# Observation of the In-plane Anomalous Hall Effect induced by Octupole in
Magnetization Space

Wenzhi Peng1,2# Zheng Liu2,3# Haolin Pan1,2 Peng Wang1,2 Yulong Chen1,2
Jiachen Zhang1,2 Xuhao Yu1,2 Jinhui Shen1,2 Mingmin Yang1,4 Qian Niu2,3∗ Yang
Gao2,3∗ Dazhi Hou1,2∗

1ICQD, School of Emerging Technology, University of Science and Technology of
China, Hefei, Anhui 230026, China
2Department of Physics, University of Science and Technology of China, Hefei,
Anhui 230026, China
3CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China
4Hefei National Laboratory, Hefei, Anhui 230088, China.

∗Corresponding author. Email: dazhi@ustc.edu.cn(D.H.);
ygao87@ustc.edu.cn(Y.G.);
niuqian@ustc.edu.cn(Q.N.)
# These authors contributed equally to this work

###### Abstract

The Anomalous Hall Effect (AHE) manifests as a transverse voltage proportional
to magnetization in ferromagnetic materials under the application of a charge
current, being an indispensable tool for probing magnetism, especially in
nanoscale devices. However, the AHE primarily sensitizes to out-of-plane
magnetization, thereby hindering its capacity to discern the in-plane
magnetization, a characteristic prevalent in ferromagnetic films. Here we
challenge this conventional understanding by demonstrating the in-plane
magnetization-induced AHE in iron and nickel, two ubiquitous ferromagnets.
This observation of the in-plane AHE is remarkable as it contradicts existing
theories that forbid such phenomena in cubic crystal systems. We trace the
origin of this unanticipated phenomenon to a hitherto unconsidered octupole of
the anomalous Hall conductivity in the magnetization space, a mechanism we
propose could enable the detection of in-plane AHE in a wide range of
ferromagnetic materials. This work realizes the in-plane AHE in common
ferromagnets by exploiting the anomalous Hall conductivity octupole, revealing
a new physical origin of the AHE and promising to revolutionize the design of
magnetic devices and sensors.

The Anomalous Hall Effect (AHE) is a phenomenon of paramount importance in
condensed matter physics, bridging deep theoretical concepts with practical
applications in magnetism sensing (?, ?, ?, ?, ?, ?, ?, ?). Particularly in
nanoscale devices, where conventional magnetometry yields to limitations, the
AHE offers unparalleled sensitivity coupled with remarkable ease of
integration (?, ?, ?, ?, ?, ?, ?).However, a significant limitation of the AHE
lies in its exclusive detection of out-of-plane magnetization (?, ?). This
inherent bias substantially restricts its application spectrum, especially in
ferromagnets where in-plane magnetization is commonly observed (?, ?, ?, ?, ?,
?, ?, ?, ?). As a result, detecting in-plane magnetization often necessitates
the use of more complex device architectures or relies on indirect
methodologies in practical scenarios (?, ?, ?). Previous theoretical studies
have suggested that the in-plane AHE might occur in ferromagnets possessing
singular mirror symmetry or strong crystalline anisotropy (?, ?, ?, ?, ?, ?),
and recent experiments demonstrated that Hall effect can be induced by an in-
plane magnetic field in low-symmetry nonmagnetic systems (?, ?, ?, ?, ?).
However, the rarity of materials fulfilling both ferromagnetism and reduced
crystalline symmetry poses significant experimental challenges in realizing
in-plane AHE in ferromagnets.

In this study, we report the observation of the in-plane AHE unexpected in
common ferromagnets, achieved by exploring the multipolar structure of the
anomalous Hall conductivity in the magnetization space hitherto neglected in
previous researches. The AHE can be characterized by a vector derived from the
antisymmetric part of the conductivity tensor, expressed as $\sigma_{\text{\rm
AHE}}^{i}=\varepsilon_{ijk}\sigma_{jk}$, where $\varepsilon_{ijk}$ is the
Levi-Civita symbol, and the Einstein summation convention applies to repeated
indices. Here we represent $\bm{\sigma}_{\rm AHE}$ by the multipoles in the
magnetization space where $\bm{M}=|\bm{M}|\hat{\bm{M}}$ with
$\hat{\bm{M}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$:

$\displaystyle\sigma_{\rm
AHE}^{i}=p_{ij}\hat{M}_{i}+\frac{1}{15}o_{ijkl}\hat{M}_{j}\hat{M}_{k}\hat{M}_{l},$
(1)

in which the first and second terms are the dipole and octupole contributions,
respectively. All the even order terms are excluded by the Onsager relation:
$\bm{\sigma}_{\text{\rm AHE}}(-\bm{M})=-\bm{\sigma}_{\text{AHE}}(\bm{M})$, and
higher order contributions are omitted here. The coefficient of the dipole
term reduces to a scalar: $p_{ij}=p_{0}\delta_{ij}$ in ferromagnetic materials
with cubic crystal structures, such as Fe and Ni. So the AHE dipole aligns
$\bm{\sigma}_{\rm AHE}$ to $\bm{M}$ as illustrated in Fig. 1A, which yields an
isotropic contribution to the anomalous Hall conductivity and reproduce the
familiar orthogonal geometry of the AHE. This indicates that first term in Eq.
(1) corresponds to $\bm{\sigma}_{\text{AHE}}$ linearly dependent on
magnetization which has been the focus of previous works on the AHE,
precluding the in-plane AHE in most ferromagnets according to the expression
of anomalous Hall current density:
$\bm{J}_{\text{AHE}}=\bm{\sigma}_{\text{AHE}}\times\bm{E}$ (?).

Distinctively, the AHE octupole in the magnetization space, the second term in
Eq. (1), can be identified as a fundamental origin of the in-plane AHE.
$o_{ijk\ell}$ is a rank-4, traceless tensor, symmetric with respect to the
last three indices. In cubic crystals, $o_{ijk\ell}$ has only one independent
parameters:
$o_{xxxx}=o_{yyyy}=o_{zzzz}=-2o_{xxyy}=-2o_{xxzz}=-2o_{yyzz}=-2o_{yyxx}=-2o_{zzxx}=-2o_{zzyy}$.
So $\bm{\sigma}_{\rm AHE}$ can be put into the following form in cubic
lattices:

$\displaystyle\sigma_{\rm AHE}^{i}=\alpha\hat{M}_{i}+\beta\hat{M}_{i}^{3},$
(2)

where the $x,y~{}\text{or}~{}z$ are along the (100), (010), and (001)
directions, respectively, $\alpha=p_{0}-\frac{3}{10}o_{xxxx}$ and
$\beta=\frac{1}{2}o_{xxxx}$. In sharp contrast to the dipole term, the
octupole term $\beta\hat{M}_{i}^{3}$, can cause a misalignment between
$\bm{\sigma}_{\rm AHE}$ and $\bm{M}$. It is due to the fact that
[$\hat{M}_{x}$,$\hat{M}_{y}$,$\hat{M}_{z}$] may not parallel
[$\hat{M}^{3}_{x}$,$\hat{M}^{3}_{y}$,$\hat{M}^{3}_{z}$], as illustrated in
Fig. 1B, thus compromising the $\bm{\sigma}_{\rm AHE}\parallel\bm{M}$ geometry
observed in previous experiments (?, ?, ?, ?, ?). For instance, $\bm{M}$ in Fe
along [103] causes a $\bm{\sigma}_{\rm AHE}$ component along [1,0,27] through
the $\beta\hat{M}_{i}^{3}$ term. Such misalignment amounts to a finite
anomalous Hall conductivity vector $\bm{\sigma}_{\text{AHE}}^{\perp}$ normal
to $\bm{M}$, which consequently induces an anomalous Hall voltage at $\bm{E}$
$\parallel$ $\bm{M}$, characteristic of the in-plane AHE.

Figure 1C illustrates the experimental setup for measuring the in-plane AHE in
Fe(103) films. Before patterned into Hall bar structures, thin films of
Fe(103) are epitaxially grown on MgO(113) substrates using magnetron
sputtering, the technical details of which was published elsewhere (?). The
Fe(103) orientation is confirmed by the X-ray Diffraction (XRD) and
transmission electron microscopy (TEM) patterns as shown in Fig. S1 of the
supplementary materials. The crystal lattice of our sample is determined to be
2.860 Å, close to that of bulk Fe at 2.866 Å (?). In this setup, when both
$\bm{M}$ and $\bm{E}$ orient along the in-plane Fe[$30\bar{1}$] direction,
$\bm{\sigma}_{\text{AHE}}^{\perp}$ is expected to induce an in-plane anomalous
Hall voltage along the [$0\bar{1}0$] direction.

Figure 1D shows the Hall resistivity ($\rho_{xy}$) as a function of magnetic
field $\bm{H}$ at 300K for $\bm{H}\parallel$ Fe$[30\bar{1}]$ and
$\bm{H}\parallel$ Fe[$0\bar{1}0$]. The inset of Fig. 1D is the
$\bm{M}$-$\bm{H}$ curve, indicating that $\bm{M}$ reaches saturation at
$\lvert H\rvert<$ 100 Oe for both $\bm{H}$ orientations. The Hall measurement
was carried out by applying charge current along the Fe$[30\bar{1}]$ direction
and the Fe film thickness is 100 nm. An evident anomalous Hall signal is
observed for $\bm{H}\parallel$ Fe$[30\bar{1}]$, similar to the $\bm{H}$
dependence observed in the $\bm{M}$-$\bm{H}$ curve. This underscores its
origin from the in-plane magnetization, consistent with the in-plane AHE as
expected in Fig. 1C. On the other hand, the anomalous Hall signal is absent
for $\bm{H}\parallel$ Fe[$0\bar{1}0$], which can be understood by the
vanishing of $\bm{\sigma}_{\text{AHE}}^{\perp}$ when $\bm{\sigma}_{\rm AHE}$
$\parallel$ $\bm{M}$. Collectively, these findings demonstrate the observation
of an anomalous Hall effect rooted in the in-plane magnetization of bcc Fe,
offering compelling evidence for the in-plane AHE.

Figure 2A depicts an alternative configuration for the Hall measurement, where
the orientations of the applied current and the Hall probes are swapped in
comparison to the arrangement shown in Fig. 1C. Figure 2B shows the transverse
resistivity $\rho_{xy}$ measured with the setup described in Fig. 2A. The
magnetic field dependencies of $\rho_{xy}$ for both $\bm{H}\parallel$
Fe$[30\bar{1}]$ and $\bm{H}\parallel$ Fe[$0\bar{1}0$] closely resemble those
observed in Fig. 1D. This indicates that the Hall signal is predominantly
influenced by the magnetization direction rather than the direction of the
applied current, which agrees with the
$\bm{\sigma}_{\text{AHE}}^{\perp}$-induced in-plane AHE scenario.

It is worth noting that if the magnetization is not fully aligned with the
magnetic field during the in-plane field sweeping process, its out-of-plane
component can also induce an anomalous Hall signal due to the linear $\bm{M}$
term in the AHE (?, ?). To discern whether the observed Hall signals in Figs.
1D and 2B are of such a trivial physical origin, we carried out Hall
measurement in both setups at magnetic field of 90000 Oe in magnitude, with
the result shown in Figs. 2C and 2D. In both figures, the Hall signals remain
constant with increasing in-plane magnetic field after saturation, distinct
from the attenuation behavior of the AHE induced by the out-of-plane component
of tilted magnetization as reported in Ref. (?, ?). When the magnetic field
was aligned normal to the sample, the signal due to the linear AHE can be
observed. Combining the measurement of the longitudinal resistivity, the
anomalous Hall conductivities of the linear AHE and in-plane AHE are
determined to be 1122 $\Omega^{-1}cm^{-1}$ and $-$34.5 $\Omega^{-1}cm^{-1}$,
respectively. The opposite signs in the anomalous Hall conductivities of the
linear AHE and the in-plane AHE further indicate their different physical
origin. Although the measured in-plane AHE signal in Fe amounts to only a
small fraction of the linear AHE signal, its magnitude in
$\bm{\sigma}_{\text{AHE}}$ is still comparable to the largest AHE values found
in ambient-temperature 2D ferromagnets (?, ?, ?). The in-plane AHE can also be
observed in thinner Fe(103) films which shows magnetic field dependence with
slightly different magnitudes (please see Fig. S2 in the supplementary
materials).

Figure 3A shows the experimental setup for the Hall measurement in a Ni(111)
film in which we also observed the in-plane AHE. Although the Ni(111) film is
a well-studied ferromagnetic system in comparison to Fe(103) (?, ?, ?), its
the Hall effect under an in-plane magnetization has not been thoroughly
investigated (?). Figure 3B shows the magnetic field dependence of $\rho_{\rm
xy}$ for $\bm{H}\parallel$ Ni$[10\bar{1}]$ and $\bm{H}\parallel$
Ni$[2\bar{1}\bar{1}]$. Despite a pronounced planar Hall signal that is
symmetric with respect to $\bm{H}$, an antisymmetric component is discernible
exclusively when $\bm{H}$ is parallel to Ni$[2\bar{1}\bar{1}]$. Figure 3C
shows the field-antisymmetric contribution extracted from Fig. 3B, where only
the red curve shows a hysteresis loop with a saturation field close to the
value yielded from the $\bm{M}$-$\bm{H}$ curve of the Ni(111) film. We can
explain the in-plane AHE presence at $\bm{H}\parallel$ Ni$[2\bar{1}\bar{1}]$
and absence for $\bm{H}\parallel$ Ni$[10\bar{1}]$ by the
$\beta\hat{M}_{i}^{3}$ term induced by the AHE octupole in Eq. (2) as well.
When $\bm{M}$ is parallel to Ni$[10\bar{1}]$, $\bm{\sigma}_{\text{AHE}}$
aligns with Ni$[10\bar{1}]$, maintaining parallelism with $\bm{M}$ and thus
lacking an out-of-plane component. Conversely, when $\bm{M}$ is parallel to
Ni$[2\bar{1}\bar{1}]$, $\beta\hat{M}_{i}^{3}$ generates a
$\bm{\sigma}_{\text{AHE}}$ component along Ni$[8\bar{1}\bar{1}]$, resulting in
an out-of-plane component $\bm{\sigma}_{\text{AHE}}^{\perp}$ which induced the
in-plane AHE. The observation of the in-plane AHE in both iron and nickel
suggests its ubiquitousness in ferromagnetic materials.

Figures 4A and 4B depict the experimental setup for measuring the
magnetization angle dependence of the Hall signal for Fe(103) and Ni(111)
films. The magnitude of $\bm{H}$ is set at 10000 Oe to ensure that the
magnetization is saturated and parallel to $\bm{H}$ throughout the rotation
process. Figure 4C shows the field angle dependence of $\rho_{xy}$ in a 100-nm
Fe(103) film, encompassing contributions from both the planar Hall effect
(PHE) and the in-plane AHE (?, ?, ?). Owing to its even parity in relation to
magnetization, the PHE signal is evident as even harmonics of the field angle
$\theta$. Conversely, the in-plane AHE, with its odd parity concerning
magnetization, should be characterized by odd harmonics of $\theta$. We employ
Fourier analysis to extract the odd and even harmonic constituents of the
$\rho_{xy}-\theta$ curve in Fig. 4C, as plotted in Figs. 4D and 4E,
respectively. The field angle dependence of the in-plane AHE significantly
diverges from the commonly observed $sin(\theta)$ characteristic in the
previous AHE studies (?, ?, ?). Figure 4F shows the field angle dependence of
$\rho_{xy}$ in an 80-nm Ni(111) film. Figure 4G shows the odd harmonic
component of the curve in Fig. 4F which manifests a three-fold angle
dependence, and Fig. 4H shows a typical planar Hall behavior as in Fig. 4E. We
further carried out the same in-plane field rotation measurement of
$\rho_{xy}$ in a 100-nm Fe(001) film, and the results are shown in Fig. S3 in
the supplementary materials in which the in-plane AHE was not found. These
findings reveal qualitative differences in the in-plane AHE behaviors across
samples with varying crystalline orientations, which calls for a unified
theoretical exploration to elucidate this unconventional behavior not observed
in prior AHE researches.

Now we are in a position to derive the magnetization angle dependence of the
in-plane AHE from Eq. (2), and it is insightful to start from the symmetry
analysis of the AHE octupole. Based on Eq. (2), when the magnetization
direction varies on a Bloch sphere, the anomalous Hall conductivity can have a
nontrivial landscape determined by the crystalline symmetry. With only the
dipole term, the AHE is always isotropic in cubic crystals and the Hall
deflection plane is always perpendicular to the magnetization. In contrast,
the octupole term allows the magnetization lying in the Hall deflection plane
to induce a Hall signal as demonstrated in this work.

In Fe(103) films, when we rotate the magnetization $\bm{M}$ within the (103)
plane by an angle $\theta$ with respect to the Fe[010], the corresponding
magnetization direction can be expressed as:
$\hat{M}_{x}=-\sin\theta\cos\phi$, $\hat{M}_{y}=\cos\theta$, and
$\hat{M}_{z}=\sin\theta\sin\phi$, with $\phi$ representing the angle between
the (103) and (001) plane. By substituting $\hat{M}_{i}$ into Eq. (2) and
isolating the component perpendicular to the (103) plane, we obtain

$\displaystyle|\bm{\sigma}_{\text{AHE}}^{\perp}|_{\rm
Fe(103)}=\frac{6}{25}\beta_{\rm Fe}\sin^{3}\theta,$ (3)

which yields an excellent match with the AHE data in Fig. 4D using just a
single adjustable parameter. Similarly, we obtain the magnetization angle
dependence of $|\bm{\sigma}_{\text{AHE}}^{\perp}|$ in Ni(111) by applying the
same derivation:

$\displaystyle|\bm{\sigma}_{\text{AHE}}^{\perp}|_{\rm
Ni(111)}=\frac{\sqrt{2}}{6}\beta_{\rm Ni}\cos(3\theta),$ (4)

which explains the three-fold symmetry of the in-plane AHE data in Fig. 4G.
Additionally, the analysis of Fe(001) reveals that
$|\bm{\sigma}_{\text{AHE}}^{\perp}|_{\text{Fe(001)}}$ vanishes, elucidating
the absence of in-plane AHE in Fe(001) films. Thus we reach a comprehensive
understanding of the different magnetization angle dependencies of the in-
plane AHE in Fe and Ni films, which are essentially different manifestation of
the AHE octupole.

Our symmetry analysis indicates that the AHE octuple in the magnetization
space is permissible across all Bravais lattices, hinting at its widespread
presence in ferromagnetic materials. For instance, in the hexagonal $D_{6}$
lattice, there exist two nonzero coefficients $o_{ijkl}$, whereas there are
three in the tetragonal $D_{4}$ and $C_{4v}$ and six in the orthorhombic
$D_{2}$ lattices. This increase in the number of nonzero $o_{ijkl}$
coefficients correlates with a more observable in-plane AHE in experiments.
Moreover, our analysis identifies several two-dimensional (2D) ferromagnets,
including $\rm CrTe_{2}$, $\rm Cr_{2}Ge_{2}Te_{6}$, and $\rm CrSiTe_{3}$ (?,
?, ?), also meet the criteria for in-plane AHE due to the influence of the AHE
octupole. It is also worthy to note that the in-plane AHE was reported
recently in the antiferromagnet $\rm Mn_{3}Ni_{0.35}Cu_{0.65}N$ and
interpreted by the effect of noncollinear magnetic structure (?), which might
be relevant to the the AHE octuple proposed in this work. So we anticipate the
in-plane AHE will emerge as a effective characterization technique for
materials with magnetic ordering.

The observation of the in-plane AHE due to the AHE octupole underscores the
significance of the distribution of geometric quantities such as the Berry
curvature within the magnetization space, highlighting a paradigm shift from
the traditional focus on geometric influences in real and momentum spaces (?,
?, ?). Based on the concept of the octupole in the magnetization space,
additional in-plane Hall-type effects could be discovered. For instance, the
anomalous Nernst effect, anomalous Ettinghausen effect, and anomalous thermal
Hall effect—each a divergent branch from the AHE (?, ?)—it is plausible to
expect their in-plane counterparts due to the octupole contribution within
suitable ferromagnets. Theoretically, there exists substantial intrigue in
discussing the criteria essential for optimizing the octupole contribution and
its extended effects across diverse systems. Collectively, these emerging
effects due to octupole in the magnetization space may constitute a novel
classification of physical phenomena, expanding our understanding and offering
new perspectives and paradigms in the study of condensed matter physics.

## References

  * 1. E. H. Hall, Proc. Phys. Soc. London 4, 325 (1880).
  * 2. N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * 3. T. Jungwirth, et al., Appl. Phys. Lett. 83, 320 (2003).
  * 4. S. Nakatsuji, N. Kiyohara, T. Higo, Nature 527, 212 (2015).
  * 5. E. K. Liu, et al., Nat. Phys. 14, 1125 (2018).
  * 6. H. Chen, Q. Niu, A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014).
  * 7. K. Wang, Y. O. Zhang, G. Xiao, Phys. Rev. Appl. 13, 064009 (2020).
  * 8. Y. Ni, Z. Zhang, I. C. Nlebedim, D. C. Jiles, IEEE Trans. Magn. 52, 1 (2016).
  * 9. M. Yamanouchi, D. Chiba, F. Matsukura, H. Ohno, Nature 428, 539 (2004).
  * 10. I. M. Miron, et al., Nature 476, 189 (2011).
  * 11. D. Chiba, et al., Nat. Mater. 10, 853 (2011).
  * 12. L. Q. Liu, et al., Science 336, 555 (2012).
  * 13. C. Tan, et al., Nat. Commun. 9, 1554 (2018).
  * 14. E. D. L. Rienks, et al., Nature 576, 423 (2019).
  * 15. B. Li, et al., Nat. Mater. 20, 818 (2021).
  * 16. M. Julliere, Phys. Lett. A 54, 225 (1975).
  * 17. M. N. Baibich, et al., Phys. Rev. Lett. 61, 2472 (1988).
  * 18. G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Phys. Rev. B 39, 4828 (1989).
  * 19. R. Allenspach, A. Bischof, Phys. Rev. Lett. 69, 3385 (1992).
  * 20. L. D. L. S. Valladares, et al., J. Vac. Sci. Technol 32, 051808 (2014).
  * 21. D. Chiba, et al., Nature 455, 515 (2008).
  * 22. C. M. Jaworski, et al., Nat. Mater. 9, 898 (2010).
  * 23. S. Fukami, T. Anekawa, C. Zhang, H. Ohno, Nat. Nanotechnol. 11, 621 (2016).
  * 24. H. X. Tan, Y. Z. Liu, B. H. Yan, Phys. Rev. B 103, 214438 (2021).
  * 25. H. B. Zhang, S. Blugel, Y. Mokrousov, Phys. Rev. B 84, 024401 (2011).
  * 26. Y. P. Zhang, C. W. Zhang, Phys. Rev. B 84, 085123 (2011).
  * 27. X. Liu, H. C. Hsu, C. X. Liu, Phys. Rev. Lett. 111, 086802 (2013).
  * 28. Y. F. Ren, et al., Phys. Rev. B 94, 085411 (2016).
  * 29. P. C. Zhong, Y. F. Ren, Y. L. Han, L. Y. Zhang, Z. H. Qiao, Phys. Rev. B 96, 241103 (2017).
  * 30. T. Liang, et al., Nat. Phys. 14, 451 (2018).
  * 31. P. Li, et al., Phys. Rev. B 100, 205128 (2019).
  * 32. J. Ge, et al., Natl. Sci. Rev. 7, 1879 (2020).
  * 33. J. D. Zhou, et al., Nature 609, 46 (2022).
  * 34. Y. Cui, et al., arXiv:2308.06651 (2023).
  * 35. R. Karplus, J. M. Luttinger, Phys. Rev. 95, 1154 (1954).
  * 36. J. M. Luttinger, Phys. Rev. 112, 739 (1958).
  * 37. L. Berger, Phys. Rev. B 2, 4559 (1970).
  * 38. W. Z. Peng, Y. L. Chen, X. H. Yu, D. Z. Hou, Materials 16, 4352 (2023).
  * 39. D. P. Rojas, L. F. Barquín, J. R. Fernández, L. R. Fernández, J. Gonzalez, Nanotechnology 21, 445702 (2010).
  * 40. Y. R. You, et al., Phys. Rev. B 100, 134441 (2019).
  * 41. J. X. Tang, et al., Appl. Phys. Lett. 117, 202402 (2020).
  * 42. M. Huang, et al., Nano Lett. 21, 4280 (2021).
  * 43. G. J. Zhang, et al., Nat. Commun. 13, 5067 (2022).
  * 44. R. Suzuki, T. H. Gao, H. Nakayama, K. Ando, AIP Adv. 13, 055311 (2023).
  * 45. H. Ibach, D. Bruchmann, Phys. Rev. Lett. 44, 36 (1980).
  * 46. A. Meyer, et al., Phys. Rev. B 82, 085424 (2010).
  * 47. B. Lalmi, et al., Phys. Rev. B 85, 245306 (2012).
  * 48. A. Philippi-Kobs, et al., Phys. Rev. Lett. 123, 137201 (2019).
  * 49. H. X. Tang, R. K. Kawakami, D. D. Awschalom, M. L. Roukes, Phys. Rev. Lett. 90, 107201 (2003).
  * 50. T. Lin, C. Tang, H. M. Alyahayaei, J. Shi, Phys. Rev. Lett. 113, 037203 (2014).
  * 51. S. Nandy, G. Sharma, A. Taraphder, S. Tewari, Phys. Rev. Lett. 119, 176804 (2017).
  * 52. W. Limmer, et al., Phys. Rev. B 74, 205205 (2006).
  * 53. Y. S. Wang, et al., Nat. Commun. 11, 216 (2020).
  * 54. T. Guillet, et al., Phys. Rev. B 103, 064411 (2021).
  * 55. X. D. Sun, et al., Nano Res. 13, 3358 (2020).
  * 56. Y. M. Fang, S. Q. Wu, Z. Z. Zhu, G. Y. Guo, Phys. Rev. B 98, 125416 (2018).
  * 57. L. D. Casto, et al., APL Mater. 3, 041515 (2015).
  * 58. A. Rajan, et al., arXiv:2304.10747 (2023).
  * 59. Z. Fang, et al., Science 302, 92 (2003).
  * 60. X. K. Li, et al., Phys. Rev. Lett. 119, 056601 (2017).
  * 61. T. Seki, R. Iguchi, K. Takanashi, K. Uchida, Appl. Phys. Lett. 112, 152403 (2018).

Acknowledgments: This work was supported by the National Key R&D Program under
grant Nos. 2022YFA1403502, the National Natural Science Foundation of China
(12234017,12074366). Y. Gao is supported by the Fundamental Research Funds for
the Central Universities (Grant No. WK2340000102). Z. Liu is supported by the
National Natural Science Foundation of China (11974327 and 12004369),
Fundamental Research Funds for the Central Universities (WK3510000010,
WK2030020032), Anhui Initiative in Quantum Information Technologies (No.
AHY170000), and Innovation Program for Quantum Science and Technology
(2021ZD0302800). The set-up of magnetic sputtering system and thin film growth
was assisted by Anhui epitaxy technology co. Ltd. The sample fabrication was
supported by the USTC Center for Micro- and Nanoscale Research and
Fabrication. The authors thank S. Yang, Z.G. Liang and L.F. Wang for the
magnetization measurement. The authors thank B. Wan for the TEM measurement.
Author contributions: W.P. and D.H. designed the experiment and analysed the
data. W.P., H.P. and M.Y. fabricated the samples and performed
characterization of the sample. W.P. carried out the electric measurement.
Z.L., D.H., Q.N. and Y.G. developed the explanation of the experiment. W.P.,
Z.L., P.W. and D.H. wrote the manuscript. Y.C., J.Z., X.Y. and J.S. helped the
experiment. All authors discussed the results and commented on the manuscript.
Competing interests: The authors declare that they have no competing
interests.

Figure. 1: Observation of the in-plane anomalous Hall effect in iron. (A)
Illustration of the alignment between magnetization $\bm{M}$ and the anomalous
Hall conductivity vector $\bm{\sigma}_{\text{AHE}}(\bm{M})$ with a linear
$\bm{M}$ response in a cubic lattice, e.g. bcc Fe, where
$\bm{\sigma}_{\text{AHE}}(\bm{M})$ always parallels $\bm{M}$. (B) Illustration
of the misalignment between $\bm{M}$ and $\bm{\sigma}_{\text{AHE}}(\bm{M})$
caused by the AHE octupole in a cubic lattice, where
$\bm{\sigma}_{\text{AHE}}(\bm{M})$ can be represented as
$(\hat{M}^{3}_{x},\hat{M}^{3}_{y},0)$. $\bm{\sigma}_{\text{AHE}}(\bm{M})$ does
not parallel $\bm{M}=(\hat{M}_{x},\hat{M}_{y},0)$ when
$\hat{M}_{x}\neq\hat{M}_{y}$. This misalignment leads to a
$\bm{\sigma}_{\text{AHE}}(\bm{M})$ component perpendicular to $\bm{M}$,
denoted as $\bm{\sigma}_{\text{AHE}}^{\perp}$. (C) Schematic of the
measurement for the in-plane AHE in a bcc Fe(103) film in which $\bm{M}$ and
current density $\bm{J}_{c}$ are both aligned in-plane along Fe$[30\bar{1}]$.
D, The in-plane AHE is observed when sweeping the magnetic field $\bm{H}$
along the Fe$[30\bar{1}]$ direction (red curve), but is absent when $\bm{H}$
is aligned with Fe[$0\bar{1}0$] (blue curve). The inset shows the
$\bm{M}$-$\bm{H}$ curves for an Fe(103) film with $\bm{H}$ sweeping along both
Fe$[30\bar{1}]$ (red curve) and Fe[$0\bar{1}0$] (blue curve). Notably, the in-
plane AHE signal shows an asymmetric magnetic-field dependence, echoing the
saturation behavior of the $\bm{M}-$$\bm{H}$ curve. Figure. 2: Systematic
measurement of the in-plane Anomalous Hall effect in Fe(103) films. (A)
Illustration of the Hall measurement for an Fe(103) thin film with a current
applied along Fe[$0\bar{1}0$]. (B) Hall resistivity measured with $\bm{H}$
$\parallel$ Fe$[30\bar{1}]$ and $\bm{H}$ $\parallel$ Fe$[0\bar{1}0]$ for
$\lvert H\rvert$ $\leq$ 200 Oe at T = 300 K, and the inset depicts the
$\bm{M}$-$\bm{H}$ curves measured for the two field orientations. An in-plane
AHE signal is observed for $\bm{H}\parallel\text{Fe}[30\bar{1}]$ but is absent
for $\bm{H}\parallel\text{Fe}[0\bar{1}0]$, mirroring the behavior shown in
Fig. 1D. (C and D) Hall resistivity measured at a magnetic field up to 90000
Oe with current applied along Fe$[30\bar{1}]$ and Fe$[0\bar{1}0]$,
respectively, where the magnetic field is swept along the perpendicular and
two in-plane directions. Figure. 3: Observation of the in-plane anomalous Hall
effect in Ni. (A) Illustration of a Hall bar of Ni(111) thin film with a
charge current along Ni$[1\bar{1}0]$. (B) Hall resistivity measured with
$\bm{H}$ $\parallel$ Ni$[2\bar{1}\bar{1}]$ and $\bm{H}$ $\parallel$
Ni$[10\bar{1}]$ for $\lvert H\rvert$ $\leq$ 10000 Oe at T = 300 K, and the
inset depicts the in-plane crystalline directions of Ni(111). Besides the
similar symmetric contributions due to the planar Hall effect (?, ?, ?), an
antisymmetric contribution is observed for $\bm{H}$ $\parallel$
Ni$[2\bar{1}\bar{1}]$ but is absent for $\bm{H}$ $\parallel$ Ni$[10\bar{1}]$.
(C) The field-asymmetric contribution extracted from (B), which reveals that a
hysteresis-loop-like Hall signal is only observed for $\bm{H}$ $\parallel$
Ni$[2\bar{1}\bar{1}]$, consistent with the in-plane AHE induced by the AHE
octupole. Figure. 4: Magnetization orientation dependences of the in-plane
Anomalous Hall effect. (A and B) Illustrations for the Hall measurement with
rotating in-plane magnetic field $\bm{H}$ for Fe(103) and Ni(111) films,
respectively. The magnitude of the magnetic field is set at 10000 Oe, which is
sufficiently strong to align $\bm{M}$ parallel to $\bm{H}$. (C) Field angle
dependence of the Hall resistivity in a 100-nm Fe(103) film. (D and E) The
contributions from odd harmonics and even harmonics derived from (C)
respectively. The odd-harmonic pattern agrees well with Eq. (3) predicted by
the AHE octupole. The even-harmonic contribution displays a sin(2$\theta$)
dependence due to the planar Hall effect. (F) Field angle dependence of the
Hall resistivity in a 80-nm Ni(111) film. (G) The odd-harmonic components in
(F), which shows a three-fold field angle dependence with a comparable
magnitude as in (D), consistent with Eq. (4) predicted by the AHE octupole.
(H) The even-harmonic components in (F), showing a sin(2$\theta$) dependence
due to the planar Hall effect.


# Canonical Saliency Maps: Decoding Deep Face Models

Thrupthi Ann John, Vineeth N Balasubramanian, and C V Jawahar Thrupthi Ann
John is and C V Jawahar are with the International Institute of Information
Technology, Hyderabad
E-mail<EMAIL_ADDRESS>Vineeth N Balasubramanian is with the
Indian Institute of Technology, Hyderabad.

###### Abstract

As Deep Neural Network models for face processing tasks approach human-like
performance, their deployment in critical applications such as law enforcement
and access control has seen an upswing, where any failure may have far-
reaching consequences. We need methods to build trust in deployed systems by
making their working as transparent as possible. Existing visualization
algorithms are designed for object recognition and do not give insightful
results when applied‘ to the face domain. In this work, we present ‘Canonical
Saliency Maps’, a new method which highlights relevant facial areas by
projecting saliency maps onto a canonical face model. We present two kinds of
Canonical Saliency Maps: image-level maps and model-level maps. Image-level
maps highlight facial features responsible for the decision made by a deep
face model on a given image, thus helping to understand how a DNN made a
prediction on the image. Model-level maps provide an understanding of what the
entire DNN model focuses on in each task, and thus can be used to detect
biases in the model. Our qualitative and quantitative results show the
usefulness of the proposed canonical saliency maps, which can be used on any
deep face model regardless of the architecture.

###### Index Terms:

Deep Neural Networks, Face Understanding,
Explainability/Accountability/Transparency, Canonical Model

##  1 Introduction

Figure 1: Are all parts of the face of equal importance for different face
classification tasks? In this work, we show that deep models do not give equal
importance to the entire face. Canonical Model Saliency (CMS) maps show parts
of the face that play a significant role in decisions made by the deep model.
CMS maps reveal how deep face models work and allow us to detect and diagnose
problems inherent to the models, such as biases. For heatmaps, red indicates a
high value while blue indicates a low value (Best viewed in color).

Deep learning achieves state-of-the-art performance in most computer vision
tasks, surpassing earlier methods by a large margin. The performance of deep
neural networks is improving in leaps and bounds for face processing tasks
such as face recognition and detection. In 2014, DeepFace [1] approached
human-like performance for the first time on the LFW benchmark [2], a dataset
of face images in unconstrained settings (DeepFace: 97.35% vs. Human: 97.53%),
using a training dataset of 4 million images. In recent years, the accuracy
has increased up to 99.8% [3], thereby surpassing human performance on the
benchmark. Deep face models are now deemed to be real-world ready. They are
used in many critical areas by government agencies including law enforcement
and access control. Currently, models for face tasks are available from major
companies like Microsoft, IBM and Amazon who claim that their models are
highly accurate. In this scenario, two crucial questions arise: Do pre-trained
models perform as well as they claim, and how do we find the weaknesses
existing in these models and improve them.

Failures of face models in critical areas have far-reaching and devastating
consequences. Inaccuracies in facial recognition technology can result in an
innocent person being misidentified as a criminal and subjected to unwarranted
police scrutiny. Big Brother Watch UK released the Face-Off report [4]
highlighting false positive match rates of over 90% for the facial recognition
technology deployed by the Metropolitan police. A recent study [5]
demonstrated that although commercial software solutions report high
accuracies (Amazon’s Rekognition reports an accuracy of 97%), they demonstrate
skin-type and gender biases that go unreported as the benchmarks themselves
are skewed. When performance is reported on public or private databases, they
are always subject to the biases inherent in these databases. The algorithms
may be then used in the real world in conditions that differ wildly from the
ones that they are tested in, causing the algorithms to produce erroneous
results. How do we catch such issues at an early stage? High reported accuracy
is not enough to determine how an algorithm will perform under real-life
conditions. We need to be able to peek inside the algorithms and understand
how they work. The opaqueness of deep models restricts its usefulness in
highly regulated environments (e.g. healthcare, autonomous driving), which may
require the reasoning of the decisions taken by the deep models to be
provided. To build trust in deployed intelligent systems, they need to be
transparent i.e. they should be able to explain why they predict what they
predict [6]. Interpretable algorithms allow us to responsibly deploy deep face
models in the real world, as they help end users be aware of these models’
characteristics and shortcomings.

Several visualization methods have been proposed to increase the
interpretability and transparency of deep neural networks. So far, most neural
network visualization methods have been created with the task of object
recognition in mind. There have been very few works that applied these
algorithms exclusively to the face domain [7, 8]. The saliency methods of
object recognition do not readily translate to the face domain, as the images
used for face tasks have different properties from those used for generic
object recognition. Face images are highly structured forms of input. The
intra-class difference is very small and face tasks are a form of fine-grained
classification. Input images to face classification models are usually pre-
processed so that they are centered around the face of interest and there is
only one face per image. Examples of current saliency methods applied to faces
are given in Figure 2. We observe that most methods highlight a large area in
the center of the face. This type of heatmap may be useful for object
recognition when there are multiple objects in a single image, but shows only
trivial information for face images. Since faces are centered in the input
image, the question ‘where in the image’ is not as relevant as ‘where on the
face’. In this work, we introduce a simple yet effective ‘standardization’
process for visualization of deep learning models for face processing, that
converts image coordinates to face coordinates and thus makes the resultant
saliency maps more effective in practice. We utilize the structure of faces
and project the saliency maps onto a standard frontal face to obtain
‘Canonical Saliency Maps’ that are independent of image coordinates. These
canonical saliency maps can be further processed to compare images or observe
trends.

To this end, we propose two types of canonical maps: Canonical Image Saliency
(CIS) maps and Canonical Model Saliency (CMS) maps. CIS maps are detailed
attention maps of input faces projected onto a standard frontal face. CMS
maps, on the other hand, globally visualize the characteristic heatmap of an
entire face network, as opposed to an input image. This shows the general
trend of facial features a network fixates on while making decisions. Such a
model-level saliency map can only be generated using a canonical approach, and
not by currently available saliency maps. CMS maps highlight areas that are of
most significance for a given face task across a dataset. Since we need only
the confidence of the classifier for this purpose, these can be generated for
any available model or architecture, even if the implementation details are
not available. Thus, this approach may even be used for analyzing commercial
models that may not reveal their architecture designs.

In order to validate our contributions comprehensively, we study our canonical
maps on five different face processing tasks: face recognition, gender
recognition, age recognition, head pose estimation and facial expression
recognition (Figure 1). We use well-known architectures in our studies and
also compare the fixation patterns of the models for human recognition of
faces. We also show that our visualization method helps discovers a bias in
gender recognition models which rely on eye make-up to make decisions.

Our key contributions can be summarized as follows:

  * •

We present a method to standardize face saliency images and project them from
image coordinates to face coordinates. This ‘standardization’ produces
canonical heat-maps that show the relevance of different facial parts to a
deep face task. The new maps are more insightful than the saliency maps
produced by current methods and can be used for comparison and observation of
trends.

  * •

We introduce two types of canonical heatmaps: (i) Canonical Image Saliency
maps which highlight the significant facial areas of a specific input image
pertinent to a prediction; and (ii) Canonical Model Saliency maps, which
capture global characteristics of an entire deep face model while making
predictions across data points, which allows us to understand the network and
potentially diagnose problems.

  * •

Our algorithms can be performed on any face model even if the implementation
is not available. We demonstrate the superior performance of our method using
extensive suite of experiments.

  * •

We explore the working of deep face models trained for various face tasks
having different architectures. We illustrate how to interpret the canonical
maps and demonstrate their diagnostic utility by detecting a bias that arises
from using a celebrity face dataset to train a deep network to classify
gender.

##  2 Related Work

Figure 2: A comparison of various saliency visualization methods on the VGG-
Face model [9] for the task of face recognition. For each image, the target
class of the visualization is the ground truth class. (a) Original image; (b)
Vanilla gradients[10]; (c) Smooth-grad [11]; (d) Guided Backpropagation [12];
(e) Guided GradCAM++ [13]; (f) GradCAM [6]; (g) GradCAM++ [13]; (h) ScoreCAM
[14]; (i) Occlusion map [15]. Images are taken from the VGG-Face dataset [9].
Rows (1, 2), (3, 4), (5, 6) and (7, 8) have the same identity. (Best viewed in
color)

There has been extensive research dedicated to saliency visualization methods
in recent years. One of the first efforts to obtain image saliency was by
Simonyan et al [10] which used the magnitude of the gradients to obtain a
noisy and scattered saliency map. Zeiler and Fergus [15] and Springernberg et
al. [12] subsequently introduced methods to highlight the important details of
the image. These visualizations were not class-sensitive. Zeiler and Fergus
[15] also proposed a method to obtain coarse class-specific saliency maps by
occluding parts of the input image and monitoring the output of the
classifier. Recent works such as CAM [16], GradCAM [6], GradCAM++ [13] and
ScoreCAM [14] proposed gradient-based methods to produce coarse, class-
sensitive saliency maps that highlights areas of the input image that were
influential in the classifier output. Smilkov et al. proposed a technique
called ‘SmoothGrad’ [11] which produced a smooth version of such maps by
averaging gradient maps after perturbing the input image with noise.

Although there have been many methods introduced for saliency visualization
for general image classification settings, such methods do not explicitly
address non-trivial fine-grained details when used on face images, as shown in
Figure 2. Columns (b) and (c) in the figure show results of methods that use
the magnitude of gradients to produce a heatmap. These heatmaps are scattered
and it is difficult to see the details and interpret classification results
using them. Guided backpropagation, shown in column (d), shows the finer
details of the face, but is not class-sensitive, thus reducing their utility
for interpretation. Columns (f), (g) and (h), corresponding to GradCAM [6],
GradCAM++ [13] and ScoreCAM [14], are class-specific, but most commonly
highlight the central area of a face making them uninformative across
different face processing tasks. Column (e) represents the results of Guided
GradCAM++, obtained by multiplying the output of guided backpropagation with
the GradCAM++ heatmap, shows fine details while highlighting the class-
discriminative area of the face. Occlusion maps in column (i) of Figure 2 seem
to give the most informative results for our use case. This method maps the
impact that each region of the image has on the classification, in effect
mapping out how representative of the class each region is. It produces a more
non-trivial heatmap showing finer details than the other heatmaps. The heatmap
resolution can also be adjusted by changing the size of the occlusion and the
stride, and the method can be used with any architecture and loss function.
Our visualization method is hence built on occlusion maps given this inference
from our studies on face images. The closest method to ours is [8], which uses
occlusion maps generated between pairs of similar-looking face images to
assist humans in telling them apart. They do this by aligning two faces using
keypoints and systematically occluding patches of both faces and recording the
change in cosine similarity between the faces on a heatmap. The resulting
heatmaps reflect the degree of difference between the face pairs. Unlike this
work, our method works on multi-class classification tasks and introduces the
face canonicalization procedure.

Our proposed Canonical Model Saliency Maps visualize saliency of face networks
w.r.t. different regions of the face for different face processing tasks.
These maps allow us to conduct useful analysis by comparing the facial areas
important to the network to the areas that are expected to be important to
classify the task. However, the challenge herein is - how do we obtain the
‘correct’ expectations to compare the network’s saliency map to? One may look
at human cognition as a benchmark for what a deep network should see.
Extensive research exists on how humans recognize faces; important results
have been presented recently in [17]. For e.g., humans are known to be good at
recognizing low-resolution and degraded faces, when compared to machines.
There is a marked difference in the recognition rate of humans when seeing
familiar faces when compared to unknown faces. The face’s top part, especially
the eyebrows, is known to be an important cue for human face recognition [17].
Comparing our face saliency maps with such insights can tell us when the
obtained saliency maps of trained networks point to wrong cues for
classification (see Section 5.2 for examples.) We now describe our
methodology.

##  3 Canonical Saliency Maps: Methodology

The key aim of our methodology is to create a visualization which highlights
the discriminative parts of a face for a given task. Our method is based on
the assumption that the discriminative importance of a part of an input image
is proportional to the drop in confidence of the classifier when the part is
occluded [15], however on a canonical face representation. Like other
occlusion-based saliency map methods, given an image
$I\in\mathbb{R}^{W_{I}\times H_{I}}$ and the coordinates $(i,j)$, the
importance of a patch $(|i-x|<\frac{sz}{2}$ $\forall x<W_{I}$,
$|j-y|<\frac{sz}{2}$ $\forall y<H_{I})$ is given as follows:

$S_{i,j}=\phi(I,c)-\phi(I\odot B_{i,j},c)$ (1)

where $\phi(I,c)$ is the confidence of class $c$ for image $I$ and
$B_{i,j}\in\\{0,1\\}^{W_{I}\times H^{I}}$ is a mask such that:

$\displaystyle B_{i,j}[x][y]$ $\displaystyle=0\text{ if
}|i-x|<\frac{sz}{2}\text{ and }|j-y|<\frac{sz}{2}$ (2) $\displaystyle=1\text{
otherwise }$ (3)

and $sz$ is the size of the patch, which is a hyperparameter.

###  3.1  Alignment to a ‘Canonical’ Face

In order to capture the finer details of the parts of an image a trained DNN
model looks at, we compute our saliency map on a standard neutral frontal face
image $F\in\mathbb{R}^{W_{F}\times H_{F}}$ called the canonical face, which
helps compare saliency maps on a standardized platform.

We find an one-to-one mapping between the input face image and the canonical
face image by fitting a 3D modular morphable model (3DMMM) [18] using the
procedure used by PR-Net [19]. In particular, we use a convolutional neural
network to regress a UV positional map from the input image, which gives the
depth for a set of fixed points on the UV map of the face. For details of this
procedure, please see [19]. Let $M\in\mathbb{R}^{N\times 3}$ be a set of $N$
3D points representing the 3DMMM. We fit it on the input image $I$ and the
canonical image $F$ to obtain the set of 2D points
$M_{I}\in\mathbb{R}^{N\times 2}$ and $M_{F}\in\mathbb{R}^{N\times 2}$ as the
projection of $M$ on $I$ and $F$ respectively. Thus, we have a 1:1 dense
mapping of points from $I$ to $F$ such that $I[M_{I}[n,1]][M_{I}[n,2]]$ refers
to the same facial feature as $F[M_{F}[n,1]][M_{F}[n,2]]$ $\forall
n\in\\{1.2,\cdots,N\\}$.

Figure 3: Procedure of computing Canonical Image Saliency (CIS) map. First,
the input face is densely aligned. Each part of the input face is occluded
with a small patch and the classification confidence is obtained. The drop in
confidence is plotted on the same face location on a neutral face image to
obtain the Canonical Image Saliency map

###  3.2 Mapping Discriminative Areas

The Canonical Image Saliency (CIS) map is generated by accumulating the drop
in confidence at each point of the dense alignment matrix $M_{I}$ and
recording it on the corresponding location of $F$ on an intermediate matrix
$P^{*}\in\mathbb{R}^{W_{F}\times H_{F}}$ as follows:

$\displaystyle P^{*}_{M_{F}[n,1],M_{F}[n,2]}$
$\displaystyle=P^{*}_{M_{F}[n,1],M_{F}[n,2]}$ (4)
$\displaystyle+S_{M_{I}[n,1],M_{I}[n,2]}$ $\displaystyle\forall n<N$

where $P^{*}_{M_{F}[n,1],M_{F}[n,2]}$ is the patch around the point
$(M_{F}[n,1],M_{F}[n,2])$ on the heatmap $P$, and $S_{M_{I}[n,1],M_{I}[n,2]}$
is the drop in confidence in the patch around point $(M_{I}[n,1],M_{I}[n,2])$
calculated according to Equation 1.

###  3.3 Density Normalization

Note that an equi-spaced grid on a 3-dimensional face may not correspond to
equi-spaced grid on a 2D projection of the face. For example, on a frontal
face image, the points on the sides of the face may be more spatially
concentrated due to the curvature of the face. The heatmap values in these
regions will hence be higher due to the concentration. We hence introduce a
normalization step that keeps track of the number of times a pixel on an image
is occluded, when performing the occlusion heatmap on the mesh. Let
$N\in\mathbb{R}^{W_{F}\times H_{F}}$ be a matrix which stores the count of
times each pixel of $P^{*}$ was updated. The final CIS map is calculated as
follows:

$P=P^{*}\oslash N$ (5)

where $\oslash$ represents element-wise division. Figure 4 shows the effect of
density normalization on the CIS map.

Figure 4: Effect of applying density normalization to the heatmap. Without
density normalization, the nose is not highlighted despite it being a
discriminative feature, mainly because the density of points on the nose is
low

###  3.4 From Image Saliency to Model Saliency

We now discuss how the CIS maps are used to understand facial features that
are important across all images for a given model trained for a specific task
(for e.g, the part of the face that may be important for gender recognition vs
another part that may be important for age recognition). We call these
Canonical Model Saliency (CMS) Maps, which are model-level saliency
visualizations to highlight facial areas that influence the model across all
test images.

Given a test set $D$ consisting of images $\\{I_{1},I_{2},I_{3},...\\}$ with
variations in factors such as pose, lighting, or expressions, we consider the
average CIS map across these test images as the CMS map, i.e.

$V=\frac{1}{N}\sum_{i}P_{i}\qquad\forall I\in D$ (6)

where $P_{i}$ is the CIS map of $I_{i}\in D$. It is possible to combine the
CIS maps in other ways, but we found that simple averaging worked well in
practice for model-level analysis. Learning CMS maps in other ways could be an
interesting direction of future work. Furthermore, in practice, we observe
that it requires only a few images to generate a stable CMS map for a complete
trained model. This suggests that face networks consistently rely on a few
facial features and the canonical visualizations are stable across images.
This is shown in Figure 5 where we see that the trends become obvious from the
first random 100 images. After 1000 images, the CMS is practically unchanged
with the addition of more images.

Figure 5: Ablation study to study the effect of the number of images used to
create a CMS map. CMS maps for recognition using 100, 500, 1000, 2000, 5000
and 10000 random CIS maps

Figure 6 shows a comparison between occlusion heatmaps of [15] and our CIS
maps. Our methodology is summarized as follows:

Algorithm 1 Canonical Image Saliency Map

Input:

  * •

input image $I$ of size $W_{I}\times H_{I}$

  * •

input mesh $M_{I}$ of size $N\times 3$

  * •

frontal image $F$ of size $W_{F}\times H_{F}$

  * •

frontal mesh $M_{F}$ of size $N\times 3$

  * •

model $\phi$: deep model to find saliency where $\phi(I,c)$ gives the
confidence of I for class c

  * •

target class $C$ of the input image I

  * •

$sz$: size of occlusion square

Output: heatmap $P$ of size $W_{F}\times H_{F}$

1:procedure CIS($I,M_{I},F,M_{F},\phi,C,sz$)

2: $P\leftarrow\\{0\\}^{W_{F}\times H_{F}}$

3: $N\leftarrow\\{0\\}^{W_{F}\times H_{F}}$

4: $fsz\leftarrow fsz\times\frac{H_{F}}{H_{I}}$

5: for $i\leftarrow 0$ to $n$ do

6: $I^{*}\leftarrow I$

7:
$I^{*}[M_{I}[i,0]-\frac{sz}{2}:M_{I}[i,0]+\frac{sz}{2}][M_{I}[i,1]-\frac{sz}{2}:M_{I}[i,1]+\frac{sz}{2}]\leftarrow
0$

8: $x_{F},y_{F}\leftarrow M_{F}[i,0],M_{F}[i,1]$

9:

10:
$P[x_{F}-\frac{fsz}{2}:x_{F}+\frac{fsz}{2}][y_{F}-\frac{fsz}{2}:y_{F}+\frac{fsz}{2}]+=\phi(I,C)-\phi(I^{*},C)$

11:
$N[x_{F}-\frac{fsz}{2}:x_{F}+\frac{fsz}{2}][y_{F}-\frac{fsz}{2}:y_{F}+\frac{fsz}{2}]+=1$

12: $P[N=0]\leftarrow 0$

13: $N[N=0]\leftarrow 1$

14: $P\leftarrow P\oslash N$

15: return $P$

###  3.5 Model Saliency for Non-classification Tasks

CMS maps can be generated for any face model which has a measure of confidence
associated with each input image. Our method can be adapted to non-
classification models by defining an appropriate confidence function. Here, we
define the confidence function for two commonly-used face tasks: zero-shot
recognition using nearest neighbour and face verification.

####  3.5.1 Zero-shot Face Recognition

Here, the query image $q$ is assigned the label of the image from the training
set whose features have the highest cosine similarity with the features of the
query image [20]. We define the confidence of classification in this setting
as follows:

$S_{q,c}=\frac{A.Q}{\lVert A\rVert\lVert Q\rVert}$ (7)

where $c$ is the ground truth label of $q$, $Q$ is the feature of $q$ and $A$
is the feature of the closest training set image with label $c$. This new
confidence function can be used in place of the class confidence $\phi$ in
Equation 1.

####  3.5.2 Face Verification

Here, a pair of face images is considered to have the same identity if the
cosine similarity between their features is more than a threshold calculated
on the training set [20]. We define the confidence in this setting as follows:

$S_{q_{1},q_{2},c}=c\times(\tau-\frac{Q_{1}.Q_{2}}{\lVert Q_{1}\rVert\lVert
Q_{2}\rVert})$ (8)

where $c\in\\{-1,1\\}$ is the verification ground truth label, $\tau$ is the
verification threshold, and $Q_{1}$ and $Q_{2}$ are the features of the image
pair $q_{1}$ and $q2$. Using this function, we generate an IMS map for each
pair of images and calculate the CMS map using Equation 6.

##  4 Experiments and Results

We now present our comprehensive experimental results, that analyze the
effectiveness of canonicalizing saliency maps for face processing tasks.
First, we explore our saliency maps through visual examples in Section 4.1.
Second, we objectively assess the ability of our visualization to highlight
discriminative parts of the face in Section 4.2. Third, we present the results
of a user survey which shows that the parts of the face highlighted by our
algorithm are important for the human perception of facial attributes in
Section 4.3. Finally, we present extensive ablation experiments and
discussions on our method in Section 5. Unless otherwise mentioned, our
experiments are conducted using the VGG-Face pre-trained model [9] based on
the VGG-16 architecture [21]. We use a random subset of the CelebA dataset
[22] consisting of 22,000 images (henceforth called CelebA-subset) for all our
experiments. (Note that these images are only used in the model’s test phase,
the model by itself is trained on all the training images in the CelebA
benchmark). See the Supplementary Section for more details.

###  4.1 Qualitative Results

Figure 6: Column (a) shows Occlusion Maps used for saliency visualization (see
Section 2); Column (b) shows Canonical Image Saliency (CIS) maps. CIS maps are
a projection of occlusion maps onto a canonical frontal face; Column (c) shows
Canonical Model Saliency (CMS) maps. These maps are generated for a model as a
whole and hence do not vary with input; Column (d) shows the CMS maps
reprojected back onto the input face.

We compare the saliency maps produced by various methods in Figure 2. As in
Section 2, most visualizations are not practically useful, and highlight a
vague central portion of the face. In Figure 6, we display the visualization
methods introduced in this work. From simple occlusion maps in column (a), we
obtain Canonical Image Saliency (CIS) maps by projecting the occlusion maps
onto a neutral frontal face, as shown in column (b). This ‘canonicalizing’
allows us to collate the CIS maps to create Canonical Model Saliency (CMS)
maps as shown in column (c). In column (d), we show that when the CMS maps are
reprojected onto the input images, the saliency maps become meaningful for
analysis.

####  4.1.1 Evaluation of Canonical Model Saliency Maps on Various Face Tasks

For this experiment, we used our algorithm on five models trained for the
tasks of classification, expression, head pose, age and gender. We used the
VGG-Face [9] pre-trained model, and finetuned it for each of the
aforementioned tasks on the CelebA [22] dataset. The ground truth labels for
gender are provided with the CelebA dataset. The head pose ground truth was
obtained by using PRNet [19], and the age ground truth was obtained using the
DEX method [23]. For expression, the ground truth for CelebA was obtained from
a model trained on the FER 2013 data set [24]. Since both head pose and age
are real-valued, we grouped the values into discrete bins to convert them into
classification tasks. For pose, the yaw and pitch values were binned into 9
bins ranging from top-left to bottom-right (see Figure 19). Similarly, the
real-valued ages obtained from the DEX model were grouped into 10 bins, each
having 10 years. More details of the networks used are given in the
Supplementary Section S1. The generated CMS maps are shown in Figure 1. We
notice how models of the same architecture trained on different tasks focus on
different face areas. For recognition, the eye-nose triangle is important and
there is less focus on the mouth or the chin. Gender models surprisingly find
the corners of the eyes to be the most discriminative facial features. We
discuss the implications of this in Section 5.2. The nose is a crucial feature
for the head pose model and the area between the eyebrows for the expression
model. The age model looks at many different facial features. We see that CMS
maps are a valuable asset to understand the nature of face tasks and the
characteristics of various deep models when addressing these tasks. We discuss
some of these results in more detail in Section 5.

####  4.1.2 Canonical Model Saliency Maps on Non-classification Tasks

Figure 7: Calculating CMS maps for non-classification tasks on the LFW
dataset: (a) CMS map for zero-shot learning of identity using nearest
neighbour; (b) CMS map for face verification

In this experiment, we show that CMS maps can be generated for non-
classification face tasks. We generated CMS maps for zero-shot learning of
face identities using nearest neighbour and face verification of VGG-Face fc1
features on the LFW [2] dataset. For the zero-shot learning task, we occluded
parts of the query image while using Equation 7 as the confidence function.
For the verification task, we occluded the same region of both images in a
verification pair and used Equation 8 as the confidence function. The results
are shown in Figure 7. In both cases, we see the highlighted facial areas are
similar to the classification task of recognition in Figure 1.

####  4.1.3 Sanity Check Using Randomization

Figure 8: Sanity check on our visualization method. We progressively
randomized the layers of the VGG-16 face model starting with the output layer
as described in [25]. We observe that the CMS map gets progressively
randomized; our method passes the sanity check. (a) Last layer randomized; (b)
Last two layers randomized; (c) Last three layers randomized; (d) Last four
layers randomized

[25] proposed a sanity check for saliency maps, where the layers of a trained
model are progressively randomized starting from the output layer, and the
changes in generated saliency maps are observed. A method is said to pass the
sanity check if progressive randomization increases the randomization of the
corresponding visualization. We perform a sanity check on our visualization
using the same procedure, and reports the results of this experiment in Figure
8. We observe that as more layers get randomized, the visualization gets more
randomized. Thus, our method passes the sanity check.

###  4.2 Quantitative Results

We conduct an objective evaluation of the faithfulness of our method on two
datasets: CelebA and LFW [2] and compare it with three popular saliency
visualizations: GradCAM [6], GradCAM++ [13] and ScoreCAM [14]. Similar to [13,
14], we measure the confidence drop of explanation images produced by pixel-
wise multiplication of the saliency heatmap with the base image. In
particular, we utilize a ‘negative explanation image’ by darkening the
relevant areas of the base image. Unlike the task of object recognition, face
images have a single object at the center of the image, and models trained on
face images focus on different parts of the face image. In this process,
saliency maps at times fail to detect the face completely (see Figure 9).
Using negative explanation maps addresses such concerns. The negative
explanation image $E$ is given by:

$E=(1-H)\otimes I$ (9)

where $H$ is the heatmap, $I$ is the base image and $\otimes$ represents
pixel-wise multiplication. The heatmaps are first normalized to a range of
[0,1] and the heatmaps for all the methods are standardized to have the same
sum of pixels for each image:

$H^{\prime}=\frac{h-\min(h)}{\max(h)-\min(h)};H=\frac{s}{\Sigma
H^{\prime}}H^{\prime}$ (10)

where $h$ is the original heatmap, $s$ is a scalar which is the same for all
heatmaps of the same image, and $H$ is the final heatmap which is used to
create negative explanation maps. Normalizing the heatmaps in this way ensures
that no visualization method gets an advantage of highlighting a large area of
the input image, as only the discriminative parts should be highlighted.

Figure 9: Since face models are trained to look holistically at the face, they
have more confidence in figure (a) than in figure (b), even though figure (b)
highlights more relevant features. Thus, we use negative saliency maps where
darkening relevant features should cause a larger drop in confidence. This
also ensures that there is enough context for the model to interpret the face
holistically. Another reason for using negative saliency maps is to take care
of cases where a visualization method does not interpret the face correctly,
as in figure (d). Here, the heatmap completely misses the face and is focused
on disparate parts of the image. Using normal explanation maps will result in
almost the original image, which will give a high score in the metrics used.
This is avoided by using negative explanation maps and normalizing the sum of
pixels

We adopt the three metrics used in [13] with negative explanation images:

_Average Drop %:_ The confidence of an image when passed through a model is
expected to decrease when the most discriminative parts are covered. We
measure the drop of confidence when compared to the unmodified image as:

$\frac{1}{N}\sum_{n=1}^{N}{\max(0,\frac{M(I_{n})-M(E_{n})}{M(I_{N})}}\times
100$ (11)

where $M(E_{n})$ and $M(I_{n})$ are the confidence values of the $n^{th}$
explanation image and original image respectively. A high value of Average
Drop % indicates that the heatmap accurately highlights the most
discriminative parts of the image.

_% Increase in confidence:_ In some images, covering the highlighted parts may
result in an undesired increase in confidence with respect to the original
image. We measure the number of such images using this measure as follows:

$\frac{1}{N}\sum_{i=1}^{N}\mathbb{I}_{M(E_{n})>M(I_{n})}\times 100$ (12)

where $\mathbb{I}$ is the indicator function which returns 1 if
$M(E_{n})>M(I_{n})$ and 0 otherwise. A low score in this metric is better.

_Win %:_ Here, we compare all the four methods and measure which method
produces the greatest drop in confidence for a given test image. For example,
Win % of CMS is calculated as follows:

$\begin{split}\frac{1}{N}\sum_{i=1}^{N}\mathbb{I}_{M(E^{CMS}_{n})<(M(E^{GradCAM}_{n}),}\qquad\qquad\qquad\\\
\qquad{}_{M(E^{GradCAM++}_{n}),M(E^{ScoreCAM}_{n}))}\times 100\end{split}$
(13)

where the indicator returns 1 if the explanation map produced by CMS has the
lowest confidence. The sum of Win % across all the visualization methods for a
single task should add up to 100.

We conduct three experiments for quantitative evaluation. First, we calculate
the above metrics on VGG-16 for the tasks of recognition, gender, age, head
pose and expression on the CelebA dataset. For fair comparison, we use our
maps projected back onto the input image (Col (d) of Figure 6). Figure 10
shows our results and a comparison with other visualization methods. Our
method outperforms all other methods in all metrics. The Win % shows that for
most images, removing the explanation map given by our method causes the
highest drop in confidence (higher the better).

Figure 10: Results for Average Drop %, % Increase in Confidence and Win % of
VGG-16 on Celeb-A for the tasks of recognition, gender, age, head pose and
expression.

Secondly, we repeat the experiment on the LFW [2] dataset using the VGG-Face
network, using the same experimental settings as above. We show the results in
Figure 11. Here too, our method outperforms all other methods by a large
margin in all quantitative metrics, showing that our method generalizes across
datasets.

Figure 11: Results for Average Drop %, % Increase in Confidence and Win % of
the explanations generated by Grad-CAM, Grad-CAM++, ScoreCAM and CMS on LFW
for the VGG-16 model.

We also compare our saliency methods on various off-the-shelf gender models.
We use pretrained models from [26, 23, 27] and evaluate our metrics on CelebA-
subset. More details about these models are given in the Supplementary Section
S1. Our results are shown in Figures 12. Once again, we see that our method
outperforms all other methods on all metrics. We show the CMS maps obtained
using the various networks in Figure 13.

Figure 12: Results for Average Drop %, % Increase in Confidence and Win % of
the explanations generated by Grad-CAM, Grad-CAM++, ScoreCAM and CMS on CelebA
for various deep face gender models Figure 13: We compare CMS maps obtained
from various off-the-shelf deep gender models

###  4.3 User Survey on Perception of Facial Attributes

We conducted a user survey to evaluate the human interpretability of our
saliency maps as compared to other visualization methods. In particular, we
explored whether the discriminative facial areas found by Canonical Model
Saliency Maps are vital for human perception of facial attributes. We focused
on the tasks of gender and expression for this study. The survey used a total
of 96 images, each of which were evaluated by 154 participants not involved in
this work. Twelve base images for each task were used, for which we generated
four negative explanation maps corresponding to the four saliency
visualization methods GradCAM, GradCAM++, ScoreCAM and reprojected CMS maps
using the Gender and Expression models mentioned in Section 4.2. We also
applied a vignette to each of the explanation images to hide the context
information (see Figure 15 for sample images). Each participant was given a
binary choice for each image (male-female or happy-sad, depending on the
task). Since a better visualization algorithm hides crucial information and
makes it more difficult to interpret an image, we use the percentage of wrong
answers as a measure of the goodness of the visualization method. We show some
sample survey images in Figure 14. See the Supplementary section for more
examples.

Figure 14: Samples of figures used in our survey (see Section 4.3

The results of our survey are given in Figure 15. We see that the percentage
of wrong answers marked by the respondents is higher for our method than other
methods, indicating that our method performed better at hiding the most
crucial and discriminative facial areas.

Figure 15: Results for user survey on the perception of gender and emotion on
explanation maps. We used 12 base images modified using GradCAM, GradCAM++,
ScoreCAM and CMS. The users had to pick binary labels for each image (male-
female, happy-sad). Each question was answered by 143 people who were not
involved in this project

##  5 Analysis and Discussion

In this section, we present analysis of the proposed method including ablation
studies and discussions.

###  5.1 Why Model-level Saliency Maps?

Figure 16: In this figure, we compare individual occlusion maps of gender
(first row) and recognition (second row) to the respective cumulative model
saliency maps on the right. Individual occlusion maps vary widely and may have
slightly different areas highlighted due to differences in pose, occlusion and
lighting. Thus, it is hard to compare these images and get the big picture
from them. Aggregating heatmaps gets rid of tiny differences caused due to the
conditions in which the photo is taken, allowing us to gain valuable insights.

Canonical Model Saliency (CMS) maps allow us to observe patterns and trends in
the functioning of deep face models by adding the simple yet powerful step of
alignment of occlusion-based saliency maps to a canonical face model. For
example, using CMS maps, we observed that the corners of the eyes are
important for gender classification (Section 5.2). This is not directly
apparent by observing individual, unaligned occlusion maps, as seen in Figure
16. The advantage of this alignment process is in allowing comparison and
aggregation of saliency maps. A single occlusion map may contain variations
caused by differences in the image setting such as pose, occlusion and
lighting, thus not allowing us to understand the whole picture. The process of
aggregation averages out the effects of variations in individual images,
showing us the parts of the face that are truly important.

###  5.2 Effect of Make-up on Gender Classification

Figure 17: Make-up matters! The figure shows the classification confidence of
a gender model on the same person with and without eye make-up. The top row
shows the confidence for ‘female’ classification and the bottom row shows the
confidence for ‘male’ classification. The ground truth label is given below
each pair of images. Figure 18: Canonical Image Saliency maps generated when
the size of the occluding patch is varied. We used a patch size of $15\times
15$ in all other experiments in this work

The CMS maps for the gender model provided interesting insights using our
method (Figure 1E). We expected the heatmap to highlight the areas around the
mouth, jaw and cheeks, as they contain facial hair cues and different bone
structure for different genders. However, the map showed that the model
fixated mostly on eye corners. We hypothesize that this is because the model
was finetuned on the CelebA dataset [22], which consists of images of
celebrities who use make-up extensively. The model picked up on the cue of eye
make-up to classify gender. We presume that such a model will not work well
for a different demographic distribution. This may be the reason why many
commercial face models fail in detecting gender for females and different
races [5]. This indicates the importance of detecting dataset biases as they
can have a significant impact on the performance of deep models. We test our
hypothesis with the following qualitative experiment. We collect a few images
of people with and without eye make-up from the Internet. These images were
passed through the gender model and the confidence for ‘male’ and ‘female’
classification was observed. Our results are presented in Figure 17. We
observed that in all cases, there was a drop of confidence in ‘male’
classification when the men wore make-up and a smaller drop in confidence of
‘female’ classification for women without make-up. In some cases, the drop in
confidence was large enough to flip the original classification result. This
was especially true for males of Asian origin, especially those from the far
East. We conclude that eye make-up has a significant effect on the performance
of such a gender model, which is skewed towards people of a certain ethnicity.

###  5.3 Head Pose Model Relies on the Nose

Figure 19: Look at the close-ups of the nose tip in this figure. Can you tell
the 3D orientation of the face with this information? The nose, along with
jawline, provide a good cue for the face pose. We also observe that the
quadrant of the face area in which the nose tip is found is consistent for the
same 3D orientation

The shape of the nose changes according to the pose of the face (Figure 19A).
Generally, the nose is positioned at the centre of the face, and its placement
on the face changes consistently with the 3D orientation of the face. The head
pose can be detected quite accurately from the shape of the nose and the
quadrant of the face in which the nose tip resides (along with the jawline),
especially when there are only nine classes, as shown in Figure 19. The nose
thus provides the strongest cue for the head pose. This is reflected in the
CMS map shown in Figure 1D.

###  5.4 Age Model Uses the Whole Face

The CMS map for age (Figure 1F) shows that the cues for age are present in
multiple areas of the face. Some of the distinctive features for age may be
the tightness of skin around the eyes and jaws, wrinkles and receding
hairline. Pre-deep learning methods used the geometry or texture of the face
for age prediction [28], thus corroborating our finding on why age-related
cues are found all over the face.

###  5.5 How Occlusion Size Affects Saliency Maps

We present a qualitative ablation study to explore the effect of the size of
the occluding patch on the generated CIS map. The number of vertices provided
by the dense face alignment algorithm is very high and the time required to
compute the heatmap at each vertex is large. Hence we use a tunable ‘stride’
parameter to omit vertices at regular intervals. As the size of the occluding
patch decreases, a smaller stride is chosen so that gaps don’t appear in the
visualization. The stride can be larger for bigger occluding patches without
affecting the visualization quality. In Figure 18, we show the result of
changing the patch size on the CIS maps generated from the same input image.
We observe that as the patch size increases, the map becomes fuzzier but
general patterns do not change. Our method provides useful information
regardless of the size of the occluding patch, although smaller patches give
better resolution. We used a patch of size $15\times 15$ for generating other
saliency maps in this work, as it provides a good balance between heatmap
resolution and computation time.

###  5.6 Why Align to Canonical Face?

Figure 20: Ablation study on the effect of different types of alignment. Shown
are the Average Drop%, % Increase in confidence and Win % for three different
types of alignment on the LFW dataset: 1. Canonical face 2. Keypoint-based
alignment 3. No alignment. Figure 21: Ablation study on the effect of
different types of alignment. Shown are the model saliency maps for three
different types of alignment on the LFW dataset: (a) No alignment,
superimposed on the average image of LFW; (b) Keypoint-based alignment,
superimposed on the average image of LFW-funneled; and (c) CMS superimposed on
the canonical face.

Here we examine the need for a canonical face instead of using keypoint-based
alignment or the image pixel positions. The main advantage of canonical face
alignment is that it ensures that the model saliency maps remain accurate
while aggregating the individual image saliency maps. If we do not align the
heatmaps precisely, the changes in position add up to produce an inaccurate
model saliency map.

We conduct an ablation study to demonstrate this effect. We use three types of
alignment and generate model saliency maps on the LFW dataset: 1) no
alignment; 2) keypoint-based alignment; and 3) canonical face alignment. For
the first case, we create image saliency maps by sliding an occlusion window
over the entire input image. We repeat the procedure for the second case, but
we used LFW images aligned with keypoint alignment [29] as the input instead
of the raw LFW images. The third case used the same setting as previous CMS
experiments. We create the model saliency maps for each case by averaging
individual image saliency maps. We generate explanation maps and calculate
quantitative metrics. The results are shown in Figure 20. Canonical alignment
performs better than keypoint-based alignment or no alignment in all cases. We
show all three model saliency maps in Figure 21.

Using canonical faces also results in lower computation cost, as we know
exactly which parts of the image we need to occlude, as opposed to sliding the
occlusion patch over the whole image.

###  5.7 Robustness in Deep Models

Robustness refers to the property of a model wherein small deviations in input
images, due to noise or natural variations, do not affect the correctness of
the model. If a model relies on a small set of cues, it is more likely to go
wrong due to input image diversity. Instead, if the model looks at many cues,
small variations are less likely to confuse the model. The CMS maps indicate
the areas from which deep models pick up cues. The maps thus also allow us to
obtain an estimate of the model’s robustness. A model that concentrates on a
few facial areas is likely to be less robust than one that focuses on many
facial areas. Less robust models are more prone to mistakes when presented
with extreme cases of occlusion, lighting and other deviations. We see an
example with our trained gender model (Section 5.2), where the model is not
robust to changes in the face due to make-up.

##  6 Conclusion

In this work, we showed that standardization of saliency maps via Canonical
Saliency Maps provides usable and interpretable results in the face domain
when compared to current saliency methods which give trivial outputs for face
images. Canonical Saliency Maps highlight the facial areas of importance by
projecting occlusion-based heatmaps onto a neutral face. Computing model-level
canonical saliency maps enable us to perceive which facial features are
important for different face tasks, thereby revealing the strengths and
weaknesses of face models. These observations can be compared to human
perception, which can show us if the model is behaving in unexpected ways. The
maps aid in detecting problems and biases inherent in the model. In
particular, by utilizing Canonical Model Saliency maps, we identified a bias
in a gender model, wherein the model was wrongly using make-up as a cue to
classify gender. We confirmed the presence of the bias with additional
studies. Such models can cause problems when used in demographics unlike the
training dataset, where the patterns of applying make-up are different.

Nowadays, deep face models are deployed in critical applications like security
and law enforcement – the proposed Canonical Saliency Maps allow such systems
to be critically analyzed before deployment, and thus increase trust. They can
also be used to predict failures during development and help improve the
models. We hope that the tools presented in this work, while simple, can be
very effective in practical use for deeper understanding of face models, their
biases and failures. In future work, we aim to study methods of mitigating the
problems and biases detected by our visualization methods.

## Acknowledgments

VNB would like to thank MHRD, Govt of India, and Honeywell for funding support
through the UAY program (UAY-IITH005).

## References

  * [1] Y. Taigman, M. Yang, M. Ranzato, and L. Wolf, “Deepface: Closing the gap to human-level performance in face verification,” in _CVPR_ , 2014.
  * [2] G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: A database for studying face recognition in unconstrained environments,” University of Massachusetts, Amherst, Tech. Rep., 2007.
  * [3] M. Wang and W. Deng, “Deep face recognition: A survey,” _Neurocomputing_ , 2021.
  * [4] B. B. W. UK. Face off. [Online]. Available: https://bigbrotherwatch.org.uk/all-campaigns/face-off-campaign/
  * [5] J. Buolamwini and T. Gebru, “Gender shades: Intersectional accuracy disparities in commercial gender classification,” in _Conference on fairness, accountability and transparency_ , 2018.
  * [6] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, “Grad-cam: Visual explanations from deep networks via gradient-based localization,” in _ICCV_ , 2017.
  * [7] M. D. Zeiler and R. Fergus, “Exploring features and attributes in deep face recognition using visualization techniques,” in _AFGR_ , 2019.
  * [8] Y. Zhong and W. Deng, “Deep difference analysis in similar-looking face recognition,” in _ICPR_ , 2018.
  * [9] O. M. Parkhi, A. Vedaldi, and A. Zisserman, “Deep face recognition,” in _BMVC_ , 2015.
  * [10] K. Simonyan, A. Vedaldi, and A. Zisserman, “Deep inside convolutional networks: Visualising image classification models and saliency maps,” _ICLR Workshop_ , 2014.
  * [11] D. Smilkov, N. Thorat, B. Kim, F. B. Viégas, and M. Wattenberg, “Smoothgrad: removing noise by adding noise,” _ICML_ , 2017.
  * [12] J. T. Springenberg, A. Dosovitskiy, T. Brox, and M. Riedmiller, “Striving for simplicity: The all convolutional net,” _ICLR_ , 2015.
  * [13] A. Chattopadhyay, A. Sarkar, P. Howlader, and V. N. Balasubramanian, “Grad-cam++: Generalized gradient-based visual explanations for deep convolutional networks,” in _WACV_ , 2018.
  * [14] H. Wang, Z. Wang, M. Du, F. Yang, Z. Zhang, S. Ding, P. Mardziel, and X. Hu, “Score-cam: Score-weighted visual explanations for convolutional neural networks,” in _CVPR_ , 2020.
  * [15] M. D. Zeiler and R. Fergus, “Visualizing and understanding convolutional networks,” in _ECCV_ , 2014.
  * [16] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba, “Learning deep features for discriminative localization,” in _CVPR_ , 2016.
  * [17] P. Sinha, B. Balas, Y. Ostrovsky, and R. Russell, “Face recognition by humans: Nineteen results all computer vision researchers should know about,” _Proceedings of the IEEE_ , 2006.
  * [18] P. Paysan, R. Knothe, B. Amberg, S. Romdhani, and T. Vetter, “A 3d face model for pose and illumination invariant face recognition,” in _AVSS_ , 2009.
  * [19] Y. Feng, F. Wu, X. Shao, Y. Wang, and X. Zhou, “Joint 3d face reconstruction and dense alignment with position map regression network,” in _ECCV_ , 2018\.
  * [20] W. Liu, Y. Wen, Z. Yu, M. Li, B. Raj, and L. Song, “Sphereface: Deep hypersphere embedding for face recognition,” in _CVPR_ , 2017.
  * [21] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _ICLR_ , 2015.
  * [22] Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in _ICCV_ , 2015.
  * [23] R. Rothe, R. Timofte, and L. V. Gool, “Dex: Deep expectation of apparent age from a single image,” in _ICCVW_ , 2015.
  * [24] A. T. Lopes, E. de Aguiar, A. F. D. Souza, and T. Oliveira-Santos, “Facial expression recognition with convolutional neural networks: Coping with few data and the training sample order,” _Pattern Recognition_ , 2017.
  * [25] J. Adebayo, J. Gilmer, M. Muelly, I. Goodfellow, M. Hardt, and B. Kim, “Sanity checks for saliency maps,” in _NeurIPS_ , 2018.
  * [26] K. Karkkainen and J. Joo, “Fairface: Face attribute dataset for balanced race, gender, and age for bias measurement and mitigation,” in _WACV_ , 2021.
  * [27] C.-Y. Hung, C.-H. Tu, C.-E. Wu, C.-H. Chen, Y.-M. Chan, and C.-S. Chen, “Compacting, picking and growing for unforgetting continual learning,” in _NeurIPS_ , 2019.
  * [28] M. Georgopoulos, Y. Panagakis, and M. Pantic, “Modelling of facial aging and kinship: A survey,” _Image and Vision Computing_ , 2018.
  * [29] G. B. Huang, V. Jain, and E. Learned-Miller, “Unsupervised joint alignment of complex images,” in _ICCV_ , 2007.
  * [30] P. Huber, G. Hu, J. R. Tena, P. Mortazavian, W. P. Koppen, W. J. Christmas, M. Rätsch, and J. Kittler, “A multiresolution 3d morphable face model and fitting framework,” in _VISIGRAPP_ , 2016.
  * [31] P.-L. Carrier, A. Courville, I. J. Goodfellow, M. Mirza, and Y. Bengio, “Fer-2013 face database,” _Technical report_ , 2013.

|  Thrupthi Ann John recieved the B.Tech in Computer Science from the National
Institute of Technology, Warangal. She is currently a PhD student in the
Computer Vision and Information Technology lab in IIIT Hyderabad. She is
advised by Prof C V Jawahar and Prof Vineeth N Balasubramanian. Her interests
include deep algorithms for face tasks, visualization of deep algorithms and
solving computer vision problems using deep learning.
---|---
|  Vineeth N Balasubramanian is an Associate Professor in the Department of
Computer Science and Engineering at the Indian Institute of Technology,
Hyderabad (IIT-H). His research interests include deep learning, machine
learning, and computer vision, with a focus on explainable deep learning and
learning with limited supervision. His research has been published at premier
peer-reviewed venues including ICML, NeurIPS, CVPR, ICCV, KDD, ICDM, IEEE
TPAMI and IEEE TNNLS. For more details, please see
https://iith.ac.in/~vineethnb/.
---|---
|  C V Jawahar is the Amazon Chair Professor at IIIT Hyderabad, India, where
he leads a group focusing on AI, computer vision, and machine learning. His
primarily research interest is in a set of problems that overlap with vision,
language and learning. Prof Jawahar is a Fellow of INAE and IAPR.
---|---

## Supplementary Section

In this section, we provide additional details that could not be added in the
paper due to space constraints. The contents are as follows:

  1. 1.

Details of the models used for experiments in the paper

  2. 2.

Details of user survey

##  S1 Details of Models

Our main experiments are conducted on five different models trained for five
tasks. The details of these models are given in Table S1. Models for
Expression, Pose, Gender and Age were obtained by finetuning the VGG-Face
model on the Celeb-A [22] dataset. The ground truth for gender was provided in
the dataset. For age, emotion and pose, we generated the ground truth using
known methods. The ground truth for age was obtained using the method DEX:
Deep EXpectation of apparent age from a single image [23]. This method uses a
VGG16 architecture and was trained on the IMDB-WIKI data set which consists of
0.5 million images of celebrities crawled from IMDB and Wikipedia. The ages
obtained using this method were binned into 10 bins, with each bin having 10
ages. Head pose was obtained by registering the face to a 3D face model using
linear pose fitting [30]. The model used is a low-resolution shape-only
version of the Surrey Morphable Face Model. The yaw and pitch values were
binned into 9 bins ranging from top-left to bottom-right. The binned pose
values are shown in Figure 19. For emotion, the ground truth was obtained
using a VGG-16 model trained on FER 2013 data set [31] with 7 classes. The
accuracy for the recognition models are reported on the LFW dataset. The
accuracy for other models are reported on a test partition of the Celeb-A
dataset. We used three additional models for our experiments on gender
detailed in Sections 4.2. The details of these models are given in Table S2.

Figure S1: Most experiments in this work are conducted on the VGG-Face network [9] shown above, which follows the architecture of VGG-16 [21]. Layers C1 to C13 stand for convolutional layers. Layers D1, D2 and D3 represent fully connected dense layers. There is a ReLU non-linearity after each convolutional and fully connected layer Task | Architecture | Training |  Accuracy %
---|---|---|---
Recognition | VGG-16 [21] | VGG-Face [9] | 98.95 on LFW [2]
Recognition | LightCNN-9 |  | Casia-WebFace
---
MS-Celeb-1M
98.8 on LFW
Expression | VGG-16 |  | Celeb-A using
---
FER13 [31]
69.01
Pose | VGG-16 |  | Celeb-A using
---
3DMM [30]
96.62
Gender | VGG-16 | Celeb-A [22] | 98.37
Age | VGG-16 |  | Celeb-A using
---
IMDB-Wiki [23]
61.72
TABLE S1: Details of deep face models used in this work Model name | Implementation | Base Architecture
---|---|---
VGG Gender | Trained by authors on CelebA | VGG-16
Fairface [26] | https://github.com/dchen236/FairFace | resnet34
DEX [23] | https://github.com/siriusdemon/pytorch-DEX | VGG-16
CPG [27] | https://github.com/ivclab/CPG | spherenet
TABLE S2: Details of the deep gender models used for Figures 12 and 13

##  S2 Details of User Survey

Figure S2: All base images used for our user survey (see Section 4.3

In this section, we give additional details on the user study conducted to
understand human perception of facial expressions in Section 4.3. The study
was conducted to determine the parts of the face that are important for human
perception of expressions and compare it to machine perception. All the base
images used in our survey are given in Figure S2. There were four negative
explanation images created from each base image by using GradCAM, GradCAM++,
ScoreCAM and CMS, as shown in Figure 14. The survey was conducted using Google
Forms. The 64 images were arranged into four pages such that each page
contained one variation of each base image. The variations were mixed equally
across the four pages.