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# Shock shaping? Nebular Spectroscopy of Nova V906 Carinae

É. J. Harvey,1 E. Aydi,2 L. Izzo,3 C. Morisset,4 M. J. Darnley,1 K.
Fitzgerald,5 P. Molaro,6 F. Murphy-Glaysher,1 M. P. Redman7 and M.
Shrestha,1,8
1 Astrophysics Research Institute, Liverpool John Moores University, IC2
Liverpool Science Park, Liverpool, L3 5RF, UK
2Center for Data Intensive and Time Domain Astronomy, Department of Physics
and Astronomy, Michigan State University, East Lansing, MI 48824, USA
3DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200
Copenhagen
4Universidad Nacional Autónoma de México, Instituto de Astronomía, AP 106,
Ensenada 22800, BC, México
5Technological University of the Shannon, Athlone Campus, Co. Westmeath,
N37HD68, Ireland
6Osservatorio Astronomico di Trieste, Vuz GB Trepolo 11 34131, Trieste, Italy
7Centre for Astronomy, National University of Ireland Galway, Galway, Ireland
8 Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson,
AZ 85721-0065, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

V906 Carinae was one of the best observed novae of recent times. It was a
prolific dust producer and harboured shocks in the early evolving ejecta
outflow. Here, we take a close look at the consequences of these early
interactions through study of high-resolution UVES spectroscopy of the nebular
stage and extrapolate backwards to investigate how the final structure may
have formed. A study of ejecta geometry and shaping history of the structure
of the shell is undertaken following a spectral line SHAPE model fit. A search
for spectral tracers of shocks in the nova ejecta is undertaken and an
analysis of the ionised environment. Temperature, density and abundance
analyses of the evolving nova shell are presented.

###### keywords:

radiation mechanisms:general – shock waves – atomic processes – methods:
observational – methods: data analysis – techniques: spectroscopic – (stars:)
binaries (including multiple): close – (stars:) circumstellar matter –
(stars:) novae, cataclysmic variables

††pubyear: 2023††pagerange: Shock shaping? Nebular Spectroscopy of Nova V906
Carinae–Shock shaping? Nebular Spectroscopy of Nova V906 Carinae

##  1 Introduction

Classical nova events are characterised by observed increases in visual
brightness of 6-12 magnitudes and are due to eruptions on the surface of a
white dwarf in a binary system. Much has been learned from the optical
photometric evolution of novae and they are normally classified by their
optical eruption spectra and light curves. The photometric decline time of a
nova is characterised by its $t_{2}$ or $t_{3}$, the time taken to decline by
2 or 3 magnitudes, respectively, from maximum optical light. Strope et al.
(2010) classify a variety of nova eruption light curves and give physical
explanations for many of their features. Many of these light curves can be
characterised simply by a fast rise and a smooth decline. However, they can
often feature transitional features such as plateaus, flat-tops, cusps,
jitters, plus other fast and slow temporary increases in brightness during the
nova decline phase.

Spectroscopic observations, however, are usually required to distinguish novae
from other transients as well as classifying them within established nova
taxonomy. For this reason, novae have been observed spectroscopically since T
Aur (1891; see Payne-Gaposchkin, 1957), and have been studied systematically
since Williams et al. (1991). A commonly adopted classification scheme for
nova spectroscopy is that known as the Tololo scheme, first presented in
(Williams et al., 1991; Williams et al., 1994). Nova spectra are characterised
by several observable stages during and post eruption. The spectral
fingerprint of a nova may be derived either if the nova shows some critical
features, or else if the nova does not show certain stages. However, a new
paradigm now exists following the detection of novae in the $\gamma$-ray
regime (Fermi-LAT-Collaboration, 2014) as well as the growing availability of
high-cadence, medium/high-resolution spectroscopy from large ground based
telescopes (Aydi et al., 2020b). Novae have also been recently unveiled as one
of the main lithium factories in the universe (Molaro et al., 2020).

The shaping mechanisms of nova shells are probes of the processes that take
place at energy scales between planetary nebulae and supernova remnants.
However, the three-dimensional structure of nova shells is difficult to
untangle when viewed on the plane of the sky. This is true even when spatially
resolved imaging of the nova shell is available, which is not the case for
recent nova events. In order to pin down the geometry, one must use the
observations that are commonly accessible to them. In the optical domain this
is mostly photometric observations and a growing wealth of spectroscopic
observations. Although, polarimetric and spectro-polarimetric observations are
increasingly being gathered for their diagnostic potential on system
orientation and scattering properties (Evans et al., 2002; Harvey et al.,
2018; Kawahita et al., 2019).

V906 Carinae (2018) was a bright nova that was observed across the
electromagnetic spectrum. It has even served as a basis for the development of
a classical nova unification theory (Aydi et al., 2020b). V906 Car was
discovered on the rise (UT 2018-03-16.32), saturating the 14cm Cassius
telescope detector, reported in Stanek et al. (2018). As it was discovered by
the ASASSN network in 2018 it was named ASASSN-18fv (Shappee et al., 2014).
Long baseline optical and NIR photometery are provided in Wee et al. (2020a)
of the DQ Her-like light curve, where the extinction towards the nova as well
as distance are derived. Wee et al. (2020a) report
$M_{\mathrm{WD}}<0.8\,\mathrm{M}_{\odot}$ (or $0.71\,\mathrm{M}_{\odot}$, from
modelling the accretion disk of the quiescent system) and a companion star
possibly of G or K type with mass $0.23-0.43\,\mathrm{M}_{\odot}$. $t_{2}$ is
difficult to define for novae with non-smooth declines. The nova was not
detected in the radio regime Ryder et al. (2018). The X-ray characteristic
behaviour is summarised in Sokolovsky et al. (2020), where solutions to column
densities, temperatures and elemental abundances are presented. Pavana et al.
(2020) find an ejected mass of $6\times 10^{-4}\,\mathrm{M}_{\odot}$ from a
detailed photoionisation analysis of the ejecta.

V906 Car is the brightest $\gamma$-ray detected nova to date (Piano et al.,
2018; Aydi et al., 2020a). With the $\gamma$-ray emission being associated
with shocks internal to the ejecta. The shocks are hypothesised to originate
from the interaction of an initial slower, equatorial bound, denser ejection
and subsequent higher velocity, but less massive ejection events. The presence
of faster subsequent ejections (or high-velocity winds of finite duration) are
seen by the appearance of high velocity features in optical spectroscopy shown
in Aydi et al. (2020b). The appearance of high velocity features in the
spectra appear coincident with short-duration flare episodes in the early
light curve. The interaction of the ejection episodes then appear to coincide
with longer-duration optical rebrightenings. This is affirmed through the
analysis of the concurrence of the optical and $\gamma$-ray emission peaks,
albeit only available at later times due to Fermi-LAT down-time.

The geometrical structure of the expelled shell from the V906 Car nova has
been studied primarily in Pavana et al. (2020), where they fit an evolving
morpho-kinematic model using the SHAPE code (Steffen & Lopez, 2006) over
several epochs. They find an inclination of the system of 53${}^{\textrm{o}}$
and arrive at a slightly asymmetric nebular morphology. The spectral lines
used for modelling are not yet in their frozen state, although are primarily
isolated lines. This is suspected to lead conclusions towards an asymmetric
structure, at odds with the population of well resolved classical nova shells
that are found to be axisymmetric structures. Before nova shell lines have
entered their frozen state they are subject to internal reddening effects in
an optically thick shell. Pavana et al. (2020) give solutions on the position
angle of the shell, however, we hold the position angle of the nova shell
constant as the object is much less extended than the slit width at time of
observation, and do not have access to polarimetric data to inform the
structure’s position angle. With this said, the shell inclination and
structural components found in Pavana et al. (2020) are similar to those found
in this work. Also, the results of the Cloudy simulations conducted in Pavana
et al. (2020) are found to complement our own, considering the base models are
for different epochs. In the work McLoughlin et al. (2021) the geometry of the
outflow of V906 Car is also discussed. The interpretation of McLoughlin et al.
(2021) suggests the spectral line signatures to be related to a slow moving
spherical ejection, jets and an accretion disk. The jet paradigm relies on the
accretion disk being, at least, partially responsible for the launching of a
jet, such that the accretion disk must re-establish before the jet is
launched. A theoretical model and explanation of the launching of gamma-rays
within this model is assumed to be in progress. Considering the shaping
mechanism of nova shells to be still open question, jets may have a role.

The origin of transient heavy element absorption (THEA) lines in nova spectra
has proven to be a point of contention in the literature. They were first
proposed by Williams et al. (2008) to originate in a circumbinary disk, fed by
an outflow from the companion star. Adding to this McLoughlin et al. (2020)
arrive at the conclusion of the existence of circumbinary Fe and O for the
V906 Car system and claim this reservoir may provide the origin of the THEA
lines. The argument for the spectral features not originating in the shell
itself is from the lower velocities observed. However, in Aydi et al. (2020b)
it is suggested that the THEA lines originate instead in the ejecta, the dense
(equatorial dominated) outflow expelled from the white dwarf i.e., the early,
slower component. The evolution of the THEA lines were found to have the same
structure as Balmer, Fe and other lines associated with the expelled nova
shell. Also, in this work we find the $`$M’ spectral line shape to be
representative of an unblended shell line at the corresponding epoch to
McLoughlin et al. (2020).

The reader is recommended to refer to Aydi et al. (2020a, b) for the evolution
of the high velocity features observed early on in the nova and to Pavana et
al. (2020) for a classical treatment of the pre-nebular phase spectral
evolution.

##  2 Observations

###  2.1 VLT-UVES optical spectroscopy.

This publication is based on a series of spectra of V906 Car obtained with the
Ultraviolet and Visual Echelle Spectrograph (UVES) (Dekker et al., 2000)
mounted at the UT2 telescope of the European Southern Observatory (ESO) in
Paranal, Chile (Obs ID. 0100.D-0621 PI: Paolo Molaro; Obs ID. 2100.D-5048 PI:
Paolo Molaro; Obs ID. 0103.D-0764, PI: Luca Izzo). Out of a set of 20 obtained
spectra, concentrated during the early evolving epochs from 2018 March 20.65
to 2018 July 5.0 (that have been covered extensively in Aydi et al., 2020a;
Molaro et al., 2020; Aydi et al., 2020b) we instead focus here on the later
spectrum from 2019 March 22.1 (day +395 post-maximum (used in this paper), or
day 405 post-eruption as used in Aydi et al. (2020a, b)), covering the nebular
phase of the nova. The instrumental configuration for all the epochs was the
DIC1 346-564, with central wavelength of 346 nm (range 305–388 nm) in the blue
arm and 564 nm (460–665 nm) in the red arm. Every observation was followed by
another with setting DIC2 437-760, which cover the 380–490 nm region and the
700–1000 nm region. The log of early UVES observations is reported in Table.
D.1 of Aydi et al. (2020b) and available at
http://www.astrosurf.com/aras/Aras_DataBase/Novae/2018_NovaCar2018.htm. A full
list of observations used in this work can be seen in Table. 1. The
observations were reduced using the reflex environment Freudling et al.
(2013).

Table 1: Diary of the UVES observations used for this work. The slit width was 0.6" for all observations using filter Her 5. A wavelength range of 3050 to 9000 Å. MJD | Day | exp (s) | R/1000 | exp (s) | R/1000
---|---|---|---|---|---
| a.m. | 346 nm |  | 564 nm |
58199.1438 | -6 | 60 | 59 | 60 | 66
58199.1452 |  | 300 | 59 | 300 | 66
58201.1412 | -4 | 300 | 59 | 60 | 66
58203.0924 | -2 | 300 | 59 | 60 | 66
58205.0687 | 0 | 300 | 59 | 60 | 66
58207.0599 | 2 | 600 | 59 | 120 | 66
58209.0218 | 4 | 600 | 59 | 120 | 66
58213.0665 | 8 | 600 | 59 | 120 | 66
58215.1292 | 10 | 600 | 59 | 120 | 66
58217.0686 | 12 | 600 | 59 | 120 | 66
58220.1292 | 15 | 600 | 59 | 120 | 66
58223.1262 | 18 | 600 | 59 | 60 | 66
58228.0002 | 22 | 150 | 59 | 30 | 66
58235.9959 | 30 | 300 | 59 | 60 | 66
58303.9851 | 98 | 1400 | 59 | 13x60 | 66
58598.0729 | 395 | 460 | 59 | 90 | 66

###  2.2 Goodman Spectrograph on SOAR

A spectrum was obtained on 12 December 2020 at 07:55:10 UTC in order to view
the shell in its pure nebular state, this corresponds to the +1001 post
eruption epoch. The spectrum was obtained using the Goodman spectrograph
mounted on the 4.1 m SOAR telescope. The spectrograph was used in its red
configuration, with a 0.95" long slit and an integration time of 60 s at an
airmass of 1.23. The 2100 lines/mm grating was used, providing a dispersion of
0.15 Å/ pixel. The spectrum was reduced, wavelength and flux calibrated using
standard routines in IRAF and can be seen overlaid in Fig. 1 .

##  3 Analysis

###  3.1 Distance

A distance estimate is given in Aydi et al. (2020a) based on the UVES data set
with Galactic reddening maps (Chen et al., 2019) and a reddening value
$E(B-V)$ = 0.36$\pm$ 0.05 mag, to derive a distance of $4.0\pm 0.5$ kpc for
the nova.

For additional constraints on the distance, we carried out Gaussian fitting of
the interstellar Na and K lines. The interstellar line profiles consist of 4
components in Na i D and K i 7699Å lines and 6 in Ca ii 3933 & 3968Å lines,
see Fig 2, which indicates that the line-of-sight absorption consists of a
number of discrete clouds each of which has some specific mean radial velocity
with respect to the local standard of rest ($V_{\mathrm{LSR}}$). In
particular, velocities derived from both Na i D1, D2 and K i lines indicate a
range of mean cloud velocities of $-37$ to $-7.5$ km s-1 whereas the Ca ii
lines indicate $-43$ to $-1$ km s-1 range. If the velocity is due to Galactic
rotation it can be used to derive a lower limit to the distance. In
particular, the fourth component in Na i D and K i indicates a distance
$d\gtrsim$ 4 kpc whereas the fifth component in Ca ii indicates distance
$d\gtrsim$ 4.5 kpc.

The velocity and intensity structure of the interstellar Na i D and Ca ii
lines in V906 Car is practically identical with those in two nearby well
studied B1 Ib stars, HD 93827 and HD 94493 (see fig. 4 of Sembach et al.
(1993)). While their line of sight are very close to each other, ($l,b$) =
(288.55, $-$1.54) and (289.01, $-$1.18) for HD 93827 and HD 94493,
respectively, as well as to that of V906 Car (286.6, $-$1.09) the distance to
HD 93827, $d$=8.3 kpc, is much larger than that for HD 94493 ($d$=3.3 kpc)
which means that the interstellar lines can offer limited constraints for
distances larger than $\sim$ 4kpc in the field/line of sight of V906 Car. The
reddening towards these two B1 Ib stars is also similar, $E(B-V)$=0.23 and
0.20 for HD 93827 and HD 94493, respectively (Sembach et al., 1993),
consistent with a lack of interstellar clouds beyond a distance of $\sim$ 4
kpc in this line of sight.

###  3.2 Unblended line evolution analysis and modelling: Morpho-kinematics
with SHAPE

The asymmetry in the lines at earlier times is opposite to those of the X-ray
lines reported in Sokolovsky et al. (2020), i.e. X-ray region is bright in the
blue, but in the optical regime spectral lines are brighter on their red side
for coincident observations. Implying the X-ray emitting region is at the
shell exterior, probably due to the X-rays being heavily absorbed by the gas
and the dust compositing the main shell.

Blending of spectral lines is common for all optical nova spectra, apart from
very late time nebular spectra where the strong nebular [O iii] and [N ii]
lines can often be orders of magnitude brighter than any remaining lines. The
strongest lines of the Balmer series also remain visible during these late
times, where H$\alpha$ may be blended with the [N ii] lines on either side.

However, during the early times, when the object is still bright enough to
acquire observations of high signal-to-noise ratio (SNR). With many Fe ii
lines in these spectra line blending is a problem. This is especially true if
the brightest spectral lines are used for line modelling, as they are more
often than not blended. This is true for even H$\alpha$, H$\beta$, [O iii]
5007,4958Å and [N ii] straddling H$\alpha$. Therefore, care must be taken when
choosing which lines to use to model the ejecta geometry.

Telescope systems such as the eMerlin network, ALMA, large ground-based
telescopes (with M1 > 8m diameter) and HST can begin to visually untangle
nearby (less than 1 kpc), fast ($V_{\mathrm{exp}}>1000$ km s-1) nova shells in
the first 100 days of the nova shell evolution, however this type of data
product is not commonly available. Additionally, imaging data needs to be
complemented by polarimetric and/or spectroscopic data to fully untangle these
early time nova shells - as line blending can also be a problem for narrow-
band imaging, especially for the faster expanding nova shell population. The
V906 Car nova shell did not settle into its frozen nebular state until > 1
year following maximum, and even at a distance $\sim$4 kpc and an expansion
velocity $\sim$500 km s-1 it could be spatially resolved by the aforementioned
space based and large telescopes.

Through visual inspection of the day 395 post-max UVES spectrum, several lines
were marked to be most likely unaffected by neighbouring lines, see Fig. 6.
The selection of isolated lines with sufficient SNR were identified.

The isolated med/high SNR lines that can be compared in this way are: 3444Å (O
iii), 3797Å (H i), 3995Å (N ii), 4686Å (He ii), 5876Å (HeI), 6087Å (N ii),
6678Å (He i), 7065Å (He i), as well as 8599Å, 8750Å, 8861Å of the Paschen
series.

The He ii 4686Å line had the highest SNR of the sample, as well as most level
blue- and red-shifted peaks, such that it was chosen as the most appropriate
line to perform SHAPE fitting of. These lines were also unblended in the SOAR
spectrum, also shown in Fig. 1, giving increased confidence in their
identification. The main differences seen between the +395 and +1001 spectra
is the cooling of the expanding shell, characterised by the growth in strength
of [OIII] lines.

Nova spectral lines, once frozen into their nebular state, can reveal the
inclination of the nova system as for any one geometry and axial ratio,
although one or two solutions may exist, see Harvey et al. (2020).
Unfortunately, we do not know the structure of a nova shell before it has been
resolved. The most one can do is estimate the shell geometry based on the
resolved nova shell population.

Figure 1: UVES day +395 spectrum with a SOAR day +1001 spectrum overlaid in
light blue.  Figure 2: UVES spectrum showing absorption components used for
distance derivation in Aydi et al. (2020a). Figure 3: Options for bipolar
shell with equatorial waists. Best fit inclinations were derived for type (A)
through to (E) assuming an axial ratio from 1.2 - 1.8. From these options Type
(E) with an axial ratio of 1.4 was chosen for subsequent investigation. Figure
4: Type (E) shell of Fig. 3 for fitting geometry at 90${}^{\textrm{o}}$ and
the best fit inclinations for the observed spectral lines, i.e.
68.5${}^{\textrm{o}}$ and 38.0${}^{\textrm{o}}$. Figure 5: He ii fit of
spectral line ’crown’. Grey is 68.5${{}^{\textrm{o}}}$ and red is
38${}^{\textrm{o}}$ inclination and. Two solutions from geometrical model
shown in Fig. 4. The models have been offset for clarity.

Figure 6: Unblended lines with SHAPE fit of at 68.5${}^{\textrm{o}}$
inclination (SHAPE model overlaid in grey), Figs. 4 and 5

.

As the fitting geometry, a simple equatorial waist and polar feature
morphology was used, as it is a commonly employed basic physical structure
associated with nova shells, e.g. Porter et al. (1998); Ribeiro et al. (2011);
Diaz et al. (2018); Harvey et al. (2016); Harvey et al. (2018, 2020), see
Figs. 3 and 4. Morpho-kinematic modelling was undertaken using the SHAPE code
(Steffen & Lopez, 2006). Although, it is normal to observe tropical rings in
nova shells also, they are left out of the model for the inclination fitting
routine in order to have fewer starting elements. As they are intermediate
between equatorial and polar features their inclusion is less important in
line shape fitting. Tropical rings are often present closer in azimuthal angle
to the equatorial portion of the shell, as such they can have the affect of
broadening the equatorial spectral components, especially for higher
inclination systems. Without knowing the true deprojected velocity of the
system inclination determinations remain first-pass. This could be solved with
high-resolution multi-epoch narrow-band imaging of the resolved shell, or
high-resolution IFU spectroscopy.

The best fit inclinations given are for an axial ratio of 1.4, see Figs. 3 and
4. These being 38.0${}^{\textrm{o}}$ and 68.5${}^{\textrm{o}}$. The
approximation is based on that on average deprojected nova shells have axial
ratios from 1.0 - 1.8, see for example Fig. 6 of Bode (2002). For a different
axial ratio the velocity of the polar versus equatorial features would vary by
the same ratio and this would give different inclination solutions. Then for
an axial ratio of 1.0 inclination solutions of 32${}^{\textrm{o}}$ and
60${}^{\textrm{o}}$ are found. Additionally, for an axial ratio of 1.8 system
inclination solutions of 44${}^{\textrm{o}}$ and 76${}^{\textrm{o}}$ are
found. The SHAPE model fits apply only to the top of the emission line since
the line is further broadened by the shell sitting on higher velocity
pedestals, i.e. lower density, higher velocity outflows also associated with
the nova event (again see Aydi et al., 2020a, b).

As seen earlier, two inclinations are possible with this given geometry of
polar caps and equatorial ring, i.e., 38${}^{\textrm{o}}$ and
68.5${}^{\textrm{o}}$. This is because it is not known if the equatorial
waist, or the polar cones are contributing as the faster component in the 1D
spectral line, as this is inclination dependent. Interestingly, the average of
our estimates is the same as that of Pavana et al. (2020). Figure 5 shows both
fits, i.e. whether the equatorial waist is contributing to the faster peaks of
the lines (68.5${}^{\textrm{o}}$, dashed grey fit), due to line of sight
effect, or whether higher velocity peaks are associated with the polar cones
are responsible for the observed higher velocity components
(38${}^{\textrm{o}}$, dashed red fit). The almost flat section of the
68.5${}^{\textrm{o}}$ fit between -100 and 100 km s-1 is due to an averaging
out of the plane-of-the-sky movement of the polar and equatorial components,
i.e. a mix or a projection effect and assumptions on geometry.

The fit in Fig. 6 is discussed here in an attempt to extrapolate the shaping
history of the shell given what we now know from Aydi et al. (2020a), where an
unequivocal relationship was observed between the $\gamma$-ray and optical
emission from the early evolving nova event. As the shocks most likely
originate from the equatorial region (at least initially; Metzger et al.,
2014), this lends itself as a likely explanation, implied by viewing angle
effects on the radial velocity measurements. For later shocks one might expect
them to consist of two components, an equatorial shock followed by shocks at
the poles between the swept up material from the previous interactions as well
as the most recent, at a given time.

Interpretation of Aydi et al. (2020a) allows to associate UVES spectroscopy
observation dates with the following nova phenomenology, as can be seen in
Figure 7 :
Max -4: Follows the first probable ejection. Average P-Cygni radial velocity:
-200 km s-1 (measured from Na i D 5889.95 A line). The initial shell ejection
has begun, understood to be more massive and slower than the subsequent
ejection episodes. For now the P-Cygni line profile is best matched by a blue
inverted fit. The blue absorption dip is characteristically narrow, due to it
coming from a single, finite ejection. The velocity matches that of the inner
peaks of the nebular lines - this is thought to be coincidental. However, the
inverted peak fit allows for the visualisation of the acceleration of the main
shell in Fig. 7. The red emission component is not yet visible.
Max -2: During second ejection. Average P-Cygni radial velocity: two
components: -200 km s-1 and -280 km s-1. The blue-shifted absorption profile
broadens and increases in velocity. This is believed to be due to the pre-
shock scenario where the slow flow visible at Max -4 and a new higher
velocity, less massive absorption component are unresolved from each other.
Max 0: During first hypothesized shock. Average P-Cygni radial velocity: two
components: -220 km s-1 -300 km s-1. An increase in blue absorption line
velocity, this is expected to be from the interaction of the line as described
for Max -2 and a 600 km s-1. Outflow traced in the Na D lines by Aydi et al.
(2020a). Now the red shifted emission component is beginning to appear in the
Paschen series. The blue profile is beginning to $`$grow’ out of its
absorption trough, with internal reddening effects becoming less important as
the shell expands.
Max +4: After 3rd, 4th ejections and during 5th ejection. The entire line is
now in emission, however due to optical depth effects the blue component is
still partially absorbed.
Max +22: Follows Fermi-LAT’s second detection, i.e., post-shock. Fermi-LAT’s
two strongest detections show a secondary lower amplitude episode. This can be
explained by the initial stronger component arising from interactions at
equatorial latitudes, followed by shocks in the system’s polar direction. This
implies that Fermi-LAT did miss an earlier shock episode during its downtime.
Fig. 7 demonstrates a change between Max +4 to Max +22 to an averaging of Max
+4’s two blue-shifted absorption components, signifying the meeting of two
ejection episodes.
Max +394: Now the shell is in the nebular stage, i.e., all major ejection and
shock phenomenology have ceased and the forbidden lines are strong. The
thinned and shaped ejecta has now frozen in structure, although continuing in
expansion. Final average shell radial velocity is 370 km s-1. All previous
absorption components have become their emission counterparts. Those that were
due to later ejections that ran into the earlier ejected bulk shell, are now
seen as the pedestal on which the line sits.

Here we examine only the lower velocity features to look at the nova shell
growth. For a summary of the high velocity features see Aydi et al. (2020a).
The time of appearance and velocity of the ejecta were traced in the Na D
lines in Aydi et al. (2020a). Three ejecta velocities from Aydi et al. (2020a)
are recorded as $v_{1}$ < 600 km s-1, $v_{2}$ $\sim$ 1200 km s-1 and $v_{3}$
$\sim$ 2500 km s-1. The post-shocked components of the higher velocity
features are eventually seen in the final nebular shell lines as the pedestals
of which the crown fits sit on, refer to H i 8863 Å.

In the context of the 38${}^{\textrm{o}}$ model it could be an outflow with an
equatorial over-density that would be seen to emerge first. Then, subsequent
ejections crash into the initial outflow and get funnelled towards the poles,
possibly creating tropical rings, akin to the $`$sweeping-up’ as described in
Porter et al. (1998). The polar material is expected to be relatively
unimpeded following the first shock episode, subsequent shock episodes are
then expected to have a different $\gamma$-ray light curve profile to account
for both equatorial and soon later the polar material being shocked. This is
expected to give double peaked gamma-ray light curves, however forward and
reverse shocks may also give this effect.

The central part of the line profile, which is stronger earlier on (and falls
following the second hypothesised shock, i.e. between day +12 to +15) may be
associated with the pseudo-photosphere, accretion disk, a section of the
equatorial disk, or even part of a more isotropic early outflow that is
subsequently shaped. Sokolovsky et al. (2020) state that this nova experienced
no distinct super soft phase, possibly due to a low mass white dwarf.

To uncover shock footprints it is best to uncover isolated lines, as well as
the Na D absorption lines, from before, during and after the shock episodes.
Each of the three Fermi-LAT detection episodes is followed by a more minor
episode. This is possibly due to (a) an equatorial shock, followed by a shocks
in the polar directions; or (b) the double peaks in the Fermi-LAT detections
could be due to forward and reverse shock signature; or (c) shocks at
different layers of the ejecta. Considering the hypothetical first shock event
(which is thought to be missed during the Fermi-LAT down-time), if (a) is
true, should only have a single peak, from a shock in the equatorial region.

Later shocks should give the investigator a solution to the system axial
ratio, provided the correct velocity components can be identified. Over the
course of an individual shock episode the faster component should lose
velocity on the average and diminish in strength. There should exist the two
pre-shock velocities, one each for the slower and faster components, the
appearance of a new shock velocity (equivalent to the difference between the
two ejecta fronts and depending on relative densities), and a post-shock
velocity. Motivated by this we seek here to find the corresponding velocity
components within the UVES spectral data set.

As usual with novae, care must be taken with line overlap. The blended
H$\beta$ line and the relatively unblended Paschen series line at 8863 Å are
used to demonstrate this exercise (Fig. 7). We find, initial outflow velocity
of -200 km s-1 and a later ejection velocity of -540 km s-1. During the first
(suspected) shock a transient velocity feature appears at -520 km s-1. This
transient velocity feature reduces to -460 km s-1 after eight days, then two
days later this feature jumps back up to -500 km s-1 before being washed out.
The arrival of this -500 km s-1 component corresponds to the second suspected
shock. For a summary of the higher velocity components observed in the UVES
spectra see Aydi et al. (2020b). A challenge encountered is in determining
when specific line components should be in emission or absorption, for this we
refer to Fig. 7. The optical depth of the region under study is dependent on
the relative placements of the various components with respect to each other
at that point in time. However, blending of even the strongest lines with weak
neighbours can lead the investigator to derive incorrect expansion velocity
values for the various components.

In Fig. 7 it is evident in the earlier spectra that the H$\beta$ line appears
artificially faster than the Paschen series counterpart due to the proximity
of a line on its blue side. Highlighting the need to explore unblended lines
for this type of analysis. However, comparing unblended lines with strong
blended lines lends information on what line features are real. This implies
that the Paschen 8863 Å line should be used to monitor the nova shell
evolution for this nova. Referring back to Fig. 7 and analysing the 8863 Å
line (dot-dash black line), a characteristic $`$step’ can be seen where in the
first panel a broad P-Cygni is observed, followed by a thinning of the P-Cygni
profile in the second panel (labelled eject (-4)) and gradual appearance of
two additional (narrow) superimposed absorption components. These then become
more apparent in the third panel (Eject (-2)) where the line appears to
broaden but instead is comprised of at least three distinguishable velocity
components. Then in the fourth panel the P-Cygni profile has increased in
velocity and has narrowed - the spectral line signature of a shock.

Following this the relative densities of the separate outflows can be
determined if the two responsible velocity components can be disentangled.
Interestingly, weak during this shock and stronger in the next two panels we
observe two sharp and apparently lower velocity absorption spikes. With the
suspected equatorial (side-on) view of the system these $`$lower’ velocity
components may indeed be the deprojected higher velocity polar components
forming. In the middle panel, the suspected second shock, these two lower
radial velocity features weaken and by the panel marked the end of the
suspected $2^{nd}$ shock when the the main equatorial component (now in
emission) narrows again. This is followed by the lower radial velocity
components becoming important once more - falling in line with the idea of
excitation of the polar components. This would make sense in a system where
following an equatorial shock, some material is swept up or $`$funnelled’ in
the polar direction. In this picture the shocks are shaping the shell. In
later shocks, as the polar components should be well formed according to the
proposed mechanism, an equatorial shock should then be followed by a polar
shock. However, a means to separate out forward and reverse shocked components
as well as equatorial and polar shocks would require similar observations at a
higher cadence with coincident $\gamma$-ray and, ideally, radio observations.

Figure 7: Evolution of H$\beta$ (solid colour lines) in comparison to the 8863
Å Paschen series line (black dot-dashed overlay) at corresponding epochs. To
highlight a particular interpretation of the optical spectra and light curve
the colour coding of the H$\beta$ follows blue for optical rises and peaks not
thought to be associated with shocks, red for spectra taken during suspected
shocks, green for pre- and post-shock spectra, cyan for the end of the primary
decline and black for nebular. As elsewhere in this paper day 0 is defined as
when the first suspected shock took place. Such that, for example, the second
panel, labelled (Eject (-4)), corresponds to a suspected ejection event four
days before the first suspected shock.

###  3.3 Density and temperature estimates

In order to refine the parameter space for deriving solutions to shell
temperatures and densities we used the photoionisation code Cloudy version
17.00 (Ferland et al., 2017), as well as several python wrappers available for
it, namely pyCloudy Morisset (2013), pyNeb Luridiana et al. (2015) for
diagnostics and, finally, pyCROSS (Fitzgerald et al., 2020) that pairs SHAPE
and Cloudy. We begin by fixing the shell inner radius by assuming a bulk shell
velocity of 450 km s-1, giving $1.25\times 10^{15}$ cm inner radius), distance
(4kpc), filling factor (1%) and log luminosity (4.2 L⊙), i.e. from spectral
fitting of the +395 day UVES observation and a sanity check to the Balmer line
ratio matching to the Mondal et al. (2019) model grid. Then, as free
parameters we iterate over $T_{eff}$ (80,000 K to 200,000 K), nH ($10^{6}$ to
$10^{9}$ cm-3) and the logarithmic shift of the metallicity (-1 to +1 dex)
relative to the Cloudy "NOVA" range exists in the early onset. For this grid
instead of using an outer radius we assume the nova shell is radiation bound.
This is since greater velocities exist in the early spectra that are weaker in
emission (the pedestals) on day +395. Instead we use a simulation stop
condition of a temperature of 100 K. Results can be seen in Fig. 8, where the
observed ratios are overplotted as boxes. Immediately problems are evident,
solutions to the observations are applicable over a large range of
temperatures and metallicities. As such Teff is not determined in our model
grid as every Teff cycled over fits within the observed box of Fig. 8. We also
find that log(U) is not really a constraint and, regardless, will not give a
reliable value of nH given uncertainties in the radius, filling factor and
luminosity of the ionising source. Interestingly, a broad set of abundances
can replicate the observed results for the often used [OIII] diagnostics, i.e.
the nova abundances as well as +1 dex and -0.5 dex changes to them. This
analysis demonstrates the challenges behind the curtain of using single
effective temperature estimates for single electron density solutions, when in
general a family of solutions exist within the nova shell parameter space,
sitting between the neat power law dependencies of lower density and lower
energy planetary nebulae and the higher energetics of supernova remnants.

When considering the reddening correction there is an issue where some claim
EBV to be 0.75 (Pavana et al., 2020). However, from our observations (i.e. the
same used in Aydi et al. (2020a)) the reddening is of the order of 0.35. This
latter value would give H$\alpha$/H$\beta$ $\sim$ 2.85, which suggests an
electron density of $<$ $1\times 10^{8}$ cm-3. On investigation of Figs. 9 and
10, products of the model grids as detailed at the beginning of this section,
we get a lower limit to ne of $\sim$ $5\times 10^{6}$ cm-3. We took
measurements of the [NII] and [OIII] line ratio diagnostics from the line
central plateaus and the secondary red shifted line peaks. The [NII]
diagnostic gave no discernible differentiation, however the [OIII] diagnostic
indicates that for any given Teff in the range a corresponding lower density
is found for the central plateau than the red peak, and/or a lower temperature
for a given density.

Looking at Figs. 9 and 10, which were created using pyNeb Luridiana et al.
(2015), we see that the typical [OIII] and [NII] diagnostics paired with the
reddening would suggest likely electron density in the region of
$(1-5)x10^{7}$ cm-3. However, the temperature is poorly constrained.

Looking at the largest radius ($10^{15}$cm) and thickest shell (also
$10^{15}$cm) covered in the results tables of Mondal et al. (2019) an n(H) of
$\leq 10^{8}$ at a blackbody temperature of $(1-5)x10^{5}$ K and bolometric
luminosity of $10^{37}$ \- $10^{38}$ ergs s-1 are found.

Figure 8: Grid showing the models run for this work in an attempt to pin down
the blackbody temperature of the ionising source. The size of the markers are
related to the shell hydrogen density, the colours related to the blackbody
temperature, the shapes to the shell abundances and deviation from in dex. The
boxes indicated the measured values from our observations. We find that a
range of abundances and temperatures can account for the observed line ratios.
Figure 9: [OIII] diagnostic, with observed ratio marked by the dashed line.
This demonstrates that the commonly used temperature diagnostic line ratio is
instead an electron density ratio for that observed here.  Figure 10: [NII]
diagnostic, again with the dashed line showing our measured line ratio.
Similarly to the [OIII] diagnostic we have a strong lower bound of the
electron density.

Despite the difficulties in constraining the system, we wanted to simulate the
ionisation structure of a purely photoionised look at the V906 Car shell at
about a year post maximum. We used the above diagnostic information, including
morphology, kinematics, temperature and densities, and with pyCross
(Fitzgerald et al., 2020) at hand we model the ionisation structure of the
nova shell, see Figure 11. Here, in this model neutral H is seen to occupy the
entire shell, with neutral C and Fe brightest in the equatorial region. For
lines associated with the Balmer series, they appear almost everywhere in the
nebula. In our simulations He II lines, such as 4686Åare inner to He I lines,
with He II near identical to the C++ and N++ panels in Figure 11. Whereas
[OIII] appears almost everywhere, showing that the observable nebula is
cooling as a whole. O I lines are the most localised, similar origin to He II
but over a more narrow area. These ionisation structure models can be used to
further interpret the spectral line profile evolution of the shell. However,
these models show photoionisation stratification, where a different stratified
structure would be expected to be observed if shocks contribute to the
ionisation budget in part or in whole.

Cloudy, as used for the parameter search, assumes spherical symmetry or an
open geometry (i.e. a small opening angle). Using pyCross it is possible to
run Cloudy over axi-symmetric geometries, such as bipolar nebulae, it also
hosts the ability to define different density laws for various regions of the
nebula. However, here this was not done as we did not have the diagnostic
information to separate components. Although possible with high resolution
line spectra, this would be better done with IFU spectroscopy of a spatially
resolved nebula. One reason why we did show our attempt here is because we
found the diagnostic lines used were not sufficient in their accuracy to
disentangle the subtle density variations between shell components as their
effect was drowned out when considering sources of error. In any case, we are
limited to axially symmetric structures with pyCross. If the nova shell were
to deviate from axial-symmetry, then our simulations would be inadequate as
they do not cover the complexities that would imply. However, near symmetric
structure of the unblended spectral lines during nebular stage, shown in Fig.
6, suggest an axi-symmetric shell. For reference we show readily identified
non Balmer lines and equivalent widths in Table. 5, which with Table. 2 makes
clear the measurable lines for the epoch under study.

In Table. 2 we show a model fit from an electron shell density of 7.5 dex. The
source distance was set at 4 kpc, covering factor of 0.3, filling factor of
0.01, radius $1.25\times 10^{15}$ cm, luminosity solar $10^{4.2}$ (in
agreement with Wee et al. (2020b)), age 395 days and Cloudy V1500 Cyg nova
abundances from Ferland & Shields (1978). This model gives an electron
temperature of 6,500 K, and an energy density temperature of 3600 K, such that
grains would survive in the environment. In this context for O,
photoionisation of upper levels of [O i] reached 7% of the total O destruction
rate. Photoionisation of upper levels of [O iii] reached 4% of the total O++
photoionisation rate. Non-collisional excitation of [O iii] 4363 Å reached 1%
of the total. This is the same fit as used to make Fig. 11. This model is
shown for reference only, given that the conditions are difficult to pin down
given the uncertainties demonstrated in Figs. 8, 9 and 10. The consequences of
shocks can be ignored at this stage as the shell has entered the nebular
phase, although as discussed in Derdzinski et al. (2017) early shocks
influence the expected shell density and clumping efficiency.

Intrinsic and emergent line intensities for 1000+ lines for all grid models
are available as supplementary data available online. Column depths and
averaged quantities of ionisation parameters are also provided within the same
data files, see Data Availability statement.

Table 2: Line ratio models and observations for H$\beta$ and the better known lines shown in Fig 6. Where lines could not be traced in the model, lines of the same species are chosen as replacements. Columns represent line i.d., wavelength, modelled and observed line ratios with reference to H$\beta$. We assume Case B recombination for H. line |  $\lambda$ (Å) | Model (L/LH$\beta$) | Obs (L/LH$\beta$) |
---|---|---|---|---
H I | 3798 | 0.06 | 0.04 |
$[$O iii$]$ | 4363 | 0.81 | 1.09 |
He I | 4471 | 0.08 | 0.07 |
He II | 4686 | 0.09 | 0.13 |
H I | 4861 | 1 | 1 |
$[$O iii$]$ | 5007 | 2.56 | 2.68 |
He I | 5876 | 0.29 | 0.21 |
He I | 7065 | 0.19 | 0.15 |
H I | 8598 | 0.02 | 0.03 |
H I | 8750 | 0.02 | 0.03 |
H I | 8863 | 0.02 | 0.04 |
Table 3: Log10 Column density ($cm^{-2}$). Fe(a) are Cloudy nova abundance (Ferland & Shields, 1978), which are based on the very fast nova V1500 Cyg (Ferland & Shields, 1978). Fe(b) = -3.77, which is in reference to the work presented here see Table. 4. level | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
H | 22.424 | 21.461 | 14.987 |  |  |  |
He | 21.449 | 19.861 | 18.521 |  |  |  |
Li | 9.38 | 13.778 | 10.79 | 9.801 |  |  |
Be | 8.558 | 11.872 | 10.294 | 8.615 | 6.559 |  |
B | 9.801 | 14.341 | 12.517 | 12.115 | 10.183 | 7.398 |
C | 15.874 | 19.431 | 17.801 | 16.373 | 15.627 | 13.05 | 5.141
N | 20.411 | 19.419 | 18.64 | 17.455 | 16.283 | 14.972 | 11.819
O | 20.659 | 19.584 | 18.748 | 17.646 | 16.548 | 15.133 | 13.427
Ca | 11.848 | 16.755 | 16.016 | 14.134 | 13.441 | 12.527 | 11.342
Fe(a) | 14.01 | 18.347 | 16.411 | 16.32 | 15.198 | 14.869 | 14.053
Fe(b) | 8.723 | 15.716 | 17.759 | 16.797 | 15.03 | 14.591 | 13.051

###  3.4 Fe abundance

The early evolving environment of novae is complex, as there is a combination
of shock- and photoionisation exciting the layered gas and dust shells. As
such there exists a large abundance discrepancy especially for Fe as
determined from X-ray and optical diagnostics, see Shchekinov (2009). As the
degree of ionisation from shock contributions has been historically difficult
to determine, with there being additional ways to hide Fe from X-ray
observations (Arimoto et al., 1997), a possible way to look at the problem
would be in understanding the correlation between temperatures derived from
coincident X-ray and optical observations (Loewenstein, 1999).

Here, we calculate the nebular stage abundance of [Fe III]. It is cautioned
that the abundance calculations depend on density, for which we assume the
best fit value from the previous section, i.e., of 7.5 dex. Line blending is
taken into account, for example the 5412 Å line could be either He II or [Fe
iii], or a blend of both and as such is not included in the calculation. We
refer to the solar abundance value of Fe/H from Grevesse et al. (2010) of
$3.16\times 10^{-5}$.

Table 4: This table shows log(Fe/H) for a list of close and unblended [Fe iii] lines. Determinations are for the lines listed and an average at the end of the table, i.e., where -3.77 = $1.70\times 10^{-04}$, or 5.4 $\times$ (Fe/H) solar. species |  $\lambda$ (Å) | log (Fe/H)
---|---|---
[Fe iii] | 4659 | -4.050
[Fe iii] | 4702 | -3.750
[Fe iii] | 4734 | -3.725
[Fe iii] | 4755 | -3.675
[Fe iii] | 4931 | -3.650
[Fe iii] | mean | -3.77

Looking at Table. 3 the column density distribution of the Fe level is shifted
towards higher levels for the higher Fe nova abundance for V906 Car, as found
in Table. 4. The large discrepancy in Fe abundances found between X-ray (< 0.1
Fe/Fe${}_{\odot})$ Sokolovsky et al. (2020)) and optical methods, found here
(i.e., 5.4 Fe/Fe⊙), implies that Fe could be hidden in the X-ray spectra. This
can be achieved by acceleration of dust grains behind shocks. This is proposed
by Shchekinov (2009) where they argue the freezing-out of heavy elements is
efficient due to betatron acceleration of dust grains behind shocks.

Figure 11: pyCross visualisation of various ionisation states of an equatorial
structure and polar blobs (the components visible at top left and bottom right
of each subplot) at 68.5${}^{\textrm{o}}$. This grid shows the expected
location of neutral H, He, C, N, O and Fe as well as the first and second
ionisation states of C, N and O. The x and y axes are normalised distances on
the plane of the sky. The colourbar represents the ionised fraction within the
species. These simulations are for a purely photoionised nebula. If partially
or wholly shock ionised then the ionisation stratification would be expected
to deviate from that shown here.

##  4 Conclusions

A methodology for uncovering blended lines has been laid out. This can be
achieved through fitting a line at the nebular stage to develop a template.
Then using line lists and ionisation codes lines can be separated, provided a
sufficient number of unblended lines for that species exists for regular
nebular diagnostics. There is scope for developing this methodology into a
future fitting routine. It is suggested in this work that it is best to wait
until a nova has entered the nebular stage in order to begin diagnostics, and
then one can work backwards to try and understand the evolution up to the
point of lines freezing into their final structure.

It was shown that the fitting of a simple 3D morpho-kinematic model to a 1D
line profile crown can aid in deciphering the evolutionary history of
unblended spectral lines as well as aiding in the detection of weak lines.
Sensitive spectroscopic observations are key to uncovering the underlying
physics of these enigmatic events. It is suggested in this work that the slope
of the broadened pedestal on which the line profile crown sits contains
information of the faster flows from the nova. This information is restricted
to velocity, as there is no discernible shape to the pedestal. Therefore if we
assume the pedestal to belong to an isotropically expanding spherical
component then we can take the 900 km -1, otherwise we can deproject its
velocity using the 38${}^{\textrm{o}}$ or 68.5${}^{\textrm{o}}$ inclination
solutions.

McLoughlin et al. (2020) state that the $`$M’ spectral line shape can only be
reproduced by a rotating disk, however we find that the sloped $`$drop-off’ is
well explained by the existence of faster moving material of lower inherent
flux and that the width of velocity components is well explained by a thicker
shell or even a thin shell/disk/ring/barrel viewed at any non-zero
inclination. The argument presented in McLoughlin et al. (2020) against the
‘M’ profile arising from the shell is that it is too narrow (of too low a
velocity) to arise from the shell. However, the observed line velocity is well
represented as a component of the final (frozen/nebular) unblended shell line
velocity.

In Aydi et al. (2020b), the initial outflow is concentrated towards the
equator, this may be due to the oblate nature of the fast spinning underlying
white dwarf (see Porter et al. (1998)) and/or the location of where material
is falling onto its surface. The subsequent ejections are expected to also
originate in an equatorial dominated outflow, however they are swept up into
conical regions as they interact with the slow, dense initial ejection. The
shocks between these outflows are the suspected origin of the $\gamma$-ray
emission (Aydi et al., 2020a; Metzger et al., 2014). The shock velocities are
expected to be the difference in velocity of the interacting regions.

It was found that during this investigation that both fast recurrent and slow
classical novae tend to inhabit a rapidly varying density and temperature
diagnostic space of commonly used nebular diagnostic lines. Meaning that one
should tread with caution when making temperature and density estimates in
novae. A larger sample of diagnostic lines of many species across the
electromagnetic spectrum should be used to better constrain densities,
abundances, reddening, filling factor, geometry and ionisation factors. It
would be advisable for the community to extend the grid parameters, in terms
of radii and resolution of models from Mondal et al. (2019) as well as lines
covered, this would allow for a concerted diagnostic effort.

In the resolved nova shell population where narrow band images of H$\alpha$ \+
$[NII]$ and $[OIII]$ are common the general ionisation stratification is
similar to that seen here, see T Aur (Gallagher et al., 1980), HR Del
(Duerbeck, 1987; Hutchings, 1972; Moraes & Diaz, 2009), DQ Her (Williams et
al., 1978; Vaytet et al., 2007), V1500 Cyg (Becker & Duerbeck, 1980; Hutchings
& McCall, 1977), V476 Cyg (Duerbeck, 1987), FH Ser (Gill & O’Brien, 2000), CP
Pup (Duerbeck, 1987), RR Pic (Gill & O’Brien, 1998), and GK Per (Liimets et
al., 2012; Harvey et al., 2016). Unfortunately high resolution IFU
observations (or channel maps) of easily resolved nova shells are rare. An
example of such though are the IFU observations of Moraes & Diaz (2009), which
also includes He II 4686Å, and the GK Per channel maps of Lawrence et al.
(1995).

We find an Fe abundance discrepancy between X-ray and optical observations,
parallel to that found in molecular clouds. The discrepancy between Fe
abundances, found from optical and X-ray studies detected should contain shock
information. It is also suggested in this work that spatially resolved narrow-
band imaging of nova shells at early times should reveal if ionisation
stratification is due to shock- or photoionisation, or both.

Prolonged and structured (e.g. waves/bumps) gamma emission and/or non-thermal
radio emission in a growing number of classical novae has been observed over
recent years (Cheung et al., 2016; Li et al., 2017; Finzell et al., 2018;
Babul et al., 2022). Supporting the observations are a series of theoretical
works, e.g. Metzger et al. (2014); Derdzinski et al. (2017); Steinberg &
Metzger (2020); Hachisu & Kato (2022). Slow evolving dust producing novae,
similar to V906 Car, have a tendency to feature the brightest (V1324 Sco
Finzell et al. (2018)) and longest duration (e.g. V5668 Sgr Cheung et al.
(2016)) gamma (>100 MeV) episodes. Where some have argued for a white dwarf
spin correlation to the gamma periodicity (ASASSN- 16ma Li et al. (2017)),
here we posit several forward and reverse shocks in both polar and equatorial
nova shell regions to give the observed structured optical and gamma-ray light
curve of V906 Car, similar to the theoretical work of Hachisu & Kato (2022),
effectively leading to the shaping of the nova shell as observed in its
nebular phase. Steinberg & Metzger (2020) discuss varying velocity outflows
over the course of a nova eruption leading to flares and complex velocity
evolution of absorption and emission components of spectral lines, which is
what we propose here being observed in V906 Car.

Shocks lead to clumping of shell material (Pittard, 2007), and the clump
centres give a safe haven for dust formation in an otherwise harsh environment
(Joiner, 1999). Lending itself linearly to infrared peaks (i.e. dust)
following gamma peaks (i.e. shocks, e.g., V5668 Sgr Banerjee et al. (2016);
Harvey et al. (2018) and V809 Cep Babul et al. (2022)) and strongly-shaped
dust-forming novae, e.g. T Aur (Gallagher et al., 1980), HR Del (Duerbeck,
1987; Hutchings, 1972; Moraes & Diaz, 2009), FH Ser (Gill & O’Brien, 2000) and
DQ Her (Williams et al., 1978; Vaytet et al., 2007). This leads us to conclude
that shocks likely shape nova shells in both their equatorial and polar shell
shapes (giving the shell covering factor) and the clumping within these
regions (the filling factor).

## Acknowledgements

ÉJH and MJD receive funding from STFC under grant ST/V00087X/1. CM
acknowledges the support from UNAM/DGAPA/PAPIIT grant IN101220. E.A.
acknowledges support by NASA through the NASA Hubble Fellowship grant HST-
HF2-51501.001-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS5-26555. EA also acknowledges NSF award
AST-1751874, NASA award 11-Fermi 80NSSC18K1746, and a Cottrell fellowship of
the Research Corporation.

## Data Availability

Reduced data and scripts to run moels are available upon reasonable request.
Raw data is publicly available through the ESO and SOAR science data archive
facilities.

Table 5: Spectral line identification of the optical spectra for lines not previously mentioned in Tables. 2 and 4 . We list the EW and integrated flux of the lines for which an estimate was possible (some lines are not strong enough or blended and therefore accurate measurements for these parameters were not feasible). The listed measurements are for +395 days after maximum. Line | $\lambda_{0}$ | EW ($\lambda$) | Flux
---|---|---|---
| (Å) | (Å) | ($10^{-13}$ erg cm-2 s-1)
$[$Ne v$]$ | 3346 | $-17\pm 2$ | 5.8 $\pm$ 0.5
$[$Ne v$]$ | 3426 | $-22\pm 2$ | 7.7 $\pm$ 0.5
O iii | 3444 | $-25\pm 2$ | 9.0 $\pm$ 0.5
$[$Ne iii$]$ | 3869 | $-150\pm 30$ | 21.3 $\pm$ 3.0
$[$Ne iii$]$ | 3968 | – | –
$[$Fe xi$]$ | 3978 | – | –
$[$Fe v$]$ | 4071 | – | –
He i | 4144 | – | –
C iii | 4187 | – | –
$[$Fe ii$]$ | 4244 | – | –
C ii | 4267 | – | –
$[$O iii$]$ | 4363 | – | –
He i | 4471 | – | –
N v | 4609 | – | –
N iii | 4638 | $-250\pm 40$ | 30 $\pm 2$
He ii | 4686 | $-100\pm 10$ | 12 $\pm 1$
He i | 4922 | – | –
$[$O iii$]$ | 4959 | $-800\pm 100$ | 134 $\pm 10$
$[$O iii$]$ | 5007 | $-2500\pm 200$ | 422 $\pm 20$
He i | 5048 | – | –
He ii | 5412 | – | –
$[$N ii$]$ | 5755 | $-950\pm 50$ | 104 $\pm 10$
C iv | 5805 | – | –
He ii | 5876 | $-180\pm 20$ | 22 $\pm 2$
N ii | 5938 | – | –
$[$O i$]$ | 6300 | $-200\pm 20$ | 22 $\pm 2$
$[$O i$]$ | 6364 | – | –
He i | 6678 | $-60\pm 5$ | 8.0 $\pm 0.5$
He i | 7065 | $-125\pm 5$ | 17 $\pm 1$
$[$Ar ii$]$ | 6136? | – | –
$[$Ar ii$]$ | 6237? | – | –
$[$O ii$]$ | 7320/30 | $-500\pm 50$ | 70 $\pm 10$

## References

  * Arimoto et al. (1997) Arimoto N., Matsushita K., Ishimaru Y., Ohashi T., Renzini A., 1997, ApJ, 477, 128
  * Aydi et al. (2020a) Aydi E., et al., 2020a, Nature Astronomy, 4, 776
  * Aydi et al. (2020b) Aydi E., et al., 2020b, ApJ, 905, 62
  * Babul et al. (2022) Babul A.-N., et al., 2022, MNRAS, 515, 3028
  * Banerjee et al. (2016) Banerjee D. P. K., Srivastava M. K., Ashok N. M., Venkataraman V., 2016, MNRAS, 455, L109
  * Becker & Duerbeck (1980) Becker H. J., Duerbeck H. W., 1980, PASP, 92, 792
  * Bode (2002) Bode M. F., 2002, AIP Conf.Proc., 637, 497
  * Chen et al. (2019) Chen B. Q., et al., 2019, MNRAS, 483, 4277
  * Cheung et al. (2016) Cheung C. C., et al., 2016, ApJ, 826, 142
  * Dekker et al. (2000) Dekker H., D’Odorico S., Kaufer A., Delabre B., Kotzlowski H., 2000, in Iye M., Moorwood A. F., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4008, Optical and IR Telescope Instrumentation and Detectors. pp 534–545, doi:10.1117/12.395512
  * Derdzinski et al. (2017) Derdzinski A. M., Metzger B. D., Lazzati D., 2017, MNRAS, 469, 1314
  * Diaz et al. (2018) Diaz M. P., Abraham Z., Ribeiro V. A. R. M., Beaklini P. P. B., Takeda L., 2018, MNRAS
  * Duerbeck (1987) Duerbeck H. W., 1987, ApSS, 131, 461
  * Evans et al. (2002) Evans A., Yudin R. V., Naylor T., Ringwald F. A., Koch Miramond L., 2002, A&A, 384, 504
  * Ferland & Shields (1978) Ferland G. J., Shields G. A., 1978, ApJ, 226, 172
  * Ferland et al. (2017) Ferland G. J., et al., 2017, Rev. Mex. Astron. Astrofis., 53, 385
  * Fermi-LAT-Collaboration (2014) Fermi-LAT-Collaboration 2014, Science, 345, 554
  * Finzell et al. (2018) Finzell T., et al., 2018, ApJ, 852, 108
  * Fitzgerald et al. (2020) Fitzgerald K., Harvey E. J., Keaveney N., Redman M. P., 2020, Astronomy and Computing, 32, 100382
  * Freudling et al. (2013) Freudling W., Romaniello M., Bramich D. M., Ballester P., Forchi V., García-Dabló C. E., Moehler S., Neeser M. J., 2013, A&A, 559, A96
  * Gallagher et al. (1980) Gallagher J. S., Hege E. K., Kopriva D. A., Butcher H. R., Williams R. E., 1980, ApJ, 237, 55
  * Gill & O’Brien (1998) Gill C., O’Brien T., 1998, MNRAS, 300, 221
  * Gill & O’Brien (2000) Gill C. D., O’Brien T. J., 2000, MNRAS, 314, 175
  * Grevesse et al. (2010) Grevesse N., Asplund M., Sauval A. J., Scott P., 2010, Ap&SS, 328, 179
  * Hachisu & Kato (2022) Hachisu I., Kato M., 2022, ApJ, 939, 1
  * Harvey et al. (2016) Harvey E., Redman M. P., Boumis P., Akras S., 2016, A&A, 595, A64
  * Harvey et al. (2018) Harvey E. J., Redman M. P., Darnley M. J., Williams S. C., Berdyugin A., Piirola V. E., Fitzgerald K. P., O’Connor E. G. P., 2018, AAP, 611
  * Harvey et al. (2020) Harvey E. J., et al., 2020, MNRAS, 499, 2959
  * Hutchings (1972) Hutchings J. B., 1972, MNRAS, 158, 177
  * Hutchings & McCall (1977) Hutchings J. B., McCall M. L., 1977, ApJ, 217, 775
  * Joiner (1999) Joiner D., 1999, PhD thesis, Rensselaer Polytechnic Institute, Rensselaer Polytechnic Inst, NY
  * Kawahita et al. (2019) Kawahita H., Shinnaka Y., Arai A., Arasaki T., Ikeda Y., 2019, AJ, 872
  * Lawrence et al. (1995) Lawrence S., MacAlpine G., Uomoto A., Woodgate B., Brown L., Oliversen R., Lowenthal J., Liu C., 1995, AJ, 109, 2635
  * Li et al. (2017) Li K.-L., et al., 2017, Nature Astronomy, 1, 697
  * Liimets et al. (2012) Liimets T., Corradi R. L. M., Santander-García M., Villaver E., Rodríguez-Gil P., Verro K., Kolka I., 2012, ApJ,, 761, 34
  * Loewenstein (1999) Loewenstein M., 1999, in Carral P., Cepa J., eds, Astronomical Society of the Pacific Conference Series Vol. 163, Star Formation in Early Type Galaxies. p. 153 (arXiv:astro-ph/9808204)
  * Luridiana et al. (2015) Luridiana V., Morisset C., Shaw R. A., 2015, A&A, 573, A42
  * McLoughlin et al. (2020) McLoughlin D., Blundell K. M., Lee S., 2020, MNRAS, 494, 743
  * McLoughlin et al. (2021) McLoughlin D., Blundell K. M., Lee S., McCowage C., 2021, MNRAS, 505, 2518
  * Metzger et al. (2014) Metzger B. D., Hascoët R., Vurm I., Beloborodov A. M., Chomiuk L., Sokoloski J. L., Nelson T., 2014, MNRAS, 442, 713
  * Molaro et al. (2020) Molaro P., Izzo L., Bonifacio P., Hernanz M., Selvelli P., della Valle M., 2020, MNRAS, 492, 4975
  * Mondal et al. (2019) Mondal A., Das R., Shaw G., Mondal S., 2019, MNRAS, 483, 4884
  * Moraes & Diaz (2009) Moraes M., Diaz M., 2009, AJ, 138, 1541
  * Morisset (2013) Morisset C., 2013, Astrophysics Source Code Library, p. 1304.020
  * Pavana et al. (2020) Pavana M., Raj A., Bohlsen T., Anupama G. C., Gupta R., Selvakumar G., 2020, MNRAS, 495, 2075
  * Payne-Gaposchkin (1957) Payne-Gaposchkin C. H., 1957, The galactic novae. New York, Dover Publications
  * Piano et al. (2018) Piano G., et al., 2018, The Astronomer’s Telegram, 11553, 1
  * Pittard (2007) Pittard J. M., 2007, ApSS proc, pp 245–268
  * Porter et al. (1998) Porter O’Brien Bode M. F., 1998, MNRAS, 296, 943
  * Ribeiro et al. (2011) Ribeiro V. A. R. M., Darnley M. J., Bode M. F., Munari U., Harman D. J., Steele I. A., Meaburn J., 2011, MNRAS, 412
  * Ryder et al. (2018) Ryder S. D., Kool E. C., Chomiuk L., 2018, The Astronomer’s Telegram, 11504, 1
  * Sembach et al. (1993) Sembach K. R., Danks A. C., Savage B. D., 1993, A&AS, 100, 107
  * Shappee et al. (2014) Shappee B., et al., 2014, in American Astronomical Society Meeting Abstracts #223. p. 236.03
  * Shchekinov (2009) Shchekinov Y. A., 2009, Astronomy Reports, 53, 605
  * Sokolovsky et al. (2020) Sokolovsky K. V., et al., 2020, MNRAS, 497, 2569
  * Stanek et al. (2018) Stanek K. Z., et al., 2018, The Astronomer’s Telegram, 11454, 1
  * Steffen & Lopez (2006) Steffen W., Lopez J. A., 2006, RMxAA, 42, 99
  * Steinberg & Metzger (2020) Steinberg E., Metzger B. D., 2020, MNRAS, 491, 4232
  * Strope et al. (2010) Strope R. J., Schaefer B. E., Henden A. A., 2010, AJ, 140, 34
  * Vaytet et al. (2007) Vaytet N. M. H., Brien T. J. O., Rushton A. P., 2007, MNRAS, 380, 175
  * Wee et al. (2020a) Wee J., et al., 2020a, ApJ, 899, 162
  * Wee et al. (2020b) Wee J., et al., 2020b, ApJ, 899, 162
  * Williams et al. (1978) Williams R. E., Woolf N. J., Hege E. K., Moore R. L., Kopriva D. A., 1978, ApJ, 224, 171
  * Williams et al. (1991) Williams R. E., Hamuy M., Phillips M. M., Heathcote S. R., Wells L., Navarrete M., 1991, ApJ, 376, 721
  * Williams et al. (1994) Williams R. E., Phillips M. M., Hamuy M., 1994, ApSS, 90, 297
  * Williams et al. (2008) Williams R., Mason E., Della Valle M., Ederoclite A., 2008, ApJ, 685, 451


# Exploring the Regularity of Sparse Structure in Convolutional Neural
Networks

Huizi Mao1, Song Han1, Jeff Pool2, Wenshuo Li3, Xingyu Liu1,
Yu Wang3, William J. Dally1,2
1Stanford University
2NVIDIA
3Tsinghua University
<EMAIL_ADDRESS>
###### Abstract

Sparsity helps reduce the computational complexity of deep neural networks by
skipping zeros. Taking advantage of sparsity is listed as a high priority in
the next generation DNN accelerators such as TPU[1]. The structure of
sparsity, i.e., the granularity of pruning, affects the efficiency of hardware
accelerator design as well as the prediction accuracy. Coarse-grained pruning
brings more regular sparsity patterns, making it more amenable for hardware
acceleration, but more challenging to maintain the same accuracy. In this
paper we quantitatively measure the trade-off between sparsity regularity and
the prediction accuracy, providing insights in how to maintain the accuracy
while having more structured sparsity pattern. Our experimental results show
that coarse-grained pruning can achieve similar sparsity ratio as unstructured
pruning given no loss of accuracy. Moreover, due to the index saving effect,
coarse-grained pruning is able to obtain better compression ratio than fine-
grained sparsity at the same accuracy threshold. Based on the recent sparse
convolutional neural network accelerator (SCNN), our experiments further
demonstrate that coarse-grained sparsity saves $\sim 2\times$ of the memory
references compared with fine-grained sparsity. Since memory reference is more
than two orders of magnitude more expensive than arithmetic operations, the
regularity of sparse structure leads to more efficient hardware design.

##  1 Introduction

Deep Neural Networks (DNNs) have many parameters, which leads to problems
related to storage, computation and energy cost. State-of-art Convolutional
Neural Network (CNN) models have hundreds of millions parameters and take tens
of billions operations[2, 3, 4]. That makes DNN models difficult to deploy on
embedded systems with limited resources.

To deal with this problem, various methods have been proposed to compress DNN
models and reduce the amount of computation. Some methods are based on
decomposition and factorization[5, 6]. These methods can preserve the regular
dense computation structure of the original models, thus are able to to
achieve both compression and acceleration on general-purpose processors.
Pruning serves as another effective method to greatly reduce the number of
parameters with no loss of accuracy[7, 8].

Pruning based methods are better at preserving accuracy as well as achieving
higher compression rates[7]. However, such improvements come at the cost of
the irregularity of the sparse computation pattern. On the other side,
structured pruning, such as pruning entire filters will cause larger accuracy
loss than pruning individual weights[9]. Those observations pose several
questions: _What is the trade-off between regularity and accuracy? Is it
possible to find a sweet spot in the range of regularity? How does the sweet
spot improve the efficiency of hardware implementation?_

We attempt to answer those questions by looking into pruning with different
granularity, as shown in Figure 1. There are existing works trying to prune
filters or channels instead of individual weights[10, 11, 12]. However, they
are individual points in the design space. Due to the various methods they
used, we cannot directly evaluate the relationship between pruning granularity
and final accuracy. We therefore apply the exact same method and experimental
setting for an effective comparison. We also want to explore a consistent
range of granularity, which includes intermediate grain size like 2D kernels
and 1D sub-kernel vectors. Based on a thorough space exploration, we are able
to analyze the storage saving and hardware efficiency at different granularity
of sparsity.

Figure 1: Different sparse structure in a 4-dimensional weight tensor. Regular
sparsity makes hardware acceleration easier.

In this work, we make the following contributions:

  * •

We explore a complete range of pruning granularity and evaluate the trade-off
between the model’s regularity and accuracy.

  * •

We demonstrate that coarse-grained pruning is able to reach similar or even
better compression rates than the fine-grained one, even though it obtains
less sparsity.

  * •

We show that coarse-grained sparsity is able to skip computations and reduce
memory references in a structured manner, which leads to more efficient
hardware accelerator implementation.

##  2 Related Works

Methods of pruning. Sparsity has been proven as an effective approach to save
parameters of Deep Neural Network models[7, 8]. A number of works have
investigated how to select the important connections and effectively recover
accuracy. Second-order derivative[13], absolute value[7], loss-approximating
Taylor expansion[10], and output sensitivity[14] are examples of importance
metrics used for pruning. There are also methods trying to better integrate
pruning and training, like iterative pruning[7] and dynamic pruning[8].

Granularity of sparsity. Among all types of sparsity, fine-grained sparsity
(vanilla sparsity) and filter-wise sparsity (very coarse-grained sparsity) are
two extreme cases that has been studied[7, 9]. Fine-grained sparsity is a type
of sparsity in which individual weights are deleted and was first proposed in
1989 by LeCun et al.[13]. Fine-grained sparsity has been proven to work well
on a wide range of popular neural network models of CNN and RNN[7, 8, 15, 16].
There is also channel reduction and filter reduction, which reduce the
dimension of input/output features as well as layers. Channel reduction can be
viewed as very coarse-grained sparsity that removes 3-dimensional sub-tensors
in convolutional layers. Such coarse-grained sparsity is beneficial for
acceleration due to regularity[17, 11]. However, it usually causes notable
reduced accuracy compared with fine-grained sparsity, as indicated by Li et
al.[9].

There is a large range of granularity between vanilla sparsity and channel
reduction. Some literature attempts to explore one or a few possibilities
among all choices. Intra-kernel strided pruning is one case investigated in
the work of Anwar et al.[12].

Accelerating sparse models. For very coarse-grained sparsity like filter-
sparsity and channel-sparsity, it is simple to achieve acceleration on
general-purpose processors because it is equivalent to obtaining a smaller
dense model[11]. For fine-grained sparsity, custom accelerators[18, 19] have
been proposed to exploit the reduction of computations.

##  3 Granularity of Sparsity

###  3.1 Notations

To simplify descriptions, we use the following notations for CNN. In a single
convolutional layer, the weights compose a 4-dimensional tensor of shape
$C\times K\times R\times S$. $C$ is the output dimension, i.e., the number of
output feature maps. $K$ is the input dimension. $R$ and $S$ are the shape of
convolution kernels ($R$=3, $S$=3 for a 3x3 kernel).

One layer’s weights consist of multiple filters (3-dimensional tensor of shape
$K\times R\times S$), each one associated with an output feature map. The
weights can also be viewed as multiple channels (3-dimensional tensor $C\times
R\times S$), each one associated with an input feature map. Filters and
channels are both composed of kernels (2-dimensional tensor $R\times S$),
which are the key element in the 2-d convolution operation. Sub-kernel vectors
(1-dimensional tensor of size $R$ or $S$) and scalar weights(0-dimensional
tensor) are lower-level elements in a convolutional layer. Figure 3
illustrates these concepts.

###  3.2 Range of Granularity

It has been stated that fine-grained sparsity and channel reduction are two
extreme cases of granularity. Among all possible grain sizes, there are a
large variety of choices.

In this paper, we investigate granularity where the grain size increases with
the number of dimensions. To be specific, we study 4 cases where the atomic
elements(grain) during pruning are scalar weights (0-D), sub-kernel vectors
(1-D), kernels (2-D) and filters (3-D). We explain these cases with numpy-like
pseudo codes as below. Figure 3 also illustrates the different granularities
of sparsity.

We select these four cases because they can be and have already been mapped
into simple computation logics. _Fine-grained sparsity_ breaks the tensor into
disjoint weights, therefore fine-grained multiplication-accumulation are
required. Its implementations has been provided in EIE[18] and SCNN[19]. _Sub-
kernel Vector sparsity_ can be mapped into 1-D convolution. Though the sparse
case has yet been studied, Eyeriss[20] deals with the dense case and treats
1-D convolution as primitive operation. _Kernel sparsity_ is related with 2-D
convolution, which is primitive operations in a variety of algorithms and
platforms, like Winograd convolution in cuDNN111Nvidia developer’s blog and
some FPGA implementations[21]. _Channel sparsity_ is equivalent to model
reduction and, thus, is easy for acceleration in all types of platforms.

Figure 2: Example of Sub-kernel Vector, Filter and Kernel.

Figure 3: Pseudo code: different granularity levels

###  3.3 Coarse-grained Pruning Method

Coarse-grained pruning deletes multiple weights together instead of individual
weights. Typically the grain can be filters, kernels, sub-kernels vectors or
anything else. Because we are interested in the effects of the grain size
rather than the pruning method, we adopt the simple magnitude-based pruning
criterion in [7]. For a grain $G_{i}$ that consists of multiple weights, the
Salience $S_{i}$ is defined as the sum of absolute values, i.e. the L1 norm in
Equation 1. Given the targeted sparsity, say we want 30% of the weights to be
zero, we sort the grains of weights according to the L1 norm defined in
Equation 1. The grains with the smallest 70% L1-norm are deleted.

$S_{i}=\sum_{w\in G_{i}}|w|$ (1)

We also adopt the iterative pruning method proposed by Han et al.[7]. It is
able to reach higher sparsity than direct pruning. The sparsity during each
pruning stage is determined by sensitivity analysis, which requires
individually pruning every layer and measure the accuracy loss on the training
dataset.

##  4 Sparsity-Accuracy Relation with Different Grain Sizes

Our goal is to study how the granularity of pruning influences the accuracy.
Specifically, we want to compare the accuracy of different pruning
granularities at the same sparsity, and the result is shown in Figure 4.
Sparsity serves as a regularizer because it lowers the model capacity by
reducing the number of parameters. Coarse-grained sparsity not only reduces
the number of parameters but also constrains the positions of parameters,
which is an even stronger regularizer. That’s why at low sparsity rate we
observed the accuracy improvement in Figure 4.

To ensure fair comparison, we enforce the same sparsity setting and training
schedule for the same model. All experiments were performed on the ImageNet
dataset[22] with Caffe[23].

For CNN models, we only count the overall sparsity of convolutional layers
since convolutional layers take up most of the computations in a typical CNN
model[24]. However, we still prune the fully-connected layers together with
convolutional layers to obtain consistent comparisons with previous works[7,
8]. For fc layers, we only use fine-grained pruning because there’s no such
hierarchy of pruning granularity in the fc layer.

We experimented on AlexNet for detailed accuracy-sparsity curves. We also
experimented with modern CNNs, including VGG-16[4], GoogLeNet[25],
ResNet-50[2], and DenseNet-121[26] and compared their accuracies at the same-
sparsity point. Their results are reported and compared in Table 1.

Figure 4: Accuracy-Sparsity Curve of AlexNet obtained by iterative pruning.

Figure 4 shows the accuracy curve of density(1-sparsity) under various
settings. In this figure there are four different types granularity of
sparsity; in each case the atomic element for pruning is

  * •

Fine-grained(0-Dim): Individual weights.

  * •

Vector(1-Dim) : Sub-kernel vectors of size $S$.

  * •

Kernel(2-Dim): Kernels of shape $R\times S$.

  * •

Filter(3-Dim): Filters of shape $K\times R\times S$.

When the grain size of pruning is very large, say, filters, we observed huge
accuracy loss during iterative pruning. AlexNet loses nearly 1% validation
accuracy at the very first pruning stage, which implies it is unsuitable for
lossless model compression. For finer-grained pruning, the accuracy loss is
much smaller; we even noticed small accuracy increases during the first
several pruning stages. Note that the results for AlexNet are better than the
original work by Han et al.[7] due to a smoother pruning process. We give a
detailed description in Section 7.

Table 1: Comparison of accuracies with the same density/sparsity. Model | Density | Granularity | Top-5
---|---|---|---
AlexNet | 24.8% | Kernel Pruning (2-D) | 79.20%
Vector Pruning (1-D) | 79.94%
Fine-grained Pruning (0-D) | 80.41%
VGG-16 | 23.5% | Kernel Pruning (2-D) | 89.70%
Vector Pruning (1-D) | 90.48%
Fine-grained Pruning (0-D) | 90.56%
GoogLeNet | 38.4% | Kernel Pruning (2-D) | 88.83%
Vector Pruning (1-D) | 89.11%
Fine-grained Pruning (0-D) | 89.40%
ResNet-50 | 40.0% | Kernel Pruning (2-D) | 92.07%
Vector Pruning (1-D) | 92.26%
Fine-grained Pruning (0-D) | 92.34%
DenseNet-121 | 30.1% | Kernel Pruning (2-D) | 91.56%
Vector Pruning (1-D) | 91.89%
Fine-grained Pruning (0-D) | 92.21%

The results in Table 1 and Figure 4 support the assumption that coarse-grained
sparsity causes greater accuracy loss than fine-grained sparsity. Pruning with
a large grain size like filters will greatly hurt accuracy. On the other hand,
pruning with a smaller grain size leads to similar accuracy-sparsity curves
with fine-grained pruning. Notice that in Figure 4, some curves appear to rise
smoothly at first. That suggests coarse-grained pruning can still reach
similar compression rates as fine-grained pruning, giving additional
advantages described in the following section.

##  5 Comparison of Storage

Model size is an important factor for real-world mobile applications. On the
one hand, it constrains the application in memory-bounded devices. On the
other hand, memory access is more than two orders of magnitude more energy
expensive during the execution of deep neural network[7]. Sparsity serves as
an effective approach to compress neural network models. Sparse neural
networks are usually stored in a similar format to Compressed Row
Storage(CRS), where both values and indices are stored. Coarse-grained
sparsity, due to its regularity, is able to save the number of indices as
illustrated in Figure 6. Therefore the coarse-grained sparse models take up
less storage than fine-grained models at the same sparsity.

Figure 5: Illustration of index saving.

Figure 6: Three curves are almost identical, indicating sparsity structure
does not impact quantization.

Figure 7: Accuracy-Storage Curve of AlexNet with different grain sizes. Notice
that vector pruning only causes 1.5% more storage and kernel pruning causes
6.7% more storage.

We want to investigate how the prediction accuracy changes with different
grain sizes of pruning at the same level of storage(instead of sparsity). We
do not use full-precision 32-bit weights, but 8-bit weights instead, as 8-bit
weights, either true 8-bit integer formats or 8-bit indices indexing to a
table of shared fp32 weights, have been proven to be sufficient in a lot of
literature[1, 18, 27]. We use 4-bit indices to store the distances between
adjacent non-zeros, following the method in Deep Compression [28]. Moreover,
as indicated in Deep Compression, the quantization method works independently
with sparsity. To check if it still works with coarse-grained sparsity, we
plot the accuracy-bits curves of different types of pruned models in Figure 6.
The results show that sparsity structure has negligible influence over
quantization.

Figure 7 shows the accuracy-storage relationship of AlexNet. We find that the
first three curves(Fine-grained, Vector and Kernel) are closer than those in
Figure 4. This figure shows the effect of index saving for coarse-grained
pruning.

To better compare the compression ratio with modern deep neural nets, we list
the results of AlexNet, VGG-16 and GoogLeNet, ResNet-50 and DenseNet-121 in
Table 2. Here the storage ratio is defined as the model size of pruned 8-bit
models(with 4-bit indices) to that of dense 8-bit models. Note that it is
almost impossible to prune a model that exactly matches the baseline accuracy,
so we use linear interpolation to obtain the estimated density and storage
ratio at a given point of accuracy.

Table 2: Comparison of storage savings at the baseline accuracy. Storage ratio is compared with the 8-bit dense model.  Model | Top-5 |  Granularity |  Density | Storage Ratio
---|---|---|---|---
Accuracy
AlexNet | 80.3% |  Kernel Pruning (2-D) |  37.8% | 39.7%
Vector Pruning (1-D) |  29.9% | 34.5%
Fine-grained Pruning (0-D) |  22.1% | 33.0%
VGG-16 | 90.6% |  Kernel Pruning (2-D) |  44.4% | 46.9%
Vector Pruning (1-D) |  30.7% | 35.8%
Fine-grained Pruning (0-D) |  27.0% | 40.6%
GoogLeNet | 89.0% |  Kernel Pruning (2-D) |  43.7% | 51.6%
Vector Pruning (1-D) |  36.9% | 47.4%
Fine-grained Pruning (0-D) |  32.3% | 48.5%
ResNet-50 | 92.3% |  Kernel Pruning (2-D) |  61.3% | 77.0%
Vector Pruning (1-D) |  40.0% | 52.7%
Fine-grained Pruning (0-D) |  37.1% | 55.7%
DenseNet-121 | 91.9% |  Kernel Pruning (2-D) |  35.5% | 48.9%
Vector Pruning (1-D) |  31.1% | 43.8%
Fine-grained Pruning (0-D) |  26.6% | 39.8%

For a sparse network, the larger the grain size is, the less storage needed.
This is due to index sharing among the weights of the kernel as shown in
Figure 6. However, AlexNet and VGG-16 in particular have much closer
density/storage results for kernel pruning than GoogLeNet, ResNet, and
DenseNet. This is caused by the small size of the convolutional kernels being
pruned: these networks have many layers of 1x1 convolutions, which do not
benefit from sharing index values. AlexNet and VGG-16, on the other hand, have
a multitude of larger convolutions.

##  6 Regular Sparsity Helps Hardware Implementation

It has been mentioned in the previous sections that filter pruning is able to
obtain acceleration on general-purpose processors like CPUs or GPUs. For
intermediate grain sizes like kernels or sub-kernel vectors, though it is
still difficult for acceleration on general-purpose processors, there are
several advantages over fine-grained sparsity on customized hardware. Those
advantages enable simpler circuit design and higher energy efficiency on
customized hardware. We qualitatively and quantitatively analyze the
advantages as follows:

Qualitative analysis. In convolutional layers, 2-D convolution is usually the
primitive operation. Kernel pruning (2-D pruning) can easily lead to
computation reduction, because the 2-D convolutions of deleted kernels can be
saved. Recent custom hardware design for CNN also use 1-D convolution as the
primitive operation[20]. In this case, sub-kernel vector pruning is
beneficial. Compared with fine-grained sparsity, coarse-grained sparsity is
able to preserve the low-level computation logic, therefore simplifying the
hardware design.

Quantitative analysis. Memory reference is a major factor of energy
consumption[7]. Recent work on custom hardware exploits both the sparsity of
weights and activations of CNNs[19]. In their implementation, the weights and
input activations are both stored in sparse format while output activations
are stored in dense format. The indices of weights and activations are used
for calculating the output address, to which the product of weight and
activation will perform a scatter-add operation. This process is illustrated
in Figure8. After one layer is finished, the output activations will then be
compressed into the sparse format for next layer.

Figure 8: A simplified dataflow of SCNN architecture. Weights and activations
are both stored in sparse format. Bypass is possible when the same output
address is referenced again.

If the same output address is referenced again, a data shortcut can be used to
avoid the expensive read/write. For example, two adjacent weights and two
adjacent activations will reference 3 addresses instead of 4. Due to the
locality, coarse-grained sparse weights have a larger probability of output
address collision. We simulated with VGG-16 on ImageNet’s validation set to
compare the number of memory references and listed the results in Table 3.
With the same density, coarse-grained sparsity saves $30\%-35\%$ of the total
output memory references.

Table 3: Output memory references for VGG-16 (convolutional layers only). Density | Fine-grained | Vector Pruning | Relative # of memory references
---|---|---|---
(0-D) | (1-D)
40.1% | 1.77B | 1.23B | 69.5%
33.1% | 1.53B | 1.03B | 67.2%
27.5% | 1.33B | 0.87B | 65.3%

##  7 Summary

In this section we compare our results with previous works on pruning[7, 8].
We select AlexNet, as its layer-wise sparsity is published in previous papers.
By using a smoother pruning process, we find the results of Han et al.[7] can
be further improved without any algorithmic change.

Table 4 gives an overall comparison of key statistics for AlexNet. Apart from
the number of parameters, there are some other factors affecting the
efficiency of DNN models. Here FLOPs is the total number of floating-point
operations. Storage is measured with of 8-bit weights and 4-bit indices as
indicated in Section 5. Due to the fact that the storage of convolutional
layers is much smaller but reused much more frequently than fully-connected
layers, we add an additional row for storage of convolutional layers. The
number of memory referenced is calculated by simulating the process of Figure
8. Here the baseline number of memory references is obtained from dense model
with sparse activations.

The results show that the our fine-grained pruned model already has advantages
over the previous state-of-art work in terms of FLOPs, storage of
convolutional layers and number of memory references. Moreover, compared with
our fine-grained baseline, the vector pruning method can further reduce the
storage of convolutional layers by 23% and the number of memory references by
43%.

Table 4: Comparison of pruned AlexNet with previous works which used fine-grained pruning.  Layer | Param. | NIPS’15 | NIPS’16 | Fine-grained | Vector | Kernel
---|---|---|---|---|---|---
[7] | [8] | Pruning | Pruning | Pruning
|  | (ours) | (ours) | (ours)
conv1 | 35K | 84% | 54% | 83% | 83% | 83%
conv2 | 307K | 38% | 41% | 26% | 26% | 26%
conv3 | 885K | 35% | 28% | 23% | 23% | 23%
conv4 | 664K | 37% | 32% | 23% | 23% | 23%
conv5 | 443K | 37% | 33% | 23% | 23% | 23%
fc6 | 38M | 9% | 3.7% | 7% | 7% | 7%
fc7 | 17M | 9% | 6.6% | 7% | 7% | 7%
fc8 | 4M | 25% | 4.6% | 18% | 18% | 18%
Total | 61M | 11% | 5.7% | 8.4% | 8.4% | 8.4%
FLOPs | 1.5B | 30% | 25.4% | 24.1% | 24.1% | 24.1%
Storage(conv) | 2.3MB | 55.6% | 48.3% | 36.4% | 28.0% | 25.5%
Storage(total) | 61MB | 16.7% | 8.5% | 12.6% | 12.3% | 12.2%
#Mem Reference | 99M | 74.4% | 71.7% | 60.5% | 34.6% | 35.2%
Top-5 Accuracy | 80.23% | 80.01% | 80.41% | 79.94% | 79.20%

##  8 Conclusion

We thoroughly explored the granularity of sparsity with experiments on
detailed accuracy-density relationship. Due to the advantage of index saving,
coarse-grained pruning is able to achieve a higher model compression ratio,
which is desirable for mobile implementation. We also analyzed the hardware
implementation advantages and show that coarse-grained sparsity saves $\sim
2\times$ output memory access compared with fine-grained sparsity, and $\sim
3\times$ compared with dense implementation. Given the advantages of
simplicity and efficiency from a hardware perspective, coarse-grained sparsity
enables more efficient hardware architecture design of deep neural networks.

## References

  * [1] Norman P. Jouppi, Cliff Young, Nishant Patil, David Patterson, Gaurav Agrawal, and et al. .  In-datacenter performance analysis of a tensor processing unit.  In 44th International Symposium on Computer Architecture, 2017.
  * [2] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 770–778, 2016.
  * [3] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Advances in neural information processing systems, pages 1097–1105, 2012.
  * [4] Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition.  arXiv preprint arXiv:1409.1556, 2014.
  * [5] Xiangyu Zhang, Jianhua Zou, Kaiming He, and Jian Sun.  Accelerating very deep convolutional networks for classification and detection.  IEEE transactions on pattern analysis and machine intelligence, 38(10):1943–1955, 2016.
  * [6] Vadim Lebedev, Yaroslav Ganin, Maksim Rakhuba, Ivan Oseledets, and Victor Lempitsky.  Speeding-up convolutional neural networks using fine-tuned cp-decomposition.  arXiv preprint arXiv:1412.6553, 2014.
  * [7] Song Han, Jeff Pool, John Tran, and William Dally.  Learning both weights and connections for efficient neural network.  In Advances in Neural Information Processing Systems, pages 1135–1143, 2015.
  * [8] Yiwen Guo, Anbang Yao, and Yurong Chen.  Dynamic network surgery for efficient DNNs.  In Advances In Neural Information Processing Systems, pages 1379–1387, 2016.
  * [9] Hao Li, Asim Kadav, Igor Durdanovic, Hanan Samet, and Hans Peter Graf.  Pruning filters for efficient convnets.  arXiv preprint arXiv:1608.08710, 2016.
  * [10] Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz.  Pruning convolutional neural networks for resource efficient transfer learning.  International Conference on Learning Representations, 2017.
  * [11] Wei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li.  Learning structured sparsity in deep neural networks.  In Advances in Neural Information Processing Systems, pages 2074–2082, 2016.
  * [12] Sajid Anwar, Kyuyeon Hwang, and Wonyong Sung.  Structured pruning of deep convolutional neural networks.  J. Emerg. Technol. Comput. Syst., 13(3):32:1–32:18, February 2017\.
  * [13] Yann LeCun, John S Denker, Sara A Solla, Richard E Howard, and Lawrence D Jackel.  Optimal brain damage.  In NIPs, volume 2, pages 598–605, 1989.
  * [14] Andries Petrus Engelbrecht.  A new pruning heuristic based on variance analysis of sensitivity information.  IEEE transactions on Neural Networks, 12(6):1386–1399, 2001.
  * [15] C Lee Giles and Christian W Omlin.  Pruning recurrent neural networks for improved generalization performance.  IEEE transactions on neural networks, 5(5):848–851, 1994.
  * [16] Song Han, Junlong Kang, Huizi Mao, Yiming Hu, Xin Li, Yubin Li, Dongliang Xie, Hong Luo, Song Yao, Yu Wang, et al.  ESE: Efficient speech recognition engine with sparse LSTM on FPGA.  In Proceedings of the 2017 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, pages 75–84. ACM, 2017.
  * [17] Vadim Lebedev and Victor Lempitsky.  Fast convnets using group-wise brain damage.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2554–2564, 2016.
  * [18] Song Han, Xingyu Liu, Huizi Mao, Jing Pu, Ardavan Pedram, Mark A Horowitz, and William J Dally.  EIE: Efficient inference engine on compressed deep neural network.  In Proceedings of the 43rd International Symposium on Computer Architecture, pages 243–254. IEEE Press, 2016.
  * [19] Angshuman Parashar, Minsoo Rhu, Anurag Mukkara, Antonio Puglielli, Rangharajan Venkatesan, Brucek Khailany, Joe Emer, Stephen Keckler, and William J. Dally.  SCNN: An accelerator for compressed-sparse convolutional neural networks.  In 44th International Symposium on Computer Architecture, 2017.
  * [20] Yu-Hsin Chen, Tushar Krishna, Joel S Emer, and Vivienne Sze.  Eyeriss: An energy-efficient reconfigurable accelerator for deep convolutional neural networks.  IEEE Journal of Solid-State Circuits, 2016.
  * [21] Jiantao Qiu, Jie Wang, Song Yao, Kaiyuan Guo, Boxun Li, Erjin Zhou, Jincheng Yu, Tianqi Tang, Ningyi Xu, Sen Song, et al.  Going deeper with embedded fpga platform for convolutional neural network.  In Proceedings of the 2016 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, pages 26–35. ACM, 2016.
  * [22] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 248–255. IEEE, 2009.
  * [23] Yangqing Jia, Evan Shelhamer, Jeff Donahue, Sergey Karayev, Jonathan Long, Ross Girshick, Sergio Guadarrama, and Trevor Darrell.  Caffe: Convolutional architecture for fast feature embedding.  In Proceedings of the 22nd ACM international conference on Multimedia, pages 675–678. ACM, 2014.
  * [24] Hessam Bagherinezhad, Mohammad Rastegari, and Ali Farhadi.  LCNN: Lookup-based convolutional neural network.  arXiv preprint arXiv:1611.06473, 2016.
  * [25] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich.  Going deeper with convolutions.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1–9, 2015.
  * [26] Gao Huang, Zhuang Liu, Kilian Q Weinberger, and Laurens van der Maaten.  Densely connected convolutional networks.  arXiv preprint arXiv:1608.06993, 2016.
  * [27] Vincent Vanhoucke, Andrew Senior, and Mark Z Mao.  Improving the speed of neural networks on CPUs.  In Proc. Deep Learning and Unsupervised Feature Learning NIPS Workshop, volume 1, page 4. Citeseer, 2011.
  * [28] Song Han, Huizi Mao, and William J Dally.  Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding.  arXiv preprint arXiv:1510.00149, 2015.