 11.95 | 94.65 | 11.59 | 93.66 | 77.54
SAL | 2.29 | 97.96 | 6.29 | 96.66 | 3.92 | 97.81 | 4.87 | 97.10 | 8.28 | 95.95 | 5.13 | 97.10 | 77.71

Table 20: OOD detection performance on Cifar-100 as ID. All methods are
trained on ResNet-34 for 100 epochs. For each dataset, we create corresponding
wild mixture distribution
$\mathbb{P}_{\text{wild}}=(1-\pi)\mathbb{P}_{\text{in}}+\pi\mathbb{P}_{\text{out}}$
for training and test on the corresponding OOD dataset. Bold numbers highlight
the best results.

Methods | OOD Datasets | ID ACC
---|---|---
Svhn | Places365 | Lsun-C | Lsun-Resize | Textures | Average
FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC | FPR95 | AUROC
With $\mathbb{P}_{\text{in}}$ only
MSP | 78.89 | 79.80 | 84.38 | 74.21 | 83.47 | 75.28 | 84.61 | 74.51 | 86.51 | 72.53 | 83.12 | 75.27 | 79.04
ODIN | 70.16 | 84.88 | 82.16 | 75.19 | 76.36 | 80.10 | 79.54 | 79.16 | 85.28 | 75.23 | 78.70 | 79.11 | 79.04
Mahalanobis | 87.09 | 80.62 | 84.63 | 73.89 | 84.15 | 79.43 | 83.18 | 78.83 | 61.72 | 84.87 | 80.15 | 79.53 | 79.04
Energy | 66.91 | 85.25 | 81.41 | 76.37 | 59.77 | 86.69 | 66.52 | 84.49 | 79.01 | 79.96 | 70.72 | 82.55 | 79.04
KNN | 81.12 | 73.65 | 79.62 | 78.21 | 63.29 | 85.56 | 73.92 | 79.77 | 73.29 | 80.35 | 74.25 | 79.51 | 79.04
ReAct | 82.85 | 70.12 | 81.75 | 76.25 | 80.70 | 83.03 | 67.40 | 83.28 | 74.60 | 81.61 | 77.46 | 78.86 | 79.04
DICE | 83.55 | 72.49 | 85.05 | 75.92 | 94.05 | 73.59 | 75.20 | 80.90 | 79.80 | 77.83 | 83.53 | 76.15 | 79.04
CSI | 44.53 | 92.65 | 79.08 | 76.27 | 75.58 | 83.78 | 76.62 | 84.98 | 61.61 | 86.47 | 67.48 | 84.83 | 77.89
KNN+ | 39.23 | 92.78 | 80.74 | 77.58 | 48.99 | 89.30 | 74.99 | 82.69 | 57.15 | 88.35 | 60.22 | 86.14 | 78.32
With $\mathbb{P}_{\text{in}}$ and $\mathbb{P}_{\text{wild}}$
OE | 2.11 | 98.23 | 60.12 | 83.22 | 6.08 | 96.34 | 3.94 | 98.13 | 30.00 | 92.27 | 20.45 | 93.64 | 75.72
Energy (w/ OE) | 1.94 | 95.03 | 68.84 | 85.94 | 7.66 | 92.04 | 6.86 | 97.63 | 40.82 | 93.07 | 25.22 | 92.74 | 78.75
WOODS | 2.08 | 97.33 | 25.37 | 88.93 | 4.26 | 97.74 | 1.05 | 97.30 | 8.85 | 96.86 | 8.32 | 95.63 | 78.97
SAL | 0.98 | 99.94 | 2.98 | 99.08 | 0.07 | 99.94 | 0.03 | 99.96 | 4.01 | 98.83 | 1.61 | 99.55 | 78.01

##  Appendix V Broader Impact

Our project aims to improve the reliability and safety of modern machine
learning models. From the theoretical perspective, our analysis can facilitate
and deepen the understanding of the effect of unlabeled wild data for OOD
detection. In Appendix F, we properly verify the necessary conditions and the
value of our error bound using real-world datasets. Hence, we believe our
theoretical framework has a broad utility and significance.

From the practical side, our study can lead to direct benefits and societal
impacts, particularly when the wild data is abundant in the models’ operating
environment, such as in safety-critical applications i.e., autonomous driving
and healthcare data analysis. Our study does not involve any human subjects or
violation of legal compliance. We do not anticipate any potentially harmful
consequences to our work. Through our study and releasing our code, we hope to
raise stronger research and societal awareness towards the problem of
exploring unlabeled wild data for out-of-distribution detection in real-world
settings.