2022/10/032022/11/07 stars:rotation – starspots – white dwarfs

# Modeling photometric variations due to a global inhomogeneity on an
obliquely rotating star: application to lightcurves of white dwarfs

Yasushi Suto11affiliation: Department of Physics, The University of Tokyo,
Tokyo 113-0033, Japan 22affiliation: Research Center for the Early Universe,
School of Science, The University of Tokyo, Tokyo 113-0033, Japan
33affiliation: Laboratory of Physics, Kochi University of Technology, Tosa
Yamada, Kochi 782-8502, Japan Shin Sasaki44affiliation: Department of Physics,
Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan Masataka
Aizawa55affiliation: Tsung-Dao Lee Institute, Shanghai Jiao Tong University,
Shengrong Road 520, 201210 Shanghai, P. R. China Kotaro
Fujisawa11affiliationmark:  and Kazumi Kashiyama66affiliation: Astronomical
Institute, Tohoku University, Sendai 980-8578, Japan 77affiliation: Kavli
Institute for the Physics and Mathematics of the Universe, The University of
Tokyo, Kashiwa 277-8583, Japan<EMAIL_ADDRESS>
###### Abstract

We develop a general framework to compute photometric variations induced by
the oblique rotation of a star with an axisymmetric inhomogeneous surface. We
apply the framework to compute lightcurves of white dwarfs adopting two simple
models of their surface inhomogeneity. Depending on the surface model and the
location of the observer, the resulting lightcurve exhibits a departure from a
purely sinusoidal curve that are observed for a fraction of white dwarfs. As a
specific example, we fit our model to the observed phase-folded lightcurve of
a fast-spinning white dwarf ZTF J190132.9+145808.7 (with the rotation period
of 419s). We find that the size and obliquity angle of the spot responsible
for the photometric variation are $\Delta\theta_{\rm s}\approx 60^{\circ}$ and
$\theta_{\star}\approx 60^{\circ}$ or $90^{\circ}$, respectively, implying an
interesting constraint on the surface distribution of the magnetic field on
white dwarfs.

##  1 Introduction

A number of white dwarfs (WDs) have exhibited periodic photometric variations
(e.g., Achilleos et al., 1992; Barstow et al., 1995; Wade et al., 2003;
Brinkworth et al., 2005, 2013; Reding et al., 2020; Caiazzo et al., 2021;
Kilic et al., 2021; Williams et al., 2022). Those variations are commonly
interpreted to originate from inhomogeneities on the stellar surface that
obliquely rotate around the spin axis. Those inhomogeneities may be produced
by magnetic spots in a convective atmosphere (e.g., Valyavin et al., 2008,
2011) or the so-called magnetic dichroism (Ferrario et al., 1997); the
continuum opacity changes with rotational phase according to the amplitude of
the magnetic field strength across the stellar disk. As a result, the
photometric flux varies as well. Hence, photometric lightcurves of WDs carry
rich information on the distribution of the surface magnetic fields, which may
be further disentangled by combining the spectro-polarimetric data if
available (e.g., Liebert et al., 1977; Donati et al., 1994; Euchner et al.,
2002; Valyavin et al., 2014).

For instance, Brinkworth et al. (2013) have performed time-series photometry
of 30 isolated magnetic WDs, and 5 (24 %) are variable with reliably measured
periods of (1 – 7) hours. Quite interestingly, all of their lightcurves are
well fitted by a monochromatic sinusoidal curve. Indeed, apart from 9 WDs
whose lightcurves are significantly contaminated by their varying comparison
stars, they find that 14 out of the remaining 21 show evidence for variability
whose lightcurves are consistent with a monochromatic sinusoidal curve. In
marked contrast to other stars for which non-sinusoidal rotation signatures
are commonly observed (e.g., Roettenbacher et al., 2013), the sinusoidal
lightcurve seems to be fairly generic for these types of WDs, which have a
relatively stable surface structure over time.

Thanks to the Kepler satellite, Transiting Exoplanet Survey Satellite (TESS),
and Zwicky Transient Facility (ZTF) (Maoz et al., 2015; Reding et al., 2020;
Hermes et al., 2021) among others, the number of photometric lightcurves for
WDs with a high accuracy and cadence is rapidly increasing. Indeed, fast-
spinning WDs with a rotation period even less than 10 minutes have been
recently discovered (Reding et al., 2020; Caiazzo et al., 2021; Kilic et al.,
2021). For such cases, however, it may be due to their pulsation, instead of
the rotation, since their frequency ranges are overlapped (e.g., Winget &
Kepler, 2008; Hollands et al., 2020). This points to an importance of
quantitative modeling of photometric lightcurves due to inhomogeneities on
rotating stars. For instance, the phase-folded lightcurve of one of the fast-
spinning WDs (ZTF J190132.9+145808.7) seems to exhibit a non-sinusoidal
feature (Caiazzo et al., 2021, and see Sec. 3.5 below), which could be
relatively common for these young massive WDs. Thus, such modeling of
photometric variations due to an inhomogeneity on the stellar surface should
become even more crucial in the coming era of the Rubin Observatory LSST
Camera (Ivezić et al., 2019).

Photometric rotational variations due to multiple circular starspots have been
studied previously (e.g., Budding, 1977; Dorren, 1987; Eker, 1994; Landolfi et
al., 1997; Kipping, 2012). More recently, Suto et al. (2022) proposed a fully
analytic model for the photometric variations due to infinitesimally small
multiple starspots on a differentially rotating star, and performed a series
of mock Lomb-Scargle analysis relevant to the Kepler data (e.g., Lu et al.,
2022). In this paper, instead, we focus on an inhomogeneous surface intensity
pattern that is axisymmetric around an axis misaligned to the stellar spin
axis. The inhomogeneity produces periodic photometric variations due to the
stellar rotation. The purpose of the present paper is to develop a general
formulation to describe the rotational variation, compute the resulting
lightcurves for a couple of physically-motivated specific models for WDs, and
apply the methodology to derive constraints on several parameters for ZTF
J190132.9+145808.7 as an example.

The rest of the paper is organized as follows. Section 2 presents our
formulation of the lightcurve modeling for a solid-body rotating star with an
inhomogeneous surface intensity distribution. We first present a general
formulation to compute the photometric modulation due to the oblique stellar
rotation by adopting a quadratic limb darkening law. While our formulation
would be essentially the same as those in the previous literature, the
resulting expressions are characterized by a set of parameters with clear
physical interpretations, and thus more useful in fitting to the observed
lightcurves. In order to show the advantage of our method, we apply the
formulation to two simple inhomogeneity models in section 3: a constant-
intensity single circular spot (cap-model), and a globally varying-intensity
surface (p-model). We find that the p-model leads to a sinusoidal lightcurve
strictly, while the cap-model exhibits a departure from a sinusoidal
lightcurve depending on the geometrical configurations between the starspot
and the observer. In both models, the photometric variations are fairly
insensitive to the limb darkening effect. In subsection 3.5, we show that the
cap-model well explains the observed lightcurve of ZTF J190132.9+145808.7.
Finally, section 4 is devoted to discussion and conclusion of the present
paper. Analytic derivations of several integrals appearing in the main text
are given in Appendix.

##  2 Basic formulation of the lightcurve modeling for an obliquely rotating
star

As shown in Figure 1, we consider a star rotating along the $z$-axis with a
spin angular frequency of $\omega_{\star}$, and the observer is located at
$(\sin\theta_{\rm o},0,\cos\theta_{\rm o})$. If we denote the obliquity angle
between the stellar spin axis and the symmetry axis of the surface
inhomogeneity distribution by $\theta_{\star}$, the angle between the
observer’s line-of-sight and the symmetry axis, $\gamma(t)$ is given by

$\cos\gamma(t)=\sin\theta_{\star}\sin\theta_{\rm
o}\cos\omega_{\star}t+\cos\theta_{\star}\cos\theta_{\rm o},$ (1)

and thus $\gamma(t)$ varies between $|\theta_{\star}-\theta_{\rm o}|$ and
$\theta_{\star}+\theta_{\rm o}$ ($<\pi$). In practice, the symmetry axis may
correspond to the magnetic dipole axis of WDs, or to the central axis of a
single spherical spot on the stellar surface. While we assume that
$\theta_{\star}$ is constant throughout the present analysis, its possible
time-dependence is easily incorporated by substituting the specific function
$\theta_{\star}(t)$ in equation (1) because the photometric variation in our
formulation is completely specified by $\gamma(t)$ alone.

Following Suto et al. (2022), the normalized photometric lightcurve of the
stellar surface is

$\displaystyle L(t)$ $\displaystyle=$ $\displaystyle\frac{\displaystyle\iint
K(\theta,\varphi)I(\theta,\varphi)I_{\rm
LD}(\theta,\varphi)\sin\theta\,d\theta\,d\varphi}{\displaystyle\iint
K(\theta,\varphi)\overline{I}\sin\theta\,d\theta\,d\varphi}$ (2)
$\displaystyle=$ $\displaystyle\frac{1}{\pi\overline{I}}\iint
K(\theta,\varphi)I(\theta,\varphi)I_{\rm
LD}(\theta,\varphi)\sin\theta\,d\theta\,d\varphi,$ (3)

where $K$ is the weighting kernel of the surface visible to the observer,
$I(\theta,\varphi)$ and $I_{\rm LD}(\theta,\varphi)$ indicate the surface
intensity distribution and the limb darkening (the edge of the stellar disk is
observed to be dimmer than its central part), respectively, and the
integration is performed over the entire stellar surface (see also, Fujii et
al., 2010, 2011; Farr et al., 2018; Haggard & Cowan, 2018; Nakagawa et al.,
2020). We introduce a constant surface intensity, $\overline{I}$, just for
normalization, but it is not directly determined from observed data and can be
chosen arbitrarily.

Figure 1: Schematic illustration of the star and the observer. The observer is
located at $(\sin\theta_{\rm o},0,\cos\theta_{\rm o})$. In this frame, the
stellar spin axis is chosen as the $Z$-axis, and the unit vector of symmetry
axis is defined to be $(\sin\theta_{\star},0,\cos\theta_{\star})$ at $t=0$.

For an isotropically emitting stellar surface, the weighting kernel $K$ is
equivalent to the visibility computed from the direction cosine between the
unit normal vector of the stellar surface ($\bm{e}_{\star}$) and the unit
vector toward the observer (${\bm{e}}_{\rm o}$). Without loss of generality,
one can define a spherical coordinate system in which the symmetry axis is
instantaneously set to be $z$-axis. In this case, one obtains

$\displaystyle\bm{e}_{\star}\cdot\bm{e}_{\rm
o}\equiv\mu_{\star}=\sin\gamma(t)\sin\theta[\cos\varphi+\Gamma(t)],$ (4)

where

$\displaystyle\Gamma(t)\equiv\cot\gamma(t)\cot\theta.$ (5)

The location $(\theta,\varphi)$ on the surface is visible to the observer if
$\mu_{\star}>0$. Therefore, the weighting kernel is simply written as

$\displaystyle
K(\theta,\varphi)=\max(\mu_{\star},0)=\sin\gamma(t)\sin\theta~{}\max[\cos\varphi+\Gamma(t),0].$
(6)

If we adopt a quadratic limb darkening law, $I_{\rm LD}$ in equation (2) is
specified by the two parameters $u_{1}$ and $u_{2}$ as

$\displaystyle I_{\rm
LD}(\theta,\varphi)=1-u_{1}(1-\mu_{\star})-u_{2}(1-\mu_{\star})^{2}=(1-u_{1}-u_{2})+(u_{1}+2u_{2})\mu_{\star}-u_{2}\mu_{\star}^{2}.$
(7)

For instance, $u_{1}=0.47$ and $u_{2}=0.23$ for the Sun at $550$nm, and
$u_{1}=0.05$ and $u_{2}=0.51$ for a typical white dwarf with $T_{\rm
eff}=10^{4}$K and log $g$=8.0 in the LSST g-band (Cox, 2000; Gianninas et al.,
2013).

We focus on a case where the surface intensity distribution is axisymmetric,
i.e., $I(\theta,\varphi)=I(\theta)$. Then, equation (2) reduces to

$L(t)=L_{0}(t)+L_{1}(t)+L_{2}(t),$ (8)

where

$\displaystyle
L_{0}(t)=(1-u_{1}-u_{2})\frac{\sin\gamma}{\pi\overline{I}}\iint\max[\cos\varphi+\Gamma(t),0]~{}\sin^{2}\theta\,I(\theta)\,d\theta
d\varphi,$ (9) $\displaystyle
L_{1}(t)=(u_{1}+2u_{2})\frac{\sin^{2}\gamma}{\pi\overline{I}}\iint\max[\cos\varphi+\Gamma(t),0]~{}\sin^{3}\theta\,[\cos\varphi+\Gamma(t)]I(\theta)\,d\theta
d\varphi,$ (10)

and

$\displaystyle
L_{2}(t)=-u_{2}\frac{\sin^{3}\gamma}{\pi\overline{I}}\iint\max[(\cos\varphi+\Gamma(t))^{3},0]~{}\sin^{4}\theta\,I(\theta)\,d\theta
d\varphi.$ (11)

The visible part of the surface location $(\theta,\varphi)$ relative to the
instantaneous observer’s line-of-sight is illustrated in Figure 2 (see also
Figure 2 of Suto et al., 2022). By defining $\varphi_{\rm c}(t)$ for
$|\Gamma(t)|\leq 1$ as

$\displaystyle\cos\varphi_{\rm
c}(t)\equiv-\Gamma(t)=-\cot\gamma(t)\cot\theta,$ (12)

the instantaneously visible part of the surface is limited to $-\varphi_{\rm
c}(t)<\varphi<\varphi_{\rm c}(t)$.

Consider first the case of $0<\gamma<\pi/2$. Then, equation (9) becomes

$\displaystyle\frac{\pi\overline{I}}{1-u_{1}-u_{2}}L_{0}(t)$ $\displaystyle=$
$\displaystyle{\sin\gamma}\int_{0}^{\pi/2-\gamma}d\theta\int_{-\pi}^{\pi}d\varphi[\cos\varphi+\cot\gamma\cot\theta]\sin^{2}\theta\,{I(\theta)}$
(15)
$\displaystyle\qquad\qquad+{\sin\gamma}\int_{\pi/2-\gamma}^{\pi/2+\gamma}d\theta\int_{-\varphi_{\rm
c}}^{\varphi_{\rm c}}d\varphi(\cos\varphi-\cos\varphi_{\rm
c})\sin^{2}\theta\,{I(\theta)}$
$\displaystyle=2\pi{\cos\gamma}\int_{0}^{\pi/2-\gamma}\sin\theta\cos\theta{I(\theta)}\,d\theta+2\sin\gamma\int_{\pi/2-\gamma}^{\pi/2+\gamma}(\sin\varphi_{\rm
c}-\varphi_{\rm c}\cos\varphi_{\rm c})\sin^{2}\theta\,{I(\theta)}\,d\theta.$

Using the following relations derived from equation (12):

$\displaystyle\sin^{2}\theta=\frac{\cot^{2}\gamma}{\cos^{2}\varphi_{\rm
c}+\cot^{2}\gamma},\qquad d\theta=-\frac{\sin\varphi_{\rm
c}\cot\gamma}{\cos^{2}\varphi_{\rm c}+\cot^{2}\gamma}d\varphi_{\rm c},$ (16)

the second term in equation (15) becomes an integral in terms of $\varphi_{\rm
c}$ alone:

$\displaystyle\int_{\pi/2-\gamma}^{\pi/2+\gamma}\sin^{2}\theta(\sin\varphi_{\rm
c}-\varphi_{\rm c}\cos\varphi_{\rm
c})\,d\theta=\int_{\pi}^{0}\cot^{3}\gamma\frac{\sin\varphi_{\rm
c}(\varphi_{\rm c}\cos\varphi_{\rm c}-\sin\varphi_{\rm
c})}{(\cos^{2}\varphi_{\rm c}+\cot^{2}\gamma)^{2}}\,d\varphi_{\rm c}.$ (17)

Finally, equation (15) reduces to

$\displaystyle\frac{\overline{I}}{1-u_{1}-u_{2}}L_{0}(t)$
$\displaystyle=2{\cos\gamma}\int_{0}^{\pi/2-\gamma}\sin\theta\cos\theta{I(\theta)}\,d\theta$
(19) $\displaystyle\hskip
28.45274pt-\frac{2\sin\gamma\cot^{3}\gamma}{\pi}\int_{0}^{\pi}F_{0}(x;\gamma)I(\theta(x))dx,$

where

$\displaystyle F_{0}(x;\gamma)\equiv\frac{\sin x(x\cos x-\sin
x)}{(\cos^{2}x+\cot^{2}\gamma)^{2}},\qquad\cos[\theta(x)]\equiv\frac{\cos
x}{\cos^{2}x+\cot^{2}\gamma}.$ (20)

Similarly, equation (10) is rewritten as

$\displaystyle\frac{\overline{I}}{u_{1}+2u_{2}}L_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{\sin^{2}\gamma}{\pi}\int_{0}^{\pi/2-\gamma}d\theta\int_{-\pi}^{\pi}d\varphi[\cos\varphi+\cot\gamma\cot\theta]^{2}\sin^{3}\theta\,{I(\theta)}$
(22)
$\displaystyle\qquad\qquad+\frac{\sin^{2}\gamma}{\pi}\int_{\pi/2-\gamma}^{\pi/2+\gamma}d\theta\int_{-\varphi_{\rm
c}}^{\varphi_{\rm c}}d\varphi(\cos\varphi-\cos\varphi_{\rm
c})^{2}\sin^{3}\theta\,{I(\theta)}$ $\displaystyle=$
$\displaystyle\sin^{2}\gamma\int_{0}^{\pi/2-\gamma}\sin^{3}\theta{I(\theta)}\,d\theta+2{\cos^{2}\gamma}\int_{0}^{\pi/2-\gamma}\sin\theta\cos^{2}\theta{I(\theta)}\,d\theta$
(24)
$\displaystyle\qquad+\frac{\sin^{2}\gamma}{2\pi}\int_{\pi/2-\gamma}^{\pi/2+\gamma}(2\varphi_{\rm
c}\cos 2\varphi_{\rm c}-3\sin 2\varphi_{\rm
c})\sin^{3}\theta\,{I(\theta)}\,d\theta$ $\displaystyle=$
$\displaystyle\sin^{2}\gamma\int_{0}^{\pi/2-\gamma}\sin^{3}\theta{I(\theta)}\,d\theta+2{\cos^{2}\gamma}\int_{0}^{\pi/2-\gamma}\sin\theta\cos^{2}\theta{I(\theta)}\,d\theta$
(26)
$\displaystyle\qquad-\frac{\sin^{2}\gamma\cot^{4}\gamma}{2\pi}\int_{0}^{\pi}F_{1}(x;\gamma)I(\theta(x))dx,$

where

$\displaystyle F_{1}(x;\gamma)\equiv\frac{\sin x(2x\cos 2x-3\sin
2x+4x)}{(\cos^{2}x+\cot^{2}\gamma)^{5/2}}.$ (27)

Also equation (11) becomes

$\displaystyle-\frac{\overline{I}}{u_{2}}L_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{\sin^{3}\gamma}{\pi}\int_{0}^{\pi/2-\gamma}d\theta\int_{-\pi}^{\pi}d\varphi[\cos\varphi+\cot\gamma\cot\theta]^{3}\sin^{4}\theta\,{I(\theta)}$
(29)
$\displaystyle\qquad+\frac{\sin^{3}\gamma}{\pi}\int_{\pi/2-\gamma}^{\pi/2+\gamma}d\theta\int_{-\varphi_{\rm
c}}^{\varphi_{\rm c}}d\varphi(\cos\varphi-\cos\varphi_{\rm
c})^{3}\sin^{4}\theta\,{I(\theta)}$ $\displaystyle=$ $\displaystyle
3\sin^{2}\gamma\cos\gamma\int_{0}^{\pi/2-\gamma}\sin^{3}\theta\cos\theta{I(\theta)}\,d\theta+2{\cos^{3}\gamma}\int_{0}^{\pi/2-\gamma}\sin\theta\cos^{3}\theta{I(\theta)}\,d\theta$
(31)
$\displaystyle+\frac{\sin^{3}\gamma}{12\pi}\int_{\pi/2-\gamma}^{\pi/2+\gamma}[27\sin\varphi_{\rm
c}+11\sin 3\varphi_{\rm c}-6\varphi_{\rm c}(9\cos\varphi_{\rm c}+\cos
3\varphi_{\rm c})]\sin^{4}\theta\,{I(\theta)}\,d\theta$ $\displaystyle=$
$\displaystyle
3\sin^{2}\gamma\cos\gamma\int_{0}^{\pi/2-\gamma}\sin^{3}\theta\cos\theta{I(\theta)}\,d\theta+2{\cos^{3}\gamma}\int_{0}^{\pi/2-\gamma}\sin\theta\cos^{3}\theta{I(\theta)}\,d\theta$
(33) $\displaystyle\hskip
99.58464pt+\frac{\sin^{3}\gamma\cot^{5}\gamma}{12\pi}\int_{0}^{\pi}F_{2}(x;\gamma)I(\theta(x))dx,$

where

$\displaystyle F_{2}(x;\gamma)\equiv\frac{\sin x[6x(9\cos x+\cos 3x)-27\sin
x-11\sin 3x]}{(\cos^{2}x+\cot^{2}\gamma)^{3}}.$ (34)

In the same manner, we rewrite equations (9) to (11) for $\pi/2<\gamma<\pi$ as
follows.

$\displaystyle\frac{\overline{I}}{1-u_{1}-u_{2}}L_{0}(t)$
$\displaystyle=2{\cos\gamma}\int_{3\pi/2-\gamma}^{\pi}\sin\theta\cos\theta{I(\theta)}\,d\theta$
(36) $\displaystyle\hskip
28.45274pt-\frac{2\sin\gamma|\cot^{3}\gamma|}{\pi}\int_{0}^{\pi}F_{0}(x;\gamma)I(\theta(x))dx,$
$\displaystyle\frac{\overline{I}}{u_{1}+2u_{2}}L_{1}(t)$ $\displaystyle=$
$\displaystyle\sin^{2}\gamma\int_{3\pi/2-\gamma}^{\pi}\sin^{3}\theta{I(\theta)}\,d\theta+2{\cos^{2}\gamma}\int_{3\pi/2-\gamma}^{\pi}\sin\theta\cos^{2}\theta{I(\theta)}\,d\theta$
(38) $\displaystyle\hskip
28.45274pt-\frac{\sin^{2}\gamma\cot^{4}\gamma}{2\pi}\int_{0}^{\pi}F_{1}(x;\gamma)I(\theta(x))dx,$
$\displaystyle-\frac{\overline{I}}{u_{2}}L_{2}(t)$ $\displaystyle=$
$\displaystyle
3\sin^{2}\gamma\cos\gamma\int_{3\pi/2-\gamma}^{\pi}\sin^{3}\theta\cos\theta{I(\theta)}\,d\theta+2{\cos^{3}\gamma}\int_{3\pi/2-\gamma}^{\pi}\sin\theta\cos^{3}\theta{I(\theta)}\,d\theta$
(40) $\displaystyle\hskip
42.67912pt+\frac{\sin^{3}\gamma\cot^{5}\gamma}{12\pi}\int_{0}^{\pi}F_{2}(x;\gamma)I(\theta(x))dx,$

Figure 2: Range of $\theta$ and $\gamma$ for the visible part of the stellar
sphere for the observer. The region with $\Gamma<-1$ corresponds to the
position $(\theta,\varphi)$ that is invisible to the observer, while that with
$\Gamma>1$ is always visible regardless of the value of $\varphi$. The region
with $|\Gamma|<1$ becomes visible periodically when $\varphi<\varphi_{\rm c}$,
where $\cos\varphi_{\rm c}\equiv-\Gamma$.

##  3 Application to simple models of the surface inhomogeneity

###  3.1 Cap-model: a constant-intensity circular spot

Let us consider a simple model (Figure 3) in which the surface intensity is
given as

$\displaystyle
I(\theta)=\left\\{\begin{array}[]{ll}I_{s}&(0<\theta<\Delta\theta_{\rm s})\\\
I_{0}&(\Delta\theta_{\rm s}<\theta<\pi)\end{array}\right..$ (43)

This corresponds to a large spot of constant intensity $I_{s}$ around the
north pole on the otherwise homogeneous surface of intensity $I_{0}$. Without
loss of generality, we assume that the size of the spot is less than half of
the entire surface ($\Delta\theta_{\rm s}<\pi/2$); the result for
$\Delta\theta_{\rm s}>\pi/2$ is obtained simply by exchanging $I_{s}$ and
$I_{0}$ and replacing $\Delta\theta_{\rm s}$ with $\pi-\Delta\theta_{\rm s}$
in the following expressions.

Figure 3: An instantaneous view of the spot configuration for the cap-model.
The spot is located at the pole region ($\theta<\Delta\theta_{\rm s}$) with
the constant intensity $I_{\rm s}$, while the intensity of the other surface
is $I_{0}$.

We compute equation (19) without limb darkening (the derivation is given in
appendix A), and summarize the final expressions below.

(a)

$0<\gamma<\pi/2$ and $\Delta\theta_{\rm s}<\pi/2-\gamma$:

$\displaystyle L_{\rm
0a}(t)=\frac{I_{0}}{\overline{I}}-\frac{I_{0}-I_{s}}{\overline{I}}\sin^{2}\Delta\theta_{\rm
s}\cos\gamma.$ (44)

(b)

$0<\gamma<\pi/2$ and $\pi/2-\gamma<\Delta\theta_{\rm s}<\pi/2$:

$\displaystyle L_{\rm
0b}(t)=\frac{I_{0}}{\overline{I}}-2\sin\gamma\cot^{3}\gamma\frac{(I_{0}-I_{s})}{\pi\overline{I}}\int_{\varphi_{\rm
c,s}}^{\pi}F_{0}(x;\gamma)dx,$ (45)

where $\cos\varphi_{\rm c,s}\equiv-\cot\gamma\cot\Delta\theta_{\rm s}$.

(c)

$\pi/2<\gamma<\pi$ and $\Delta\theta_{\rm s}<\gamma-\pi/2$:

$\displaystyle L_{\rm 0c}(t)=\frac{I_{0}}{\overline{I}}.$ (46)

(d)

$\pi/2<\gamma<\pi$ and $\gamma-\pi/2<\Delta\theta_{\rm s}<\pi/2$:

$\displaystyle L_{\rm
0d}(t)=\frac{I_{0}}{\overline{I}}-2\sin\gamma\cot^{3}\gamma\frac{(I_{0}-I_{s})}{\pi\overline{I}}\int_{0}^{\varphi_{\rm
c,s}}F_{0}(x;\gamma)dx.$ (47)

The case (a) corresponds to the situation that the whole spot is visible to
the observer, and thus the modulation is proportional to the projected area of
the spot ($\propto\cos\gamma$). This clearly indicates that the resulting
modulation is proportional to $\cos\omega_{\star}t$; see equation (1). In
contrast, the spot is completely invisible and there is no observed modulation
in the case (c). In the other two cases (b) and (d), the part of the spot is
periodically visible, and the corresponding lightcurves become a sinusoidal
form truncated at upper/lower amplitudes.

We note that the cap-model can be generalized to a star with multiple large
spots in a straightforward manner by summing up the photometric variations due
to the $i$-th spot with $I_{{\rm s},i}$, $\theta_{{\rm s},i}(t)$,
$\Delta\theta_{{\rm s},i}$, and the initial phase of $\varphi_{i}$.

###  3.2 p-model: an inhomogeneous surface with a globally varying intensity

Another simple model that we consider here assumes the parameterized surface
intensity $I(\theta)$:

$\displaystyle I(\theta)=I_{0}-\Delta I\cos^{p}\theta,$ (48)

where $I_{0}$ is the intensity at the equator ($\theta=\pi/2$), and $\Delta I$
is the amplitude of the intensity variation on the surface.

If $p$ is an odd integer, $I(\theta)$ varies monotonically from $I_{0}-\Delta
I$ (at $\theta=0$) to $I_{0}+\Delta I$ (at $\theta=\pi$), which is roughly
consistent with previous data (e.g., Landolfi et al., 1993; Valyavin et al.,
2008, 2011, 2014). In this case, equation (19) is rewritten as

$\displaystyle L_{\rm 0p}(t)$
$\displaystyle=\frac{I_{0}}{\overline{I}}-\frac{\Delta
I}{\overline{I}}\frac{2(1-\sin^{p+2}\gamma)\cos\gamma}{p+2}$ (50)
$\displaystyle\qquad\qquad-\frac{\Delta
I}{\overline{I}}\frac{2\sin\gamma\cot^{3}\gamma}{\pi}\int_{0}^{\pi}F_{0}(x;\gamma)\left(\frac{\cos
x}{\sqrt{\cos^{2}x+\cot^{2}\gamma}}\right)^{p}dx,$

where the second line is obtained from equation (76). Interestingly, after
integrating by parts, we find the analytical result:

$\displaystyle\int_{0}^{\pi}F_{0}(x;\gamma)\left(\frac{\cos
x}{\sqrt{\cos^{2}x+\cot^{2}\gamma}}\right)^{p}dx=\frac{\pi}{p+2}\frac{\sin^{p+2}\gamma}{\cot^{2}\gamma},$
(51)

if $p$ is an odd integer. Thus, equation (50) reduces to the following simple
expression:

$\displaystyle L_{\rm 0p}(t)=\frac{I_{0}}{\overline{I}}-\frac{\Delta
I}{\overline{I}}\frac{2}{p+2}\cos\gamma.$ (52)

As in the cap-model, it has a modulation component proportional to
$\cos\gamma$, and thus to $\cos\omega_{\star}t$, but the parameter $p$ cannot
be measured because it is completely degenerate with the modulation amplitude
$\Delta I$, as long as it is an odd integer.

###  3.3 Lightcurve variations without limb darkening

In our modeling, the time-dependence of the lightcurve $L(t)$ is completely
specified by that of $\gamma(t)$. Thus we define the scaled lightcurve
variation in terms of $\gamma$:

$\displaystyle\Delta\tilde{L}(\gamma)\equiv\frac{L(\gamma)-L(\gamma=0)}{L(\gamma=\pi)-L(\gamma=0)},$
(53)

which is plotted in Figure 4 for our cap-model and p-model without limb
darkening. For the p-model, $\Delta\tilde{L}(\gamma)$ simply reduces to
$(1-\cos\gamma)/2$ regardless of the value of $p$, as shown in equation (52).
In contrast, $\Delta\tilde{L}(\gamma)$ depends on the value of
$\Delta\theta_{\rm s}$ as well; see equations (44) to (47).

Figure 4: Scaled lightcurve variation $\Delta\tilde{L}(\gamma)$ against
$\gamma$ for $I_{\rm s}<I_{0}$. The four solid lines correspond to the cap-
model with $\Delta\theta_{\rm s}=15^{\circ}$, $30^{\circ}$, $60^{\circ}$, and
$75^{\circ}$, while the dashed line corresponds to the p-model. The bottom
panel shows a schematic illustration of the stellar surface at different
phases for the p-model.

In reality, however, the value of $\gamma$ is not known to the observer, and
$\Delta\tilde{L}(\gamma)$ is not directly observable. Instead, we define the
scaled lightcurve variation in terms of $t$:

$\displaystyle\tilde{L}(t)\equiv\frac{L(\omega_{\star}t)-L_{\rm
0a}(\omega_{\star}t=0)}{L_{\rm 0a}(\omega_{\star}t=\pi)-L_{\rm
0a}(\omega_{\star}t=0)},$ (54)

which can be evaluated by fitting equation (44) to the sinusoidal portion of
the measured data. Figure 5 plots $\tilde{L}(t)$ for the cap-model against
$\omega_{\star}t$ without limb darkening.

Figure 5: Scaled lightcurve variation $\tilde{L}(t)$ for the cap-model against
the orbital phase ($\omega_{\star}t$) with $I_{\rm s}<I_{0}$. The top, middle,
and bottom panels correspond to $\theta_{\star}=15^{\circ}$, $60^{\circ}$, and
$120^{\circ}$ respectively, for $\theta_{\rm o}=30^{\circ}$, $60^{\circ}$, and
$90^{\circ}$. Each panel has four curves corresponding to the spot size of
$\Delta\theta_{\rm s}=15^{\circ}$, $30^{\circ}$, $60^{\circ}$, and
$75^{\circ}$.

There is a certain degeneracy among parameters in the cap-model. If the
lightcurve becomes saturated corresponding to equation (46), we can measure
the threshold value of $\omega_{\star}t=\theta_{\rm crit}$ which gives

$\displaystyle\sin\Delta\theta_{\rm s}=|\cos\gamma_{\rm
crit}|\equiv|\sin\theta_{\star}\sin\theta_{\rm o}\cos\theta_{\rm
crit}+\cos\theta_{\star}\cos\theta_{\rm o}|.$ (55)

Then fitting to equation (44) with equation (55) provides a relation among
$I_{s}/I_{0}$, $\theta_{\star}$ and $\theta_{\rm o}$:

$\displaystyle L_{\rm 0a}(t)$
$\displaystyle=\frac{I_{0}}{\overline{I}}-\frac{I_{0}-I_{s}}{\overline{I}}\sin^{2}\Delta\theta_{\rm
s}\left(\sin\theta_{\star}\sin\theta_{\rm
o}\cos\omega_{\star}t+\cos\theta_{\star}\cos\theta_{\rm o}\right)$ (58)
$\displaystyle=\left(\frac{I_{0}}{\overline{I}}-\frac{I_{0}-I_{s}}{\overline{I}}\sin^{2}\Delta\theta_{\rm
s}\cos\theta_{\star}\cos\theta_{\rm o}\right)$
$\displaystyle\qquad\times\left[1-\frac{(1-I_{s}/I_{0})\sin^{2}\Delta\theta_{\rm
s}\sin\theta_{\star}\sin\theta_{\rm
o}}{1-(1-I_{s}/I_{0})\sin^{2}\Delta\theta_{\rm
s}\cos\theta_{\star}\cos\theta_{\rm o}}\cos\omega_{\star}t\right].$

The above equation explicitly indicates the degeneracy of the parameters
involved in the amplitude of the sinusoidal curve. Therefore, the departure
from the purely sinusoidal curve (see Figure 5) corresponding to equations
(45) to (47) is important for the accurate determinations of the model
parameters. We will discuss the parameter degeneracy in §3.5 below.

###  3.4 Limb darkening effect for the p-model

We compute the effect of limb darkening for the p-model. Substituting equation
(48) to equations (22), (29), (38), and (40), we find

$\displaystyle L_{\rm 1p}(\gamma)$ $\displaystyle=$
$\displaystyle(u_{1}+2u_{2})\left[\frac{2}{3}\frac{I_{0}}{\overline{I}}-\frac{\Delta
I}{\overline{I}}J_{\rm 1p}(\gamma)\right],$ (59) $\displaystyle L_{\rm
2p}(\gamma)$ $\displaystyle=$ $\displaystyle-
u_{2}\left[\frac{1}{2}\frac{I_{0}}{\overline{I}}-\frac{\Delta
I}{\overline{I}}J_{\rm 2p}(\gamma)\right],$ (60)

where

$\displaystyle J_{\rm 1p}(\gamma)$
$\displaystyle=\frac{2+2p\cos^{2}\gamma-\sin^{p+3}\gamma[3(p+1)\cos^{2}\gamma+2]}{(p+1)(p+3)}\frac{\cos\gamma}{|\cos\gamma|}$
(62)
$\displaystyle\qquad\qquad\qquad-\frac{\sin^{2}\gamma\cot^{5}\gamma}{2\pi|\cot\gamma|}\int_{0}^{\pi}F_{1}(x;\gamma)\left(\frac{\cos
x}{\sqrt{\cos^{2}x+\cot^{2}\gamma}}\right)^{p}dx$ $\displaystyle J_{\rm
2p}(\gamma)$
$\displaystyle=\frac{6+2(p-1)\cos^{2}\gamma-\sin^{p+4}\gamma[5(p+2)\cos^{2}\gamma+6]}{(p+2)(p+4)}\cos\gamma$
(65)
$\displaystyle\qquad\qquad\qquad+\frac{\sin^{3}\gamma\cot^{5}\gamma}{12\pi}\int_{0}^{\pi}F_{2}(x;\gamma)\left(\frac{\cos
x}{\sqrt{\cos^{2}x+\cot^{2}\gamma}}\right)^{p}dx$
$\displaystyle=\frac{6+2(p-1)\cos^{2}\gamma}{(p+2)(p+4)}\cos\gamma.$

Thus, the normalized lightcurve for the p-model including quadratic limb
darkening is given by

$\displaystyle L_{\rm p}(\gamma)$
$\displaystyle=\frac{I_{0}}{\overline{I}}\left[(1-u_{1}-u_{2})+\frac{2(u_{1}+2u_{2})}{3}-\frac{u_{2}}{2}\right]$
(67) $\displaystyle\qquad-\frac{\Delta
I}{\overline{I}}\left[\frac{2(1-u_{1}-u_{2})}{p+2}\cos\gamma+(u_{1}+2u_{2})J_{\rm
1p}(\gamma)-u_{2}J_{\rm 2p}(\gamma)\right],$

where $p$ is an odd integer.

Figure 6: Effect of limb darkening for the p-model. The upper two panels are
$J_{\rm 1p}$ and $J_{\rm 2p}$ against $\gamma$ with different values of $p$.
The lower two panels plot the corresponding scaled lightcurve variations
$\Delta\tilde{L}_{\rm p}$ for $p=1$ and 3, adopting the different sets of the
limb darkening coefficients $(u_{1},\;u_{2})$.

The upper two panels in Figure 6 show $J_{\rm 1p}$ and $J_{\rm 2p}$ against
$\gamma$ for the p-model with $p=1$ (blue), 3 (orange) and 5 (green). The
corresponding scaled lightcurve variations $\Delta\tilde{L}_{\rm p}(\gamma)$,
computed from equations (53) and (67) are plotted in the lower two panels for
$p=1$ and 3, adopting the limb darkening coefficients of
$(u_{1},\;u_{2})=(0.47,\;0.23)$ for the Sun, and $(0.19,\;0.43)$ and
$(0.05,\;0.15)$ for WDs with $(T_{\rm eff},\;\log g)=(8\times 10^{3}{\rm
K},\;9.2)$, and $(4.6\times 10^{4}{\rm K},\;9.5)$, respectively (Kilic et al.,
2021; Caiazzo et al., 2021). For reference, $\Delta\tilde{L}_{\rm p}(\gamma)$
without limb darkening is plotted as well, while it is barely distinguishable
from the other curves.

In the case of $p=1$, the scaled lightcurves seem to be identical to
$(1-\cos\gamma)/2$, independently of the limb-darkening coefficients. This
implies that both $J_{\rm 1p}$ and $J_{\rm 2p}$ are proportional to
$\cos\gamma$ for $p=1$. Equation (65) indeed shows that it is the case for
$J_{\rm 2p}$(p=1). We believe that it is also the case for $J_{\rm 1p}(p=1)$,
while we are not yet able to prove it analytically. The scaled lightcurves
with $p=3$, on the other hand, show very weak dependence on the limb-darkening
coefficients, but they are still very well approximated by $(1-\cos\gamma)/2$.
Therefore, we conclude that the limb-darkening effect is practically
negligible in the p-model, and expect that it is the case for the cap-model as
well.

###  3.5 Fitting the cap-model to the phase-folded lightcurves of a fast-
spinning white dwarf ZTF J190132.9+145808.7

As an application of our current modeling, we attempt to fit the lightcurve of
a fast-spinning white dwarf ZTF J190132.9+145808.7. We download the reduced
photometric lightcurves111https://github.com/ilac/ZTF-J1901-1458 that are made
publicly available by Caiazzo et al. (2021). The total duration of the data is
$\sim 80$ minutes, and the cadence is $\sim 1$ second. We use the data in the
blue band alone, and omit the last 200 data points because they seem to be
significantly contaminated by outliers. The long-term trend is then removed
using the Savitzky-Golay filter (Savitzky & Golay, 1964), implemented as
LightCurve.flatten in lightkurve modules (Lightkurve Collaboration et al.,
2018). We adopt a window width of $1500$ seconds, and the data are folded
according to the rotational period of $416.097$ seconds. Finally, we average
the data over the bin width of $1$ second.

Figure 7: Best-fits to the phase-folded normalized lightcurve of ZTF
J190132.9+145808.7, and their residuals for p-model (left) and cap-model
(right).

As shown in the above subsection, the limb darkening effect is practically
negligible. Thus, we ignore it in the present analysis. Figure 7 shows the
best-fits and the residuals for the p-model (left) and the cap-model (right)
from the nonlinear least square regression. While a purely sinusoidal curve
predicted from the p-model does not fit the data, the cap-model is consistent
with the feature of the observed lightcurve. Thus, we proceed to perform a
Markov Chain Monte Carlo (MCMC) ensemble sampler for the cap-model using EMCEE
(Foreman-Mackey et al., 2013).

Figure 8: Posterior distribution among the four parameters, $I_{s}/I_{o}$,
$\Delta\theta_{\rm s}$, $\theta_{\rm o}$ and $\theta_{\star}$, from the MCMC
analysis in the cap-model. The observed lightcurve is averaged over a bin size
of 5 seconds. The filled circle and triangle indicate solutions obtained from
the nonlinear least square fit. Different curve correspond to the contours of
$0.5\sigma$ (cyan) , $1.0\sigma$ (red), $1.5\sigma$ (orange), and $2.0\sigma$
(green) levels for two-dimensional histograms, respectively. The magenta
symbols with error-bars indicate our best-fit parameters. The figure is
generated using the CORNER package (Foreman-Mackey, 2016).  Figure 9: Same of
Figure 8, but the posterior distribution of $\theta_{\star}-\theta_{\rm o}$
and $\theta_{\star}+\theta_{\rm o}$ in the cap-model.

The likelihood function is taken to be proportional to
$\mathrm{exp}(-\chi^{2})$:

$\chi^{2}=\sum_{i}\frac{(L_{\mathrm{cap},i}-L_{\mathrm{obs},i})^{2}}{\sigma_{i}^{2}},$
(68)

where $L_{\mathrm{cap},i}$ is the cap-model prediction, $L_{\mathrm{obs},i}$
is the observed data, and $\sigma_{i}^{2}$ is the variance of the data in the
$i$-th bin. The predicted lightcurve of the cap-model is normalized to fit the
amplitude of the observed light curve. We adopt uniform priors for the four
free parameters over the ranges of $0.8<I_{s}/I_{o}<1.2$,
$0^{\circ}<\Delta\theta_{\rm s}<90^{\circ}$, and $0^{\circ}<\theta_{\rm
o},~{}\theta_{\star}<180^{\circ}$.

The posterior distributions for $I_{s}/I_{o}$, $\Delta\theta_{\rm s}$,
$\theta_{\rm o}$ and $\theta_{\star}$ from the MCMC analysis are shown in
Figure 8, together with their marginalized probability density functions. The
degree of the parameter degeneracy is clearly visible in Figure 8. The
amplitude of the photometric variation and the departure from the sinusoidal
curve are specified by $I_{s}/I_{o}$ and $\Delta\theta_{\rm s}$, respectively.
Thus, these two parameters are well constrained from the fit. We define the
best-fit parameters of $I_{s}/I_{o}$ and $\Delta\theta_{\rm s}$ by the mode
and the corresponding $1\sigma$ errors from the histograms in Figure 8.

In contrast, the remaining two parameters, $\theta_{\star}$ and $\theta_{\rm
o}$, are intrinsically degenerate, which can be understood from the fact that
$\gamma$ is invariant by exchanging $\theta_{\rm o}$ and $\theta_{\star}$.
Therefore, we put a constraint of $\theta_{\star}>\theta_{\rm o}$ in Figure 8.
Even under the constraint, there are two peaks, roughly corresponding to
$(\theta_{\rm o},\theta_{\star})$ $\approx(60^{\circ},90^{\circ})$ and
$(90^{\circ},120^{\circ})$. In reality, however, these two are the same
solution because $\theta_{\star}=60^{\circ}$ and $120^{\circ}$ lead to the
identical lightcurve if $\theta_{\rm o}=90^{\circ}$ as shown in Figure 5.
Thus, the peak at $(\theta_{\rm o},\theta_{\star})$
$\approx(90^{\circ},120^{\circ})$ is the same at $(90^{\circ},60^{\circ})$,
which is simply equivalent to the other peak at $(\theta_{\rm
o},\theta_{\star})$ $\approx(60^{\circ},90^{\circ})$.

This parameter degeneracy is shown more clearly in Figure 9, where we plot the
posterior distribution on the $\theta_{\star}-\theta_{\rm o}$ vs
$\theta_{\star}+\theta_{\rm o}$ plane. Therefore, the best-fits values of
$\theta_{\rm o}$ and $\theta_{\star}$ are not easy to specify, and we decide
to adopt their modes and the associated $1\sigma$ errors from the two-
dimensional contours on the $\theta_{\rm o}$ – $\theta_{\star}$ plane in
Figure 8. Having such degeneracy in mind, we summarize the best-fit parameters
of the cap-model for ZTF J190132.9+145808.7 as follows:

$\displaystyle I_{s}/I_{0}$
$\displaystyle=0.971^{+0.001}_{-0.008},\qquad\Delta\theta_{\rm
s}=64^{+6}_{-14}~{}\mathrm{deg.},$ (69) $\displaystyle(\theta_{\rm
o},\theta_{\star})$
$\displaystyle=(55^{+11}_{-31}~{}\mathrm{deg.},85^{+31}_{-7}~{}\mathrm{deg.})~{}{\rm
or}~{}(90^{+18}_{-12}~{}\mathrm{deg.},120^{+24}_{-14}~{}\mathrm{deg.}).$ (70)

The above solutions are indeed quite interesting. If $\theta_{\rm o}\approx
60^{\circ}$, the spot is located along the equatorial plane of the spin axis
($\theta_{\star}=90^{\circ}$). Even if we are viewing the WD from the edge-on
direction ($\theta_{\rm o}\approx 90^{\circ}$), the obliquity of the spot is
$\theta_{\star}\approx 60^{\circ}$, implying that the spot region is
significantly away from the polar regions of the spin axis of the rotating WD.

We conclude that ZTF J190132.9+145808.7 has a relatively large dark spot. As
inferred from the phase-resolved spectrum (Caiazzo et al., 2021), the dark
spot is likely to be explained in terms of the magnetic dichroism. The spot
structure may be also consistent with an off-centered dipole field as in the
case of EUVE J0317-855 (Ferrario et al., 1997); the off-centered dipole field
is an empirical configuration of the magnetic field, and assumes that its
center differs from the stellar center by some offset along the dipole
direction, while the surface brightness is still axisymmetric with respect to
the dipole axis. Ferrario et al. (1997) found that the off-centered dipole
field explains the lightcurve and polarization of EUVE J0317-855. If ZTF
J190132.9+145808.7 has a similar configuration, its obliquity angle between
the rotation and magnetic axis should be very large ($\theta_{\star}\approx
60^{\circ}$ or $90^{\circ}$), implying that the spot region, probably
challenging a formation mechanism of such spots.

##  4 Discussion and conclusion

We have presented a formulation of photometric variations of rotating stars
with an inhomogeneous surface intensity. We have provided a set of general
expressions for the photometric lightcurve due to the oblique stellar rotation
under the assumption that the inhomogeneity pattern is symmetric around an
axis misaligned to the stellar spin direction. We have applied the formulation
to two simple inhomogeneity models that are supposed to approximate the
surface of WDs, and showed that they are able to reproduce the monochromatic
sinusoidal lightcurves observed for a majority of variable WDs. Therefore, we
conclude that the monochromatic sinusoidal lightcurve is a fairly generic
prediction for obliquely rotating stars.

Furthermore, one of our surface models (the cap-model: §3.1) can model the
departure from a purely sinusoidal lightcurve that are observed for a fraction
of WDs including ZTF J190132.9+145808.7 (Caiazzo et al., 2021). The degree of
the parameter degeneracy has been discussed extensively in §3.5. Our precise
model prediction for the lightcurve shape will be also useful to distinguish
between the oblique rotation model and the pulsation signature together with
their difference in the frequency power spectra.

We note that the present method is not limited to white dwarfs alone, and
applicable to predict photometric variations of a wide class of obliquely
rotating stars, once their surface inhomogeneity distribution is specified.
Indeed, our current models provide a very useful parameterization that
characterizes a physical model of the surface distribution. Thus it is
complementary to the inversion method (e.g., Kawahara & Fujii, 2010, 2011;
Fujii & Kawahara, 2012; Aizawa et al., 2020; Luger et al., 2021) to infer the
surface inhomogeneity distribution that does not require a priori model but is
challenging to reconstruct the surface model from the limited signal-to-noise
ratio of the observational data.

Even in purely sinusoidal lightcurves, the parameter degeneracy may be broken
by combining spectro-polarimetric data. Since the surface inhomogeneity is
likely originated from the stellar magnetic field, the measurements of the
longitudinal and surface components of the time-varying magnetic fields may
reveal the orientation of the dipolar magnetic fields, i.e., $\theta_{\rm o}$
and $\theta_{\star}$ (Landolfi et al., 1993) in a complementary manner. Then
we may break the parameter degeneracy from the joint fits to the spectro-
polarimetric and photometric data. This methodology has been successfully
applied to WD 1953-011 (Valyavin et al., 2008) for instance, and will become
applicable more precisely using our quantitative modeling of the photometric
lightcurve.

According to our result, an appreciable non-sinusoidal signature appears on a
lightcurve when both the obliquity angle $\theta_{\star}$ and the stellar spin
inclination angle relative to the observer $\theta_{\rm o}$ is larger than the
spot size $\Delta\theta_{\rm s}$. In this case, the spot becomes invisible
from the observer for a certain rotation phase, and the photometric lightcurve
in the phase exhibits plateaus. The plateaus appear either at the tops or the
bottoms of the lightcurve when the spot has a lower or a higher intensity than
the average of the stellar disk, respectively. The former corresponds to a
dark spot ($I_{s}<I_{0}$) as in the case of ZTF J190132.9+145808.7. The latter
cases with a bright spot ($I_{s}>I_{0}$), or a facula, could also occur for a
strongly-magnetized fast-spinning WD, or a WD pulsar with a force-free
magnetosphere, as inferred for neutron-star pulsars e.g., PSR J0030+0451
(Bilous et al., 2019; Miller et al., 2019; Riley et al., 2019) and PSR
J0740+6620 (Riley et al., 2021; Miller et al., 2021). The non-sinusoidal
fraction of lightcurves of WDs in a certain temperature and mass range should
be a useful statistical measure of their magnetospheric structure. We plan to
perform systematic comparison of our lightcurve models against the photometric
data obtained by Kepler, TESS, ZTF, and Tomo-e Gozen (Aizawa et al., 2022),
which will be reported elsewhere in due course.

## Acknowledgements

Y.S. thanks Kei Iida for his hospitality at Kochi University where the
manuscript of the present paper was completed. Simulations and analyses in
this paper made use of a community-developed core Python package for
Astronomy, Astropy (Price-Whelan et al., 2018), and a Python package for
Kepler and TESS data analysis, Lightkurve (Lightkurve Collaboration et al.,
2018). This work is supported by Grants-in Aid for Scientific Research by the
Japan Society for Promotion of Science, Nos.18H012 (YS), 19H01947 (YS),
20K14512 (KF), 17K14248 (KK), and 18H04573 (KK).

## References

  * Achilleos et al. (1992) Achilleos, N., Wickramasinghe, D. T., Liebert, J., Saffer, R. A., & Grauer, A. D. 1992, ApJ, 396, 273
  * Aizawa et al. (2020) Aizawa, M., Kawahara, H., & Fan, S. 2020, ApJ, 896, 22
  * Aizawa et al. (2022) Aizawa, M., Kawana, K., Kashiyama, K., Ohsawa, R., Kawahara, H., Naokawa, F., Tajiri, T., Arima, N., Jiang, H., Hartwig, T., Fujisawa, K., Shigeyama, T., Arimatsu, K., Doi, M., Kasuga, T., Kobayashi, N., Kondo, S., Mori, Y., Okumura, S.-i., Takita, S., & Sako, S. 2022, PASJ
  * Barstow et al. (1995) Barstow, M. A., Jordan, S., O’Donoghue, D., Burleigh, M. R., Napiwotzki, R., & Harrop-Allin, M. K. 1995, MNRAS, 277, 971
  * Bilous et al. (2019) Bilous, A. V., Watts, A. L., Harding, A. K., Riley, T. E., Arzoumanian, Z., Bogdanov, S., Gendreau, K. C., Ray, P. S., Guillot, S., Ho, W. C. G., & Chakrabarty, D. 2019, ApJ, 887, L23
  * Brinkworth et al. (2013) Brinkworth, C. S., Burleigh, M. R., Lawrie, K., Marsh, T. R., & Knigge, C. 2013, ApJ, 773, 47
  * Brinkworth et al. (2005) Brinkworth, C. S., Marsh, T. R., Morales-Rueda, L., Maxted, P. F. L., Burleigh, M. R., & Good, S. A. 2005, MNRAS, 357, 333
  * Budding (1977) Budding, E. 1977, Ap&SS, 48, 207
  * Caiazzo et al. (2021) Caiazzo, I., Burdge, K. B., Fuller, J., Heyl, J., Kulkarni, S. R., Prince, T. A., Richer, H. B., Schwab, J., Andreoni, I., Bellm, E. C., Drake, A., Duev, D. A., Graham, M. J., Helou, G., Mahabal, A. A., Masci, F. J., Smith, R., & Soumagnac, M. T. 2021, Nature, 595, 39
  * Cox (2000) Cox, A. N. 2000, Allen’s astrophysical quantities, 4th edn. (Springer–Verlag)
  * Donati et al. (1994) Donati, J. F., Achilleos, N., Matthews, J. M., & Wesemael, F. 1994, A&A, 285, 285
  * Dorren (1987) Dorren, J. D. 1987, ApJ, 320, 756
  * Eker (1994) Eker, Z. 1994, ApJ, 420, 373
  * Euchner et al. (2002) Euchner, F., Jordan, S., Beuermann, K., Gänsicke, B. T., & Hessman, F. V. 2002, A&A, 390, 633
  * Farr et al. (2018) Farr, B., Farr, W. M., Cowan, N. B., Haggard, H. M., & Robinson, T. 2018, AJ, 156, 146
  * Ferrario et al. (1997) Ferrario, L., Vennes, S., Wickramasinghe, D. T., Bailey, J. A., & Christian, D. J. 1997, MNRAS, 292, 205
  * Foreman-Mackey (2016) Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Fujii & Kawahara (2012) Fujii, Y. & Kawahara, H. 2012, ApJ, 755, 101
  * Fujii et al. (2011) Fujii, Y., Kawahara, H., Suto, Y., Fukuda, S., Nakajima, T., Livengood, T. A., & Turner, E. L. 2011, ApJ, 738, 184
  * Fujii et al. (2010) Fujii, Y., Kawahara, H., Suto, Y., Taruya, A., Fukuda, S., Nakajima, T., & Turner, E. L. 2010, ApJ, 715, 866
  * Gianninas et al. (2013) Gianninas, A., Strickland, B. D., Kilic, M., & Bergeron, P. 2013, ApJ, 766, 3
  * Haggard & Cowan (2018) Haggard, H. M. & Cowan, N. B. 2018, MNRAS, 478, 371
  * Hermes et al. (2021) Hermes, J. J., Putterman, O., Hollands, M. A., Wilson, D. J., Swan, A., Raddi, R., Shen, K. J., & Gänsicke, B. T. 2021, ApJ, 914, L3
  * Hollands et al. (2020) Hollands, M. A., Tremblay, P. E., Gänsicke, B. T., Camisassa, M. E., Koester, D., Aungwerojwit, A., Chote, P., Córsico, A. H., Dhillon, V. S., Gentile-Fusillo, N. P., Hoskin, M. J., Izquierdo, P., Marsh, T. R., & Steeghs, D. 2020, Nature Astronomy, 4, 663
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., Abel, B., Acosta, E., Allsman, R., Alonso, D., AlSayyad, Y., Anderson, S. F., Andrew, J., Angel, J. R. P., Angeli, G. Z., Ansari, R., Antilogus, P., Araujo, C., Armstrong, R., Arndt, K. T., Astier, P., Aubourg, É., Auza, N., Axelrod, T. S., Bard, D. J., Barr, J. D., Barrau, A., Bartlett, J. G., Bauer, A. E., Bauman, B. J., Baumont, S., Bechtol, E., Bechtol, K., Becker, A. C., Becla, J., Beldica, C., Bellavia, S., Bianco, F. B., Biswas, R., Blanc, G., Blazek, J., Blandford, R. D., Bloom, J. S., Bogart, J., Bond, T. W., Booth, M. T., Borgland, A. W., Borne, K., Bosch, J. F., Boutigny, D., Brackett, C. A., Bradshaw, A., Brandt, W. N., Brown, M. E., Bullock, J. S., Burchat, P., Burke, D. L., Cagnoli, G., Calabrese, D., Callahan, S., Callen, A. L., Carlin, J. L., Carlson, E. L., Chandrasekharan, S., Charles-Emerson, G., Chesley, S., Cheu, E. C., Chiang, H.-F., Chiang, J., Chirino, C., Chow, D., Ciardi, D. R., Claver, C. F., Cohen-Tanugi, J., Cockrum, J. J., Coles, R., Connolly, A. J., Cook, K. H., Cooray, A., Covey, K. R., Cribbs, C., Cui, W., Cutri, R., Daly, P. N., Daniel, S. F., Daruich, F., Daubard, G., Daues, G., Dawson, W., Delgado, F., Dellapenna, A., de Peyster, R., de Val-Borro, M., Digel, S. W., Doherty, P., Dubois, R., Dubois-Felsmann, G. P., Durech, J., Economou, F., Eifler, T., Eracleous, M., Emmons, B. L., Fausti Neto, A., Ferguson, H., Figueroa, E., Fisher-Levine, M., Focke, W., Foss, M. D., Frank, J., Freemon, M. D., Gangler, E., Gawiser, E., Geary, J. C., Gee, P., Geha, M., Gessner, C. J. B., Gibson, R. R., Gilmore, D. K., Glanzman, T., Glick, W., Goldina, T., Goldstein, D. A., Goodenow, I., Graham, M. L., Gressler, W. J., Gris, P., Guy, L. P., Guyonnet, A., Haller, G., Harris, R., Hascall, P. A., Haupt, J., Hernandez, F., Herrmann, S., Hileman, E., Hoblitt, J., Hodgson, J. A., Hogan, C., Howard, J. D., Huang, D., Huffer, M. E., Ingraham, P., Innes, W. R., Jacoby, S. H., Jain, B., Jammes, F., Jee, M. J., Jenness, T., Jernigan, G., Jevremović, D., Johns, K., Johnson, A. S., Johnson, M. W. G., Jones, R. L., Juramy-Gilles, C., Jurić, M., Kalirai, J. S., Kallivayalil, N. J., Kalmbach, B., Kantor, J. P., Karst, P., Kasliwal, M. M., Kelly, H., Kessler, R., Kinnison, V., Kirkby, D., Knox, L., Kotov, I. V., Krabbendam, V. L., Krughoff, K. S., Kubánek, P., Kuczewski, J., Kulkarni, S., Ku, J., Kurita, N. R., Lage, C. S., Lambert, R., Lange, T., Langton, J. B., Le Guillou, L., Levine, D., Liang, M., Lim, K.-T., Lintott, C. J., Long, K. E., Lopez, M., Lotz, P. J., Lupton, R. H., Lust, N. B., MacArthur, L. A., Mahabal, A., Mandelbaum, R., Markiewicz, T. W., Marsh, D. S., Marshall, P. J., Marshall, S., May, M., McKercher, R., McQueen, M., Meyers, J., Migliore, M., Miller, M., Mills, D. J., Miraval, C., Moeyens, J., Moolekamp, F. E., Monet, D. G., Moniez, M., Monkewitz, S., Montgomery, C., Morrison, C. B., Mueller, F., Muller, G. P., Muñoz Arancibia, F., Neill, D. R., Newbry, S. P., Nief, J.-Y., Nomerotski, A., Nordby, M., O’Connor, P., Oliver, J., Olivier, S. S., Olsen, K., O’Mullane, W., Ortiz, S., Osier, S., Owen, R. E., Pain, R., Palecek, P. E., Parejko, J. K., Parsons, J. B., Pease, N. M., Peterson, J. M., Peterson, J. R., Petravick, D. L., Libby Petrick, M. E., Petry, C. E., Pierfederici, F., Pietrowicz, S., Pike, R., Pinto, P. A., Plante, R., Plate, S., Plutchak, J. P., Price, P. A., Prouza, M., Radeka, V., Rajagopal, J., Rasmussen, A. P., Regnault, N., Reil, K. A., Reiss, D. J., Reuter, M. A., Ridgway, S. T., Riot, V. J., Ritz, S., Robinson, S., Roby, W., Roodman, A., Rosing, W., Roucelle, C., Rumore, M. R., Russo, S., Saha, A., Sassolas, B., Schalk, T. L., Schellart, P., Schindler, R. H., Schmidt, S., Schneider, D. P., Schneider, M. D., Schoening, W., Schumacher, G., Schwamb, M. E., Sebag, J., Selvy, B., Sembroski, G. H., Seppala, L. G., Serio, A., Serrano, E., Shaw, R. A., Shipsey, I., Sick, J., Silvestri, N., Slater, C. T., Smith, J. A., Smith, R. C., Sobhani, S., Soldahl, C., Storrie-Lombardi, L., Stover, E., Strauss, M. A., Street, R. A., Stubbs, C. W., Sullivan, I. S., Sweeney, D., Swinbank, J. D., Szalay, A., Takacs, P., Tether, S. A., Thaler, J. J., Thayer, J. G., Thomas, S., Thornton, A. J., Thukral, V., Tice, J., Trilling, D. E., Turri, M., Van Berg, R., Vanden Berk, D., Vetter, K., Virieux, F., Vucina, T., Wahl, W., Walkowicz, L., Walsh, B., Walter, C. W., Wang, D. L., Wang, S.-Y., Warner, M., Wiecha, O., Willman, B., Winters, S. E., Wittman, D., Wolff, S. C., Wood-Vasey, W. M., Wu, X., Xin, B., Yoachim, P., & Zhan, H. 2019, ApJ, 873, 111
  * Kawahara & Fujii (2010) Kawahara, H. & Fujii, Y. 2010, ApJ, 720, 1333
  * Kawahara & Fujii (2011) —. 2011, ApJ, 739, L62
  * Kilic et al. (2021) Kilic, M., Kosakowski, A., Moss, A. G., Bergeron, P., & Conly, A. A. 2021, ApJ, 923, L6
  * Kipping (2012) Kipping, D. M. 2012, MNRAS, 427, 2487
  * Landolfi et al. (1997) Landolfi, M., Bagnulo, S., Landi Degl’Innocenti, M., Landi Degl’Innocenti, E., & Leroy, J. L. 1997, A&A, 322, 197
  * Landolfi et al. (1993) Landolfi, M., Landi Degl’Innocenti, E., Landi Degl’Innocenti, M., & Leroy, J. L. 1993, A&A, 272, 285
  * Liebert et al. (1977) Liebert, J., Angel, J. R. P., Stockman, H. S., Spinrad, H., & Beaver, E. A. 1977, ApJ, 214, 457
  * Lightkurve Collaboration et al. (2018) Lightkurve Collaboration, Cardoso, J. V. d. M., Hedges, C., Gully-Santiago, M., Saunders, N., Cody, A. M., Barclay, T., Hall, O., Sagear, S., Turtelboom, E., Zhang, J., Tzanidakis, A., Mighell, K., Coughlin, J., Bell, K., Berta-Thompson, Z., Williams, P., Dotson, J., & Barentsen, G. 2018, Lightkurve: Kepler and TESS time series analysis in Python, Astrophysics Source Code Library
  * Lu et al. (2022) Lu, Y., Benomar, O., Kamiaka, S., & Suto, Y. 2022, arXiv e-prints, arXiv:2206.05411
  * Luger et al. (2021) Luger, R., Foreman-Mackey, D., Hedges, C., & Hogg, D. W. 2021, AJ, 162, 123
  * Maoz et al. (2015) Maoz, D., Mazeh, T., & McQuillan, A. 2015, MNRAS, 447, 1749
  * Miller et al. (2019) Miller, M. C., Lamb, F. K., Dittmann, A. J., Bogdanov, S., Arzoumanian, Z., Gendreau, K. C., Guillot, S., Harding, A. K., Ho, W. C. G., Lattimer, J. M., Ludlam, R. M., Mahmoodifar, S., Morsink, S. M., Ray, P. S., Strohmayer, T. E., Wood, K. S., Enoto, T., Foster, R., Okajima, T., Prigozhin, G., & Soong, Y. 2019, ApJ, 887, L24
  * Miller et al. (2021) Miller, M. C., Lamb, F. K., Dittmann, A. J., Bogdanov, S., Arzoumanian, Z., Gendreau, K. C., Guillot, S., Ho, W. C. G., Lattimer, J. M., Loewenstein, M., Morsink, S. M., Ray, P. S., Wolff, M. T., Baker, C. L., Cazeau, T., Manthripragada, S., Markwardt, C. B., Okajima, T., Pollard, S., Cognard, I., Cromartie, H. T., Fonseca, E., Guillemot, L., Kerr, M., Parthasarathy, A., Pennucci, T. T., Ransom, S., & Stairs, I. 2021, ApJ, 918, L28
  * Nakagawa et al. (2020) Nakagawa, Y., Kodama, T., Ishiwatari, M., Kawahara, H., Suto, Y., Takahashi, Y. O., Hashimoto, G. L., Kuramoto, K., Nakajima, K., Takehiro, S.-i., & Hayashi, Y.-Y. 2020, ApJ, 898, 95
  * Price-Whelan et al. (2018) Price-Whelan, A. M., Sipőcz, B., Günther, H., Lim, P., Crawford, S., Conseil, S., Shupe, D., Craig, M., Dencheva, N., Ginsburg, A., et al. 2018, The Astronomical Journal, 156, 123
  * Reding et al. (2020) Reding, J. S., Hermes, J. J., Vanderbosch, Z., Dennihy, E., Kaiser, B. C., Mace, C. B., Dunlap, B. H., & Clemens, J. C. 2020, ApJ, 894, 19
  * Riley et al. (2019) Riley, T. E., Watts, A. L., Bogdanov, S., Ray, P. S., Ludlam, R. M., Guillot, S., Arzoumanian, Z., Baker, C. L., Bilous, A. V., Chakrabarty, D., Gendreau, K. C., Harding, A. K., Ho, W. C. G., Lattimer, J. M., Morsink, S. M., & Strohmayer, T. E. 2019, ApJ, 887, L21
  * Riley et al. (2021) Riley, T. E., Watts, A. L., Ray, P. S., Bogdanov, S., Guillot, S., Morsink, S. M., Bilous, A. V., Arzoumanian, Z., Choudhury, D., Deneva, J. S., Gendreau, K. C., Harding, A. K., Ho, W. C. G., Lattimer, J. M., Loewenstein, M., Ludlam, R. M., Markwardt, C. B., Okajima, T., Prescod-Weinstein, C., Remillard, R. A., Wolff, M. T., Fonseca, E., Cromartie, H. T., Kerr, M., Pennucci, T. T., Parthasarathy, A., Ransom, S., Stairs, I., Guillemot, L., & Cognard, I. 2021, ApJ, 918, L27
  * Roettenbacher et al. (2013) Roettenbacher, R. M., Monnier, J. D., Harmon, R. O., Barclay, T., & Still, M. 2013, ApJ, 767, 60
  * Savitzky & Golay (1964) Savitzky, A. & Golay, M. J. E. 1964, Analytical Chemistry, 36, 1627
  * Suto et al. (2022) Suto, Y., Sasaki, S., Nakagawa, Y., & Benomar, O. 2022, PASJ, 74, 857
  * Valyavin et al. (2011) Valyavin, G., Antonyuk, K., Plachinda, S., Clark, D. M., Wade, G. A., Fox Machado, L., Alvarez, M., Lopez, J. M., Hiriart, D., Han, I., Jeon, Y.-B., Bagnulo, S., Zharikov, S. V., Zurita, C., Mujica, R., Shulyak, D., & Burlakova, T. 2011, ApJ, 734, 17
  * Valyavin et al. (2014) Valyavin, G., Shulyak, D., Wade, G. A., Antonyuk, K., Zharikov, S. V., Galazutdinov, G. A., Plachinda, S., Bagnulo, S., Fox Machado, L., Alvarez, M., Clark, D. M., Lopez, J. M., Hiriart, D., Han, I., Jeon, Y.-B., Zurita, C., Mujica, R., Burlakova, T., Szeifert, T., & Burenkov, A. 2014, Nature, 515, 88
  * Valyavin et al. (2008) Valyavin, G., Wade, G. A., Bagnulo, S., Szeifert, T., Landstreet, J. D., Han, I., & Burenkov, A. 2008, ApJ, 683, 466
  * Wade et al. (2003) Wade, G. A., Bagnulo, S., Szeifert, T., Brinkworth, C., Marsh, T., Landstreet, J. D., & Maxted, P. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 307, Solar Polarization, ed. J. Trujillo-Bueno & J. Sanchez Almeida, 569
  * Williams et al. (2022) Williams, K. A., Hermes, J. J., & Vanderbosch, Z. P. 2022, AJ, 164, 131
  * Winget & Kepler (2008) Winget, D. E. & Kepler, S. O. 2008, ARA&A, 46, 157

##  Appendix A Analytic derivation of integrals appearing in the expressions
for photometric variations

Lightcurves for the cap-model without limb darkening are computed from
equation (19), and rewritten as follows, depending on the four different
parameter ranges of $\gamma$ and $\Delta\theta_{\rm s}$.

(a)

$0<\gamma<\pi/2$ and $\Delta\theta_{\rm s}<\pi/2-\gamma$:

$\displaystyle L_{\rm 0a}(t)$ $\displaystyle=$
$\displaystyle\frac{\cos\gamma}{2\overline{I}}\left[I_{s}(1-\cos
2\Delta\theta_{\rm s})+I_{0}(\cos 2\Delta\theta_{\rm s}+\cos 2\gamma)\right]$
(72)
$\displaystyle\qquad\qquad+\frac{2I_{0}\sin\gamma\cot^{3}\gamma}{\pi\overline{I}}\int_{0}^{\pi}F_{0}(x;\gamma)dx$

(b)

$0<\gamma<\pi/2$ and $\pi/2-\gamma<\Delta\theta_{\rm s}<\pi/2$:

$\displaystyle L_{\rm 0b}(t)$ $\displaystyle=$
$\displaystyle\frac{I_{s}\cos^{3}\gamma}{\overline{I}}+\frac{2\sin\gamma\cot^{3}\gamma}{\pi\overline{I}}\left({I_{s}}\int_{\varphi_{\rm
c,s}}^{\pi}F_{0}(x;\gamma)dx+{I_{0}}\int_{0}^{\varphi_{\rm
c,s}}F_{0}(x;\gamma)dx\right),$ (73)

where $\cos\varphi_{\rm c,s}\equiv-\cot\gamma\cot\Delta\theta_{\rm s}$.

(c)

$\pi/2<\gamma<\pi$ and $\Delta\theta_{\rm s}<\gamma-\pi/2$:

$\displaystyle L_{\rm
0c}(t)=\frac{I_{0}\cos^{3}\gamma}{\overline{I}}+\frac{2I_{0}\sin\gamma\cot^{3}\gamma}{\pi\overline{I}}\int_{0}^{\pi}F_{0}(x;\gamma)dx.$
(74)

(d)

$\pi/2<\gamma<\pi$ and $\gamma-\pi/2<\Delta\theta_{\rm s}<\pi/2$:

$\displaystyle L_{\rm 0d}(t)$ $\displaystyle=$
$\displaystyle\frac{I_{0}\cos^{3}\gamma}{\overline{I}}+\frac{2\sin\gamma\cot^{3}\gamma}{\pi\overline{I}}\left({I_{s}}\int_{0}^{\varphi_{\rm
c,s}}F_{0}(x;\gamma)dx+{I_{0}}\int_{\varphi_{\rm
c,s}}^{\pi}F_{0}(x;\gamma)dx\right).$ (75)

Since equation (74) corresponds to a case where the spot is invisible to the
observer, it should reduce to $I_{0}/\overline{I}$. This implies that the
following definite integral of $F_{0}(x;\gamma)$ is given as

$\displaystyle\int_{0}^{\pi}F_{0}(x;\gamma)dx=\int_{0}^{\pi}\frac{\sin x(x\cos
x-\sin
x)}{(\cos^{2}x+\cot^{2}\gamma)^{2}}dx=-\frac{\pi}{2}\frac{\sin^{2}\gamma}{|\cos\gamma|^{3}}(1-|\cos\gamma|^{3}).$
(76)

Indeed, one can explicitly prove equation (76) by decomposing it in three
integrals as follows.

$\displaystyle\int_{0}^{\pi/2}\frac{x\sin x\cos
x}{(\cos^{2}x+\cot^{2}\gamma)^{2}}dx$
$\displaystyle=\sin^{4}\gamma\int_{0}^{\pi/2}\frac{x\sin x\cos
x}{(\cos^{2}\gamma\sin^{2}x+\cos^{2}x)^{2}}dx$ (78)
$\displaystyle=\frac{\pi}{4}\frac{\sin^{4}\gamma}{\cos^{2}\gamma(1+|\cos\gamma|)}.$
$\displaystyle\int_{\pi/2}^{\pi}\frac{x\sin x\cos
x}{(\cos^{2}x+\cot^{2}\gamma)^{2}}dx$
$\displaystyle=-\pi\int_{0}^{\pi/2}\frac{\sin x\cos
x}{(\cos^{2}x+\cot^{2}\gamma)^{2}}dx+\int_{0}^{\pi/2}\frac{x\sin x\cos
x}{(\cos^{2}x+\cot^{2}\gamma)^{2}}dx$ (80)
$\displaystyle=-\frac{\pi}{2}\frac{\sin^{4}\gamma}{\cos^{2}\gamma}+\frac{\pi}{4}\frac{\sin^{4}\gamma}{\cos^{2}\gamma(1+|\cos\gamma|)}.$
$\displaystyle-\int_{0}^{\pi}\frac{\sin^{2}x}{(\cos^{2}x+\cot^{2}\gamma)^{2}}dx$
$\displaystyle=\int_{0}^{\pi}\frac{dx}{\cos^{2}x+\cot^{2}\gamma}-(1+\cot^{2}\gamma)\int_{0}^{\pi}\frac{dx}{(\cos^{2}x+\cot^{2}\gamma)^{2}}$
(82)
$\displaystyle=\pi\frac{\sin^{2}\gamma}{|\cos\gamma|}-\frac{\pi}{2}\sin^{2}\gamma\frac{1+\cos^{2}\gamma}{|\cos\gamma|^{3}}.$

Summing up the above three equations leads to equation (76). Finally,
substituting equation (76) into equations (72) to (75) yields equations (44)
to (47) shown in §3.1.

One can repeat the above argument that derives equation (76) for integrals of
$F_{1}(x;\gamma)$ and $F_{2}(x;\gamma)$. When we set $I(\theta)=I_{0}$,
equations (22) and (29) should be independent of $\gamma$. So we evaluate
their right-hand-sides for $\gamma=0$, and find they reduce to $2I_{0}/3$ and
$I_{0}/2$, respectively. Since the result should hold for an arbitrary value
of $\gamma$, we obtain the following results:

$\displaystyle\int_{0}^{\pi}F_{1}(x;\gamma)dx=\frac{2\pi}{3}\frac{\sin^{5}\gamma}{\cos^{4}\gamma}(5-3\sin^{2}\gamma),$
(83)
$\displaystyle\int_{0}^{\pi}F_{2}(x;\gamma)dx=\pi\frac{\sin^{2}\gamma}{|\cos^{5}\gamma|}(21|\cos^{5}\gamma|-15|\cos^{7}\gamma|-6).$
(84)

These results are used in deriving equations (59) and (60).

We also find that a long but straightforward algebra reveals

$\displaystyle\int_{0}^{\pi}F_{2}(x;\gamma)\left(\frac{\cos
x}{\sqrt{\cos^{2}x+\cot^{2}\gamma}}\right)^{p}dx=\frac{12\pi\sin^{p+6}\gamma\cos^{4}\gamma}{(p+2)(p+4)}[6+5(p+2)\cos^{2}\gamma],$
(85)

if $p$ is an odd integer. Thus, we obtain an analytic expression as the last
line in equation (65). We were not able to find an analytic expression for
equation (62), which is evaluated numerically in plotting Figure 6.


# Weighted Sampled Split Learning (WSSL): Balancing Privacy, Robustness, and
Fairness in Distributed Learning Environments

Manish Osti1, Aashray Thakuri 2, Basheer Qolomany3, and Aos Mulahuwaish1  1
Department of Computer Science and Information Systems, Saginaw Valley State
University, University Center, MI, USA  2 College of Engineering and Computer
Science, University of Michigan- Dearborn, Dearborn, USA  3 Cyber Systems
Department, University of Nebraska at Kearney, Kearney, USA

###### Abstract

This study presents Weighted Sampled Split Learning (WSSL), an innovative
framework tailored to bolster privacy, robustness, and fairness in distributed
machine learning systems. Unlike traditional approaches, WSSL disperses the
learning process among multiple clients, thereby safeguarding data
confidentiality. Central to WSSL’s efficacy is its utilization of weighted
sampling. This approach ensures equitable learning by tactically selecting
influential clients based on their contributions. Our evaluation of WSSL
spanned various client configurations and employed two distinct datasets:
Human Gait Sensor and CIFAR-10. We observed three primary benefits: heightened
model accuracy, enhanced robustness, and maintained fairness across diverse
client compositions. Notably, our distributed frameworks consistently
surpassed centralized counterparts, registering accuracy peaks of 82.63% and
75.51% for the Human Gait Sensor and CIFAR-10 datasets, respectively. These
figures contrast with the top accuracies of 81.12% and 58.60% achieved by
centralized systems. Collectively, our findings champion WSSL as a potent and
scalable successor to conventional centralized learning, marking it as a
pivotal stride forward in privacy-focused, resilient, and impartial
distributed machine learning.

###### Index Terms:

Weighted Sampled Split Learning (WSSL), Privacy-Preserving Machine Learning,
Unbiased Learning, Robust Learning, and Distributed Learning.

##  I Introduction

In today’s world, characterized by data ubiquity, the imperative for privacy-
preserving machine learning amidst an explosion of data has made distributed
learning essential [1]. Split learning stands out among distributed
techniques, allowing deep neural networks to train on decentralized data
sources without compromising data privacy [2, 3, 4]. By dividing models into
front-end and back-end portions, split learning allows client devices to
handle initial processing. In contrast, servers take on the bulk of
computational duties [5]. This approach, which emphasizes sending intermediate
model representations over raw data, offers numerous advantages. It’s
especially relevant in sectors with high privacy demands, such as healthcare
and finance [6, 7, 8]. Moreover, this transmission strategy reduces
communication burdens, making it bandwidth-efficient and particularly valuable
in networks with limited connectivity.

Our contributions in this space are:

  1. 1.

WSSL Algorithm: We present the Weighted Sampled Split Learning (WSSL)
algorithm, spotlighting the role of data distribution in distributed learning.
This approach tackles the challenge of uneven data distribution in
decentralized contexts and proposes a technique to enhance model training
efficiency.

  2. 2.

Dynamic Client Importance-based Selection: Instead of conventional static
client choices, our model employs a dynamic system that varies client
participation based on their contribution to the learning. This adaptability
ensures consistent training and leads to better model results. This method,
paired with global model averaging, fosters cooperative learning among
clients, yielding improved training performance.

  3. 3.

Privacy, Robustness, and Fairness in Depth: WSSL advances beyond typical
distributed learning frameworks. By blending inputs from various client
models, our technique strengthens model robustness, reduces the risk of
overfitting, and promotes impressive generalization. Our unique client
selection algorithm, which determines weights from validation performance,
ensures that all clients have a balanced influence on learning, maintaining
fairness. Empirical studies validate our method’s superiority over centralized
systems in robustness, fairness, and privacy.

  4. 4.

Practical Implications and Performance Metrics: Our findings have tangible
real-world applications, especially in resource-limited settings. For example,
using the Human Gait Sensor dataset, WSSL achieved an impressive accuracy of
82.63% in a 2-client setting over 20 communication rounds. This surpassed the
81.12% peak accuracy of centralized methods. When tested with CIFAR-10, our
distributed approach consistently outdid its centralized counterpart. A
10-client setup, for instance, attained an accuracy of 75.51%, significantly
outpacing the centralized peak of 58.60%. These results emphasize the
potential and adaptability of WSSL in a range of dataset conditions.

With these advances, our work moves beyond traditional distributed learning
limits, emphasizing data distribution’s critical role. It paves the way for a
refined WSSL methodology set to redefine model training efficiency.

##  II Related Work

This section identifies the research related to our work, spanning different
areas. We also pinpoint the existing research gap and elucidate how our
approach addresses and bridges this gap.

###  II-A Split Learning in Healthcare

Both [3] and [4] advocate for split learning, emphasizing its applicability in
collaborative training on limited healthcare data with a focus on data
privacy. The challenges of this approach become evident when [9] introduces a
method aimed at curtailing data leakage in distributed health data learning.
Furthermore, [10] identifies potential privacy leakages within split learning.
Interestingly, this is observed even when the split learning models achieve
similar accuracy as their non-split counterparts in tasks such as heart
abnormality detection using 1D CNNs. While mitigation techniques have been
explored, they appear to be insufficient in isolation. Collectively, these
studies highlight the evolution of split learning and the imminent need to
explore data privacy assurance further.

###  II-B Integration of Federated and Split Learning

The synthesis of federated learning (FL) and split learning (SL) has been
explored by multiple studies. [11] presents splitfed learning (SFL), which
combines the strengths of both FL and SL. This approach promises enhanced
model privacy, quicker training, and accuracy levels on par with traditional
SL. Delving deeper into SL’s intricacies, [12] details its advantages and
potential pitfalls, notably resource efficiency and data leakage. Similarly,
[13] showcases an architecture that seamlessly blends FL and SL, offering
enhanced privacy, improved accuracy, and efficient communication for
sequentially partitioned data. Together, these studies champion the harmonious
fusion of FL and SL, emphasizing their collective strengths in ensuring
privacy, boosting efficiency, and maintaining accuracy.

###  II-C IoT and Edge-device Machine Learning

In the Internet of Things (IoT) context, [14] compares SplitNN and FL. The
findings suggest that while SplitNN excels in handling imbalanced data, it
falters with extreme non-IID data. In contrast, FL emerges as a more fitting
solution for IoT environments, especially when deploying 1D CNN models.
However, both methodologies encounter challenges, especially when dealing with
intricate models on resource-constrained devices. Broadening the scope of
split learning, [15] introduces a multi-modal framework tailored for enhancing
mmWave RF signal power predictions. Additionally, [16] shifts the focus to
edge learning in 5G networks, underscoring resource efficiency and inventive
incentive mechanisms. These contributions collectively reinforce the potential
of split learning in IoT settings, promising heightened accuracy, stringent
data privacy, optimal resource utilization, and collaborative incentives.

###  II-D Security and Efficiency in Distributed Learning

Security concerns associated with split learning are brought to the fore by
[17]. This study not only underscores potential vulnerabilities but also
illustrates inference attacks on client data orchestrated by malicious
servers. A detailed comparison between the communication efficiencies of split
learning and federated learning is presented by [18]. Their findings suggest
that split learning takes the lead in scenarios characterized by larger client
bases or expansive model sizes. In contrast, federated learning showcases
superior performance with abundant data samples and more constrained client or
model sizes. Further, [19] provides insights into distributed deep learning
models, emphasizing those that operate without direct client data access, and
evaluates the associated merits, challenges, and trade-offs.

###  II-E Identifying the Gap

Despite the comprehensive insights provided by the aforementioned studies, a
significant research gap is evident regarding the distribution of data in
distributed learning, particularly concerning non-IID datasets. Many strides
have been made in split learning and federated learning methodologies.
However, the nuances of data distribution across various clients and the
inherent challenges presented by non-IID datasets have often been overlooked
or underemphasized. This oversight is especially salient in scenarios where
data is intrinsically diverse and distributed unevenly across participants.
The current literature is noticeably bereft of specialized methodologies that
adeptly address these challenges.

It’s this very gap that the proposed Weighted Sampled Split Learning (WSSL)
intends to fill, positioning itself as a pioneering approach to refine
distributed learning practices. By accentuating dynamic client selection based
on importance weights, WSSL adeptly tackles the multifaceted issue of non-
uniform data distribution in decentralized settings. This innovative strategy
ensures that all client contributions to the global model training are
optimal, regardless of any disparities in individual data distributions.
Through careful design and consideration, WSSL aims to elevate the overall
performance and efficiency of distributed learning systems, providing a
framework that not only fills the existing gap but also promises to redefine
best practices in the realm of distributed machine learning.

##  III Proposed Framework

This section unveils our innovative framework, Weighted Sampling Split
Learning (WSSL). This framework adeptly integrates the concepts of weighted
sampling with split learning, promising to be a game-changer in distributed
machine learning by amplifying its efficiency and performance. Central to WSSL
is a re-envisioned model training process that disseminates tasks between a
myriad of clients and a pivotal server. This distribution takes into account
the significance of each client, astutely calibrating their respective
contributions. The detailed mechanics and procedures of our proposed structure
are delineated in the subsequent sections.

###  III-A System Model and Problem Objective

We consider a distributed learning system consisting of a central server and
multiple edge clients. In this setup, we aim to achieve a collaborative model
training. A significant constraint is that the server does not have direct
access to the raw data of the clients, which is essential to ensure data
privacy and meet certain regulatory requirements.

Given:

  * •

$S$ \- Central server

  * •

$C_{i},i=1,...,N$ \- N clients

  * •

$D_{i}$ \- Local dataset for client $C_{i}$

  * •

$f_{i}()$ \- Local model for client $C_{i}$

  * •

$L_{i}$ \- Loss for client $C_{i}$

We aim to train individual client models $f_{i}()$, such that their associated
losses are minimized, and their outputs are consistent with a global model
$f_{\text{global}}()$ residing at the server. Mathematically, our objective
can be formulated as follows:

$\min_{f_{1},...,f_{N}}\sum_{i=1}^{N}L_{i}(f_{i}(D_{i}),Y)$

Subject to:

$f_{i}(D_{i})=f_{\text{global}}(D)$

Where $f_{\text{global}}()$ represents the global model at the server.

###  III-B Importance-based Client Selection

In the Weighted Sampling Split Learning (WSSL) framework, client selection for
each training round is paramount. Traditional federated learning often
neglects variations in data distributions, data quality, and past performance
across different clients, treating them uniformly. In contrast, our method
employs an importance-based weighting scheme to differentiate clients,
ensuring the most suitable clients are selected for model refinement. To
operationalize this philosophy into a systematic approach, we introduce the
Weighted Sampled Split Learning (WSSL) Client Selection Algorithm.

  1. 1.

Weighted Sampled Split Learning (WSSL) Client Selection Algorithm (Algorithm
1): This algorithm outlines the specifics of our importance-based selection.
The central server employs it to handpick a set of clients for each epoch,
focusing on those that have the greatest potential to refine the global model.
The steps of this algorithm are as follows:

    1. (a)

Initialization: Begin by setting the total number of clients, denoted as
$\alpha$, and iterate through the epochs.

    2. (b)

Importance Weight Calculation: For every epoch, the server calculates the
importance weight ($\beta_{i}$) of each client using the function compute
importance. This function incorporates factors such as data quality, alignment
with the global model, and past performance to determine the potential impact
of a client.

    3. (c)

Normalization and Client Selection: Normalize the importance weights to
produce $\gamma_{i}$, ensuring their sum equals 1. These normalized weights
quantify the likelihood of each client’s valuable contribution in the current
epoch. The number of clients to engage, represented by $\alpha^{\prime}$, is
deduced from the normalized weights’ weighted average, with at least one
client selected to maintain diversity.

    4. (d)

Weighted Sampling: Employing principles from probabilistic sampling, clients
with elevated normalized weights are more likely to be selected. The list,
$\delta$, encapsulates the chosen clients who will partake in the next
training iteration.

  2. 2.

Integrating Algorithm 1 into WSSL, this importance-driven client selection
approach not only ensures fairness by recognizing unique client attributes but
also augments the proficiency of the global model. By attributing distinct
importance weights that resonate with each client’s data quality, expertise,
and past performance, WSSL fosters a balanced, data-centric learning strategy,
amplifying the merits of distributed machine learning.

Algorithm 1 Weighted Sampled Split Learning (WSSL) Client Selection

Notations: $\alpha$ = Total number of clients, $\beta$ = Importance weights,
$\gamma$ = Normalized weights, $\delta$ = Selected clients, $\epsilon$ =
Server model, $\zeta$ = Validation dataset.

1:Initialize: $\alpha\leftarrow$ Total number of clients.

2:for each $i$ in number of epochs do

3: if $i==0$ then

4: $\delta\leftarrow[0,1,...,\alpha-1]$

5: else

6: Compute importance weights of each client: $\beta_{i}$ $\leftarrow$ compute
importance$(client_{i},\epsilon,\zeta)$

7: Compute sum of weights: $\beta_{sum}$ $\leftarrow$
$\sum_{i=1}^{\alpha}\beta_{i}$

8: Normalize weights: $\gamma_{i}$ $\leftarrow$
$\frac{\beta_{i}}{\beta_{sum}}$

9: Determine number of clients to select: $\alpha$ $\leftarrow$
$max(\alpha\times average(\gamma),1)$

10: Handpick clients: $\delta$ $\leftarrow$ weighted sampling
$([0,1,...,\alpha-1],\gamma_{i},\alpha)$

11: end if

12:end for

###  III-C Training and Model Updates

The Weighted Sampling Split Learning (WSSL) architecture stands as a testament
to the power of collaboration. Seamlessly weaving the central server with
individual clients, it showcases a harmonized dance of training processes and
model updates, detailed in Algorithm 2. More than just an exchange, this
methodology amplifies knowledge enhancement throughout the system. Let’s delve
deeper:

  1. 1.

Algorithm 2 \- Weighted Sampled Split Learning (WSSL): Shedding light on the
WSSL framework’s training mechanics, this algorithm underscores the
synchronized endeavors of the server and client entities, all striving towards
refining the global model. The steps are as follows:

    1. (a)

Importance Computation and Sampling: Each epoch kickstarts with the server
determining the importance weights ($w[i]$) for every client, guided by
Algorithm 1. Based on these weights, clients are chosen for the current
training round, ensuring that influential ones play a central role in global
optimization.

    2. (b)

Client-Side Training: Selected clients embark on local training using their
datasets. As their model ($C[i]$) parameters ($\theta[i]$) adapt through
gradient descent, their intermediate representations ($a[i]$) are sent to the
server for global integration.

    3. (c)

Server-Side Training: Armed with the intermediate representations ($a[i]$),
the server evaluates the loss ($L[i]$) for each client and processes gradients
via backpropagation. These gradients are subsequently dispatched to the
corresponding clients.

    4. (d)

Client-Side Update: Clients, upon receiving gradients from the server, refine
their model parameters, aligning more closely with global objectives.

    5. (e)

Global Model Update: By amalgamating all client contributions, the server
computes a weighted average of their parameters ($\theta_{\text{global}}$),
encapsulating the shared wisdom.

  2. 2.

Stability of Importance Weights: Beyond the algorithm’s iterative updates,
it’s essential to underscore the consistent nature of importance weights.
Contrasting dynamically updated model parameters, these weights offer
stability across training rounds, ensuring undeterred influence over the
learning curve.

For a visually enriched grasp of this distributed machine learning paradigm,
readers are directed to Figure 1. This methodology, which orchestrates a dual
backpropagation between client and server models, is further exemplified using
two case studies: the human gait sensor and the CIFAR-10 dataset, dissected
further in Section IV.

###  III-D Addressing Trustworthiness

To bolster trustworthiness within our framework, we integrate the following
measures:

  1. 1.

Data Integrity: Prior to training, a hashing mechanism is implemented on the
client’s data. This not only ensures the preservation of data privacy but also
allows the server to verify data integrity without accessing the raw data.

  2. 2.

Model Validity: After training, the server can optionally deploy techniques
like model watermarking. This ensures that the model updates retrieved from
clients are authentic and remain unaltered.

###  III-E Comparison with Existing Methods

What makes our approach superior is twofold:

  1. 1.

Fairness in Client Participation: Unlike traditional methods that randomly
select clients or prioritize clients with larger datasets, our approach
guarantees fairness by giving every client a chance to be selected based on
their importance.

  2. 2.

Efficiency in Communication: By focusing on important clients and not
mandating participation from all, we reduce the communication overhead, making
training faster.

Algorithm 2 Weighted Sampled Split Learning (WSSL)

Notations:

  1. 1.

$C[i]$ \- the $i-th$ client model,

  2. 2.

$S$ \- server model,

  3. 3.

$D[i]$ \- local dataset for client $C[i]$,

  4. 4.

$\omega$ \- parameters of server model $S$,

  5. 5.

$\theta[i]$ \- parameters of the $i-th$ client model $C[i]$,

  6. 6.

$a[i]$ \- intermediate representation from client model $C[i]$,

  7. 7.

$L[i]$ \- loss for the $i-th$ client,

  8. 8.

$w[i]$ \- importance weights for the $i-th$ client,

  9. 9.

$\theta_{\text{global}}$ \- weighted average of client models’ parameters.

Input: Local datasets $D[i]$ for each client $C[i]$, server model $S$ with
parameters $\omega$, client models $C[i]$ with parameters $\theta[i]$. Output:
Updated server model $S$ and client models $C[i]$.

1:Procedure:

2:1\. Compute Importance and Sampling:

3:for each client $C[i]$ do

4: Compute $w[i]$ to represent $C[i]$’s importance.

5: Normalize weights: $w[i]\leftarrow\frac{w[i]}{\sum(w)}$.

6: Select $C[i]$ for current training round based on its normalized weight.

7:end for

8:2\. Client-Side Training:

9:for selected client $C[i]$ in the current round do

10: Start with initial parameters $\theta[i]$.

11: for each data batch $(x,y)$ from $D[i]$ do

12: Obtain $a[i]=C[i](x)$.

13: Detach $a[i]$ from computation graph and forward to server.

14: end for

15:end for

16:3\. Server-Side Training:

17:Receive $a[i]$ from each active client $C[i]$.

18:for client $C[i]$ sending $a[i]$ do

19: Calculate loss $L[i]=L(S(a[i]),y)$.

20: Obtain gradients $\frac{\partial L[i]}{\partial a[i]}$ via backpropagation
on $S$.

21: Return gradients to respective $C[i]$.

22:end for

23:4\. Client-Side Model Update:

24:for client $C[i]$ receiving server gradients do

25: Update $C[i]$ using received $\frac{\partial L[i]}{\partial a[i]}$.

26: Refine $\theta[i]$ using gradient descent.

27:end for

28:5\. Global Model Update:

29:Compute:  $\theta_{\text{global}}=\sum_{i}w[i]\times\theta[i]$

30:Return: Updated server model $S$ and client models $C[i]$.

Figure 1: Overview of Weighted Sampling Split Learning System

##  IV Datasets

The subsequent subsections provide a description of the datasets utilized for
experimental evaluations and the preprocessing steps applied to the data. The
research employs two distinct datasets, namely the Human Gait Sensor dataset
[20] and the CIFAR-10 dataset [21], to showcase the adaptability and
robustness of Weighted Sampling Split Learning (WSSL). The Human Gait sensor
dataset demonstrates WSSL’s potential in processing real-world sensor data,
particularly in domains such as health monitoring and IoT, while the CIFAR-10
dataset evaluates its performance in image data classification, highlighting
its applicability to vision-based tasks. Through the use of diverse datasets,
this research offers a comprehensive evaluation of WSSL’s performance in
distributed machine learning scenarios, underscoring its broad applicability
and effectiveness.

###  IV-A Human Gait Sensor Case Study

We utilized a comprehensive dataset comprising 2,803,999 observations, with
each observation containing 28 attributes derived from various sensor
readings. The dataset used in this research study comprises 30 participants,
with an equal distribution of 15 males and 15 females, ensuring gender balance
and unbiased representation. In the preprocessing stage, the dataset was
segregated into features and labels (Gender), enabling supervised learning.
The features matrix encompassed all columns except the target variable, which
served as the label. Following this, an 80-20 split strategy was employed,
allocating 80% of the data for model training and reserving the remaining 20%
for model validation and testing. This division ensured that the model had a
significant amount of data to learn from while providing an independent set
for evaluating its generalization. To mitigate any disparities in the
magnitude of values across columns, standard scaling was applied to normalize
the data. This procedure transformed the features to have a mean of 0 and a
standard deviation of 1, ensuring a consistent scale and preventing dominant
influences of variables with larger values. Finally, the data was converted
into PyTorch tensors, a crucial step considering the reliance on PyTorch, a
widely used open-source machine learning library, for subsequent model
building and training stages.

###  IV-B CIFAR-10 Case Study

The dataset used in this study is the CIFAR-10 dataset, which consists of
60,000 color images of 32x32 pixels, categorized into 10 distinct classes such
as airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck.
The dataset is split into a training set of 50,000 images and a test set of
10,000 images. The data undergoes preprocessing, including conversion to
tensors and normalization, to facilitate learning and convergence. To
partition the dataset for each client in our weighted sampled split learning
approach, we employ a stratified sampling technique, ensuring that each
client’s subset maintains the same class distribution as the original dataset.
Each client is then provided with a data loader, allowing for batch processing
and data shuffling to expose the model to diverse samples and minimize
overfitting.

##  V Experimental Setup and Evaluation

###  V-A System Specifications

Our experimental environment was provisioned on a machine boasting 32 GB of
RAM paired with a 6 GB Nvidia 2060 RTX GPU. For crafting and training the
neural network models, we employed PyTorch, a renowned deep learning framework
prized for its comprehensive ecosystem and intuitive design capabilities.

###  V-B Centralized Training

Contrastingly different from the split learning approach, centralized training
amasses all data and computational muscle on one unified server. Here, the
entirety of the neural network model gets trained to utilize centrally stored
data. Though it provides the luxury of accessing the complete data
simultaneously, this method could grapple with scalability, data privacy, and
bandwidth consumption.

For clarity, the centralized systems in our experiments retain the unabridged
architectures of the models, eliminating any division between client and
server roles. Their specifications can be further consulted in Table I.

###  V-C Datasets and Model Architectures

####  V-C1 Human Gait Sensor Dataset

The dataset incorporated a dual-segment feedforward neural network design:

  * •

Client-Side Model: Positioned on the edge device, this segment is a simple
fully connected neural network spanning two layers. Post accepting the input
data, it carries out a linear transformation (leveraging layer-specific
weights and biases) and applies the ReLU activation function. Its second
layer’s output acts as the intermediate representation, forwarded to the
server model.

  * •

Server-Side Model: This counterpart is designed for the server, which
processes the intermediate data sent by the client model. After undergoing a
linear transformation and ReLU activation across its first two layers, the
final layer employs a sigmoid activation function—ideal for binary
classification challenges—to restrict the output between 0 and 1.

####  V-C2 CIFAR-10 Dataset

For this dataset, a variant of the famed ResNet-18 convolutional neural
network architecture was adapted, split strategically for both client and
server sides to classify 10 distinct image classes:

  * •

Client-Side Model: Occupying the early layers of the ResNet-18 architecture up
to a defined cut-off, this segment primarily transmutes raw input into
intermediate feature sets, dispatched to the server model.

  * •

Server-Side Model: Situated on the server, this portion picks up from the
client-defined cut-off and stretches till the ResNet-18’s final layer. It
processes the intermediate data, culminating in the final output. Its terminal
layer is a fully connected layer, followed by a Softmax function, fitting for
rendering class probabilities in multi-class challenges.

The cut-off point becomes a pivotal element, balancing the client-side
computational demands with the data transmission volume to the server.
Adjusting it judiciously is crucial, factoring in the client device’s
computational prowess and network bandwidth.

###  V-D Model Evaluation Strategy

For rigorous model evaluations, the StratifiedShuffleSplit function from the
scikit-learn library was our tool of choice. By ensuring a proportional
representation of each class in every fold, this cross-validation technique is
indispensable for skewed datasets. This method promotes a balanced and robust
data split, augmenting the veracity of our evaluation metrics.

Lastly, Table I enumerates the architectural nuances of the models in our
experiments, juxtaposing the centralized frameworks with the weighted sampled
split learning configurations.

###  V-E Human Gait Sensor Evaluation Results

We gauged the efficiency of the weighted sampled split learning approach on
the Human Gait Sensor dataset by varying the number of clients from 2 to 10
across 20 communication rounds or epochs. With each configuration, a
significant uptrend in performance was evident. As displayed in Figure 2(a)
and 2(b), the system initialized with 2 clients began at an accuracy of
65.12%, surging to 82.63% by the 20th epoch. Comparable upward trends
materialized for setups with 4, 6, 8, and 10 clients, culminating in
accuracies of 83.11%, 79.12%, 82.12%, and 82.06%, respectively. Notably, the
4, 8, and 10 client systems surpassed the centralized learning’s peak accuracy
of 81.12%. These observations underline the potency and scalability of our
distributed approach, which not only thrives with an increased client base but
also provides a significant edge over conventional centralized learning.

Architecture | Dataset | #Parameters | Layers | Kernel Size | Batch Size
---|---|---|---|---|---
Feedforward Neural Network | Human Gait Sensor | 32,000.00 | 5 | - | 128
ResNet18 | CIFAR-10 | 11.7 million | 18 | (7 × 7), (3 × 3) | 128
TABLE I: Models’ Architecture

(a) Accuracy vs. Communication Rounds for Human Gait Sensor Data

(b) Accuracy vs. Number of Clients for Human Gait Sensor Data

(c) Accuracy vs. Communication Rounds for CIFAR-10 Data

(d) Accuracy vs. Number of Clients for CIFAR-10 Data

Figure 2: Accuracy vs. Communication Rounds and Number of Clients for the Two
Datasets

###  V-F CIFAR-10 Evaluation Results

The CIFAR-10 dataset served as another testbed for our weighted sampled split
learning system. As portrayed in Figure 2(c) and 2(d), the model, when started
with 2 clients, had an inception accuracy of 35.96%, which rose to 58.96% by
the concluding epoch. Systems incorporating 4, 6, 8, and 10 clients showcased
similar growth patterns, ending with accuracies of 68.34%, 72.30%, 73.77%, and
75.51%, respectively. For perspective, the traditional centralized approach
commenced at 30.50% and plateaued at 58.60% by the 20th epoch. Crucially,
every distributed setup with varying client counts eclipsed the centralized
model in performance. These findings bolster the case for weighted sampled
split learning, especially when considering its heightened adaptability as
client numbers grow. This trend remained consistent even with the intricacies
of the CIFAR-10 dataset, where our method continually overshadowed the
standard centralized learning paradigm.

##  VI Findings and Lessons Learned

Through detailed analysis and exploration of Weighted Sampled Split Learning,
we distilled the following vital insights:

  1. 1.

Robustness and Fairness: Our study underlines the enhanced robustness and
fairness offered by weighted sampled split learning compared to conventional
centralized methods. The strategic weighting prevents undue dominance by
specific datasets or clients, fostering a balanced environment where every
client’s contributions are valued and recognized.

  2. 2.

Privacy and Accuracy: Weighted sampled split learning adeptly navigates the
delicate balance between data privacy and model performance. By confining data
to its original location and only exchanging model updates, privacy is upheld.
The amalgamated efforts of the clients, shaped by their importance weights,
fortify the model’s precision without compromising its reliability.

These insights validate weighted sampled split learning as a powerful approach
in distributed machine learning, adeptly turning challenges into assets.

##  VII Conclusions and Future Directions

In this work, we’ve spotlighted Weighted Sampled Split Learning (WSSL) as a
groundbreaking approach to addressing the intricacies of privacy, robustness,
and fairness in decentralized machine learning environments.

From our research, we highlight:

  1. 1.

Enhanced Model Performance: WSSL, through importance-based client selection
and harmonized learning, consistently outperforms traditional centralized
models.

  2. 2.

Upholding Privacy and Fairness: WSSL inherently protects individual data and
champions equitable client participation, marking a significant leap in
distributed learning.

  3. 3.

Sturdy Decentralized Learning: The strength of WSSL lies in its combined
emphasis on cooperative learning and prioritized client involvement.

Looking forward, intriguing areas of exploration are:

  1. 1.

Client Dynamics and Weighting Impact: Delving deeper into the interplay
between the number of clients and their respective weightings.

  2. 2.

Addressing Imbalances and Scalability: Adapting WSSL to scenarios with
imbalanced data and scaling to expansive environments.

  3. 3.

Refining Split Points: Fine-tuning the junctures in neural network structures
to adeptly navigate between localized and centralized processing.

In essence, WSSL offers a pioneering pathway in distributed machine learning.
With its demonstrated advantages and the potential it opens up for further
innovation, it stands poised to redefine decentralized learning, emphasizing
privacy, robustness, and fairness.

## References

  * [1] R. Shokri and V. Shmatikov, “Privacy-preserving deep learning,” in _Proceedings of the 22nd ACM SIGSAC conference on computer and communications security_ , 2015, pp. 1310–1321.
  * [2] O. Gupta and R. Raskar, “Distributed learning of deep neural network over multiple agents,” _Journal of Network and Computer Applications_ , vol. 116, pp. 1–8, 2018.
  * [3] M. G. Poirot, P. Vepakomma, K. Chang, J. Kalpathy-Cramer, R. Gupta, and R. Raskar, “Split learning for collaborative deep learning in healthcare,” _arXiv preprint arXiv:1912.12115_ , 2019.
  * [4] P. Vepakomma, O. Gupta, T. Swedish, and R. Raskar, “Split learning for health: Distributed deep learning without sharing raw patient data,” _arXiv preprint arXiv:1812.00564_ , 2018.
  * [5] M. Langer, Z. He, W. Rahayu, and Y. Xue, “Distributed training of deep learning models: A taxonomic perspective,” _IEEE Transactions on Parallel and Distributed Systems_ , vol. 31, no. 12, pp. 2802–2818, 2020.
  * [6] K. Bonawitz, H. Eichner, W. Grieskamp, D. Huba, A. Ingerman, V. Ivanov, C. Kiddon, J. Konečnỳ, S. Mazzocchi, B. McMahan _et al._ , “Towards federated learning at scale: System design,” _Proceedings of machine learning and systems_ , vol. 1, pp. 374–388, 2019.
  * [7] J. Konečnỳ, H. B. McMahan, D. Ramage, and P. Richtárik, “Federated optimization: Distributed machine learning for on-device intelligence,” _arXiv preprint arXiv:1610.02527_ , 2016.
  * [8] J. Konečnỳ, H. B. McMahan, F. X. Yu, P. Richtárik, A. T. Suresh, and D. Bacon, “Federated learning: Strategies for improving communication efficiency,” _arXiv preprint arXiv:1610.05492_ , 2016.
  * [9] P. Vepakomma, O. Gupta, A. Dubey, and R. Raskar, “Reducing leakage in distributed deep learning for sensitive health data,” _arXiv preprint arXiv:1812.00564_ , vol. 2, 2019.
  * [10] S. Abuadbba, K. Kim, M. Kim, C. Thapa, S. A. Camtepe, Y. Gao, H. Kim, and S. Nepal, “Can we use split learning on 1d cnn models for privacy preserving training?” in _Proceedings of the 15th ACM Asia Conference on Computer and Communications Security_ , 2020, pp. 305–318.
  * [11] C. Thapa, P. C. M. Arachchige, S. Camtepe, and L. Sun, “Splitfed: When federated learning meets split learning,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 36, no. 8, 2022, pp. 8485–8493.
  * [12] C. Thapa, M. A. P. Chamikara, and S. A. Camtepe, “Advancements of federated learning towards privacy preservation: from federated learning to split learning,” _Federated Learning Systems: Towards Next-Generation AI_ , pp. 79–109, 2021.
  * [13] A. Abedi and S. S. Khan, “Fedsl: Federated split learning on distributed sequential data in recurrent neural networks,” _arXiv preprint arXiv:2011.03180_ , 2020.
  * [14] Y. Gao, M. Kim, S. Abuadbba, Y. Kim, C. Thapa, K. Kim, S. A. Camtepe, H. Kim, and S. Nepal, “End-to-end evaluation of federated learning and split learning for internet of things,” _arXiv preprint arXiv:2003.13376_ , 2020\.
  * [15] Y. Koda, J. Park, M. Bennis, K. Yamamoto, T. Nishio, and M. Morikura, “One pixel image and rf signal based split learning for mmwave received power prediction,” in _Proceedings of the 15th International Conference on emerging Networking EXperiments and Technologies_ , 2019, pp. 54–56.
  * [16] W. Y. B. Lim, J. S. Ng, Z. Xiong, D. Niyato, C. Leung, C. Miao, and Q. Yang, “Incentive mechanism design for resource sharing in collaborative edge learning,” _arXiv preprint arXiv:2006.00511_ , 2020.
  * [17] D. Pasquini, G. Ateniese, and M. Bernaschi, “Unleashing the tiger: Inference attacks on split learning,” in _Proceedings of the 2021 ACM SIGSAC Conference on Computer and Communications Security_ , 2021, pp. 2113–2129.
  * [18] A. Singh, P. Vepakomma, O. Gupta, and R. Raskar, “Detailed comparison of communication efficiency of split learning and federated learning,” _arXiv preprint arXiv:1909.09145_ , 2019.
  * [19] P. Vepakomma, T. Swedish, R. Raskar, O. Gupta, and A. Dubey, “No peek: A survey of private distributed deep learning,” _arXiv preprint arXiv:1812.03288_ , 2018.
  * [20] Y. Luo, S. M. Coppola, P. C. Dixon, S. Li, J. T. Dennerlein, and B. Hu, “A database of human gait performance on irregular and uneven surfaces collected by wearable sensors,” _Scientific data_ , vol. 7, no. 1, p. 219, 2020.
  * [21] A. Krizhevsky, G. Hinton _et al._ , “Learning multiple layers of features from tiny images,” 2009.