#  Reflecting on naked singularities: iron line fitting as a probe of the
cosmic censorship conjecture

Andrew Mummery1, Adam Ingram2
1Oxford Theoretical Physics, Beecroft Building, Clarendon Laboratory, Parks
Road, Oxford, OX1 3PU, United Kingdom
2School of Mathematics, Statistics and Physics, Newcastle University, Herschel
Building, Newcastle upon Tyne, NE1 7RU, UK E-mail:
<EMAIL_ADDRESS>
###### Abstract

We demonstrate that the X-ray iron line fitting technique can be leveraged as
a powerful probe of the cosmic censorship conjecture. We do this by extending
existing emission line models to arbitrary spin parameters of the Kerr metric,
no longer restricted to black hole metrics with $|a_{\bullet}|<1$. We show
that the emission lines from naked singularity metrics ($|a_{\bullet}|>1$)
show significant differences to their black hole counterparts, even for those
metrics with identical locations of the innermost stable circular orbit, i.e.,
emission line fitting does not suffer from the degeneracy which affects
continuum fitting approaches. These differences are entirely attributable to
the disappearance of the event horizon for $|a_{\bullet}|>1$. We highlight
some novel emission line features of naked singularity metrics, such as
“inverted” emission lines (with sharp red wings and extended blue wings) and
“triple lines”. The lack of detection of any of these novel features provides
support of the cosmic censorship conjecture. We publicly release XSPEC
packages skline and skconv which can now be used to probe the cosmic
censorship conjecture in Galactic X-ray binaries and Active Galactic Nuclei.
The inclusion of super-extremal spacetimes can be alternatively posed as a way
of stress-testing conventional models of accretion.

###### keywords:

accretion, accretion discs — black hole physics

††pagerange: Reflecting on naked singularities: iron line fitting as a probe
of the cosmic censorship conjecture–References††pubyear: 2023

##  1 Introduction

Asymmetric and broadened iron emission lines are observed in the X-ray
spectrum of accreting compact objects across the entire mass scale (e.g.
Miller, 2007). This includes supermassive accretors in Active Galactic Nuclei
(Tanaka et al., 1995; Fabian et al., 2000) and stellar-mass accretors in
Galactic X-ray binary systems (Miller & Miller, 2015), including neutron star
systems (Cackett et al., 2008; Ludlam et al., 2019). The line results from the
irradiation of the accretion disk by hard X-rays from hot plasma located close
to the central object. This process is typically referred to as ‘reflection’,
although it is more accurately described as reprocessing and re-emission
(Bambi et al., 2021). The iron K$\alpha$ fluorescence line at $\sim 6.4$ keV
is a prominent feature of the reflection spectrum, which also includes
emission lines and absorption edges from all astrophysically abundant elements
and a broad Compton scattering feature at $\sim 20-30$ keV known as the
‘Compton hump’ (Matt et al., 1991; Ross & Fabian, 2005; García et al., 2013).
The iron line is a particularly powerful diagnostic because it is narrow in
the emission rest frame, whereas the observed line profile is heavily
distorted by a combination of the relativistic orbital motion of the emitting
material, and the gravitational energy-shifting of the emitted photons over
their trajectory to the observer (Fabian et al., 1989; Laor, 1991; Dauser et
al., 2010).

The line profile is sensitive to the location of the inner edge of the
accretion disc, as this sets the fastest orbital frequency and the largest
gravitational redshift. As such, it provides a powerful probe of the
dimensionless spin parameter of the Kerr metric, $a_{\bullet}$, under the
assumption that the disc inner edge is set by the innermost stable circular
orbit (ISCO), which is itself a function of spin. The spin parameter also
influences the line profile via the spin-dependence of orbital frequency,
gravitational redshift, and photon trajectories, although these are more
subtle effects. As such, analysis of observed broad iron emission lines has
been used to place constraints on numerous Kerr metric spin parameters in the
literature, under the physical assumption that these objects are black holes
(Middleton, 2016; Reynolds, 2021). The black hole assumption amounts to
restricting the spin parameter to values $|a_{\bullet}|<1$. The Kerr metric,
however, remains a mathematically valid solution of Einstein’s field equations
for all spin parameters $a_{\bullet}$, including metrics with
$|a_{\bullet}|>1$, which instead describe the spacetime external to a naked
singularity.

While remaining a valid solution in General Relativity, the conventional
assumption is that these naked singularity solutions will not be present in
nature. This assumption (more generally that all solutions of Einstein’s
equations which occur in nature and which host singularities have those
singularities hidden behind an event horizon) is known as the Weak Cosmic
Censorship Conjecture (Penrose, 1969), and while unproven does have supporting
evidence (Wald, 1997). Principally this supporting evidence comes from
theoretical modelling of the gravitational collapse and formation of
singularities, but it seems likely that X-ray spectral information (which
probes the spacetime close to the singularity) has much to offer the study of
this conjecture.

The properties of super-spinning Kerr metrics have received renewed interest
in recent years, owing to the realisation that string theory may allow for an
external spacetime described by a super-extremal Kerr metric, which is grafted
onto an interior string-theory metric which shrouds the singularity (see Gimon
& Hořava, 2009, for a detailed discussion). While no exact interior solution
in three spatial (and one temporal) dimensions is known, a four space (one
time) dimension interior solution with these properties is known (Gimon &
Horava, 2004), suggesting that such a solution may exist for three spatial
dimensions. If such a string theory “resolution” to the naked singularities of
super-spinning Kerr metrics exists, then it remains possible that these
objects may exist in our Universe without breaking the Cosmic Censorship
Conjecture, and can in principle be probed by X-ray observations. While an
interior string theory solution is expected to shroud the singularity (and as
such breaks the strict association between $|a_{\bullet}|>1$ and the cosmic
censorship hypothesis), it is not as of yet clear on what spatial scale $R$
this shrouding will occur. In this paper we assume $R\ll r_{g}$, where the
gravitational radius $r_{g}\equiv GM/c^{2}$, and do not curtail the Kerr
metric at any finite radius.

Inspired by this string theory argument Stuchlík & Schee (2010) computed
photon energy-shifts for observers in the super-extremal Kerr metric, before
extending this work in Schee & Stuchlík (2013) to examine some emission line
properties of super-spinning Kerr singularities, focusing on line profiles
produced from a disc with uniform emissivity. Similarly, Ranea-Sandoval &
García (2015) studied the properties of both black hole and super-extremal
iron lines, but focused on the effects which strong magnetic fields have on
the properties of these lines. In this work we consider more general
treatments of super-extremal emission lines, both by examining a wider range
of spins, but also more realistic disc rest-frame emissivity profiles.

Other related work involves iron line calculations in non-Kerr metrics, which
are in some ways similar to the Kerr metric. These metrics are typically
either solutions of modified theories of gravity, or which include additional
beyond the standard model physics. Liu et al. (2018) examine iron line
profiles about the naked singularity Janis-Newman-Winicour metric (which
assumes that a charged and massless scalar field $\Phi$ is present throughout
space). While Zhou et al. (2020) compute iron line profiles in a modified
gravity version of the Kerr metric, where black hole metrics (i.e., metrics
with an event horizon) can be constructed with spin parameters
$|a_{\bullet}|>1$. These modified gravity metrics have additional free
parameters which allow event horizons to be present in metrics with
$a_{\bullet}>1$. The philosophy of these papers differs somewhat from our
approach, in that these analyses seek to probe whether General Relativity is
the correct theory of gravity, while we are seeking to probe whether accreting
compact objects have event horizons.

As far as the authors are aware, no existing publicly available relativistic
reflection models (e.g. Laor, 1991; Dauser et al., 2013) can be used in the
super-extremal ($|a_{\bullet}|>1$) limit of the Kerr metric (although some
publicly available models are based on modified gravity metrics in place of
the Kerr metric e.g., Bambi et al., 2017; Abdikamalov et al., 2019). As such,
existing models must be extended, which is the purpose of this paper.

This is not just an extension of theoretical interest however. The need for
this extension can be motivated entirely from a data-driven perspective, since
measurements preferring very rapid spin parameters can run into the prior
upper bound imposed by the black hole assumption. This often occurs in studies
of both Active Galactic Nuclei (e.g. Vasudevan et al., 2016) and X-ray
binaries (e.g. Draghis et al., 2023). For example, a recent analysis of the
X-ray spectrum of the X-ray binary Cyg X-1 (which contained an iron line
reflection component, in addition to a thermal continuum) found an extremely
rapid rotation rate with a 3$\sigma$ lower bound of $a_{\bullet}>0.9985$, and
a best fit limited by the prior of $a_{\bullet}\leq 0.9999$ (Zhao et al.,
2021). Cyg-X1 has repeatedly been found to have a high spin from numerous iron
line studies (e.g., Fabian et al., 2012; Tomsick et al., 2014; Duro et al.,
2016; Walton et al., 2016). It is of critical importance that such inferences
of high spins are robust to the removal of the spin prior placed on
conventional models. This can only be tested with the extended model we
present here. In addition to probing the cosmic censorship conjecture, the
inclusion of super-extremal Kerr spacetimes in fitting procedures can equally
well be thought of as stress-testing conventional models of accretion.

In this paper, we focus principally on the properties of line profiles
resulting from a $\delta$-function fluorescence line being emitted in the
plasma rest-frame. This enables us to examine the observable properties that
the event horizon (or lack thereof) imprints onto a single emission line. We
release this line profile model publicly as the XSPEC model skline. We
additionally define a convolution model skconv, that can be convolved with a
model of the full rest-frame reflection spectrum containing numerous lines,
edges and scattering features to provide a much more complete description of
relativistic reflection. Coupled with our recent extension of disc-continuum
fitting models to the super-extremal regime (Mummery et al., 2024), all
conventional X-ray spectral modelling techniques can now be performed in the
extended Kerr metric.

The layout of this paper is the following. In section 2 we recap the
properties of the Kerr metric in both the black hole and naked singularity
regimes. In section 3 we introduce the X-ray line fitting formalism, and
describe the fundamental equations which we solve. In section 4 we present an
analysis of the structure of black hole and naked singularity spacetimes from
the perspective of observed photon energy shifts, the relevant quantity for
emission line analysis. In section 5 we present our analysis of emission line
profiles in naked singularity metrics, highlighting their potential for
probing the cosmic censorship conjecture. In section 6 we show how the
peculiar features of iron lines observed in naked singularity metrics are
observable when a more complex rest frame spectrum is taken into account,
before concluding in section 7.

##  2 The general Kerr spacetime

A detailed discussion of the properties of the super-extremal Kerr metric was
presented in Mummery et al. (2024), to which we refer the reader for full
details, providing only a summary of the key properties here. Our notation is
as follows. We use physical units in which we denote the speed of light by
$c$, and the Newtonian gravitational constant by $G$. In coorindates
$x^{\mu}$, the invariant line element ${\rm d}\tau$ is given by

${\rm d}\tau^{2}=-g_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu},$ (1)

where $g_{\mu\nu}$ is the metric tensor with spacetime indices $\mu,\nu$. The
coordinates are standard $(t,r,\phi,\theta)$ Boyer-Lindquist, where $t$ is
time as measured at infinity and the other symbols have their usual quasi-
spherical interpretation. We shall assume that the disc exists exclusively in
the Kerr midplane $\theta=\pi/2$. For singularity mass $M$, and angular
momentum parameter $a$ (which has dimensions of length and is unrestricted in
this work), the line element is:

${\rm d}s^{2}=-\left(1-{2r_{g}r\over
r^{2}+a^{2}\cos^{2}\theta}\right)c^{2}{\rm
d}t^{2}-{4r_{g}rac\sin^{2}\theta\over r^{2}+a^{2}\cos^{2}\theta}\,{\rm
d}t\,{\rm d}\phi\\\ +{r^{2}-2r_{g}r+a^{2}\over
r^{2}+a^{2}\cos^{2}\theta}\,{\rm d}r^{2}+(r^{2}+a^{2}\cos^{2}\theta)\,{\rm
d}\theta^{2}\\\ +\left(r^{2}+a^{2}+{2r_{g}a^{2}r\sin^{2}\theta\over
r^{2}+a^{2}\cos^{2}\theta}\right)\sin^{2}\theta\,{\rm d}\phi^{2}.$ (2)

Note that our choice of coordinates has asymptotic (large radius) metric
signature $(-1,+1,+1,+1)$.

###  2.1 (A lack of an) Event horizon

The outer event horizon of the Kerr metric $r_{+}$ is defined as the larger of
the two radii at which

${1\over g_{rr}(r_{+})}=0\to r_{+}^{2}-2r_{g}r_{+}+a^{2}=0.$ (3)

The solution of this condition is trivial

$r_{+}=r_{g}+\sqrt{r_{g}^{2}-a^{2}},$ (4)

but highlights a crucial point of physics, namely that if

$|a|>r_{g},$ (5)

then the Kerr metric has no event horizon. This of course does not mean that
the metric ceases to be a solution of the governing equations of general
relativity, merely that the singularity at $r=0$ is not “hidden” behind an
event horizon. As discussed in the introduction, this “naked singularity”
behaviour is hypothesised to be forbidden in astrophysically realistic
settings (Penrose, 1969).

###  2.2 Orbital motion and super-extremal ISCOs

The key orbital component relevant for the study of accretion flows is the
angular momentum of a fluid element undergoing circular motion, as this is an
excellent approximation to the dynamical behaviour of the fluid in the stable
disc regions. For the Kerr metric standard methods (e.g., Hobson et al., 2006)
lead to the following circular angular momentum profile

$U_{\phi}=(GMr)^{1/2}{(1+a^{2}/r^{2}-2ar_{g}^{1/2}/r^{3/2})\over\sqrt{1-3r_{g}/r+2ar_{g}^{1/2}/r^{3/2}}}.$
(6)

This orbital solution is valid in both the sub ($|a|<r_{g}$) and super
($|a|>r_{g}$) extremal regimes. An accretion flow will persist in its
classical form wherever these orbits are stable. The stability of circular
orbits requires $\partial_{r}U_{\phi}>0$, the relativistic analogue of the
Rayleigh fluid-stability criterion. Evaluating this gradient, and setting
$\partial_{r}U_{\phi}=0$, results in the following expression for the location
of the innermost stable circular orbit (ISCO, e.g. Bardeen et al., 1972)

$r_{I}^{2}-6r_{g}r_{I}+8a\sqrt{r_{g}r_{I}}-3a^{2}=0.$ (7)

All spin parameters of the super-extremal Kerr metric have an ISCO, with the
ISCO radius given by (cf. Bardeen et al., 1972)

${r_{I}/r_{g}}=3+Z_{2}-{\rm
sgn}(a_{\bullet})\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})},$ (8)

where $a_{\bullet}\equiv a/r_{g}$ is the dimensionless spin, and we have
defined

$Z_{1}=\begin{cases}1+(1-a_{\bullet}^{2})^{1\over
3}\left((1+a_{\bullet})^{1\over 3}+(1-a_{\bullet})^{1\over
3}\right),\quad|a_{\bullet}|\leq 1\\\ 1-(a_{\bullet}^{2}-1)^{1\over
3}\left((1+|a_{\bullet}|)^{1\over 3}-(|a_{\bullet}|-1)^{1\over
3}\right),\quad|a_{\bullet}|>1,\end{cases}$ (9)

and $Z_{2}=\sqrt{3a_{\bullet}^{2}+Z_{1}^{2}}$ . The ISCO location plotted as a
function of Kerr metric spin is displayed in Fig. 1.

Figure 1: The ISCO as a function of Kerr metric spin parameter. The ISCO
exists for all spin parameters, and is a double valued function of
$a_{\bullet}$.

It will be important to note for future sections that the ISCO is a double
valued function of Kerr metric spin parameter $a_{\bullet}$. This was shown in
(Mummery et al., 2024) to lead to degeneracy in the spin parameter inferred
from the continuum fitting method between black hole and naked singularity
solutions that share a common ISCO. In this paper we shall demonstrate that
the same degeneracy does not plague the emission line fitting technique.

##  3 The X-ray line fitting formalism

Figure 2: The disc-observer photon energy shifts of two Kerr metrics, the
upper row describing a black hole with $a_{\bullet}=0.998$, the lower row a
naked singularity with $a_{\bullet}=1.968$ (both energy shift colour bars are
normalised to show the same scale). Both of these metrics have ISCO radii at
the same location $r_{I}/r_{g}\simeq 1.237$. On the left we show the innermost
30 $r_{g}$ of both discs, while on the right only the very innermost region of
the image plane is displayed, where differences are maximal. Both systems are
observed at $\theta_{\rm obs}=75^{\circ}$.

The specific flux density $F_{E}$ from a source, as observed by a distant
observer at rest (subscript ${\rm obs}$), is given by

$F_{E}(E_{\rm obs})=\iint I_{E}(E_{\rm obs})\,\text{d}\Omega_{{{\rm obs}}}.$
(10)

Here, $E_{\rm obs}$ is the observed photon energy and $I_{E}(E_{\rm obs})$ the
specific intensity, both measured at the location of the distant observer. The
differential element of solid angle subtended on the observer’s sky by a disc
area element is $\text{d}\Omega_{{{\rm obs}}}$. Since $I_{E}/E^{3}$ is a
relativistic invariant (e.g., Misner et al., 1973), we may write

$F_{E}(E_{\rm obs})=\iint g^{3}I_{E}(E_{\rm emit})\,\text{d}\Omega_{{{\rm
obs}}},$ (11)

where the photon energy ratio factor $g$ is the ratio of $E_{\rm obs}$ to the
emitted local rest frame photon energy $E_{\rm emit}$:

$g(r,\phi)\equiv{E_{\rm obs}\over E_{\rm
emit}}={-\left(U^{\mu}p_{\mu}\right)_{\rm
obs}\over-\left(U^{\nu}p_{\nu}\right)_{\rm disc}}={1\over
U^{t}}\left(1+{p_{\phi}\over p_{t}}{U^{\phi}\over U^{t}}\right)^{-1}.$ (12)

In this final expression $p_{\phi}$ and $-p_{t}$ are the angular momentum and
energy of the emitted photon (conveniently constants of motion in the Kerr
metric), $U^{t}$ and $U^{\phi}$ are the time and azimuthal 4-velocity
components of the orbiting disc fluid, which are given by

$\displaystyle U^{t}$
$\displaystyle={1+a\sqrt{r_{g}/r^{3}}\over\left(1-3r_{g}/r+2a\sqrt{r_{g}/r^{3}}\right)^{1/2}},$
(13) $\displaystyle U^{\phi}$
$\displaystyle={\sqrt{GM/r^{3}}\over\left(1-3r_{g}/r+2a\sqrt{r_{g}/r^{3}}\right)^{1/2}}.$
(14)

For a stationary observer at a large distance $D$ from the source, the
differential solid angle into which the radiation is emitted is

$\text{d}\Omega_{{{\rm obs}}}=\frac{\text{d}\alpha\,\text{d}\beta}{D^{2}},$
(15)

where $\alpha$ and $\beta$ are (Cartesian) photon impact parameters at
infinity (e.g., Li et al., 2005). Therefore, determining the observed specific
flux from an astrophysical source breaks down into knowing the specific
intensity of the source in its own rest frame, and calculating the energy
shifts of the photons on their path to the observer. This second step must in
general be done numerically.

In the simplified case whereby the emergent spectrum is assumed to be only a
$\delta-$function emission line, the rest-frame specific intensity can be
expressed as (e.g. Dauser et al., 2010)

$I_{E}(E_{\rm emit})=I_{\rm line}\,\delta(E_{\rm emit}-E_{\rm
line})\,\epsilon(r,\phi),$ (16)

where $\epsilon(r,\phi)$ the the disc emissivity function. In this case, the
observed specific flux is

$F_{E}(E_{\rm obs})={I_{\rm line}\over D^{2}}\iint
g^{3}\epsilon(r,\phi)\,\delta(E_{\rm obs}-gE_{\rm
line})\,{\text{d}\alpha\,\text{d}\beta}.$ (17)

Following e.g., Dauser et al. (2013), we represent the emissivity as a broken
power-law function of radius

$\epsilon(r,\phi)\propto\begin{cases}r^{-\gamma_{\rm in}},\quad r\leq r_{\rm
br}\\\ r^{-\gamma_{\rm out}},\quad r>r_{\rm br},\end{cases}$ (18)

where the inner and outer emissivity indices, $\gamma_{\rm in}$ and
$\gamma_{\rm out}$, and the break radius, $r_{\rm br}$, are model parameters
in skline and skconv. This parameterisation is expected theoretically for a
compact illuminating source (Dauser et al., 2013; Ingram et al., 2019).
Whereas an extended source can instead require a twice broken power law
(Wilkins & Fabian, 2012). While twice broken power laws are indeed inferred
observationally (Wilkins & Fabian, 2011), a once broken power law provides a
reasonable approximation and is widely used. The outer emissivity index is
always expected to tend to $\gamma_{\rm out}=3$ for a flat accretion disc,
whereas the inner index is $\gamma_{\rm in}\approx 2$ in Newtonian gravity
(Zycki et al., 1999) and $\gamma_{\rm in}>3$ in General Relativity (Wilkins &
Fabian, 2012). Values as high as $\gamma_{\rm in}\approx 5-10$ are routinely
inferred observationally for the inner emissivity index (e.g., Wilms et al.,
2001; Miller et al., 2002; Wilkins & Fabian, 2011; Dauser et al., 2012). For
the rest of this paper we simply employ $\gamma_{\rm in}=\gamma_{\rm
out}\equiv\gamma$ (i.e., no break in the emissivity) to explore the line
profile properties, and note that the effects driven by the absence of an
event horizon will be stronger for a steeper emissivity profile. We shall also
assume that the disc extends down to $r_{\rm in}=r_{I}$ for the remainder of
this paper. Extensions of these models beyond the ISCO is naturally of
interest, but the rapidly changing density of the intra-ISCO region (e.g.,
Mummery & Balbus, 2023) makes a more detailed treatment of the rest-frame
spectrum important, and beyond the scope of the analysis of this paper. We
will discuss the intra-ISCO region further in the conclusions of this paper.

Figure 3: The difference in the energy shifts of photons observed in the image
plane for a naked singularity system with $a_{\bullet}=1.968$ and a black hole
system with $a_{\bullet}=0.998$. Both of these metrics have ISCO radii at the
same location $r_{I}/r_{g}\simeq 1.237$, and both systems are observed at
$\theta_{\rm obs}=75^{\circ}$.

Figure 4: Image plane structure of the $a_{\bullet}=1.002$ Kerr metric,
observed at $\theta_{\rm obs}=75^{\circ}$ (upper) and $\theta_{\rm
obs}=25^{\circ}$ (lower).

##  4  Photon energy shift maps

The computation of $g(\alpha,\beta)$, which we shall henceforth refer to as
determining the “energy-shift map”, must be performed numerically. One starts
with a finely spaced grid $(\alpha,\beta)$ of photon impact parameters, at
some large distance $d$ from the disc. This image plane is orientated at an
angle $\theta_{\rm obs}$ from the singularity’s spin-axis. For each photon,
the ray is traced back from the observer to the disc plane by solving the
null-geodesics of the Kerr metric. At the location at which the photon crosses
the $\theta=\pi/2$ plane (i.e., we assume that the disc has infinitesimal
thickness perpendicular to the spin axis), the radial coordinate $r$ and
photon 4-momenta components $p_{\phi},p_{t}$ are recorded. These three numbers
fully specify the energy shift factor $g$. Any photons lost into the event
horizon (for black hole spacetimes), or which pass between the singularity and
ISCO are not included.

To perform these calculations we use a modified version of the code `YNOGK`
(Yang & Wang, 2013), which is based on the code `GEOKERR`: (Dexter & Agol,
2009). We modify the `YNOGK` code so as to compute null geodesics in the
super-extremal Kerr spacetime (see Appendix B of Mummery et al., 2024).

In Fig. 2 we display two photon energy shift maps, computed for a black hole
spacetime with $a_{\bullet}=0.998$ (upper row), and for a naked singularity
metric with $a_{\bullet}=1.968$ (lower row). This second spin value was chosen
as it has an ISCO radius equal to that of the $a_{\bullet}=0.998$ black hole
spacetime $r_{I}/r_{g}\simeq 1.237$. Both systems are observed at an
inclination of $\theta_{\rm obs}=75^{\circ}$. We include those photons which
cross the disc plane with radii $r_{I}<r<25r_{g}$. The colour-bar in both
cases displays the disc-observer energy shift of a photon observed at a given
$\alpha,\beta$. Note that the normalisation of the two (upper and lower)
colour-bars are identical.

While on larger scales $|\alpha|,|\beta|\gtrsim 5r_{g}$ the energy shifts of
the two metrics are comparable, there is a clear difference in the central
regions, which we highlight explicitly in the right hand column. There are
subtle differences in the “normal” region of the image plane, with the naked
singularity metric having systematically bluer energy-shifts (because there is
less gravitational red-shifting in the absence of an event horizon), but the
main difference between the two is the obvious inner structure of the naked
singularity image plane which is absent for black hole metrics.

This inner structure is entirely a result of the disappearance of the event
horizon for Kerr metrics satisfying $|a_{\bullet}|>1$. The structure occurs
due to photons from the disc passing within $r\sim 1r_{g}$ of the singularity
and, instead of being swallowed by an event horizon, are “spat out” of the
region and redirected towards the observer. The photons in this region of the
image were never in the $\theta<\pi/2$ region (no such photons contribute to
the image, since we assume they will be absorbed by optically thick material),
they were emitted from the upper surface of the disc and “bounced” off the
naked singularity. We will show that this image plane structure has a profound
impact on observed iron line structures of naked singularity metrics.

The exact difference in the energy shift maps, defined as

$\Delta g(\alpha,\beta)\equiv g_{\rm naked\,singularity}-g_{\rm black\,hole},$
(19)

is displayed in Fig. 3. Despite sharing a common ISCO location, these two
metrics are very different in the innermost regions of the image plane.

This additional image plane structure is not limited to super-extremal Kerr
metrics with $a_{\bullet}\gg 1$, it in fact occurs for any Kerr metric with
$a_{\bullet}>1$ and for all observing inclinations. To verify this see Fig. 4,
where we display the image plane structure of the $a_{\bullet}=1.002$ Kerr
metric, observed at $\theta_{\rm obs}=75^{\circ}$ (upper panel) and
$\theta_{\rm obs}=25^{\circ}$ (lower panel). The upper panel shows that for
larger inclinations the change in image plane structure is pronounced across
the extremal $a_{\bullet}=1$ limit. This change is much less pronounced for
near face-on inclinations (lower panel), but nonetheless a new inner ring of
observable photons still exists. In the following section we examine the
effects of these results on the observed line profiles originating from Kerr
metric discs.

##  5  Observed line profiles

The differing image plane structures of black hole and naked singularity Kerr
metrics result in extreme differences in the properties of observed line
profiles, for much of parameter space. Here we explore these differences under
the assumption that the disc inner radius is at the ISCO.

As a first example of this, see Fig. 5. In this Figure we plot line profiles
of two discs which share a common ISCO location $r_{I}/r_{g}\simeq 1.237$, one
for a black hole metric $a_{\bullet}=0.998$ (blue solid curve) and another for
a naked singularity metric $a_{\bullet}=1.968$ (red dashed curve). The line
profile is plotted as $E^{2}{\rm d}N_{\gamma}/{\rm d}E$, where $E$ is the
photon energy and $N_{\gamma}$ is the number of detected photons, the
normalisation of these profiles is arbitrary. Both systems were observed at an
inclination $\theta_{\rm obs}=75^{\circ}$ (i.e., they had photon energy shift
maps of Fig. 2), and an emissivity index $\gamma=3$. The rest-frame line
energy was taken to be that of the Fe K$\alpha$ transition $E_{\rm line}=6.4$
keV, and we include emission from the inner 30$r_{g}$ of the disc. The naked
singularity metric results in a significantly broader observed line, with red
and blue wings spreading to lower and higher energies than the black hole
metric.

The extension to higher energies (the blue wing) results directly from the
lack of an event horizon. Generically it is gravitational red-shifting which
works against the Doppler boosting of the disc fluids rotation and ultimately
limits the extent of the blue wing. The lack of event horizon reduces the
effects of gravitational red-shifting, and allows for an extended blue wing
despite the rotational disc velocities of the two metrics being similar.

The observed extension of the red wing results from the larger observed area
of the Doppler red-shifted disc regions (Fig. 2). Again, this is a direct
result of the lack of an event horizon.

Figure 5: The very different observed line profiles of two Kerr metric discs
which share a common ISCO location. The solid blue line profile is produced
for a black hole metric with $a_{\bullet}=0.998$, with ISCO $r_{I}/r_{g}\simeq
1.237$, while the red dashed line profile is for a naked singularity metric
with $a_{\bullet}=1.968$, and identical ISCO $r_{I}/r_{g}\simeq 1.237$
location. The other parameters of the system are: $\theta_{\rm
obs}=75^{\circ}$, $\gamma=3.0$, $E_{\rm line}=6.4$ keV (black dot-dashed line)
and $r_{\rm out}=30r_{g}$. The differing properties of the observed line
profiles can be understood with reference to the observer image plane
properties of the two metrics (Fig. 2; see text).  Figure 6: Similar observed
line profiles of two Kerr metric discs which share a common ISCO location. All
parameters are identical to Figure 5, except the emissivity index which now
equals $\gamma=0.0$. The shallower illumination profile ($\gamma=0$) reduces
the importance of the event horizon, resulting in similar results for the two
metrics. Emissivity profiles this shallow are likely unphysical, and inferred
indices from spectral fitting are always much steeper.  Figure 7:  The
observed line profiles of the Schwarzschild metric, and a naked singularity
metric (with $a_{\bullet}=6.53$) which shares the same ISCO location. The
other parameters of the system are: $\theta_{\rm obs}=60^{\circ}$,
$\gamma=4.0$, $E_{\rm line}=6.4$ keV (black dot-dashed line). The two metrics
produce noticeably different line profiles.  Figure 8:  Observed line profiles
from two Kerr metrics, one describing a rapidly rotating black hole
$a_{\bullet}=0.998$ (solid curves), and another for a just-super extremal
$a_{\bullet}=1.002$ metric (dashed curves), for differing emissivity indices
$\gamma$ (denoted by different line colours). For shallow emissivity profiles
$\gamma\lesssim 5$ the line profiles are similar, but for steep emissivity
profiles they are clearly distinguishable due to the emergence of double-lines
in the naked singularity spectrum. For these line profiles $\theta_{\rm
obs}=75^{\circ}$.

It is only for relatively steep (i.e., $\gamma\gtrsim 2$) emissivity profiles
that these differences become apparent. This can be seen with reference to
Fig. 6, which shows lines produced with identical system parameters to Fig. 5
except the emissivity index $\gamma$, which is now set to $\gamma=0$ (i.e.,
$\epsilon(r)={\rm cst}.$). The line profiles in this case are dominated by
Doppler shifting of photons produced at larger radii (with larger emitting
areas). At these large radii the two metrics are near identical in the image
plane (see Fig. 5), and the importance of the event horizon is minimal.
Fortunately for the purposes of probing the cosmic censorship conjecture,
emissivity profiles this shallow are likely unphysical, and inferred indices
from spectral fitting are typically much steeper (e.g. Wilms et al., 2001;
Miller et al., 2002; Wilkins & Fabian, 2011; Dauser et al., 2012).

It is not just for black hole metrics with large spins that the differences
from the naked singularity metrics with shared ISCO locations become apparent,
the same is true for all black hole metrics. This can be seen in Fig. 7, where
we plot the observed line profiles of a Schwarzschild $a_{\bullet}=0$ metric,
and a naked singularity metric with $a_{\bullet}=8\sqrt{6}/3\approx 6.53$,
which has an identical ISCO location at $r_{I}=6r_{g}$. The other parameters
of the system are: $\theta_{\rm obs}=60^{\circ}$, $\gamma=4.0$, $E_{\rm
line}=6.4$ keV (black dot-dashed line). The two metrics produce noticeably
different line profiles.

Even Kerr metrics described by $a_{\bullet}=1\pm\delta,(\delta\ll 1)$ can be
distinguished for sufficiently steep emissivity indices $\gamma$. In Fig. 8 we
plot the observed line profiles from two Kerr metrics, one describing a
rapidly rotating black hole $a_{\bullet}=0.998$ (solid curves), and another
for a just-super extremal $a_{\bullet}=1.002$ metric (dashed curves), for
differing emissivity indices $\gamma$ (denoted by different line colours). For
shallow emissivity profiles $\gamma\lesssim 5$ the line profiles are similar,
but for steep emissivity profiles they are clearly distinguishable due to the
emergence of double-lines in the naked singularity spectrum.

It is noteworthy that the continuum emission of thin discs evolving in metrics
with $a_{\bullet}=1\pm\delta$ are radically different (as shown in Mummery et
al., 2024), and so X-ray binaries with both observed line profiles and
continuum emission should easily be able to distinguish between black hole and
naked singularity metrics.

The emergence of “double-lines” in naked singularity spectra are generic
properties of Kerr metrics with $a_{\bullet}=1+\delta$ ($\delta\ll 1$)111Of
course, black hole metrics with emission dominated from large radii also show
double lines, driven entirely from Doppler shifting (e.g., Fig. 6). These
double lines show some symmetry about $E_{\rm line}$ however, which is very
different from the double gravitationally shifted lines we are describing
here. . In fact, it is even possible to generate “triple” observed lines (from
a single delta-function rest-frame line), as demonstrated in Fig. 9. The
lowest panel of this Figure shows an observed emission line from a system with
physical parameters $a_{\bullet}=1.05,\theta_{\rm obs}=60^{\circ},\gamma=4$.
The inner cluster of photons (see upper two panels) originate from the inner
edges of the disc (photon emission radius is displayed in the top panel), and
contribute in effect a second red-shifted line (energy shifts displayed in
middle panel), in addition to the “normal” disc line. This second line is then
itself split by Doppler boosting of the disc regions which produce the inner
photon cluster, resulting in a triple line. For reference in the lowest panel
we display in blue an $a_{\bullet}=0.998$ line, with other parameters
identical. The fact that the inner cluster of photons originate from the very
innermost disc regions (upper panel Fig. 9) highlights why these doublet (and
triplet) lines appear for steep emissivity profiles $\gamma$, when the lines
are entirely dominated by emission from the inner $\sim r_{g}$ of the disc.

Figure 9:  Lowest panel, red curve: an example of a “triple” naked singularity
iron line. Physical parameters $a_{\bullet}=1.05,\theta_{\rm
obs}=60^{\circ},\gamma=4$. The central cluster of photons in the image plane
originate from the inner edges of the disc (photon emission radius displayed
in top panel), and contributes in effect a second red-shifted line (energy
shifts displayed in middle panel), in addition to the “normal” disc line. This
second line is then itself split by Doppler boosting of the inner photon
cluster, resulting in a triple line. For reference in the lowest panel we
display in blue an $a_{\bullet}=0.998$ line, with other parameters identical.

Another bizarre feature of naked singularity X-ray spectral lines is
highlighted in Fig. 10. For face-on discs evolving in naked singularity Kerr
metrics one can observe “inverted” iron lines. We use the nomenclature
“inverted” as a typical black hole metric iron line has a sharp blue edge with
an extended red tail (e.g., Figs. 5, 7, 8, 9). However, if a naked singularity
disc is observed near face on, the observed line has a sharp red edge and an
extended blue tail (lower right panel Fig. 10). Once again, this unique
feature is a result of the effect of the disappearance of the event horizon on
the energy shift maps of the naked singularity metric. In the upper left panel
we display the energy shift map of the $a_{\bullet}=0.998$ black hole metric,
observed at $\theta_{\rm obs}=5^{\circ}$, while in the upper right panel we
display the energy shift map of the $a_{\bullet}=1.968$ naked singularity
metric, also observed at $\theta_{\rm obs}=5^{\circ}$. While the two energy
shift maps look similar, the lack of an event horizon means that the
gravitational energy shifts of the $a_{\bullet}=1.968$ metric are
substantially less than the $a_{\bullet}=0.998$ metric, despite them sharing a
common ISCO location. The difference in the two energy shifts is displayed in
the lower left panel of Fig. 10, where we see a systematic energy shift
difference for the inner disc radii of order $\Delta g\approx 0.3$. This,
coupled with the larger observed area of the inner disc in the
$a_{\bullet}=1.968$ metric, results in the spectral inversion seen in the
lower right panel.

Figure 10:  Lower right panel: inverted iron lines from a near face-on
observation of a naked singularity disc. In red (dashed) we display the
observed emission line from a $a_{\bullet}=1.986$ metric, and in blue (solid)
we show a $a_{\bullet}=0.998$ metric. Other physical parameters $\theta_{\rm
obs}=5^{\circ},\gamma=4$. The much reduced gravitational red-shifting of the
naked singularity metric results in this inverted profile. Upper left panel:
energy-shift map of $a_{\bullet}=0.998$ metric. Upper right panel: energy-
shift map of $a_{\bullet}=1.968$ metric (both energy shift colour bars are
normalised to show the same scale). Lower left panel: the difference in energy
shift maps, defined by $\Delta g(\alpha,\beta)\equiv g_{\rm
naked\,singularity}-g_{\rm black\,hole}$.

##  6  Convolved X-ray spectra

Consider a general rest-frame X-ray spectrum $I_{E}(E)$ emitted from a disc
location $(r,\phi)$. We will assume for simplicity that this rest frame
spectrum factorises into a disc-location independent shape function $I_{E}(E)$
and a disc location dependent amplitude (or emissivity) $\epsilon(r,\phi)$.
Under this assumption, the emitted rest frame spectrum can always be written
as an integral over a delta function

$I_{E}(E,r,\phi)\propto\epsilon(r,\phi)\int_{0}^{\infty}I_{E}(E^{\prime})\,\delta(E-E^{\prime})\,{\rm
d}E^{\prime},$ (20)

and, following the procedure spelt out in section 3, the flux observed by a
distant observer at rest will be

$F_{E}\propto\iint
g^{3}\left[\epsilon(r,\phi)\int_{0}^{\infty}I_{E}(E^{\prime})\,\delta(E_{\rm
obs}/g-E^{\prime})\,{\rm d}E^{\prime}\right]\text{d}\Omega.$ (21)

Swapping the order of integration, we have

$F_{E}\propto\int_{0}^{\infty}I_{E}(E^{\prime})\left[\iint
g^{3}\epsilon(r,\phi)\,\delta(E_{\rm
obs}/g-E^{\prime})\,\text{d}\Omega\right]{\rm d}E^{\prime}.$ (22)

This can be represented as a convolution between the line profile (like that
calculated above) and the rest-frame reflection spectrum (see e.g., Ingram et
al., 2019, for further details). Such a “Green’s function” approach is
employed by the skconv model. Fig 11 shows an example. The rest frame spectrum
(blue dotted line) is calculated with xillver (García et al., 2013) with
default parameters ($\Gamma=2$, $\log_{10}\xi=3.1$, $A_{\rm Fe}=1$, $E_{\rm
cut}=300$ keV). We convolve with skconv for two spin values,
$a_{\bullet}=0.998$ (black solid line) and $a_{\bullet}=1.968$ (red dashed
line). The other parameters are $\theta_{\rm obs}=75^{\circ}$, $\gamma=4$. We
see that these two examples, which have the same ISCO radius, are still very
different from one another. Capturing the additional physics of (for example)
a radial dependence of the rest-frame reflection spectrum requires extra
computational expense and will be considered in the future. For now, we note
that it is an important result that more complex convolved X-ray spectra can
be distinguished even for discs with the same ISCO location.

Figure 11: Two convolved X-ray spectra produced from observing a single rest
frame spectrum in two Kerr metrics with differing spin parameters. The rest
frame spectrum (blue dotted line) is calculated with xillver with default
parameters ($\Gamma=2$, $\log_{10}\xi=3.1$, $A_{\rm Fe}=1$, $E_{\rm cut}=300$
keV). We convolve with skconv for two spin values, $a_{\bullet}=0.998$ (black
solid line) and $a_{\bullet}=1.968$ (red dashed line). The other parameters
are $\theta_{\rm obs}=75^{\circ}$, $\gamma=4$. We see that these two examples,
which have the same ISCO radius, are still very different from one another.
The inset in the upper right shows a zoom-in around the FE K$\alpha$ line,
showing that these two metrics are distinguishable.

In Figure 12 we display convolved X-ray spectra for two sets of parameters
which produce notable line profiles in the super-extremal Kerr metric
$a_{\bullet}=1.05$, $\theta=60^{\circ}$ (upper panel, see also Fig. 9) and
$a_{\bullet}=1.968$, $\theta_{\rm obs}=5^{\circ}$ (lower panel, see also Fig.
10). The rest frame spectrum (blue dotted curve) is identical to Fig. 11. The
emissivity index is $\gamma=4$. Both of the extreme line profiles translate
into clearly observable features in a more detailed reflection spectrum. The
green dashed curves show the $\delta$-function line profiles, produced
assuming a line energy $E_{\rm line}=6.4$ keV. The normalisation for this line
is arbitrary and is shown only for pedagogical purposes.

Figure 12: Two convolved X-ray spectra produced from observing a single rest
frame spectrum in Kerr metrics with differing spin parameters. The rest frame
spectrum (blue dotted line) is calculated with xillver with default parameters
($\Gamma=2$, $\log_{10}\xi=3.1$, $A_{\rm Fe}=1$, $E_{\rm cut}=300$ keV). We
convolve with skconv for two sets of parameters which produce notable line
profiles in the super-extremal Kerr metric $a_{\bullet}=1.05$,
$\theta=60^{\circ}$ (upper panel, see also Fig. 9) and $a_{\bullet}=1.968$,
$\theta_{\rm obs}=5^{\circ}$ (lower panel, Fig. 10). The emissivity index is
$\gamma=4$. Both of the extreme line profiles translate into clearly
observable features in a more detailed reflection spectrum. The green dashed
curves show the $\delta$-function line profiles, produced assuming a line
energy $E_{\rm line}=6.4$ keV. The normalisation for this line is arbitrary
and is shown only for pedagogical purposes.

##  7 Discussion and conclusions

In this paper we have highlighted how iron lines observed in the spectra of
accreting X-ray binaries and Active Galactic Nuclei can be leveraged as
powerful probes of the (weak) cosmic censorship conjecture (Penrose, 1969).
While disc-continuum fitting (the other chief approach to inferring Kerr
metric spin parameters) suffers from degeneracy between sub- and super-
extremal metrics for some regions of parameter space (e.g., Takahashi &
Harada, 2010; Mummery et al., 2024), the same is not true for iron line
fitting. The reason for this difference is that iron line fitting typically
probes much smaller physical scales than continuum fitting, where the
differences between black hole and naked singularity metrics are most
pronounced.

The small-radii image plane differences between black hole and naked
singularity metrics can be traced directly to the disappearance of the Kerr
metric event horizon for $|a_{\bullet}|>1$. There are subtle differences in
the “normal” region of the image plane, with naked singularity metrics having
systematically bluer energy-shifts (because there is less gravitational red-
shifting in the absence of an event horizon), but the main difference between
black hole and naked singularity metrics is the appearance of an inner
structure for naked singularity image planes which is absent for black hole
metrics.

In naked singularity metrics, photons are observed in a warped image plane
structure which is interior to the inner image plane edge of black hole
metrics. These photons have passed within $1r_{g}$ of the singularity at the
centre of the Kerr metric, but as there is no event horizon for
$|a_{\bullet}|>1$, these photons are “spat out” from the region around
$r\approx 0$ and then make their way to the observer. The photons in this
central cluster originate from the very inner regions of the accretion flow
(Fig. 9), and so produce a leading order contribution to observed spectral
lines for steep emissivity profiles, like those routinely inferred from
spectral fitting (e.g., Wilms et al., 2001; Miller et al., 2002; Wilkins &
Fabian, 2011; Dauser et al., 2012). This means that even metrics with
$a_{\bullet}=1+\delta,(\delta\ll 1)$ can be distinguished from black hole
metrics. For certain observation angles naked singularity metrics produce
“triple” spectral lines (Fig. 9) which would be easily distinguishable from
black hole metrics.

For naked singularities with larger spins $a_{\bullet}\gtrsim 2$, which have
ISCO radii at the same location as black hole metrics with
$-0.998<a_{\bullet}<0.998$, it is still possible to distinguish between black
hole and naked singularity metrics. This is trivial for near face-on
orientations, as naked singularity metrics produce inverted spectral lines in
this limit (Fig. 10), with a sharp red-edge, and extended blue tail. For more
edge-on inclinations it is still possible to distinguish the two metrics,
providing the emissivity profile of the disc is sufficiently steep
($\gamma\gtrsim 2$; Figs. 5, 7, 6; as is universally required from
observations). This is because naked singularity metrics produce broader lines
than black hole metrics, owing to larger blue energy shifts (no event horizon)
and brighter red wings (larger observed area of red disc regions). This is
true for both highly spinning black hole metrics (and their complimentary
naked singularity metrics; Fig. 5) as well as for the Schwarzschild metric
(Fig. 7).

In this analysis we have assumed that the disc extends down to the ISCO, with
no significant reflection component contributed from inside of the ISCO. Of
course, in a real accretion system there is non-zero density within the ISCO
which could produce a potentially observable spectral component (e.g.,
Reynolds, 2013; Wilkins et al., 2020). As the density $\rho$ falls rapidly
upon crossing the ISCO (a result of the rapid increase in the radial velocity
and fixed radial mass flux e.g., Mummery & Balbus, 2023), the disc ionization
fraction $\xi\propto 1/\rho$ grows rapidly, and for typical parameters the
flow quickly becomes over ionized within the ISCO and this region therefore
typically contributes minimally to the time averaged X-ray spectrum (Wilkins
et al., 2020). It is therefore a poor approximation to treat the intra- and
extra-ISCO regions as having identical rest frame spectra, and for that reason
we have neglected these effects in this work.

This is of course not true for all choices of the free parameters of the
theory, and in principle neglecting this region could lead to systematic
effects, particularly for naked singularity metrics (which have a
significantly larger observable intra-ISCO region). We have computed the
energy shift maps (e.g., Fig. 2) for naked singularity metrics including the
intra-ISCO region, which are well behaved even in the near-singularity region,
and in future will include these effects in a more complete treatment which
includes radius dependent rest-frame spectra, which is vital in this low
density region.

We make the code used to produce all of the figures in this paper publicly
available as new XSPEC models, skline and skconv. These models take as input 7
parameters, the Kerr metric spin $a_{\bullet}$, the observer inclination
$\theta_{\rm obs}$, the inner and outer disc radii $r_{\rm in/out}$, a break
radius $r_{\rm br}$ and two power law emissivity indices $\gamma_{\rm
in/out}$. skline takes as input a line energy $E_{\rm line}$, and computes
observed delta-function line profiles such as in Figs. 5, 6, 7, 8, 9 and 10.
The model skconv takes as input a rest-frame X-ray spectrum, and outputs a
convolved spectrum like Figs. 11, 12.

Coupled with our naked singularity continuum fitting XSPEC package superkerr
(Mummery et al., 2024), all conventional X-ray spectral fitting procedures can
now be performed in the fully general Kerr metric, and the cosmic censorship
conjecture can be probed with astronomical X-ray observations. We reiterate
the philosophy put forward in (Mummery et al., 2024), that the “measurement”
of spin parameters $a_{\bullet}>1$ at high significance may well be telling us
of deficiencies in conventional disc modelling. We believe that the inclusion
of super-extremal Kerr spacetimes in conventional data analysis pipelines is a
win-win addition, as it either increases the confidence we have in
conventional approaches (in the result of a rejection of $a_{\bullet}>1$ at
high significance), or points towards gaps in our understanding of either
accretion physics or the nature of compact objects if $a_{\bullet}>1$ is
required by the data at high significance.

## Acknowledgments

We would like to thank the reviewer, Chris Reynolds, for a constructive and
very helpful review. AM would like to thank Pedro Ferreira and Georges Obied
for stimulating discussions which initiated this work. This work was supported
by a Leverhulme Trust International Professorship grant [number LIP-202-014].
For the purpose of Open Access, AM has applied a CC BY public copyright
licence to any Author Accepted Manuscript version arising from this
submission. AI acknowledges support from the Royal Society.

## Data availability

No observational data was used in producing this manuscript. The XSPEC models
skline and skconv are available at the following github repository:
https://github.com/andymummeryastro/superkerrline. The continuum fitting XSPEC
model superkerr is available at the following github repository:
https://github.com/andymummeryastro/superkerr

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11institutetext: Department of Computer Science, University of California, Los
Angeles

# Interpreting Generative Adversarial Networks for Interactive Image
Generation

Bolei Zhou  0000-0003-4030-0684

###### Abstract

Significant progress has been made by the advances in Generative Adversarial
Networks (GANs) for image generation. However, there lacks enough
understanding of how a realistic image is generated by the deep
representations of GANs from a random vector. This chapter gives a summary of
recent works on interpreting deep generative models. The methods are
categorized into the supervised, the unsupervised, and the embedding-guided
approaches. We will see how the human-understandable concepts that emerge in
the learned representation can be identified and used for interactive image
generation and editing.

###### Keywords:

Interpretable machine learning Generative Adversarial Networks Image
generation

##  1 Introduction

Over the years, great progress has been made in image generation by the
advances in Generative Adversarial Networks (GANs) [12, 6]. As shown in Fig.1
the generation quality and diversity have been improved substantially from the
early DCGAN [16] to the very recent Alias-free GAN [11]. After the adversarial
training of the generator and the discriminator, we can have the generator as
a pretrained feedforward network for image generation. After feeding a vector
sampled from some random distribution, this generator can synthesize a
realistic image as the output. However, such an image generation pipeline
doesn’t allow users to customize the output image, such as changing the
lighting condition of the output bedroom image or adding a smile to the output
face image. Moreover, it is less understood how a realistic image can be
generated from the layer-wise representations of the generator. Therefore, we
need to interpret the learned representation of deep generative models for
understanding and the practical application of interactive image editing.

Figure 1: Progress of image generation made by different GAN models over the
years.

This chapter will introduce the recent progress of the interpretable machine
learning for deep generative models. I will show how we can identify the
human-understandable concepts in the generative representation and use them to
steer the generator for interactive image generation. Readers might also be
interested in watching a relevant tutorial talk I gave at CVPR’21 Tutorial on
Interpretable Machine Learning for Computer
Vision111https://youtu.be/PtRU2B6Iml4. A more detailed survey paper on GAN
interpretation and inversion can be found in [21].

This chapter focuses on interpreting the pretrained GAN models, but a similar
methodology can be extended to other generative models such as VAE. Recent
interpretation methods can be summarized into the following three approaches:
the supervised approach, the unsupervised approach, and the embedding-guided
approach. The supervised approach uses labels or classifiers to align the
meaningful visual concept with the deep generative representation; the
unsupervised approach aims to identify the steerable latent factors in the
deep generative representation through solving an optimization problem; the
embedding-guided approach uses the recent pretrained language-image embedding
CLIP [15] to allow a text description to guide the image generation process.

In the following sections, I will select representative methods from each
approach and briefly introduce them as primers for this rapidly growing
direction.

##  2 Supervised Approach

Figure 2: GAN dissection framework and interactive image editing interface.
Images are extracted from [3]. The method aligns the unit activation with the
semantic mask of the output image, thus by turning up or down the unit
activation we can include or remove the corresponding visual concept in the
output image.

The supervised approach uses labels or trained classifiers to probe the
representation of the generator. One of the earliest interpretation methods is
the GAN Dissection [4]. Derived from the previous work Network Dissection [3],
GAN Dissection aims to visualize and understand the individual convolutional
filters (we term them as units) in the pretrained generator. It uses semantic
segmentation networks [24] to segment the output images. It then calculates
the agreement between the spatial location of the unit activation map and the
semantic mask of the output image. This method can identify a group of
interpretable units closely related to object concepts, such as sofa, table,
grass, buildings. Those units are then used as switches where we can add or
remove some objects such as a tree or lamp by turning up or down the
activation of the corresponding units. The framework of GAN Dissection and the
image editing interface are shown in Fig.2. In the interface of GAN
Dissection, the user can select the object to be manipulated and brush the
output image where it should be removed or added.

Besides steering the filters at the intermediate convolutional layer of the
generator as the GAN Dissection does, the latent space where we sample the
latent vector as input to the generator is also being explored. The underlying
interpretable subspaces aligning with certain attributes of the output image
can be identified. Here we denote the pretrained generator as $G(.)$ and the
random vector sampled from the latent space as z, and then the output image
becomes $I=G(\textbf{z})$. Under different vectors, the output images become
different. Thus the latent space encodes various attributes of images. If we
can steer the vector z through one relevant subspace and preserve its
projection to the other subspaces, we can edit one attribute of the output
image in a disentangled way.

Figure 3: We can use classifier to predict various attributes from the output
image then go back to the latent space to identify the attribute boundaries.
Images below show the image editing results achieved by [22].

To align the latent space with the semantic space, we can first apply off-the-
shelf classifiers to extract the attributes of the synthesized images and then
compute the causality between the occurring attributes in the generated images
and the corresponding vectors in the latent space. The HiGAN method proposed
in [22] follows such a supervised approach as illustrated in Fig.3: (1)
Thousands of latent vectors are sampled, and the images are generated. (2)
Various levels of attributes are predicted from the generated images by
applying the off-the-shelf classifiers. (3) For each attribute $a$, a linear
boundary $\textbf{n}_{a}$ is trained in the latent space using the predicted
labels and the latent vectors. We consider it a binary classification and
train a linear SVM to recognize each attribute. The weight of the trained SVM
is $\textbf{n}_{a}$. (4) a counterfactual verification step is taken to pick
up the reliable boundary. Here we follow a linear model to shift the latent
code as

$I^{\prime}=G(\textbf{z}+\lambda\textbf{n}_{a}),$ (1)

where the normal vector of the trained attribute boundary is denoted as
$\textbf{n}_{a}$ and $I^{\prime}$ is the edited image compared to the original
image $I$. Then the difference between predicted attribute scores before and
after manipulation becomes,

$\Delta
a=\frac{1}{K}\sum_{k=1}^{K}\max(F(G(\textbf{z}_{k}+\textbf{n}_{a}))-F(G(\textbf{z}_{k})),0),$
(2)

here $F(.)$ is the attribute predictor with the input image, and $K$ is the
number of synthesized images. Ranking $\Delta a$ allows us to identify the
reliable attribute boundaries out of the candidate set $\\{\textbf{n}_{a}\\}$,
where there are about one hundred attribute boundaries trained from step 3 of
the HiGAN method. After that, we can then edit the output image from the
generator by adding or removing the normal vector of the target attribute on
the original latent code. Some image manipulation results are shown in Fig.3.

Similar supervised methods have been developed to edit the facial attributes
[17, 18] and improve the image memorability [5]. Steerability of various
attributes in GANs has also been analyzed [9]. Besides, the work of StyleFlow
[1] replaces the linear model with a nonlinear invertible flow-based model in
the latent space with more precise facial editing. Some recent work uses a
differentiable renderer to extract 3D information from the image GANs for more
controllable view synthesis [23]. For the supervised approach, many challenges
remain for future work, such as expanding the annotation dictionary, achieving
more disentangled manipulation, and aligning latent space with image region.

##  3 Unsupervised Approach

As generative models become more and more popular, people start training them
on a wide range of images, such as cats and anime. To steer the generative
models trained for cat or anime generation, following the previous supervised
approach, we have to define the attributes of the images and annotate many
images to train the classifiers. It is a very time-consuming process.

Alternatively, the unsupervised approach aims to identify the controllable
dimensions of the generator without using labels/classifiers.

SeFa [19] is an unsupervised approach for discovering the interpretable
representation of a generator. It directly decomposes the pre-trained weights.
More specifically, in the pre-trained generator of the popular StyleGAN [12]
or PGGAN [10] model, there is an affine transformation between the latent code
and the internal activation. Thus the manipulation model can be simplified as

$\displaystyle\textbf{y}^{\prime}\triangleq G_{1}(\textbf{z}^{\prime})$
$\displaystyle=G_{1}(\textbf{z}+\alpha\textbf{n})$
$\displaystyle=\textbf{A}\textbf{z}+\textbf{b}+\alpha\textbf{A}\textbf{n}=\textbf{y}+\alpha\textbf{A}\textbf{n},$
(3)

where y is the original projected code and $\textbf{y}^{\prime}$ is the
projected code after manipulation by n. From Eq. (3) we can see that the
manipulation process is instance independent. In other words, given any latent
code z together with a particular latent direction n, the editing can always
be achieved by adding the term $\alpha\textbf{A}\textbf{n}$ onto the projected
code after the first step. From this perspective, the weight parameter A
should contain the essential knowledge of the image variation. Thus we aim to
discover important latent directions by decomposing A in an unsupervised
manner. We propose to solve the following optimization problem:

$\displaystyle\textbf{N}^{*}=\mathop{\arg\max}_{\\{\textbf{N}\in R^{d\times
k}:\textbf{n}_{i}^{T}\textbf{n}_{i}=1\ \forall
i=1,\cdots,k\\}}\sum_{i=1}^{k}||\textbf{A}\textbf{n}_{i}||_{2}^{2},$ (4)

where $\textbf{N}=[\textbf{n}_{1},\textbf{n}_{2},\cdots,\textbf{n}_{k}]$
correspond to the top $k$ semantics sorted by their eigenvalues, and A is the
learned weight in the affine transform between the latent code and the
internal activation. This objective aims at finding the directions that can
cause large variations after the projection of A. The resulting solution
becomes the eigenvectors of the matrix $\textbf{A}^{T}\textbf{A}$. Those
resulting directions at different layers control different attributes of the
output image, thus pushing the latent code z on the important directions
$\\{\textbf{n}_{1},\textbf{n}_{2},\cdots,\textbf{n}_{k}\\}$ facilitates the
interactive image editing. Fig.4 shows some editing result.

Figure 4: Manipulation results from SeFa [19] on the left and the interface
for interactive image editing on the right. On the left, each attribute
corresponds to some $\textbf{n}_{i}$ in the latent space of the generator. In
the interface, user can simply drag each slider bar associating with certain
attribute to edit the output image

Many other methods have been developed for the unsupervised discovery of
interpretable latent representation. Härkönen et al. [7] perform PCA on the
sampled data to find primary directions in the latent space. Voynov and
Babenko [20] jointly learn a candidate matrix and a classifier such that the
classifier can properly recognize the semantic directions in the matrix.
Peebles et al. [14] develops a Hessian penalty as a regularizer for improving
disentanglement in training. He et al. [8] designs a linear subspace with an
orthogonal basis in each layer of the generator to encourage the decomposition
of attributes. Many challenges remain for the unsupervised approach, such as
how to evaluate the result from unsupervised learning, annotate each
discovered dimension, and improve the disentanglement in the GAN training
process.

##  4 Embedding-guided Approach

The embedding-guided approach aligns language embedding with generative
representations. It allows users to use any free-form text to guide the image
generation. The difference between the embedding-guided approach and the
previous unsupervised approach is that the embedding-guided approach is
conditioned on the given text to manipulate the image to be more flexible,
while the unsupervised approach discovers the steerable dimensions in a
bottom-up way thus it lacks fine-grained control.

Recent work on StyleCLIP [13] combines the pretrained language-image embedding
CLIP [15] and StyleGAN generator [12] for free-form text-driven image editing.
CLIP is a pretrained embedding model from 400 million image-text pairs. Given
an image $I_{s}$, it first projects it back into the latent space as
$\textbf{w}_{s}$ using existing GAN inversion method. Then StyleCLIP designs
the following optimization objective

$\textbf{w}^{*}=\mathop{\arg\min}D_{CLIP}(G(\textbf{w}),t)+\lambda_{L2}||\textbf{w}-\textbf{w}_{s}||_{2}+\lambda_{ID}L_{ID}(\textbf{w},\textbf{w}_{s}),$
(5)

where $D_{CLIP}(.,.)$ measure the distance between an image and a text using
the pre-trained CLIP model, the second and the third terms are some
regularizers to keep the similarity and identity with the original input
image. Thus this optimization objective results in a latent code
$\textbf{w}^{*}$ that generates an image close to the given text in the CLIP
embedding space as well as similar to the original input image. StyleCLIP
further develops some architecture design to speed up the iterative
optimization. Fig.5 shows the text driven image editing results.

Figure 5: Text driven image editing results from a) StyleCLIP [13] and b)
Paint by Word [2].

Some concurrent work called Paint by Word from Bau et al. [2] combines CLIP
embedding with region-based image editing. It has a masked optimization
objective that allows the user to brush the image to provide the input mask.

##  5 Concluding Remarks

Interpreting deep generative models leads to a deeper understanding of how the
learned representations decompose images to generate them. Discovering the
human-understandable concepts and steerable dimensions in the deep generative
representations also facilitates the promising applications of interactive
image generation and editing. We have introduced representative methods from
three approaches: the supervised approach, the unsupervised approach, and the
embedding-guided approach. The supervised approach can achieve the best image
editing quality when the labels or classifiers are available. It remains
challenging for the unsupervised and embedding-guided approaches to achieve
disentangled manipulation. More future works are expected on the accurate
inversion of the real images and the precise local and global image editing.

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