# Towards a Dynamical Collision Model of Highly Porous Dust Aggregates

Carsten Güttler  Maya Krause  Ralf Geretshauser  Roland Speith  Jürgen Blum

###### Abstract

In the recent years we have performed various experiments on the collision
dynamics of highly porous dust aggregates and although we now have a
comprehensive picture of the micromechanics of those aggregates, the
macroscopic understanding is still lacking. We are therefore developing a
mechanical model to describe dust aggregate collisions with macroscopic
parameters like tensile strength, compressive strength and shear strength. For
one well defined dust sample material, the tensile and compressive strength
were measured in a static experiment and implemented in a Smoothed Particle
Hydrodynamics (SPH) code. A laboratory experiment was designed to compare the
laboratory results with the results of the SPH simulation. In this experiment,
a mm-sized glass bead is dropped into a cm-sized dust aggregate with the
previously measured strength parameters. We determine the deceleration of the
glass bead by high-speed imaging and the compression of the dust aggregate by
x-ray micro-tomography. The measured penetration depth, stopping time and
compaction under the glass bead are utilized to calibrate and test the SPH
code. We find that the statically measured compressive strength curve is only
applicable if we adjust it to the dynamic situation with a “softness”
parameter. After determining this parameter, the SPH code is capable of
reproducing experimental results, which have not been used for the calibration
before.

###### Keywords:

dust collisions, cohesive powder, modelling, planet formation

###### :

96.10.+i

##  1 Introduction

Planets form from micrometer-sized dust grains, colliding at low velocities
and sticking to each other due to attracting van-der-Waals forces Blum and
Wurm (2008). From the interplay between laboratory experiments Blum and Wurm
(2000) and molecular dynamic simulations Paszun and Dominik (2008); Dominik
and Tielens (1997) we have a good picture of the microphysics of those
processes, but as aggregates grow larger (e.g. 1 mm), the understanding of the
collision mechanics is severely lacking. For these sizes (and the
corresponding velocities), collisions between equal sized aggregates rather
lead to bouncing or fragmentation than to sticking.

Millimeter-sized aggregates – as precursors of planets – are expected to be
highly porous, having a volume filling factor (the volume fraction of
material) of only few percent up to few ten percent. This strongly cohesive
material is comparable to millimeter-sized dust clumps present in conventional
dry powders (e.g. cocoa, confectioners’ sugar) but probably being more porous.
One well analyzed analog material are dust aggregates formed by the random
ballistic deposition (RBD) method, experimentally introduced in Ref. Blum and
Schräpler (2004). These dust aggregates have a volume filling factor of
$\phi=0.15$ (for monospheric dust) and are produced in our laboratory in
macroscopic 2.5 cm samples. Using these dust samples, we performed various
collision experiments to study the further growth and evolution of planetary
bodies (e.g. Blum and Wurm, 2008; Langkowski et al., 2008). However, as
experiments cannot be performed for any relevant set of parameters (e.g.
collisions of meter-sized bodies), a collision model is required to cover the
wide parameter space occurring for protoplanetary dust-aggregate collisions,
i.e. dust aggregate sizes of up to 1 km and collision velocities in the range
of $10^{-3}-10^{2}\;{\rm m\;s^{-1}}$. The approach is therefore to measure
macroscopic parameters of the laboratory dust samples and implement them in a
numeric simulation of a dust aggregate collision, which is concurrently
performed in a laboratory experiment. Comparing experiment and simulation and
defining and determining the free parameters, yields a calibrated collision
model to perform collision simulations with parameters unaccessible to
laboratory experiments.

##  2 Smoothed Particle Hydrodynamics for Dust Collisions

For the simulation of dust aggregate collisions we use the Smoothed Particle
Hydrodynamics (SPH) method with extensions for the treatment of solid and
porous media. A comprehensive description of the meshless Lagrangian particle
method SPH can for example be found in Monaghan (2005). In this scheme the
continuous solid objects are discretized into interacting mass packages
(“particles”) carrying all relevant continuous quantities. Time evolution is
computed according to the Lagrangian equations of continuum mechanics:

$\displaystyle\frac{\mathrm{d}\varrho}{\mathrm{d}t}+\varrho\sum_{\alpha=1}^{D}\frac{\partial
v_{\alpha}}{\partial x_{\alpha}}=0$ (1)
$\displaystyle\frac{\mathrm{d}v_{\alpha}}{\mathrm{d}t}=\frac{1}{\varrho}\sum_{\beta=1}^{D}\frac{\partial\sigma_{\alpha\beta}}{\partial
x_{\beta}}$ (2)

Here, $\varrho$ denotes the density, $v$ the velocity, $D$ the dimension and
$\sigma_{\alpha\beta}$ the stress tensor, defined as

$\sigma_{\alpha\beta}=-p\delta_{\alpha\beta}+S_{\alpha\beta}.$ (3)

It consists of a pressure part with pressure $p$ and and a shear part given by
the traceless deviatoric stress tensor $S_{\alpha\beta}$. Its time evolution
is modelled according to Ref. Benz and Asphaug (1994). This set of equations
is closed by a suitable equation of state and describes the elastic behavior
of a solid body. Together with a suitable damage model, which we do not adopt,
the authors of Ref. Benz and Asphaug (1994) have modelled collisions between
brittle basaltic rocks using this scheme.

In contrast to this, we simulate the plastic behavior of porous bodies.
Therefore we adopt a modified version of the porosity model by Sirono Sirono
(2004). According to this approach, plasticity is modelled within the equation
of state and porosity is given by $1-\varrho/\varrho_{0}$, where $\varrho$
denotes the actual and $\varrho_{0}$ the bulk density of the material. The
pressure is limited by the compressive strength $\Sigma(\varrho)$ as upper
bound and the (negative) tensile strength $T(\varrho)$ as lower bound. In
between, the solid body experiences elastic deformation, whereas outside this
regime it is deformed plastically. Thus, the full equation of state reads

$p(\varrho)=\left\\{\begin{array}[]{lc}\Sigma(\varrho)&\varrho>\varrho_{\rm
c}^{+}\\\
K({\varrho^{\prime}}_{0})(\varrho/{\varrho^{\prime}}_{0}-1)&\varrho_{\rm
c}^{-}\leq\varrho\leq\varrho_{\rm c}^{+}\\\ T(\varrho)&\varrho<\varrho_{\rm
c}^{-}\\\ \end{array}\right.$ (4)

The quantity ${\varrho^{\prime}}_{0}$ denotes the density of the material at
zero external stress. $\rho_{\rm c}^{+}$ and $\rho_{\rm c}^{-}$ are limiting
quantities, where the transition between the elastic and plastic regime for
compression and tension, respectively, takes place. Once a limit is exceeded,
the material leaves the elastic path where energy is conserved, and loses
internal energy by following the paths of the compressive and tensile
strength.

In a previous work, Sirono Sirono (2004) adopted power laws from measurements
with toner particles for $\Sigma(\varrho)$, $T(\varrho)$ and the bulk modulus
$K(\varrho)$ in order to simulate porous ice, which is a crude extrapolation.
For our approach, we used the material properties, measured for well defined
RBD dust aggregates Blum and Schräpler (2004).

Figure 1: Compressive strength curve for unidirectional (dashed line),
omnidirectional (solid line), and dynamic (dotted line) compression. The inset
illustrates the setup for the omnidirectional compression measurement.

For the tensile strength we adopted the measurement of Ref. Blum and Schräpler
(2004) who measured the tensile strength for porous ($\phi=0.15$) and compact
($\phi=0.54$) aggregates and found an agreement with a linear dependence
between tensile strength and numbers of contact per cross-sectional area. This
yields a tensile strength of

$T(\phi)=-\left(10^{2.8+1.48\phi}\right)\;{\rm Pa.}$ (5)

For the compressive strength we started with the compression curve
$\phi(\Sigma)$ measured in Ref. Blum and Schräpler (2004) (unidirectional, 1D
compression) and also made a new compression measurement (omnidirectional, 3D,
see inset in Fig. 1, Güttler et al. (2009)). Both compressive strength curves
are displayed in Fig. 1. They resemble in shape and can be described by

$\Sigma(\phi)=p_{\rm
m}\cdot\left(\frac{\phi_{2}-\phi_{1}}{\phi_{2}-\phi}-1\right)^{\Delta\cdot\ln
10}\;,$ (6)

with four free parameters $\phi_{1}$, $\phi_{2}$, $\Delta$, and $p_{\rm m}$.
For unidirectional compression we found $p_{\rm m}=5.6$ kPa, $\phi_{1}=0.15$,
$\phi_{2}=0.33$, and $\Delta=0.33$, while for ominidirectional compression
measurements performed in this work the parameters are identified as $p_{\rm
m}=13$ kPa, $\phi_{1}=0.12$, $\phi_{2}=0.58$, and $\Delta=0.58$. For
unidirectional, static compression, the material can creep sideways, releasing
pressure. As we did not observe this in the dynamic experiments (next section,
Fig. 2), we expect omnidirectional compression, defining the parameters
$\phi_{1}$ and $\phi_{2}$. For low pressures, $\Delta\cdot\ln 10$ is the slope
of a power law found by divers authors Blum and Schräpler (2004); Valverde et
al. (2004) and does not leave very much margin. Thus, we take $p_{\rm m}$ as
the only free parameter with the most influence on material softness, shifting
the compression curve towards lower pressures and determine $p_{\rm m}$ within
the calibration procedure. As the the shear strength was not measured so far,
we follow Sirono Sirono (2004) and take $Y=\sqrt{\Sigma|T|}$.

##  3 Calibration Experiment

Figure 2: The density plot reveals the compaction of the dust sample under the
glass bead (red, saturated) with a volume filling factor of 0.20 to 0.25
(green) in a well confined volume of approximately one sphere volume. The
original dust material has a volume filling factor of 0.15 (light blue).

In the experiments, a solid projectile was dropped into a 2.5 cm diameter,
highly porous dust aggregate consisting of $1.5\;\mu$m SiO2 monomers (RBD dust
aggregate from Blum and Schräpler (2004)). The experiments were performed in
vacuum ($0.1$ mbar) such that gas effects did not play a role. The projectile
was either a glass bead of 1 mm diameter or a cylindrical plastic tube with a
1 or 3 mm diameter epoxy droplet at the bottom (representing a 1 or 3 mm glass
bead). The epoxy projectile had a mass corresponding to a glass bead and
therefore was longer and could be observed for a penetration deeper than its
diameter. For 15 experiments with elongated projectiles, the deceleration
curve and, thus, the penetration depth and the stopping time were measured
with a high-speed camera. The stopping time for one projectile diameter was
found to be rather constant, ($3.0\pm 0.1$) ms for 1 mm projectiles and
($6.2\pm 0.1$) ms for 3 mm projectiles, while the penetration depth depended
on the projectile size and the impact velocity (drop height). Details on the
full deceleration curve can be found in Güttler et al. (2009). The penetration
depth could well be approximated by

$D=\left(8\cdot 10^{-4}\;\frac{\rm m^{2}\;s}{\rm
kg}\right)\cdot\frac{mv}{A}\;,$ (7)

where $v$, $m$, and $A=\pi R^{2}$ are the projectile velocity at the time of
first contact, the projectile mass, and its maximum cross-sectional area (see
also Fig. 4 in the next section). The stopping time and the relation for the
penetration depth will be used for the calibration later on.

Two experiments with a 1 mm glass bead impacting with $(0.8\pm 0.1)$ m s-1
were analyzed using an x-ray micro-tomograph (Micro-CT SkyScan 1074). In this
method, the dust sample with the embedded glass bead was positioned between an
x-ray source and a detector and rotated stepwise. Based on the 400 resulting
transmission images, a 3-dimensional density reconstruction was calculated
using the SkyScan Cone-Beam Reconstruction Software. Assuming cylindrical
symmetry in the axis of penetration, the density was averaged to one vertical
section and displayed in the contour plot in Figure 2. The compaction of the
dust is clearly visible in a confined volume under the sphere. The green color
marking this volume denotes a volume filling factor of 0.20 to 0.25, while the
material around this compaction zone is virtually unaffected with an original
volume filling factor of $\phi\approx 0.15$. The distribution of compacted
material will be used for the verification of the SPH code.

##  4 Calibration and Verification of the SPH Code

In the 2D SPH simulation, an infinite cylinder with 1.1 mm diameter
($\rho=2540$ kg m-3) impacts into a dust sample with a cross section of
$8\times 5\;{\rm mm}^{2}$ at a velocity of 0.65 m s-1. For comparison with the
3D experiments a correction factor of $\frac{8}{3\pi}$ for the penetration
depth is required (see Güttler et al. (2009)). For the calibration we will use
the penetration depth, which has to be 0.82 mm (Eq. 7 and correction factor),
the stopping time of 3 ms, and the compaction under the glass bead from the
x-ray experiments.

Figure 3: Varying the parameter $p_{\rm m}$ in the simulation changes the
softness of the material and yields different penetration depths and stopping
times. The best agreement with the experiments was found for $p_{\rm m}=1.3$
kPa.

Using the shear strength formalism introduced by Sirono Sirono (2004), e.g.
$Y=\sqrt{\Sigma|T|}$, we have one free parameter with influence on the outcome
of the simulations, i.e. the adjustment of the compressive strength curve for
dynamic compression by the parameter $p_{\rm m}$ (see Eq. 6). A detailed study
on the shear model can be found in Güttler et al. (2009).

The parameter $p_{\rm m}$ defines the softness of the material and decreasing
this parameter yields deeper penetrations. Figure 3 describes the calibration
of this parameter. The horizontal line denotes the expected penetration depth
from Eq. 7, while the vertical line represents a mean stopping time of 3 ms
for 1 mm spheres. A value of $p_{\rm m}=1.3$ kPa yields the best agreement
with the experiments and thus we will use this value for further tests.

One test for the validation of the SPH code is the penetration depth relation
for different sizes and different velocities, reproducing Eq. 7.

Figure 4: The penetration depth as a function of momentum per cross-sectional
area can well be reproduced with the SPH code.

The penetration depths of the experiments and the penetration depths of the
simulations (with correction factor) are plotted in Fig. 4. For
$mvA^{-1}\mathchar 13350\relax 1$ kg m-1 s-1 experiment and simulation are in
very good agreement and the SPH code succeeds in the scaling of radius and
velocity. A second validation test is the comparison of the compressed volume
with the results of the x-ray measurement (Fig. 2).

Figure 5: Experimental results (symbols) and simulation (solid line) of the
compressed volume.

From the 3-dimensional density data, one can determine the volume fraction
that is compressed to a given volume filling factor, which is plotted in Fig.
5. The values for $\phi\mathchar 13358\relax 0.2$ represent the original
material being distributed around the mean filling factor of 0.15. Also for
this test, we find a good agreement between experiment and simulation.

##  5 Conclusion

We have developed an SPH code for the description of collisional interaction
between high-porosity dust aggregates. The code was calibrated by static and
dynamic laboratory experiments using macroscopic RBD aggregates with an
uncompressed filling factor of $\phi=0.15$. The calibrated SPH code correctly
predicts the size and velocity dependence of the penetration depth for an
impacting solid projectile as well as the compressed volume.

We thank M.-B. Kallenrode and the University of Osnabrück for providing access
to the XRT setup. Simulations were performed on clusters of the computing
center (ZDV) of the University of Tübingen. This project is funded by the
Deutsche Forschungsgemeinschaft within the Forschergruppe 759 “The Formation
of Planets: The Critical First Growth Phase” under grants Bl 298/7-1, Bl
298/8-1, and Kl 650/8-1.

## References

  * Blum and Wurm (2008) J. Blum, and G. Wurm, _Annual Review of Astronomy and Astrophysics_ 46, 21–56 (2008).
  * Blum and Wurm (2000) J. Blum, and G. Wurm, _Icarus_ 143, 138–146 (2000).
  * Paszun and Dominik (2008) D. Paszun, and C. Dominik, _Astronomy and Astrophysics_ 484, 859–868 (2008), 0802.1832.
  * Dominik and Tielens (1997) C. Dominik, and A. G. G. M. Tielens, _The Astrophysical Journal_ 480, 647 (1997).
  * Blum and Schräpler (2004) J. Blum, and R. Schräpler, _Physical Review Letters_ 93, 115503 (2004).
  * Langkowski et al. (2008) D. Langkowski, J. Teiser, and J. Blum, _The Astrophysical Journal_ 675, 764–776 (2008).
  * Monaghan (2005) J. J. Monaghan, _Reports of Progress in Physics_ 68, 1703–1759 (2005).
  * Benz and Asphaug (1994) W. Benz, and E. Asphaug, _Icarus_ 107, 98 (1994).
  * Sirono (2004) S.-I. Sirono, _Icarus_ 167, 431–452 (2004).
  * Güttler et al. (2009) C. Güttler, M. Krause, R. Geretshauser, R. Speith, and J. Blum, The Physics of Protoplanetesimal Dust Agglomerates. IV. Towards a Dynamical Collision Model (2009), submitted.
  * Valverde et al. (2004) J. M. Valverde, M. A. S. Quintanilla, and A. Castellanos, _Physical Review Letters_ 92, 258303 (2004).


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open mouth | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on booth | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
howling | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
brick | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
khaki | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
at dock | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on head | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
around cloud | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
climbing | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
stone | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on web | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on a stick | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
leather | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
cutting | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
concrete | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 29 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
at park | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on iceberg | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on shoulder | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
at night | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
with flower | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
grazing | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
paper | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
spotted | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
at yard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
with bee | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
hanging | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
clean | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
square | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
subway | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
jumping | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on flower | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
landing | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
porcelain | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 66 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
at sunset | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on desk | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
fighting | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
purple | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 19 | 0 | 0
off | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 45 | 0
gold | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 37 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
dirty | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
resting | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
beige | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
rectangular | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
thin | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0
thick | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0
in bucket | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
city | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
in pouch | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
in hole | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 51 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on bird feeder | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on shelves | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on wall | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
on post | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
in box | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
shiny | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
sinking | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
with cargo | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
bright | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
in shell | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
aside traffic light | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
empty | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
cross tunnel | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
full | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
playing | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
light brown | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
staring | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
colorful | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
dark brown | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
young | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
dark blue | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0

Contexts
---
Concepts | grass | water | outdoor | dim | white | rock | autumn | black | brown | wood | blue | green | gray | on snow | large | eating | red | metal | lying | with people | on beach | small | in forest
t-shirt | 0 | 0 | 0 | 0 | 186 | 0 | 0 | 112 | 0 | 0 | 80 | 32 | 51 | 0 | 0 | 0 | 49 | 0 | 0 | 0 | 0 | 0 | 0
sweater | 0 | 0 | 0 | 0 | 62 | 0 | 0 | 101 | 36 | 0 | 90 | 33 | 116 | 0 | 0 | 0 | 47 | 0 | 0 | 0 | 0 | 0 | 0
surfboard | 0 | 0 | 0 | 0 | 265 | 0 | 0 | 18 | 11 | 0 | 111 | 0 | 0 | 0 | 24 | 0 | 21 | 0 | 0 | 0 | 0 | 12 | 0
tie | 0 | 0 | 0 | 0 | 44 | 0 | 0 | 113 | 19 | 0 | 104 | 21 | 36 | 0 | 0 | 0 | 75 | 0 | 0 | 0 | 0 | 0 | 0
fork | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 14 | 0 | 0 | 0 | 0 | 10 | 0 | 0 | 0 | 6 | 118 | 0 | 0 | 0 | 0 | 0
couch | 0 | 0 | 0 | 0 | 81 | 0 | 0 | 41 | 160 | 0 | 46 | 28 | 43 | 0 | 0 | 0 | 38 | 0 | 0 | 0 | 0 | 0 | 0
keyboard | 0 | 0 | 0 | 0 | 197 | 0 | 0 | 215 | 0 | 0 | 3 | 0 | 50 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
curtain | 0 | 0 | 0 | 0 | 236 | 0 | 0 | 20 | 50 | 0 | 53 | 29 | 18 | 0 | 0 | 0 | 42 | 0 | 0 | 0 | 0 | 0 | 0
Contexts
Concepts | on road | silver | running | in cage | open | yellow | standing | in city | sitting | tall | in river | in water | dark | at home | walking | glass | orange | on tree | in hand | flying | plastic | long | closed
t-shirt | 0 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 0
sweater | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
surfboard | 0 | 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 21 | 0 | 0 | 0 | 0 | 0 | 0
tie | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 | 0
fork | 0 | 274 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 48 | 0 | 0
couch | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
keyboard | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
curtain | 0 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Contexts
Concepts | on ground | aside mountain | pink | round | bare | tan | baby | in garage | yacht | on track | at station | wooden | in farm | on branch | leafy | shared | velodrome | at wharf | double decker | on | cross bridge | in sunset | at heliport
t-shirt | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
sweater | 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
surfboard | 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
tie | 0 | 0 | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
fork | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
couch | 0 | 0 | 0 | 0 | 0 | 39 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
keyboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
curtain | 0 | 0 | 0 | 0 | 0 | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Contexts
Concepts | on bridge | in burrow | aside tree | in zoo | in pot | at airport | on sea | sleeping | in desert | eating grass | in race | taking off | on power line | in circus | mountain | short | open mouth | on booth | howling | brick | khaki | at dock | on head
t-shirt | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
sweater | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
surfboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
tie | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
fork | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
couch | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
keyboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
curtain | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Contexts
Concepts | around cloud | climbing | stone | on web | on a stick | leather | cutting | concrete | at park | on iceberg | on shoulder | at night | with flower | grazing | paper | spotted | at yard | with bee | hanging | clean | square | subway | jumping
t-shirt | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
sweater | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
surfboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
tie | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
fork | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
couch | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
keyboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
curtain | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Contexts
Concepts | on flower | landing | porcelain | at sunset | on desk | fighting | purple | off | gold | dirty | resting | beige | rectangular | thin | thick | in bucket | city | in pouch | in hole | on bird feeder | on shelves | on wall | on post
t-shirt | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
sweater | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
surfboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
tie | 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
fork | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
couch | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
keyboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
curtain | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Contexts
Concepts | in box | shiny | sinking | with cargo | bright | in shell | aside traffic light | empty | cross tunnel | full | playing | light brown | staring | colorful | dark brown | young | dark blue |  |  |  |  |  |
t-shirt | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  |  |  |  |  |
sweater | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  |  |  |  |  |
surfboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  |  |  |  |  |
tie | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 24 | 0 | 0 | 0 |  |  |  |  |  |
fork | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  |  |  |  |  |
couch | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  |  |  |  |  |
keyboard | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  |  |  |  |  |
curtain | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  |  |  |  |  |

Table 6: Experiments of state-of-the-art few-shot classification methods under
5-way 1- and 5-shot setting. Three common backbones are used and the 95%
confidence intervals are displayed.

Method | Backbone | Type |  $5$-way $1$-shot |  $5$-way $5$-shot
---|---|---|---|---
Versa (Gordon et al., 2018) | Conv64F | Meta | 20.00 $\pm$ 0.00 | 51.95 $\pm$ 0.35
R2D2 (Bertinetto et al., 2019) | Conv64F | Meta | 39.56 $\pm$ 0.36 | 51.29 $\pm$ 0.36
ANIL (Raghu et al., 2020) | Conv64F | Meta | 37.24 $\pm$ 0.36 | 48.04 $\pm$ 0.39
CAN (Hou et al., 2019) | Conv64F | Metric | 43.02 $\pm$ 0.42 | 55.48 $\pm$ 0.40
ProtoNet (Snell et al., 2017) | Conv64F | Metric | 37.92 $\pm$ 0.35 | 51.39 $\pm$ 0.38
DN4 (Li et al., 2019a) | Conv64F | Metric | 38.87 $\pm$ 0.36 | 52.72 $\pm$ 0.38
Baseline (Chen et al., 2019) | Conv64F | Fine-tuning | 37.97 $\pm$ 0.32 | 52.08 $\pm$ 0.35
Baseline++ (Chen et al., 2019) | Conv64F | Fine-tuning | 40.45 $\pm$ 0.36 | 53.42 $\pm$ 0.37
[1pt/1pt] Versa (Gordon et al., 2018) | ResNet12 | Meta | 39.64 $\pm$ 0.39 | 53.06 $\pm$ 0.40
R2D2 (Bertinetto et al., 2019) | ResNet12 | Meta | 45.25 $\pm$ 0.42 | 60.14 $\pm$ 0.37
ANIL (Raghu et al., 2020) | ResNet12 | Meta | 36.58 $\pm$ 0.38 | 50.54 $\pm$ 0.40
CAN (Hou et al., 2019) | ResNet12 | Metric | 48.93 $\pm$ 0.42 | 62.36 $\pm$ 0.40
ProtoNet (Snell et al., 2017) | ResNet12 | Metric | 42.69 $\pm$ 0.39 | 59.50 $\pm$ 0.40
DN4 (Li et al., 2019a) | ResNet12 | Metric | 45.04 $\pm$ 0.39 | 57.68 $\pm$ 0.39
Baseline (Chen et al., 2019) | ResNet12 | Fine-tuning | 46.78 $\pm$ 0.41 | 60.78 $\pm$ 0.40
Baseline++ (Chen et al., 2019) | ResNet12 | Fine-tuning | 46.95 $\pm$ 0.42 | 58.50 $\pm$ 0.36
[1pt/1pt] Versa (Gordon et al., 2018) | ResNet18 | Meta | 38.88 $\pm$ 0.36 | 53.18 $\pm$ 0.36
R2D2 (Bertinetto et al., 2019) | ResNet18 | Meta | 45.26 $\pm$ 0.38 | 58.53 $\pm$ 0.38
ANIL (Raghu et al., 2020) | ResNet18 | Meta | 37.26 $\pm$ 0.36 | 49.42 $\pm$ 0.39
CAN (Hou et al., 2019) | ResNet18 | Metric | 48.18 $\pm$ 0.42 | 62.38 $\pm$ 0.41
ProtoNet (Snell et al., 2017) | ResNet18 | Metric | 43.14 $\pm$ 0.37 | 57.84 $\pm$ 0.39
DN4 (Li et al., 2019a) | ResNet18 | Metric | 43.73 $\pm$ 0.38 | 56.56 $\pm$ 0.39
Baseline (Chen et al., 2019) | ResNet18 | Fine-tuning | 46.49 $\pm$ 0.41 | 60.22 $\pm$ 0.39
Baseline++ (Chen et al., 2019) | ResNet18 | Fine-tuning | 47.26 $\pm$ 0.40 | 60.10 $\pm$ 0.38

Table 7: Experiments of state-of-the-art few-shot classification methods under
10-way 1- and 5-shot setting. Three common backbones are used and the 95%
confidence intervals are displayed.

Method | Backbone | Type |  $10$-way $1$-shot |  $10$-way $5$-shot
---|---|---|---|---
Versa (Gordon et al., 2018) | Conv64F | Meta | 10.00 $\pm$ 0.00 | 36.88 $\pm$ 0.21
R2D2 (Bertinetto et al., 2019) | Conv64F | Meta | 24.80 $\pm$ 0.20 | 35.98 $\pm$ 0.21
ANIL (Raghu et al., 2020) | Conv64F | Meta | 22.39 $\pm$ 0.19 | 33.05 $\pm$ 0.22
CAN (Hou et al., 2019) | Conv64F | Metric | 28.19 $\pm$ 0.23 | 39.06 $\pm$ 0.23
ProtoNet (Snell et al., 2017) | Conv64F | Metric | 23.72 $\pm$ 0.19 | 35.49 $\pm$ 0.22
DN4 (Li et al., 2019a) | Conv64F | Metric | 26.26 $\pm$ 0.21 | 37.54 $\pm$ 0.22
Baseline (Chen et al., 2019) | Conv64F | Fine-tuning | 23.71 $\pm$ 0.18 | 35.70 $\pm$ 0.21
Baseline++ (Chen et al., 2019) | Conv64F | Fine-tuning | 25.55 $\pm$ 0.20 | 36.34 $\pm$ 0.21
[1pt/1pt] Versa (Gordon et al., 2018) | ResNet12 | Meta | 10.00 $\pm$ 0.00 | 40.21 $\pm$ 0.22
R2D2 (Bertinetto et al., 2019) | ResNet12 | Meta | 31.16 $\pm$ 0.23 | 42.10 $\pm$ 0.22
ANIL (Raghu et al., 2020) | ResNet12 | Meta | 22.94 $\pm$ 0.20 | 33.77 $\pm$ 0.22
CAN (Hou et al., 2019) | ResNet12 | Metric | 31.92 $\pm$ 0.24 | 44.78 $\pm$ 0.23
ProtoNet (Snell et al., 2017) | ResNet12 | Metric | 29.59 $\pm$ 0.22 | 42.38 $\pm$ 0.24
DN4 (Li et al., 2019a) | ResNet12 | Metric | 30.69 $\pm$ 0.22 | 40.31 $\pm$ 0.21
Baseline (Chen et al., 2019) | ResNet12 | Fine-tuning | 31.88 $\pm$ 0.24 | 43.40 $\pm$ 0.23
Baseline++ (Chen et al., 2019) | ResNet12 | Fine-tuning | 31.01 $\pm$ 0.23 | 39.89 $\pm$ 0.21
[1pt/1pt] Versa (Gordon et al., 2018) | ResNet18 | Meta | 10.00 $\pm$ 0.00 | 36.51 $\pm$ 0.21
R2D2 (Bertinetto et al., 2019) | ResNet18 | Meta | 29.73 $\pm$ 0.22 | 39.74 $\pm$ 0.21
ANIL (Raghu et al., 2020) | ResNet18 | Meta | 10.47 $\pm$ 0.10 | 33.83 $\pm$ 0.23
CAN (Hou et al., 2019) | ResNet18 | Metric | 33.16 $\pm$ 0.25 | 43.47 $\pm$ 0.23
ProtoNet (Snell et al., 2017) | ResNet18 | Metric | 29.34 $\pm$ 0.22 | 41.34 $\pm$ 0.22
DN4 (Li et al., 2019a) | ResNet18 | Metric | 29.05 $\pm$ 0.21 | 30.12 $\pm$ 0.78
Baseline (Chen et al., 2019) | ResNet18 | Fine-tuning | 33.12 $\pm$ 0.25 | 44.63 $\pm$ 0.24
Baseline++ (Chen et al., 2019) | ResNet18 | Fine-tuning | 32.19 $\pm$ 0.23 | 43.32 $\pm$ 0.22