Aspects of AdS flux vacua with integer conformal dimensions

Fien Apers
Rudolf Peierls Centre for Theoretical Physics, Beecroft Building,
Clarendon Laboratory, Parks Road, University of Oxford, OX1 3PU, UK
<EMAIL_ADDRESS>

Abstract

The DGKT vacua are a class of AdS4 flux vacua showing full moduli
stabilization, parametric control, and a parametric separation of scales. The
particular masses of the moduli remarkably give rise to integer conformal
dimensions in the light spectrum of the would-be holographic duals. In this
note, we comment on two properties for AdS flux vacua with integer conformal
dimensions. First, there are polynomial spacetime-dependent shift symmetries
for the moduli. Secondly, the leading scalings of the central charge and the
moduli can be directly deduced from the near-horizon geometry of stacks of
orthogonally-intersecting D-brane domain walls dual to the unbounded fluxes.
We illustrate this in a couple of examples of AdS4 and AdS3 parametric flux
vacua.

###### Contents

  1. 1 Introduction
  2. 2 Polynomial Shift Symmetries
  3. 3 Large $N$ scalings from D-brane domain walls
    1. 3.1 General flux vacua
    2. 3.2 AdS4 vacua
      1. 3.2.1 Massive DGKT
      2. 3.2.2 Massless DGKT
      3. 3.2.3 Other parametric AdS4 vacua
    3. 3.3 AdS3 vacua
      1. 3.3.1 Scale-separated AdS3 in massive IIA
      2. 3.3.2 Parametric AdS3 in IIB
  4. 4 Conclusions

##  1 Introduction

Realistic string theory vacua should have full moduli stabilization and a
separation of scales between the Hubble scale $L_{H}$ and the Kaluza-Klein
length scale $L_{KK}$, corresponding to the diameter of the internal manifold,

$\displaystyle\dfrac{L_{KK}}{L_{H}}\ll 1.$ (1.1)

The DGKT vacua [1, 2] are a class of $\mathcal{N}=1$ AdS4 flux vacua with full
moduli stabilization, obtained from compactifying massive type IIA string
theory on a Calabi-Yau threefold with fluxes and intersecting O6-planes. As
there is unbounded flux $\lvert F_{4}\rvert\sim N$, parametric control

$\displaystyle g_{s}\sim N^{-3/4},\quad vol_{S}\sim N^{3/2},$ (1.2)

with $g_{s}$ the string coupling and $vol_{S}$ the internal volume in string
frame, and parametric scale separation

$\displaystyle\dfrac{L_{KK}}{L_{H}}\sim N^{-1/2}$ (1.3)

can be achieved by sending this flux to infinity.

There is however no full 10-dimensional understanding of the backreaction of
the O6-planes on the internal geometry (for recent progress on this, see e.g.
[3, 4, 5]), and scale-separated supersymmetric AdS vacua are conjectured to be
in the swampland [6]. Studying the holographic CFT duals is a promising
alternative approach for checking the consistency of these AdS vacua [7, 8,
9].

The would-be CFT duals of the DGKT vacua do have some very peculiar
properties. First, scale-separated AdS vacua in general should be dual to CFTs
with a large gap in the spectrum of single-trace primaries, and such CFTs are
unknown [10]. Also, the central charge should scale like

$\displaystyle c_{4d}\sim N^{9/2},$ (1.4)

which is a higher scaling than in any known holographic set-up[11, 12], which
goes up to $N^{3}$ [13, 14]. Moreover, the spectrum of light operators, dual
to the moduli, would consist fully of integer conformal dimensions [15] (see
also [16, 17, 18, 19]). The Kähler moduli and dilaton are dual to
$h^{1,1}_{-}$ operators with $\Delta=6$ and one operator with $\Delta=10$.
These integers are universal [20, 21]: independent of the values of the fluxes
and microscopic details of the Calabi-Yau internal manifold. As there is no
extended supersymmetry in these CFTs (3d $\mathcal{N}=1$), the presence of
integers is very surprising and requires an explanation.

In this work we argue that integer dimensions are interesting, with two
remarkable properties for AdS flux vacua with integers:

  * •

Moduli fields in AdS flux vacua, dual to light operators with integer
conformal dimensions, do have polynomial spacetime-dependent shift symmetries.

  * •

It is possible to obtain the leading scalings of the central charge, string
coupling and internal volume from the near-horizon geometry of a large number
$N$ of orthogonally-intersecting D-brane domain walls.

We heavily lean on [22] about a domain wall - flux correspondence in AdS4 flux
vacua. It is explained there how the fluxes ($F_{4}$, $H_{3}$, $F_{0}$) in
DGKT can be interchanged for domain walls (wrapped D4-, NS5- and D8-branes),
interpolating between AdS and flat spacetime, and how a AdS${}_{4}\times
T_{6}$ geometry can be recovered as the near-horizon limit of these domain
walls. We will just focus on the three stacks of $N$ orthogonally intersecting
D4-branes of this set-up, dual to the unbounded flux $F_{4}\sim N$, and
observe how the large $N$ scalings in DGKT

$\displaystyle c\sim N^{9/2},\quad g_{s}\sim N^{-3/4},\quad vol_{S}\sim
N^{3/2},$ (1.5)

can be directly deduced from the near-horizon geometry of these domain walls.
For other parametric AdS4 flux vacua with an integer $\Delta=6$ in the
spectrum, we observe how the large $N$ scalings can be derived in a similar
way.

Because the space of 2d CFTs is better understood, it might be easier to
bootstrap the holographic duals of AdS3 vacua. Scale-separated AdS3 vacua with
minimal supersymmetry were found in [23] from compactification of massive IIA
string theory on G2 holonomy spaces. The ingredients are very similar as for
the DGKT vacua, with unbounded $F_{4}$-fluxes, and bounded $H_{3}$\- and
$F_{0}$-fluxes in combination with intersecting O6-planes. Now O2-planes are
needed as well. Thanks to the unbounded $F_{4}$-flux, parametric control

$\displaystyle g_{s}\sim N^{-3/4},\quad vol_{S}\sim N^{7/4},$ (1.6)

and parametric scale separation (1.3) are achieved in the $N\rightarrow\infty$
limit. The central charge of the CFT2 duals should now be

$\displaystyle c_{3d}\sim N^{4},$ (1.7)

and the light spectrum consists of irrational dimensions [24]. In this case,
it is not possible to obtain the scalings (1.6), (1.7) from the near-horizon
geometry of a set of orthogonally intersecting D4-brane domain walls.

There are however other AdS3 vacua in IIB string theory [25] for which the
holographic brane set-up might be much simpler and more alike the 4d examples
with $\Delta=6$ we discuss. The internal manifold is a G2-structure manifold,
and $F_{7}$\- and $F_{3}$-fluxes are used [25]. Parametric scale separation
can not be achieved for these vacua, but by taking $F_{7}\sim N$ to be large,
parametric control can be obtained. Interestingly, in a two-moduli truncation,
the conformal dimensions are integer and rational ($\Delta_{1}=4$ and
$\Delta_{2}=20/7$). Moreover, the leading scalings are now obtainable from the
near-horizon limit of a stack of $N$ D1-branes, dual to the $F_{7}$-flux. So
despite not showing a separation of scales, an exploration the holographic
duals of these vacua, instead of the ones with irrational dimensions [23]
mentioned above, might bring us closer to an understanding of the DGKT duals.
The properties of the different AdS vacua we discuss are summarized in Table
1.

| IIA AdS4 [1] | IIA AdS3 [23] | IIB AdS3 [25]
---|---|---|---
parametric control | yes | yes | yes
parametric scale separation | yes | yes | no
integer dimension $\Delta=2d$ | yes | no | yes
simple brane set-up | yes | no | yes
Table 1: Properties of different parametric AdS vacua in $(d+1)$ dimensions.

The outline is as follows. In Section 2, the relation between shift symmetries
and integer dimensions is explained. Then, in Section 3, we first describe how
to derive large $N$ scalings from D-brane domain walls in general in Section
3.1. We apply this in Section 3.2 to the DGKT vacua, both in massive
($F_{0}\neq 0$) and in a double T-dual massless ($F_{0}=0$) form. We also
briefly touch upon more general parametric AdS4 vacua which have a $\Delta=6$
in the spectrum. In Section 3.3 the large $N$ scalings in IIA and IIB AdS3
vacua are discussed, and we conclude in Section 4.

##  2 Polynomial Shift Symmetries

The conformal dimensions of the light fields in the DGKT CFT duals are [10,
21]

$\displaystyle\Delta_{\phi}=(6,\dots 6,10),\quad\Delta_{a}=(5,\dots 5,11),$
(2.1)

for the $h^{1,1}_{-}+1$ saxions and axions respectively, and

$\displaystyle\Delta_{\phi}=(2,\dots 2),\quad\Delta_{a}=(3,\dots 3),$ (2.2)

for the $h^{2,1}$ complex structure moduli and their axions. All dimensions
are integer.

As conformal dimensions in AdSd+1/CFTd are obtained from the masses via a
quadratic equation

$\displaystyle\Delta(\Delta-d)=m^{2}R_{AdS}^{2},$ (2.3)

it is surprising to get integers and one expects there to be a symmetry at
work. Massless scalars, with $\Delta=3$, for example, enjoy a constant shift
symmetry. Could there be different shift symmetries for other integers?

For a free massless field in Minkowski space, there is not only the constant
shift symmetry

$\displaystyle\phi\rightarrow\phi+c,\quad c\in\mathbb{R},$ (2.4)

but also an infinite sequence of shift symmetries that are polynomial in the
spacetime coordinates $X^{\mu}$,

$\displaystyle\phi\rightarrow\phi+c+c_{\mu}X^{\mu}+c_{\mu\nu}X^{\mu}X^{\nu}+\dots,$
(2.5)

where $c_{\mu_{1}\dots\mu_{k}}$ are rank-$k$ symmetric traceless constant
tensors [26]. In AdS space, this is not true anymore, but still, each
polynomial shift symmetry of _level_ $k$ can be kept separately if the free
field $\phi$ has a particular mass depending on $k$ [27, 28]. More precisely,
in AdSd+1 a free field $\phi$ 111The interactions in DGKT are $1/N$
suppressed, and so the fields are free in the large flux $N\rightarrow\infty$
limit [15]. has a symmetry

$\displaystyle\phi\rightarrow\phi+c_{\mu_{1}\dots\mu_{k}}X^{\mu_{1}}\dots
X^{\mu_{k}}|_{AdS},$ (2.6)

where $X^{\mu}$ are coordinates on an embedding $(d+2)$-dimensional flat
spacetime, if the mass of the field equals

$\displaystyle m_{\phi}^{2}=\dfrac{k(k+d)}{R_{AdS}^{2}},$ (2.7)

with $R_{AdS}$ the radius of AdS. In a dual CFT, such masses correspond to
integer dimensions

$\displaystyle\Delta_{+}=k+d,\quad\text{or}\quad\Delta_{-}=-k.$ (2.8)

We conclude that there are indeed polynomial shift symmetries for the moduli
and axions in DGKT related to the integer dimensions:

$\displaystyle\Delta_{\phi}=\Delta_{+}=k+3,\quad k=3,7,$ (2.9)
$\displaystyle\Delta_{a}=\Delta_{+}=k+3,\quad k=2,8.$

A microscopic explanation of these shift symmetries, possibly related to the
domain walls discussed in the next section or the discrete symmetries of [29]
in the large $N$ limit, would be very interesting.

To check what these symmetries look like on the boundary of AdS, we consider
Poincaré coordinates

$\displaystyle
ds_{AdS}^{2}=\dfrac{R_{AdS}^{2}}{z^{2}}\left(-dx_{0}^{2}+\sum_{i=1}^{d-1}dx_{i}^{2}+dz^{2}\right),$
(2.10)

the boundary being at $z=\epsilon$, $\epsilon\rightarrow 0$. Then the
polynomial shifts on the boundary reduce to

$\displaystyle c_{\mu_{1}\dots\mu_{k}}X^{\mu_{1}}\dots
X^{\mu_{k}}|_{AdS}\xrightarrow{z\rightarrow
0}\left(\dfrac{R_{AdS}}{z}\right)^{k}c_{i_{1}\dots i_{k}}X^{i_{1}}\dots
X^{i_{k}}|_{boundary},$ (2.11)

with $X_{i}$ coordinates of the flat embedding restricted to the boundary.
Considering the near-boundary behaviour of the scalar field

$\displaystyle\phi\sim\phi_{0}z^{\Delta_{-}}+\phi_{1}z^{\Delta_{+}}=\phi_{0}z^{-k}+\phi_{1}z^{k+d},$
(2.12)

we see that the shift acts only on $\phi_{0}$ as

$\displaystyle\phi_{0}\rightarrow\phi_{0}+R_{AdS}^{k}c_{i_{1}\dots
i_{k}}X^{i_{1}}\dots X^{i_{k}}|_{boundary}.$ (2.13)

In standard quantization $\phi_{0}$ is fixed, and this means that the shift
symmetry is broken on the boundary [30]. This should be investigated more
precisely when more is known about an explicit DGKT CFT dual.

##  3 Large $N$ scalings from D-brane domain walls

Parametric control, and possibly scale separation, can be obtained in AdS flux
vacua in the $N\rightarrow\infty$ limit, where $N$ is proportional to some
unbounded flux. This flux, and also the bounded fluxes needed for moduli
stabilization, may be interchanged for brane domain walls interpolating
between the AdS vacuum and flat spacetime, as in [22]. Interestingly, an
AdS$\times(\text{torus})$ geometry may be found in the near-horizon region of
this entire brane-system and the dilaton approaches a finite constant when
approaching the horizon [22].

In this section, we focus on the branes dual to the unbounded fluxes and
explain how one can derive the scalings of the central charge, string
coupling, and internal volume for most AdS flux vacua by just looking at the
near-horizon geometry of these branes. We first do this for a single stack of
domain walls in general flux vacua, and afterwards apply it to some concrete
examples including the DGKT and scale-separated AdS3 vacua.

###  3.1 General flux vacua

If there is one unbounded flux, $F_{8-p}\sim N$, we can change this for $N$
D$p$-branes, wrapped on $(p+2-D)$-cycles, in a 10d flat spacetime consisting
of $D$ non-compact directions, the other directions forming a toroidal
geometry. These branes can be considered as domain walls in the
$D$-dimensional spacetime, with coordinates ($t,x_{1},x_{2},\dots x_{D-2},x$),
and where $x$ will be the direction transverse to the branes. We denote the
wrapped internal coordinates by ($y_{1},y_{2},\dots y_{p+2-D}$), and the
unwrapped ones by ($z_{1},z_{2},\dots z_{8-p}$), as shown in Table 2.

| $t$ | $x^{1}$ | $\dots$ | $x^{D-2}$ | $x$ | $y_{1}$ | $\dots$ | $y_{p+2-D}$ | $z_{1}$ | $\dots$ | $z_{8-p}$
---|---|---|---|---|---|---|---|---|---|---|---
N D$p$ | $\otimes$ | $\otimes$ | $\otimes$ | $\otimes$ |  | $\otimes$ | $\otimes$ | $\otimes$ |  |  |
Table 2: $N$ D$p$-brane domain walls in $D$ dimensions

This system is described by [14]

$\displaystyle
ds_{10}^{2}=f_{p}^{-\frac{1}{2}}\left[-dt^{2}+dx_{1}^{2}+\dots+dx_{D-2}^{2}\right]+f_{p}^{\frac{1}{2}}dx^{2}+f_{p}^{-\frac{1}{2}}dy_{i}dy^{i}+f_{p}^{\frac{1}{2}}dz_{j}dz^{j},$
(3.1)

and the string coupling is

$\displaystyle g_{s}=f_{p}^{\frac{3-p}{4}},$ (3.2)

with

$\displaystyle f_{p}\sim 1+\dfrac{N}{r^{7-p}},\quad
r^{2}=\sum_{i}y_{i}^{2}+\sum_{j}z_{j}^{2},$ (3.3)

a harmonic function in the transverse coordinates that scales like $f_{p}\sim
N$ in the near-horizon region.

Letting $N\rightarrow\infty$, the near-horizon geometry will be of the
following schematic form,

$\displaystyle
ds_{NH}^{2}=\alpha^{\prime}\left[N^{\frac{1}{2}}ds_{X_{D}}^{2}+N^{-\frac{1}{2}}dy_{i}dy^{i}+N^{\frac{1}{2}}dz_{j}dz^{j}\right],$
(3.4)

where $X_{D}$ is the $D$-dimensional spacetime and

$\displaystyle\boxed{g_{s}\sim N^{\frac{3-p}{4}}}.$ (3.5)

We see that there is a separation of scales between the wrapped internal
dimensions and the scale of $X_{D}$, and we can read off the internal volume
in string units:

$\displaystyle\boxed{vol_{S}\sim(N^{-\frac{1}{2}})^{\frac{p+2-D}{2}}\cdot(N^{\frac{1}{2}})^{\frac{8-p}{2}}\cdot
l_{s}^{10-D}=N^{\frac{D+6-2p}{4}}\cdot l_{s}^{10-D}}.$ (3.6)

From this, we deduce the D-dimensional Planck length

$\displaystyle
l_{p,D}=(g_{s}^{2}vol_{S}^{-1})^{1/(D-2)}l_{s}=N^{-\frac{D}{4\,\left(D-2\right)}}l_{s}.$
(3.7)

Hence, the metric in Planck units is

$\displaystyle
ds_{NH}^{2}=N^{\frac{1}{D-2}}\left[Nds_{X_{D}}^{2}+dy_{i}dy^{i}+Ndz_{j}dz^{j}\right]l_{p,D}^{2}.$
(3.8)

Then, the AdS radius is

$\displaystyle R_{AdS}\sim N^{\frac{D-1}{2\,\left(D-2\right)}}l_{p,D},$ (3.9)

and so the central charge will be

$\displaystyle\boxed{c\sim(R_{AdS}/l_{p,D})^{D-2}\sim N^{\frac{D-1}{2}}}.$
(3.10)

Repeating the same calculation for domain walls from NS5-branes, we find

$\displaystyle g_{s}\sim N^{\frac{1}{2}},\quad vol_{S}\sim
N^{\frac{3}{2}},\quad c\sim N^{\frac{D-1}{2}},$ (3.11)

or for F1-strings (only for $D=3$),

$\displaystyle g_{s}\sim N^{-\frac{1}{2}},\quad vol_{S}\sim
N^{\frac{D-3}{2}},\quad c\sim N^{\frac{D-1}{2}}.$ (3.12)

In the examples that we discuss below, there will often be multiple stacks
like in Table 2 of D-branes dual to unbounded flux. If these stacks are
orthogonally-intersecting, they must satisfy certain composition rules,
summarised in [31]. For the D$p$/D$q$-brane intersections that we consider,
each pair of these branes must have ($0$ mod $4$) relative transverse
coordinates. These are coordinates that are orthogonal to one of the two
branes, but not both. Then, we can assign a function $f_{p_{i}}$ to each stack
of D${}_{p_{i}}$-branes, that only depends on the overall transverse
directions. For each of the coordinate directions $\zeta=t,x,y$ or $z$, we
then multiply the appropriate powers of these functions as follows,

$\displaystyle\prod_{i}f_{p_{i}}^{1/2}\cdot\prod_{j}f_{q_{j}}^{-1/2}\cdot(d\zeta)^{2},$
(3.13)

where the $p_{i}$-branes are orthogonal, and the $q_{j}$-branes are parallel
to the $\zeta$-direction. With these harmonic superposition rules [32], we
obtain the supergravity description of the brane systems. The string coupling
will be given by

$\displaystyle g_{s}=\prod_{i}f_{p_{i}}^{\frac{3-p_{i}}{4}},$ (3.14)

where the multiplication is over all branes involved, and we will deduce the
central charge and internal volumes from the near-horizon geometries of
(3.13).

###  3.2 AdS4 vacua

####  3.2.1 Massive DGKT

The DGKT [1] potential is given by

$V=\dfrac{1}{s^{3}}\left[\dfrac{A_{F_{4}}}{us}+\dfrac{A_{F_{0}}u^{3}}{s}+\dfrac{A_{H_{3}}s}{u^{3}}-A_{O6}\right],$
(3.15)

where $u^{3}=vol_{S}$ and $s=e^{-D}=e^{-\phi}\sqrt{vol_{S}}$, and the $A$’s
are coefficients depending on the values of the fluxes or the orientifold
charge, with $A_{O6}^{2}=16A_{F_{0}}A_{H_{3}}$. The Kähler potential equals

$\displaystyle K=-3\log u-4\log s.$ (3.16)

The conformal dimensions of the dual fields are

$\displaystyle\Delta_{1}=6,\quad\Delta_{2}=10.$ (3.17)

Imposing the scaling $A_{F_{4}}\sim N^{2}$, we find that, at the minimum of
the potential

$\displaystyle V\sim N^{-\frac{9}{2}},\quad s\sim N^{\frac{3}{2}},\quad u\sim
N^{\frac{1}{2}},$ (3.18)

which means that

$\displaystyle c\sim N^{9/2},\quad g_{s}\sim N^{-3/4},\quad vol_{S}\sim
N^{3/2}.$ (3.19)

If the internal manifold is a toroidal orientifold $T^{6}/\mathbb{Z}_{3}^{2}$,
$F_{4,i}\sim N_{i}$ ($i=1,2,3$) flux is turned on along each of the 3
sub-2-tori. We can interchange these 3 fluxes for 3 stacks domain walls
consisting of respectively $N_{i}$ D4-branes wrapped on the different
2-cycles, as shown in Table 3.

| $t$ | $x^{1}$ | $x^{2}$ | $x$ | $y_{1}$ | $y_{2}$ | $y_{3}$ | $y_{4}$ | $y_{5}$ | $y_{6}$
---|---|---|---|---|---|---|---|---|---|---
$N_{1}$ D4 | $\otimes$ | $\otimes$ | $\otimes$ |  | $\otimes$ | $\otimes$ |  |  |  |
$N_{2}$ D4 | $\otimes$ | $\otimes$ | $\otimes$ |  |  |  | $\otimes$ | $\otimes$ |  |
$N_{3}$ D4 | $\otimes$ | $\otimes$ | $\otimes$ |  |  |  |  |  | $\otimes$ | $\otimes$
Table 3: D4-brane domain walls in DGKT

As explained in [22], we can also change the other bounded $H_{3}$\- and
$F_{0}$-fluxes for NS5-branes wrapped on 3-cycles and D8-branes on 6-cycles
respectively. The resulting near-horizon geometry of this brane set-up will be
AdS${}_{4}\times T_{6}$. In the following, we will not explicitly take the
presence of these sub-leading branes, dual to bounded fluxes, into account,
but just assume they are there to make the near-horizon geometry of the right
form and look only at the large $N$ dependence of this geometry.

As $N_{i}\sim N\rightarrow\infty$, using the harmonic superposition rules, we
find

$\displaystyle ds_{NH}^{2}$
$\displaystyle=\alpha^{\prime}\big{[}N_{1}^{\frac{1}{2}}N_{2}^{\frac{1}{2}}N_{3}^{\frac{1}{2}}ds_{X_{4}}^{2}+N_{1}^{-\frac{1}{2}}N_{2}^{\frac{1}{2}}N_{3}^{\frac{1}{2}}(dy_{1}^{2}+dy_{2}^{2})$
(3.20)
$\displaystyle+N_{1}^{\frac{1}{2}}N_{2}^{-\frac{1}{2}}N_{3}^{\frac{1}{2}}(dy_{3}^{2}+dy_{4}^{2})+N_{1}^{\frac{1}{2}}N_{2}^{\frac{1}{2}}N_{3}^{-\frac{1}{2}}(dy_{5}^{2}+dy_{6}^{2})\big{]},$

with

$\displaystyle g_{s}\sim
N_{1}^{-\frac{1}{4}}N_{2}^{-\frac{1}{4}}N_{3}^{-\frac{1}{4}}\sim
N^{-\frac{3}{4}}.$ (3.21)

The internal volume can be read off,

$\displaystyle vol_{S}\sim
N_{1}^{\frac{1}{2}}N_{2}^{\frac{1}{2}}N_{3}^{\frac{1}{2}}\sim
N^{\frac{3}{2}},$ (3.22)

along with the degree of scale separation,

$\displaystyle\dfrac{L_{KK}^{2}}{L_{H}^{2}}\sim\text{max}_{i}(N_{i}^{-1})\sim
N^{-1}.$ (3.23)

The 4d Planck scale is given by

$\displaystyle l_{p,4}=\dfrac{g_{s}}{\sqrt{vol_{S}}}l_{s}\sim
N_{1}^{-\frac{1}{2}}N_{2}^{-\frac{1}{2}}N_{3}^{-\frac{1}{2}}l_{s},$ (3.24)

and so it follows that the central charge is given by

$\displaystyle c\sim\dfrac{R_{X_{4}}^{2}}{l_{p,4}^{2}}\sim
N_{1}^{\frac{3}{2}}N_{2}^{\frac{3}{2}}N_{3}^{\frac{3}{2}}\sim
N^{\frac{9}{2}}.$ (3.25)

We obtain the same scalings as the ones (3.19) from the scalar potential.

####  3.2.2 Massless DGKT

We can proceed similarly for DGKT vacua without Romans mass. These are
obtained by performing two T-dualities on massive DGKT and then re-scaling
fluxes to get a solution under control [3, 31]. The internal manifold will now
have curvature becoming a more general SU(3) structure manifold. The flux
distribution is anisotropic, with only unbounded flux on 2 out of 3 2-tori in
the toroidal case. That is why we now write explicitly
$vol_{s}=u_{1}u_{2}u_{3}$. There is unbounded $F_{6}$-flux, both bounded
$F_{2}$-flux ($F_{2,1}$) and unbounded $F_{2}$-flux ($F_{2,2},F_{2,3}$), a
curvature $R$ contribution and again there are O6-planes. The scalar potential
is given by

$\displaystyle
V=\dfrac{1}{s^{3}}\left[A_{F_{6}}\dfrac{1}{su_{1}u_{2}u_{3}}+A_{F_{2,1}}\dfrac{u_{2}u_{3}}{su_{1}}+A_{F_{2,2}}\dfrac{u_{1}u_{3}}{su_{2}}+A_{F_{2,3}}\dfrac{u_{1}u_{2}}{su_{3}}+A_{R}\dfrac{su_{1}}{u_{2}u_{3}}-A_{O6}\right],$
(3.26)

with Kähler potential

$\displaystyle K=-\log u_{1}u_{2}u_{3}-4\log s.$ (3.27)

Conformal dimensions do not change under T-duality:

$\displaystyle\Delta_{1,2,3}=6,\quad\Delta_{4}=10.$ (3.28)

Letting $F_{6}\sim N$ , $F_{2,2}\sim M_{1}$ and $F_{2,3}\sim M_{2}$, it
follows that

$\displaystyle c\sim
N^{\frac{3}{2}}M_{1}^{\frac{3}{2}}M_{2}^{\frac{3}{2}},\quad g_{s}\sim
N^{\frac{1}{4}}M_{1}^{-\frac{3}{4}}M_{2}^{-\frac{3}{4}},\quad vol_{S}\sim
N^{\frac{3}{2}}M_{1}^{-\frac{1}{2}}M_{2}^{-\frac{1}{2}}.$ (3.29)

For $N$ large enough compared to $M_{1}$ and $M_{2}$, the coupling will be
strong and this solution can be uplifted to M-theory. The first dimension-6
operator can then be interpreted as being dual to the (7d) internal volume.

Now we change the $F_{6}$-flux for D2-branes, and the unbounded $F_{2}$-fluxes
for D6-branes wrapped on 4-cycles, as in Table 4.

| $x^{0}$ | $x^{1}$ | $x^{2}$ | $x$ | $y_{1}$ | $y_{2}$ | $y_{3}$ | $y_{4}$ | $y_{5}$ | $y_{6}$
---|---|---|---|---|---|---|---|---|---|---
$N$ D2 | $\otimes$ | $\otimes$ | $\otimes$ |  |  |  |  |  |  |
$M_{1}$ D6 | $\otimes$ | $\otimes$ | $\otimes$ |  | $\otimes$ | $\otimes$ | $\otimes$ | $\otimes$ |  |
$M_{2}$ D6 | $\otimes$ | $\otimes$ | $\otimes$ |  | $\otimes$ | $\otimes$ |  |  | $\otimes$ | $\otimes$
Table 4: D2- and D6-brane domain walls in massless DGKT

The following near-horizon geometry can be read off:

$\displaystyle ds_{NH}^{2}$
$\displaystyle=\alpha^{\prime}\big{[}N^{\frac{1}{2}}M_{1}^{\frac{1}{2}}M_{2}^{\frac{1}{2}}ds_{X_{4}}^{2}+N^{\frac{1}{2}}M_{1}^{-\frac{1}{2}}M_{2}^{-\frac{1}{2}}(dy_{1}^{2}+dy_{2}^{2})$
(3.30)
$\displaystyle+N^{\frac{1}{2}}M_{1}^{-\frac{1}{2}}M_{2}^{\frac{1}{2}}(dy_{3}^{2}+dy_{4}^{2})+N^{\frac{1}{2}}M_{1}^{\frac{1}{2}}M_{2}^{-\frac{1}{2}}(dy_{5}^{2}+dy_{6}^{2})\big{]},$

with

$\displaystyle g_{s}\sim
N^{\frac{1}{4}}M_{1}^{-\frac{3}{4}}M_{2}^{-\frac{3}{4}},\quad\dfrac{L_{KK}^{2}}{L_{H}^{2}}\sim\text{max}_{i}(M_{i}^{-1}),$
(3.31)

fully in accordance with the scalings from the scalar potential (3.29).

####  3.2.3 Other parametric AdS4 vacua

These large $N$ scalings from domain walls do not only work out for scale-
separated vacua as in the examples above, but also for more general 4d
examples with parametric control without scale separation. A simple example is
obtained by imposing isotropy $u_{1}=u_{2}=u_{3}$ in the potential of the last
section, so that

$\displaystyle
V=\dfrac{1}{s^{3}}\left[A_{F_{6}}\dfrac{1}{su^{3}}+A_{F_{2}}\dfrac{u}{s}+A_{R}\dfrac{s}{u}-A_{O6}\right],$
(3.32)

and then the only unbounded flux will be $F_{6}\sim N$. The conformal
dimensions are now

$\displaystyle\Delta_{1}=6,\quad\Delta_{2}=11/3.$ (3.33)

The scalings are

$\displaystyle c\sim N^{\frac{3}{2}},\quad g_{s}\sim N^{\frac{1}{4}},\quad
vol_{S}\sim N^{\frac{3}{2}},$ (3.34)

which coincide with the scalings (3.5), (3.6), (3.10) for one stack of $N$
D2-branes in 4d. There is clearly no scale separation, as can be seen from the
near-horizon geometry as well,

$\displaystyle
ds_{NH}^{2}=\alpha^{\prime}[N^{\frac{1}{2}}ds_{X_{4}}^{2}+N^{\frac{1}{2}}(dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2}+dy_{4}^{2}+dy_{5}^{2}+dy_{6}^{2})].$
(3.35)

Reference [33] classifies 2-moduli scalar potentials for asymptotic 4d
$\mathcal{N}=1$ flux vacua. For the AdS vacua they obtain near infinite
distance singularities, we observe that

$\displaystyle\Delta_{1}=6,\quad\Delta_{2}\ \text{is rational}.$ (3.36)

Moreover, the central charge is always given by

$\displaystyle c\sim N^{\frac{3k}{2}},$ (3.37)

with $N$ an unbounded flux and $k$ some positive integer. With (3.10), this
suggests that these vacua can be obtained in the near-horizon limit of $k$
stacks of D-brane domain walls. The string coupling and internal volume scale
accordingly. We will report on this in more detail in the future.

###  3.3 AdS3 vacua

####  3.3.1 Scale-separated AdS3 in massive IIA

The vacua of [23] are scale-separated AdS3 vacua, from compactification of
massive IIA string theory on a G2 holonomy manifold. There are $F_{4}$-,
$H_{3}$\- and $F_{0}$\- fluxes, together with O6-planes, and the scalar
potential is:

$V=\dfrac{A_{F_{4}}}{u^{\frac{1}{2}}s^{3}}+\dfrac{A_{F_{0}}u^{\frac{7}{2}}}{s^{3}}+\dfrac{A_{H_{3}}}{u^{3}s^{2}}-\dfrac{A_{O6}u^{\frac{1}{4}}}{s^{\frac{5}{2}}}.$
(3.38)

with $A_{O6}^{2}=12\cdot A_{f_{0}}\cdot A_{h_{0}}$, and where
$u^{7/2}=vol_{S}$ and $s=e^{-2\phi}vol_{S}$ 222To make contact with the
notation in [23]: $u=e^{\frac{\phi}{2}+2\beta v}$,
$s=e^{-\frac{\phi}{4}+7\beta v}$.. The conformal dimensions are irrational

$\displaystyle\Delta=1+\sqrt{\dfrac{191\pm 8\sqrt{277}}{7}}.$ (3.39)

With $F_{4}\sim N$, we find from (3.38) that

$\displaystyle c\sim N^{4},\quad g_{s}\sim N^{-3/4},\quad vol_{S}\sim
N^{7/4}.$ (3.40)

There are such $F_{4}$-fluxes on seven different 4-cycles, and interchanging
them for D4-branes wrapped on the dual 3-cycles, we find the system of Table
5.

| $t$ | $x^{1}$ | $x$ | $y_{1}$ | $y_{2}$ | $y_{3}$ | $y_{4}$ | $y_{5}$ | $y_{6}$ | $y_{7}$
---|---|---|---|---|---|---|---|---|---|---
$N_{1}$ D4 | $\otimes$ | $\otimes$ |  | $\otimes$ | $\otimes$ |  |  |  |  | $\otimes$
$N_{2}$ D4 | $\otimes$ | $\otimes$ |  |  |  | $\otimes$ | $\otimes$ |  |  | $\otimes$
$N_{3}$ D4 | $\otimes$ | $\otimes$ |  |  |  |  |  | $\otimes$ | $\otimes$ | $\otimes$
$N_{4}$ D4 | $\otimes$ | $\otimes$ |  | $\otimes$ |  | $\otimes$ |  |  | $\otimes$ |
$N_{5}$ D4 | $\otimes$ | $\otimes$ |  |  | $\otimes$ | $\otimes$ |  | $\otimes$ |  |
$N_{6}$ D4 | $\otimes$ | $\otimes$ |  | $\otimes$ |  |  | $\otimes$ | $\otimes$ |  |
$N_{7}$ D4 | $\otimes$ | $\otimes$ |  |  | $\otimes$ |  | $\otimes$ |  | $\otimes$ |
Table 5: D4-brane domain walls for scale-separated AdS3

There should be further domain walls from NS5-branes and D8-branes, dual to
the bounded $H_{3}$ \- and $F_{0}$-fluxes. If there is indeed an
AdS${}_{3}\times T_{7}$ geometry in the near-horizon limit of this set of
domain walls, the schematic flux dependence should be like

$\displaystyle ds_{NH}^{2}$
$\displaystyle=\alpha^{\prime}\big{[}(N_{1}\cdot\dots\cdot
N_{7})^{\frac{1}{2}}ds_{X_{3}}^{2}+(N_{1}N_{4}N_{6})^{-\frac{1}{2}}(N_{2}N_{3}N_{5}N_{7})^{\frac{1}{2}}dy_{1}^{2}+\dots\big{]}$
(3.41)
$\displaystyle=\alpha^{\prime}\big{[}N^{\frac{7}{2}}ds_{X_{3}}^{2}+N^{\frac{1}{2}}(dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2}+dy_{4}^{2}+dy_{5}^{2}+dy_{6}^{2}+dy_{7}^{2})\big{]},$

with $N_{i}=N,i=1,\dots 7$. The string coupling (3.5) would be

$\displaystyle g_{s}=(N_{1}\cdot\dotsc\cdot
N_{7})^{-\frac{1}{4}}=N^{-\frac{7}{4}},$ (3.42)

and the internal volume is

$\displaystyle vol_{S}=(N_{1}\cdot\dotsc\cdot
N_{7})^{\frac{1}{4}}=N^{\frac{7}{4}}.$ (3.43)

Finally, the central charge would be

$\displaystyle c=N_{1}\cdot\dotsc\cdot N_{7}=N^{7}.$ (3.44)

Remarkably, the scalings of the central charge and the string coupling,
derived from the brane system do not match with (3.40). Given that each of the
seven internal lengths should scale in the same way with $N$ [23], and taking
into account that each pair of stacks of D4-branes should have $0$ or $4$
relative transverse directions [31], the set-up of Table 5 is the only
possible set-up consisting of orthogonally-intersecting D4-branes.

####  3.3.2 Parametric AdS3 in IIB

To stress that this failure to obtain the scalings from the near-horizon
geometry of a simple system of orthogonally-intersecting branes is not generic
to asymptotic vacua in 3d, we mention another example: AdS3 vacua from
compactification of IIB string theory on G2-structure manifolds (so with
curvature $R$) [25] with $F_{7}$-, $F_{3}$-fluxes and O5-planes and possibly
D5-branes. These do not allow scale separation, but the moduli can take
parametric values. The scalar potential equals

$V=\dfrac{A_{F_{7}}}{u^{\frac{7}{2}}s^{3}}+\dfrac{A_{F_{3}}u^{\frac{1}{2}}}{s^{3}}+\dfrac{A_{R}}{us^{2}}-\dfrac{A_{O5/D5}}{u^{\frac{1}{4}}s^{\frac{5}{2}}},$
(3.45)

with $A_{O5/D5}^{2}=16A_{F3}\cdot A_{R}/3$. The conformal dimensions are now
integer and rational

$\displaystyle\Delta_{1}=4,\quad\Delta_{2}=\dfrac{20}{7}.$ (3.46)

Choosing $F_{7}\sim N$, the scalings

$\displaystyle c\sim N,\quad g_{s}\sim N^{1/2},\quad vol_{S}\sim N^{7/4}.$
(3.47)

from the scalar potential (3.45). These coincide with the scalings (3.5),
(3.6), (3.10) for $N$ D1-domain walls in 3d, see Table 6.

| $t$ | $x^{1}$ | $x$ | $y_{1}$ | $y_{2}$ | $y_{3}$ | $y_{4}$ | $y_{5}$ | $y_{6}$ | $y_{7}$
---|---|---|---|---|---|---|---|---|---|---
$N$ D1 | $\otimes$ | $\otimes$ |  |  |  |  |  |  |  |
Table 6: D1-brane domain walls for parametric AdS3

##  4 Conclusions

The holographic duals of the DGKT vacua are remarkable, in the sense that
their spectrum consists of integer dimensions and that the scaling of the
central charge $c\sim N^{9/2}$ is unseen in any well-understood holographic
set-up. We have observed, however, that the DGKT scalings are consistent with
the large $N$ near-horizon limit of three intersecting stacks of $N$ D4-branes
wrapped on 2-cycles. Similarly, the scalings in massless DGKT, obtained after
twice T-dualizing the original DGKT vacua [3], are consistent with the near-
horizon geometry of one stack of D2-branes and two stacks of D6-branes wrapped
on 4-cycles.

More generally, AdS vacua in $(d+1)$ dimensions with minimal supersymmetry and
an unbounded flux proportional to $N$, and where the conformal dimensions take
specific values

$\displaystyle\Delta_{1}=2d,\quad\Delta_{2}\quad\text{is \ rational}$ (4.1)

seem to agree with a near-horizon geometry of $\mathcal{O}(N)$ D-brane domain
walls as $N\rightarrow\infty$, whereas this does not seem the case for vacua
with irrational dimensions. The integer dimension $\Delta_{1}=2d$ signals the
presence of a level $d$ polynomial shift symmetry for the corresponding field.

The scale-separated AdS3 vacua are unlike their 4-dimensional DGKT
counterparts, despite being built from similar ingredients, with irrational
dimensions and scalings that cannot be obtained from orthogonal D4-domain
walls. Moreover, a massless version ($F_{0}\neq 0$) does not exist here [34],
preventing an M-theory uplift [35]. Future research will clarify the origin of
the failed scalings for the AdS3 vacua and whether they imply any
inconsistency.

### Acknowledgements

I am grateful to Thomas Van Riet, Vincent Van Hemelryck, Miguel Montero, Irene
Valenzuela, George Tringas, and especially Joe Conlon for useful discussions
and comments on the draft. I am supported by the Clarendon Scholarship in
partnership with the Scatcherd European Scholarship, Saven European
Scholarship, and the Hertford College Peter Howard Scholarship.

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# MHFC: Multi-Head Feature Collaboration for Few-Shot Learning

Shuai Shao†,1, Lei Xing†,2, Yan Wang3, Rui Xu1, Chunyan Zhao4, Yanjiang
Wang∗,1, Baodi Liu∗,1 1College of Control Science and Engineering, China
University of Petroleum (East China) 2College of Oceanography and Space
Informatics, China University of Petroleum (East China) 3Beihang University
4Suzhou Centennial College<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(2021)

###### Abstract.

Few-shot learning (FSL) aims to address the data-scarce problem. A standard
FSL framework is composed of two components: (1) Pre-train. Employ the base
data to generate a CNN-based feature extraction model (FEM). (2) Meta-test.
Apply the trained FEM to acquire the novel data’s features and recognize them.
FSL relies heavily on the design of the FEM. However, various FEMs have
distinct emphases. For example, several may focus more attention on the
contour information, whereas others may lay particular emphasis on the texture
information. The single-head feature is only a one-sided representation of the
sample. Besides the negative influence of cross-domain (e.g., the trained FEM
can not adapt to the novel class flawlessly), the distribution of novel data
may have a certain degree of deviation compared with the ground truth
distribution, which is dubbed as distribution-shift-problem (DSP). To address
the DSP, we propose Multi-Head Feature Collaboration (MHFC) algorithm, which
attempts to project the multi-head features (e.g., multiple features extracted
from a variety of FEMs) to a unified space and fuse them to capture more
discriminative information. Typically, first, we introduce a subspace learning
method to transform the multi-head features to aligned low-dimensional
representations. It corrects the DSP via learning the feature with more
powerful discrimination and overcomes the problem of inconsistent measurement
scales from different head features. Then, we design an attention block to
update combination weights for each head feature automatically. It
comprehensively considers the contribution of various perspectives and further
improves the discrimination of features. We evaluate the proposed method on
five benchmark datasets (including cross-domain experiments) and achieve
significant improvements of 2.1%-7.8% compared with state-of-the-arts.

Multi-head feature collaboration (MHFC); few-shot learning (FSL);
distribution-shift-problem (DSP); subspace learning

† indicates equal contribution. ∗ indicates corresponding author.

††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the
29th ACM International Conference on Multimedia; October 20–24, 2021; Virtual
Event, China††booktitle: Proceedings of the 29th ACM International Conference
on Multimedia (MM ’21), October 20–24, 2021, Virtual Event, China††price:
15.00††doi: 10.1145/3474085.3475553††isbn: 978-1-4503-8651-7/21/10††ccs:
Computing methodologies Image representations††ccs: Computing methodologies
Object identification

(a) SS-R-Fea

(b) SS-M-Fea

(c) Multi-Head-Fea

Figure 1. t-SNE visualization of features on mini-ImageNet. ”SS-R-Fea” and
”SS-M-Fea” indicate two categories of different single-head features (for the
details, please refer to Section 4.4). ”Multi-Head-Fea” denotes the fusion
feature after the proposed MHFC. Multi-Head-Fea based distribution is more
helpful to classification tasks.

##  1\. Introduction

Machine learning has help machines achieve outstanding performances in
computer vision tasks, such as person re-identification (Wang et al., 2020a;
Wang and Zhang, 2020; Fan et al., 2020), image classification (Yan et al.,
2016; Lin et al., 2020; Shao et al., 2020). One indispensable factor is
attributed to the large-scale labeled data. However, as the limitation of
actual circumstances, it may be infeasible to collect large amounts of labeled
data in the real world. Thus, few-shot learning (FSL) has attracted growing
attention recent year. It targets to help machines achieve or even surpass
human beings’ level with scarce labeled samples. Generally, in FSL-based
classification tasks, the current popular model usually includes two
components: (1) Pre-train. Employ the base data $\mathcal{D}_{base}$ to
generate a convolutional neural network (CNN) based feature extraction model
(FEM). (2) Meta-test. First, extract the features of novel data
$\mathcal{D}_{novel}=\\{\mathcal{S},\mathcal{U},\mathcal{Q}\\}$, where
$\mathcal{S}$, $\mathcal{U}$ and $\mathcal{Q}$ denote support set, unlabeled
set and query set. Next, design a classifier to recognize the query samples
$\mathcal{Q}$. Notably, $\mathcal{D}_{novel}$ has totally different categories
from $\mathcal{D}_{base}$. For more details, please refer to Section 3.

Figure 2.  The framework of MHFC on inductive few-shot learning (IFSL). Assume
we have two heads of feature extraction models (FEMs), e.g.,
$\mathcal{M}_{\theta}^{h}$, where $h=[1,2]$ denotes the $h_{th}$ head.
Different colors correspond to various features. There are a total of $5$
steps. (1) Input images to FEMs and obtain the support features
$\mathbf{X}_{s}^{h}$ and query feature $\mathbf{x}_{q}^{h}$. (2) Transform
multi-head features to a unified space to obtain the novel features
$\mathbf{P}_{s}^{h}$, $\mathbf{p}_{q}^{h}$. (3) Employ the transformed multi-
head support features (e.g., $\mathbf{P}_{s}^{h}$) to learn the combination
weights $\mathbf{\Omega}$ for each head, and assign them to the corresponding
heads of features. (4) Collaboratively represent the samples by concatenating
the weighted multi-head features. (5) Exploit the collaborative support
features to construct classifier and recognize query sample.

In the pre-train phase, researchers have designed a variety of classical FEMs
for base samples. Each kind of model has a distinct emphasis. As examples,
several may focus more attention on the contour information, whereas others
may lay particular emphasis on the texture information. Thus, the description
based on one category of features is often one-sided for samples, not
sufficiently accurate. Besides, due to the negative influence of cross-domain
(e.g., $\mathcal{D}_{base}\rightarrow\mathcal{D}_{novel}$, the trained FEM can
not adapt to the novel class flawlessly), the novel data distribution may have
a certain degree of deviation compared with the ground truth distribution. We
dub this fundamental problem as a distribution-shift-problem (DSP).

Recent efforts on reducing the impact of DSP are generally based on designing
a more robust and adaptive FEM to generate better features, such as
introducing meta-learning strategy (Finn et al., 2017) (Bertinetto et al.,
2019); self-supervised learning strategy (Mangla et al., 2020) (Rodríguez et
al., 2020); knowledge distillation strategy (Tian et al., 2020) (Rizve et al.,
2021). Such methods merely weaken the influence of the DSP to a certain
extent. This paper proposes to tackle the issue from a multi-modal learning
perspective. We denote the multiple features extracted from different FEMs as
multi-head features. Since the single-head feature is only one-sided for the
sample, why do not attempt to conduct multi-head features fusion? We
illustrate a t-SNE (Van der Maaten and Hinton, 2008) visualization of
different kinds of features in Figure 1.

However, two challenges prevent the idea. (1) Since the independent FEMs, the
extracted multi-head features are in separate spaces, which exists a problem
of inconsistent measurement scales. Thus, the first challenge is how to align
the multi-head features. (2) Different features are suitable for various
tasks. For example, features focusing on the contour information should be
more crucial than features concentrating on texture information for
classification tasks. Therefore, the second challenge is how to weight multi-
head features reasonably.

To tackle the DSP and overcome these two challenges, we propose a novel Multi-
Head Feature Collaboration (MHFC) algorithm to improve the discrimination of
sample’s feature. MHFC is a simple non-parametric model that can directly fuse
multi-head features extracted from the existing FEMs, such as ICI-Net (Wang et
al., 2020b), MetaOpt-Net (Bertinetto et al., 2019). To solve the first
challenge, we introduce a subspace learning method to transform the raw multi-
head features to a unified space with reconstructed low-dimensional
representation. It is also helpful to reduce redundant information. Next, we
design an attention block to automatically update combination weights for each
head of the feature to solve the second challenge efficiently. Finally, we
generate the collaborative representation of the sample by concatenating the
processed multi-head features. We illustrate the flowchart in Figure 2.

Besides, according to the data adopted in the design of the classifier,
researchers categorize the FSL-based approaches as three sorts: (1) Inductive
few-shot learning (IFSL). (2) Transductive few-shot learning (TFSL). (3) Semi-
supervised few-shot learning (SSFSL). This paper extends the proposed MHFC to
these three settings.

In summary, the main contributions focus on:

(1) We propose a novel method for FSL, dubbed as Multi-Head Feature
Collaboration (MHFC) and extend it to three FSL-based settings. It addresses
the DSP by comprehensively considering the multi-head features to improve the
discrimination of sample’s representation.

(2) Compared to tuning the network to solve DSP, the proposed MHFC is more
straightforward and effective. It is a simple non-parametric method that
directly fuses multi-head features extracted from the existing FEMs. Besides,
the more robust feature representation enables the proposed MHFC to achieve
amazing performance especially when the samples are extremely scarce (e.g.,
inductive setting on $5$-way $1$-shot case, please refer to Table 1, 2). Thus,
we consider this paper may be meaningful to be applied in reality.

(3) We evaluate the proposed method on four benchmark datasets (mini-ImageNet,
tiered-ImageNet, CIFAR-FS, FC100) and achieve significant improvements of
2.1%-7.8% compared with other state-of-the-art methods. Besides, to prove that
the proposed method can overcome DSP, we design the cross-domain experiments
(e.g., mini-ImageNet $\rightarrow$ CUB) and achieve far better performance
than state-of-the-art methods of at least 7.8%.

##  2\. Related Work

###  2.1. Few-Shot Learning

In the past decade, FSL based works have attracted lots of attention.
Researchers have proposed various classical frameworks to solve this problem.
We list the two most popular types, including (1) Meta-learning based methods,
such as MAML (Finn et al., 2017), Reptile (Nichol et al., 2018), LEO (Rusu et
al., 2019), which purpose to obtain a universal model to rapidly adapt to new
tasks. (2) Metric learning based methods, focusing on looking for ideal
distance metrics to strengthen model’s robustness, including ProtoNet (Snell
et al., 2017), MetaOpt (Bertinetto et al., 2019), TADAM (Oreshkin et al.,
2018) $et$ $al.$ In addition, all these methods can be split into another
taxonomy, e.g. inductive few-shot learning (IFSL), transductive few-shot
learning (TFSL), and semi-supervised few-shot learning (SSFSL). For example,
MAML (Finn et al., 2017), LEO (Rusu et al., 2019), S2M2 (Mangla et al., 2020)
$et$ $al.$ are based on inductive setting; DPGN (Yang et al., 2020), TEAM
(Qiao et al., 2019), SIB (Hu et al., 2020) $et$ $al.$ are TFSL methods; and
LST (Li et al., 2019b), EPNet (Rodríguez et al., 2020), ICI (Wang et al.,
2020b) $et$ $al.$ are based on semi-supervised setting.

###  2.2. Multi-Modal Learning

Just as every coin has two sides, it would be incomplete to define objects
from a single perspective. Therefore, multi-modal learning has received wide
attention in recent years. There exist lots of classical methods and
corresponding applications. For example, Liu $et$ $al.$ proposed a sparse
coding based multi-modal method MHDSC (Liu et al., 2014) for image annotation
task; Liu $et$ $al.$ proposed SPM-CRC (Liu et al., 2019b), which improves the
collaborative representation model from multi-modal learning to classify
remote sensing images; Jan $et$ $al.$ proposed MVCCA (Rupnik and Shawe-Taylor,
2010) and employed it in natural language processing. Liu $et$ $al.$ proposed
MHL (Liu et al., 2017) to solve Alzheimer’s Disease Predicting problem; Zhang
$et$ $al.$ proposed IMHL (Zhang et al., 2018), which is an inductive
hypergraph learning from multi-modal and applied it for $3D$ object
recognition. All these methods may help FSL, and some multi-model based FSL
methods were proposed, including (Dvornik et al., 2019) (introduced ensemble
strategy in pretrain), (Yue et al., 2020) (adopted multi-head classifier in
fine-tuning ), (Dvornik et al., 2020) (fused multi-domain representation).
This paper makes an orthogonal contribution towards efficient semantic-
preserving pretraining.

###  2.3. Subspace Learning

Subspace learning is capable of transferring the samples to another
representation. It is an efficient dimension reduction method. Here, we list
several classical subspace learning methods, which have been employed in the
proposed MHFC. The first one is Principal Component Analysis (PCA) (Tipping
and Bishop, 1999). It projects the raw features to a low-dimensional space by
using singular value decomposition. The second one is Locally Linear Embedding
(LLE) (Roweis and Saul, 2000), which tries to preserve distances within local
according to seeking a low-dimensional projection of the data. The last one is
Laplacian Eigenmap (LE) (Belkin and Niyogi, 2002). It exploits a spectral
decomposition of the graph Laplacian to project the raw data to a low-
dimensional representation. All these methods are helpful for our MHFC, and we
will show the comparison results in Section4.2.1.

##  3\. Problem Formulation

In this section, we introduce the complete procedure in details. It is
composed of two phases, including pre-train and meta-test. i) In pre-train
phase, given base data $\mathcal{D}_{base}=\\{(x_{i},y_{i})|{\kern
1.0pt}y_{i}\in\mathcal{C}_{base}\\}_{i=1}^{N_{base}}$, where $x$ and $y$
denote the sample and corresponding label, respectively. $N_{base}$ denotes
the total number of base data. $\mathcal{C}_{base}$ indicates the base
category set. We train the CNN-based FEM $\mathcal{M}_{\theta}(\cdot)$ on
$\mathcal{D}_{base}$, where $\theta$ indicates the parameters in CNN. In this
paper, divers FEMs are employed to extract features for different heads, and
we define the FEM on the $h_{th}$ head as $\mathcal{M}_{\theta}^{h}(\cdot)$,
where $h=1,2,\cdots,H$, more details please refer to Section 4.4.

ii) For the meta-test phase, the novel data
$\mathcal{D}_{novel}=\\{(x_{j},y_{j})|{\kern
1.0pt}y_{j}\in\mathcal{C}_{novel}\\}_{j=1}^{N_{novel}}$, where
$\mathcal{C}_{novel}$ denotes the novel category set, $N_{novel}$ denotes the
number of novel data. $C_{base}{\kern 2.0pt}\cap{\kern
2.0pt}C_{novel}=\emptyset$. $\mathcal{D}_{novel}$ includes three components,
e.g., $\mathcal{D}_{novel}=\\{\mathcal{S},\mathcal{U},\mathcal{Q}\\}$, where
$\mathcal{S}$, $\mathcal{U}$ and $\mathcal{Q}$ denote support set, unlabeled
set and query set, $\mathcal{S}{\kern 2.0pt}\cap{\kern
2.0pt}\mathcal{U}=\emptyset$, $\mathcal{S}{\kern 2.0pt}\cap{\kern
2.0pt}\mathcal{Q}=\emptyset$, $\mathcal{Q}{\kern 2.0pt}\cap{\kern
2.0pt}\mathcal{U}=\emptyset$. In this phase, we first utilize the trained FEM
to extract $\mathcal{D}_{novel}$’s feature, then design a classifier to
classify $\mathcal{Q}$. There exist three settings to design the classifier,
e.g., inductive setting, transductive setting, and semi-supervised setting.
Section 4.3 show more details. We follow standard $C$-$way$-$T$-$shot$ per
episode as (Wang et al., 2020b) for classification, where $C$-$way$ denotes
$C$ classes, and $T$-$shot$ indicates $T$ samples per class. We average the
accuracies of all the episodes with $95\%$ confidence intervals as the final
result.

##  4\. Methodology

In this section, we first briefly review the linear regression classifier as
an example to introduce our method. Then, we propose the novel Multi-Head
Feature Collaboration (MHFC) algorithm to collaboratively represent the
samples with processed multi-head features. Next, we extend the proposed MHFC
to three kinds of few-shot learning settings. Finally, we introduce the
employed multi-head feature extraction model.

###  4.1. Review of Linear Regression Classifier

This proposed method pays attention to the various features that can integrate
all types of conventional classification strategy (such as linear regression,
support vector machine, logistic regression). In this paper, we employ a
simple regularized linear regression model as an example to show the details
of our method. We formulate the objective function as:

(1)
$\begin{split}\mathop{\arg\min}\limits_{\mathbf{W}}\mathcal{F}=\left\|\mathbf{Y}-\mathbf{W}\mathbf{X}\right\|_{F}^{2}+\mu\left\|\mathbf{W}\right\|_{F}^{2}\end{split}$

where $\left\|\cdot\right\|_{F}$ represents $\left(\cdot\right)$’s Frobenius-
norm. $\mu$ is the hyperparameter.
$\mathbf{X}=[{\mathbf{x}}_{1},{\mathbf{x}}_{2},\dots,{\mathbf{x}}_{N}]\in\mathbb{R}^{dim1\times
N}$,

$\mathbf{Y}=[{\mathbf{y}}_{1},{\mathbf{y}}_{2},\dots,{\mathbf{y}}_{N}]\in\mathbb{R}^{C\times
N}$, $dim1$ and $N$ indicate the dimension and number of labeled samples,
respectively. $C$ denotes the number of categories.
${\mathbf{x}}_{n},{\mathbf{y}}_{n}$ ($n=1,2,\dots,N$) denote the feature
embedding vector and one-hot label vector of the $n_{th}$ sample.
$\mathbf{W}\in\mathbb{R}^{C\times dim1}$ represents the to-be-learned
classifier. We directly optimize the objective function and obtain the
$\mathbf{W}$ as:

(2)
$\begin{split}\mathbf{W}=\mathbf{Y}{\mathbf{X}}^{T}\left(\mathbf{X}{\mathbf{X}}^{T}+\mu\mathbf{I}\right)^{-1}\end{split}$

where $\mathbf{I}$ is the identity matrix. Following, given a testing sample
embedding $\mathbf{x}_{ts}\in\mathbb{R}^{dim1}$, we predict the
$\mathbf{x}_{ts}$’s category by:

(3)
$\begin{split}\mathcal{A}(\mathbf{x}_{ts})=max\left\\{\mathbf{W}\mathbf{x}_{ts}\right\\}\end{split}$

where $max$ denotes an operator to obtain the index of the max value in the
vector.

###  4.2. Multi-Head Feature Collaboration

####  4.2.1. Multi-Head Feature Transformation

Since one kind of incomplete feature cannot reflect a sample well, we
introduce various features to represent samples from different heads
collaboratively. Assume that we have $H$ heads in total. Each head corresponds
to one kind of feature $\mathbf{X}^{h}$, where $h=1,2,\cdots,H$. As mentioned
in Section 1, the multi-head features are in separate spaces, which exists a
problem of inconsistent measurement scales.

To address this challenge, we introduce a conventional subspace learning
method (denoted as $\mathcal{J}(\cdot)$) to transform the raw features into a
unified space with reconstructed low-dimensional representation. Specifically,
we treat $H$ heads of the same sample as $H$ samples, and denote the features
of the expanded dataset as
$\mathbf{X}_{exp}=[\mathbf{X}^{1},\mathbf{X}^{2},\cdots,\mathbf{X}^{H}]\in\mathbb{R}^{dim1\times(N\times
H)}$. We conduct subspace learning operation $\mathcal{J}(\mathbf{X}_{exp})$
and obtain the novel features
$\mathbf{X}_{exp}^{\prime}=[\mathbf{P}^{1},\mathbf{P}^{2},\cdots,\mathbf{P}^{H}]\in\mathbb{R}^{dim2\times(N\times
H)}$, where $\mathbf{P}^{h}\in\mathbb{R}^{dim2\times N}$ denotes the feature
on the $h_{th}$ head after subspace transformation. $dim2$ denotes the novel
dimension.

####  4.2.2. Multi-Head Feature Attention Block

Consider that the significance of these features is different per episode. We
try to find the optimal combination weights
$\mathbf{\Omega}=[\Omega^{1},\Omega^{2},\dots,\Omega^{H}]^{T}$ to let these
features have different influences for the final decision, where
$\mathbf{\Omega}$ is a weight vector, $\Omega^{h}(h=1,2,\cdots,H)$ denotes the
$h_{th}$ element in $\mathbf{\Omega}$. We first employ the transformed feature
$\mathbf{P}^{h}$ to replace $\mathbf{X}^{h}$ and obtain a novel classifier
$\mathbf{W}_{p}^{h}\in\mathbb{R}^{C\times dim2}$ according to Equation 2,
where $\mathbf{W}_{p}^{h}$ indicates the trained novel classifier on the
$h_{th}$ head, which can be formulated as:

(4)
$\begin{split}\mathbf{W}_{p}^{h}=\mathbf{Y}{\mathbf{P}^{h}}^{T}\left(\mathbf{P}^{h}{\mathbf{P}^{h}}^{T}+\mu\mathbf{I}\right)^{-1}\end{split}$

Then use $\mathbf{P}^{h}$ and $\mathbf{W}_{p}^{h}$ to re-calculate the
objective function’s loss on the $h_{th}$ head $\mathcal{F}_{p}^{h}$ by
Equation 1, which can be formulated as:

(5)
$\begin{split}\mathcal{F}_{p}^{h}=\left\|\mathbf{Y}-\mathbf{W}_{p}^{h}\mathbf{P}^{h}\right\|_{F}^{2}+\mu\left\|\mathbf{W}_{p}^{h}\right\|_{F}^{2}\end{split}$

Next, exploit the $\mathcal{F}_{p}^{h}$ to compute the combination weights.
The objective function can be formulated as:

(6)
$\begin{split}\mathop{\arg\min}\limits_{\mathbf{\Omega}}&=\sum_{h=1}^{H}\left(\Omega^{h}\mathcal{F}_{p}^{h}\right)+\eta\left\|\mathbf{\Omega}\right\|_{2}^{2}\\\
\text{s.t.}{\kern 4.0pt}&\sum_{h=1}^{H}\Omega^{h}=1{\kern 4.0pt}\Omega^{h}\geq
0\end{split}$

where $\Omega^{h}$ indicates the weight of $h_{th}$ head.
$\left\|\cdot\right\|_{2}$ represents $\left(\cdot\right)$’s $\ell_{2}$-norm.
$\eta$ is the parameter. We introduce the Lagrangian to solve the problem, the
Equation 6 can be rewritten as:

(7)
$\begin{split}\mathop{\arg\min}\limits_{\mathbf{\Omega,\zeta,\Lambda}}\mathcal{G}=\sum_{h=1}^{H}\left(\Omega^{h}\mathcal{F}_{p}^{h}\right)+\eta\left\|\mathbf{\Omega}\right\|_{2}^{2}-\zeta\left(\sum_{h=1}^{H}\Omega^{h}-1\right)-\mathbf{\Lambda}^{T}\mathbf{\Omega}\end{split}$

where $\zeta$ is a constant,
$\mathbf{\Lambda}=[\Lambda^{1},\Lambda^{2},\dots,\Lambda^{H}]^{T}$ is a
vector. Assume $\hat{\mathbf{\Omega}}$, $\hat{\zeta}$,
$\hat{\mathbf{\Lambda}}$ are the optimal solutions, we solved this problem as:

(8)
$\begin{split}\hat{\Omega}^{h}=\frac{1}{2\eta}max\left\\{\frac{\sum_{h=1}^{H}\mathcal{F}_{p}^{h}}{H}+\frac{2\eta}{H}-\mathcal{F}_{p}^{h}-\hat{\Lambda}_{avg},0\right\\}\\\
\end{split}$

where $\hat{\Lambda}_{avg}$ is a constant, denotes the average of
$\hat{\mathbf{\Lambda}}$. For the detailed optimization process, please refer
to Appendix A.2.

####  4.2.3. Multi-Head Feature Collaborative Classifier

After giving muli-head features the combination weights, we obtain the final
collaborative feature
$\mathbf{Z}=[\mathbf{z}_{1},\mathbf{z}_{2},\cdots,\mathbf{z}_{N}]\in\mathbb{R}^{dim3\times
N}$ by:

(9)
$\begin{split}\mathbf{Z}\leftarrow\mathbf{z}_{n}=Concat\left(\hat{\Omega}^{1}{\kern
1.0pt}\mathbf{p}_{n}^{1},\hat{\Omega}^{2}{\kern
1.0pt}\mathbf{p}_{n}^{2},\cdots,\hat{\Omega}^{H}{\kern
1.0pt}\mathbf{p}_{n}^{H}\right)\end{split}$

where $\mathbf{p}_{n}^{h},\mathbf{z}_{n}^{h}(n=1,2,\cdots,N)$ denote the
$n_{th}$ vector of $\mathbf{P}^{h}$ and $\mathbf{Z}^{h}$.

Next, according to Equation 2, we use $\mathbf{Z}$ to replace $\mathbf{X}$ and
obtain the final collaborative classifier
$\mathbf{W}_{z}\in\mathbb{R}^{C\times dim3}$, which can be formulated as:

(10)
$\begin{split}\mathbf{W}_{z}=\mathbf{Y}{\mathbf{Z}}^{T}\left(\mathbf{Z}{\mathbf{Z}}^{T}+\mu\mathbf{I}\right)^{-1}\end{split}$

Finally, given a testing sample feature
$\mathbf{x}_{ts}^{h},(h=1,2,\cdots,H)$, we first obtain the collaborative
feature $\mathbf{z}_{ts}\in\mathbb{R}^{dim3}$ by Equation 8, 9, then predict
the label by:

(11)
$\begin{split}\mathcal{A}(\mathbf{z}_{ts})=max\left\\{\mathbf{W}_{z}\mathbf{z}_{ts}\right\\}\end{split}$

###  4.3. MHFC for Few-Shot Learning

Define the feature of $\mathcal{D}_{novel}$ on the $h_{th}$ head as
${\mathbf{X}_{novel}^{h}}=[{\mathbf{X}_{s}^{h}},{\mathbf{X}_{u}^{h}},\\\
{\mathbf{X}_{q}^{h}}]$, where
${\mathbf{X}_{s}^{h}}=\mathcal{M}_{\theta}^{h}(\mathcal{S})$,
${\mathbf{X}_{u}^{h}}=\mathcal{M}_{\theta}^{h}(\mathcal{U})$, and
${\mathbf{X}_{q}^{h}}=\mathcal{M}_{\theta}^{h}(\mathcal{Q})$ denote the
features of support, unlabeled, and query data on the $h_{th}$ head. Thus, the
$\left(\mathbf{X}_{novel}^{h}\right)$’s feature after transformation can be
defined as
$\mathbf{P}_{novel}^{h}=[{\mathbf{P}_{s}^{h}},{\mathbf{P}_{u}^{h}},{\mathbf{P}_{q}^{h}}]$.
Researchers employ different data to design the classifier, and these methods
can be split into three settings, e.g., inductive setting, semi-supervised
setting, and transductive setting.

####  4.3.1. Semi-Supervised MHFC

Semi-supervised setting in few-shot learning adopts the support set
$\mathcal{S}$ and unlabeled set $\mathcal{U}$ to train the classifier, and
then predict the query label. In this paper, we extend our MHFC to the semi-
supervised setting by introducing a simple self-training strategy. It can
employ the unlabeled data to strengthen the classifier. We show the detailed
steps as:

(1) Exploit the support data to train the basic classifier on each head,
Equation 4 can be rewritten as:

(12)
$\begin{split}\mathbf{W}_{p}^{h}=\mathbf{Y}_{s}{\mathbf{P}_{s}^{h}}^{T}\left(\mathbf{P}_{s}^{h}{\mathbf{P}_{s}^{h}}^{T}+\mu\mathbf{I}\right)^{-1}\end{split}$

where $\mathbf{Y}_{s}$ denotes the one-hot label matrix of support data.

(2) Calculate the combination weight on each head by:

(13)
$\begin{split}\left\\{\begin{array}[]{lllllll}\mathcal{F}_{p}^{h}=\left\|\mathbf{Y}_{s}-\mathbf{W}_{p}^{h}\mathbf{P}_{s}^{h}\right\|_{F}^{2}+\mu\left\|\mathbf{W}_{p}^{h}\right\|_{F}^{2}\\\
\hat{\Omega}^{h}=\frac{1}{2\eta}max\left\\{\frac{\sum_{h=1}^{H}\mathcal{F}_{p}^{h}}{H}+\frac{2\eta}{H}-\mathcal{F}_{p}^{h}-\hat{\Lambda}_{avg},0\right\\}\\\
\end{array}\right.\end{split}$

(3) Obtain the collaborative feature of support set and classifier by:

(14)
$\begin{split}\left\\{\begin{array}[]{lllllll}\mathbf{Z}_{s}\leftarrow\mathbf{z}_{sn}=Concat\left(\hat{\Omega}^{1}{\kern
1.0pt}\mathbf{p}_{sn}^{1},\hat{\Omega}^{2}{\kern
1.0pt}\mathbf{p}_{sn}^{2},\cdots,\hat{\Omega}^{H}{\kern
1.0pt}\mathbf{p}_{sn}^{H}\right)\\\
\mathbf{W}_{z}=\mathbf{Y}_{s}{\mathbf{Z}_{s}}^{T}\left(\mathbf{Z}_{s}{\mathbf{Z}_{s}}^{T}+\mu\mathbf{I}\right)^{-1}\\\
\end{array}\right.\end{split}$

where $\mathbf{p}_{sn}^{h}(n=1,2,\cdots,N)$ indicates the $n_{th}$ vector in
$\mathbf{P}_{s}^{h}$. $\mathbf{Z}_{s}$ represents the collaborative feature of
support set. $\mathbf{z}_{sn}$ denotes the $n_{th}$ vector in
$\mathbf{Z}_{s}$.

(4) Utilize the trained classifier $\mathbf{W}_{z}$ to predict the unlabeled
data $\mathcal{U}$ by:

(15)
$\begin{split}\left\\{\begin{array}[]{lllllll}\mathbf{Z}_{u}\leftarrow\mathbf{z}_{un}=Concat\left(\hat{\Omega}^{1}{\kern
1.0pt}\mathbf{p}_{un}^{1},\hat{\Omega}^{2}{\kern
1.0pt}\mathbf{p}_{un}^{2},\cdots,\hat{\Omega}^{H}{\kern
1.0pt}\mathbf{p}_{un}^{H}\right)\\\
\mathbf{Y}_{pseudo}=\mathbf{W}_{z}\mathbf{Z}_{u}\\\
\end{array}\right.\end{split}$

where $\mathbf{p}_{un}^{h}(n=1,2,\cdots,N)$ indicates the $n_{th}$ vectors in
$\mathbf{P}_{u}^{h}$. $\mathbf{Z}_{u}$ represents the collaborative feature
embedding of the unlabeled set. $\mathbf{z}_{un}$ denotes the $n_{th}$ vector
in $\mathbf{Z}_{u}$. $\mathbf{Y}_{pseudo}$ represents the predicted soft-
pseudo-label of unlabeled data.

(5) Select one most confidence sample through the ${\mathbf{Y}_{pseudo}}$
without putting back, the corresponding one-hot-pseudo-label and feature
(after transformation) are defined as ${\mathbf{y}_{select}}$ and
${\mathbf{p}^{h}_{select}}$. Then, expand it to the support set by:

(16)
$\begin{split}\left\\{\begin{array}[]{ll}{\mathbf{P}^{h}_{s}}=\left[{\mathbf{P}^{h}_{s}},{\mathbf{p}^{h}_{select}}\right]\\\
{\mathbf{Y}_{s}}=\left[{\mathbf{Y}_{s}},{\mathbf{y}_{select}}\right]\end{array}\right.\end{split}$

(6) Repeat (1), (2) (3), (4), (5) until the performance of classifier is
stable.

(7) Obtain the collaborative feature embedding of query data by:

(17)
$\begin{split}\mathbf{Z}_{q}\leftarrow\mathbf{z}_{qn}=Concat\left(\hat{\Omega}^{1}{\kern
1.0pt}\mathbf{p}_{qn}^{1},\hat{\Omega}^{2}{\kern
1.0pt}\mathbf{p}_{qn}^{2},\cdots,\hat{\Omega}^{H}{\kern
1.0pt}\mathbf{p}_{qn}^{H}\right)\\\ \end{split}$

where $\mathbf{p}_{qn}^{h}(n=1,2,\cdots,N)$ indicates the $n_{th}$ vector in
$\mathbf{P}_{q}^{h}$. $\mathbf{Z}_{q}$ represents the collaborative feature of
query data. $\mathbf{z}_{qn}$ denotes the $n_{th}$ vector in $\mathbf{Z}_{q}$.

(8) Finally, employ the optimal classifier to predict the query label by:

(18)
$\begin{split}\mathcal{A}(\mathbf{Z}_{q})=max\left\\{\mathbf{W}_{z}\mathbf{Z}_{q}\right\\}\end{split}$

We summarize the Algorithm in Appendix A.1.

####  4.3.2. Inductive MHFC

Unlike semi-supervised few-shot learning (SSFSL), inductive few-shot learning
(IFSL) based methods only employ the support data to train the classifier and
predict the query’s category. IFSL can be viewed as the special case of SSFSL
(e.g., there not exist unlabeled data). Thus, we can use the steps (e.g., (1),
(2), (3), (7), (8)) to implement the inductive MHFC.

####  4.3.3. Transductive MHFC

In transductive few-shot learning (TFSL), besides the support data’s features
and label information, researchers also apply the features of query data to
construct the classifier and then predict the query label. To implement the
tansductive MHFC, we need to make some adjustments to the step (4) in Section
4.3.1. We re-represent (4) as:

(9) Utilize the trained classifier $\mathbf{W}_{z}$ to predict the query data
$\mathcal{Q}$ by:

(19)
$\begin{split}\left\\{\begin{array}[]{lllllll}\mathbf{Z}_{q}\leftarrow\mathbf{z}_{qn}=Concat\left(\hat{\Omega}^{1}{\kern
1.0pt}\mathbf{p}_{qn}^{1},\hat{\Omega}^{2}{\kern
1.0pt}\mathbf{p}_{qn}^{2},\cdots,\hat{\Omega}^{H}{\kern
1.0pt}\mathbf{p}_{qn}^{H}\right)\\\
\mathbf{Y}_{pseudo}=\mathbf{W}_{z}\mathbf{Z}_{q}\\\
\end{array}\right.\end{split}$

where $\mathbf{p}_{qn}^{h}(n=1,2,\cdots,N)$ indicates the $n_{th}$ vectors in
$\mathbf{P}_{q}^{h}$. $\mathbf{Z}_{q}$ represents the collaborative feature
embedding of the unlabeled set. $\mathbf{z}_{qn}$ denotes the $n_{th}$ vector
in $\mathbf{Z}_{q}$. $\mathbf{Y}_{pseudo}$ represents the predicted soft-
pseudo-label of query data.

After that, we achieve transductive MHFC by steps (1), (2), (3), (9), (5),
(6), (8).

###  4.4. Multi-Head Feature Extraction Model

The multi-head features we adopted come from different feature extraction
models (FEMs). As examples: (1) Standard feature (Std-Fea), the FEM utilizes a
standard CNN-based classification structure, such as (Wang et al., 2020b). (2)
Meta feature (Meta-Fea), the FEM introduces the meta-learning strategy to the
network, just like (Bertinetto et al., 2019). (3) Self-supervised-feature (SS-
Fea), the FEM adds auxiliary losses to the standard CNN-based classification
structure from a self-supervised perspective to strengthen the robustness of
the network, similar as (Mangla et al., 2020). We discuss the results of all
kinds of stacking ways in Appendix B.2.

In this paper, we merely fuse two kinds of SS-Feas for most of the experiments
as an example for convenience. For the first category, we design the FEM by
introducing standard classification loss $\mathcal{L}_{c}$ and auxiliary
rotation loss $\mathcal{L}_{r}$. $\mathcal{L}_{c}$ can be formulated as:

(20) $\begin{split}\mathcal{L}_{c}=-\sum_{c}y_{(c,x)}log(p_{(c,x)})\\\
\end{split}$

where $c\in\mathcal{C}_{base}$ denotes the $c_{th}$ class. $y_{(c,x)}$,
$p_{(c,x)}$ indicate the probabilities that the truth label and predicted
label of $x_{th}$ sample belongs to $c_{th}$ class. Then, we rotate each
sample to $r$ degree and
$r\in\mathcal{C_{R}}=\\{0^{\circ},90^{\circ},180^{\circ},270^{\circ}\\}$. We
define rotation loss as:

(21) $\begin{split}\mathcal{L}_{r}=-\sum_{r}y_{(r,x)}log(p_{(r,x)})\\\
\end{split}$

where $y_{(r,x)}$, $p_{(r,x)}$ indicate the probabilities that the truth label
and predicted label of $x_{th}$ sample belongs to $r_{th}$ class. Thus, the
first loss function is defined as $\mathcal{L}_{c}+\mathcal{L}_{r}$, and the
feature based on this kind of FEM is dubbed as SS-R-Fea.

The second feature is denoted as SS-M-Fea, which extracted from another
category of self-supervised FEM. Specifically, this FEM adds the loss
$\mathcal{L}_{c}$ and auxiliary mirror loss $\mathcal{L}_{m}$ to the neural
network to predict image mirrors. Assume that there are $m$ ways and
$m\in\mathcal{C_{M}}=\\{vertically,horizontally,diagonally\\}$, we define the
mirror loss as:

(22) $\begin{split}\mathcal{L}_{m}=-\sum_{m}y_{(m,x)}log(p_{(m,x)})\\\
\end{split}$

where $y_{(m,x)}$, $p_{(m,x)}$ indicate the probabilities that the truth label
and predicted label of $x_{th}$ sample belongs to $m_{th}$ class. Next, we
summarize the loss function as $\mathcal{L}_{c}+\mathcal{L}_{m}$.

##  5\. Experiments

In this section, we first briefly review the benchmark datasets and show the
implementation details. Then, we list the experimental results in Table 1, 2
and analyse them. Next, we perform ablation studies to discuss the factors
that influence MHFC’s performance, e.g., multi-head feature fusion, subspace
transformation, and attention block that calculates the combination weights.
In the following, we take a cross-domain experiment to further evaluate the
ability and robustness of the proposed method. We conduct all the experiments
on a Tesla-$V100$ GPU with $32G$ memory. All the source codes will be made
available to the public.

Table 1. The $5$-way few-shot classification accuracies on mini-ImageNet and tiered-ImageNet with $95\%$ confidence intervals over $600$ episodes. $(\cdot)^{\star}$, $(\cdot)^{\dagger}$, and $(\cdot)^{\ddagger}$ in Table 1, 2, 4 indicate inductive, transductive, and semi-supervised settings, respectively. $(\cdot)^{\star\star}$ in Table 1, 2 and 3 denotes the non-standardized inductive setting, which adopts the query feature when reducing the feature’s dimension. 4CONV, ResNet12, ResNet18 and WRN are the exploited FEM’s architectures. The $(80)$, $(100)$ in the semi-supervised setting indicate the number of employed unlabeled samples per class. The top two results are shown in red and blue, respectively. Method | Backbone | mini-ImageNet | tiered-ImageNet
---|---|---|---
$5$-way $1$-shot |  $5$-way $5$-shot |  $5$-way $1$-shot |  $5$-way $5$-shot
Baseline⋆ (Chen et al., 2019) (ICLR,2019) | ResNet18 | $51.75\pm 0.80$ | $74.27\pm 0.63$ | - | -
Baseline++⋆ (Chen et al., 2019) (ICLR,2019) | ResNet18 | $51.87\pm 0.77$ | $75.68\pm 0.63$ | - | -
TapNet⋆ (Yoon et al., 2019) (ICML,2019) | ResNet12 | $61.65\pm 0.15$ | $76.36\pm 0.10$ | $63.08\pm 0.15$ | $80.26\pm 0.12$
LEO⋆ (Rusu et al., 2019) (ICLR,2019) | WRN | $61.76\pm 0.08$ | $77.59\pm 0.12$ | $66.33\pm 0.05$ | $81.44\pm 0.09$
AM3⋆ (Xing et al., 2019) (NIPS,2019) | ResNet12 | $65.30\pm 0.49$ | $78.10\pm 0.36$ | $69.08\pm 0.47$ | $82.58\pm 0.31$
CTM⋆ (Li et al., 2019a) (CVPR,2019) | ResNet18 | $64.12\pm 0.82$ | $80.51\pm 0.13$ | - | -
MABAS⋆ (Kim et al., 2020) (ECCV,2020) | ResNet12 | $64.21\pm 0.82$ | $81.01\pm 0.57$ | - | -
MELR⋆ (Fei et al., 2021) (ICLR,2021) | ResNet12 | $\textbf{67.40}\pm 0.43$ | $\textbf{83.40}\pm 0.28$ | $\textbf{72.14}\pm 0.51$ | $\textbf{87.01}\pm 0.35$
Our MHFC⋆⋆ | ResNet12 | $\textbf{73.10}\pm 1.00$ | $\textbf{81.75}\pm 0.56$ | $\textbf{82.10}\pm 1.03$ | $\textbf{87.99}\pm 0.60$
TPN† (Liu et al., 2019a) (ICLR,2019) | 4CONV | $55.51\pm 0.86$ | $69.86\pm 0.65$ | $59.91\pm 0.94$ | $73.30\pm 0.75$
TEAM† (Qiao et al., 2019) (ICCV,2019) | ResNet12 | $60.07\pm 0.63$ | $75.90\pm 0.52$ | - | -
Fine-tuning† (Dhillon et al., 2020) (ICLR,2020) | WRN | $65.73\pm 0.68$ | $78.40\pm 0.52$ | $\textbf{73.34}\pm 0.71$ | $85.50\pm 0.50$
DPGN† (Yang et al., 2020) (CVPR,2020) | ResNet12 | $\textbf{67.77}\pm 0.32$ | $\textbf{84.60}\pm 0.43$ | $72.45\pm 0.51$ | $\textbf{87.24}\pm 0.39$
ODE† (Xu et al., 2021) (CVPR,2021) | ResNet12 | $67.76\pm 0.46$ | $82.71\pm 0.31$ | $71.89\pm 0.52$ | $85.96\pm 0.35$
Our MHFC† | ResNet12 | $\textbf{74.81}\pm 1.12$ | $\textbf{85.58}\pm 0.61$ | $\textbf{83.95}\pm 1.13$ | $\textbf{90.75}\pm 0.58$
TPN‡ (Liu et al., 2019a) (ICLR,2019) | 4CONV | $52.78\pm 0.27$ | $66.42\pm 0.21$ | $55.74\pm 0.29$ | $71.01\pm 0.23$
LST‡ (100) (Li et al., 2019b) (NIPS,2019) | ResNet12 | $70.10\pm 1.90$ | $78.70\pm 0.80$ | $77.70\pm 1.60$ | $85.20\pm 0.80$
EPNet‡ ($100$) (Rodríguez et al., 2020) (ECCV,2020) | ResNet12 | $\textbf{75.36}\pm 1.01$ | $\textbf{84.07}\pm 0.60$ | $81.79\pm 0.97$ | $88.45\pm 0.61$
TransMatch‡ ($100$) (Yu et al., 2020) (CVPR,2020) | WRN | $63.02\pm 1.07$ | $81.19\pm 0.59$ | - | -
ICI‡ ($80$) (Wang et al., 2020b) (CVPR,2020) | ResNet12 | $71.41$ | $81.12$ | 85.44 | 89.12
Our MHFC‡ ($80$) | ResNet12 | $79.26\pm 1.14$ | $87.30\pm 0.55$ | $\textbf{87.57}\pm 1.03$ | $91.80\pm 0.56$
Our MHFC‡ ($100$) | ResNet12 | $\textbf{79.76}\pm 1.16$ | $\textbf{87.64}\pm 0.53$ | $87.56\pm 1.04$ | $\textbf{91.90}\pm 0.56$

###  5.1. Datasets

We carry out experiments on five benchmark datasets, including mini-ImageNet
(Vinyals et al., 2016), tiered-ImageNet (Ren et al., 2018), CIFAR-FS
(Bertinetto et al., 2019), FC100 (Oreshkin et al., 2018), and CUB (Wah et al.,
2011). Both mini-ImageNet and tiered-ImageNet are the subsets of ImageNet
dataset (Russakovsky et al., 2015). mini-ImageNet consists of $100$ classes
and tiered-ImageNet contains $608$ classes. For both datasets, the number of
images for each class is $600$ and the size of each image is $84\times 84$. We
follow standard split as (Wang et al., 2020b), selecting $64$ classes as the
base set, $16$ classes as the validation set, $20$ classes as the novel set
for mini-ImageNet, and selecting $351$ classes as the base set, $97$ classes
as the validation set, $160$ classes as the novel set for tiered-ImageNet.
Both CIFAR-FS and FC100 are the subsets of CIFAR-100 dataset (Krizhevsky et
al., 2009), and consist of $100$ classes. We follow the split introduced in
(Bertinetto et al., 2019) to divide CIFAR-FS into $64$ classes as base set,
$16$ classes as validation set, $20$ classes as novel set, and divide FC100
into $60$ classes as base set, $20$ classes as validation set, $20$ classes as
novel set. All the image size is $32\times 32$. CUB totally includes $11,788$
images with $200$ categories. We follow the setting in ICI (Wang et al.,
2020b) to split it into $100$ classes as base set, $50$ classes as validation
set and $50$ classes as novel set. The images are cropped into $84\times 84$.

Table 2. The $5$-way few-shot classification accuracies on CIFAR-FS and FC100 with $95\%$ confidence intervals over $600$ episodes. The top two results are shown in red and blue, respectively. Method | Backbone | CIFAR-FS | FC100
---|---|---|---
$5$-way $1$-shot |  $5$-way $5$-shot |  $5$-way $1$-shot |  $5$-way $5$-shot
ProtoNet⋆ (Snell et al., 2017) (NIPS,2017) | 4CONV | $55.50\pm 0.70$ | $72.00\pm 0.60$ | $35.30\pm 0.60$ | $48.60\pm 0.60$
MAML⋆ (Finn et al., 2017) (ICML,2018) | 4CONV | $58.90\pm 1.90$ | $71.50\pm 1.00$ | - | -
MABAS⋆ (Kim et al., 2020) (ECCV,2020) | ResNet12 | $\textbf{73.24}\pm 0.95$ | $\textbf{85.65}\pm 0.65$ | $\textbf{41.74}\pm 0.73$ | $\textbf{57.11}\pm 0.75$
Our MHFC⋆⋆ | ResNet12 | $\textbf{80.00}\pm 1.02$ | $\textbf{86.33}\pm 0.61$ | $\textbf{46.44}\pm 0.93$ | $\textbf{59.41}\pm 0.72$
TEAM† (Qiao et al., 2019) (ICCV,2019) | ResNet12 | $70.43\pm 1.03$ | $81.25\pm 0.92$ | - | -
Fine-tuning† (Dhillon et al., 2020) (ICLR,2020) | WRN | $\textbf{76.58}\pm 0.68$ | $\textbf{85.79}\pm 0.50$ | $\textbf{43.16}\pm 0.59$ | $\textbf{57.57}\pm 0.55$
Our MHFC† | ResNet12 | $\textbf{81.83}\pm 1.16$ | $\textbf{89.27}\pm 0.63$ | $\textbf{47.02}\pm 1.05$ | $\textbf{61.88}\pm 0.80$
ICI‡ (80) (Wang et al., 2020b) (CVPR,2020) | ResNet12 | 78.07 | 84.76 | - | -
Our MHFC‡ ($80$) | ResNet12 | $84.74\pm 1.14$ | $90.19\pm 0.63$ | $\textbf{ 50.95}\pm 1.11$ | $64.05\pm 0.82$
Our MHFC‡ ($100$) | ResNet12 | $\textbf{85.76}\pm 1.08$ | $\textbf{90.42}\pm 0.64$ | $50.72\pm 1.09$ | $\textbf{64.45}\pm 0.83$

###  5.2. Implementation Details

In this paper, all the FEMs on different heads adopt the ResNet12 (He et al.,
2016) backbone, consisting of four residual blocks ($3\times 3$ convolution
layer, batch normalization layer, LeakyReLU layer), four $2\times 2$ max
pooling layers, and four dropout layers. We adopt stochastic gradient descent
(SGD) optimizer with Nesterov momentum ($0.9$) for the optimizer. For the
parameter $\eta$ in Equation 6, we fix it to $1.4$ for convenience. We set the
training epochs to $120$ and test over $600$ episodes with $15$ query samples
per class for all the models. Besides, the selected subspace learning methods
all follow the default implementation of scikit-learn (Pedregosa et al.,
2011). And there has no fine-tuning process when classifying the novel data.
For other settings, such as the learning rate, data augmentation, filters’
number, we follow the ICI (Wang et al., 2020b).

###  5.3. Experimental Results

We compare the proposed MHFC (only fuse SS-F-Fea and SS-M-Fea) with several
state-of-the-art methods, the results are listed in Table 1 and 2. Here, we
list several observations.

(1) Researchers split the few-shot learning methods into three settings, e.g.,
inductive, transductive, and semi-supervised. While, in the related works of
FSL, the results on all settings are usually compared together. Compared with
the methods proposed recently, our MHFC has achieved state-of-the-art
performance. It has far surpassed other models on the four datasets,
especially on $5$-way $1$-shot case, the MHFC outperforms other methods at
least 4.4%, 2.1%, 7.7% and 7.8% on mini-ImageNet, tiered-ImageNet, CIFAR-FS,
FC100 datasets. Our results on the $5$-way $1$-shot case are even better than
many other methods on the $5$-way $5$-shot case. And on the $5$-way $5$-shot
case, the MHFC also exceeds others at least 3.0%, 2.8%, 4.8% and 6.9% on mini-
ImageNet, tiered-ImageNet, CIFAR-FS, FC100 datasets.

(2) Compared with the methods of each setting, our proposed MHFC has achieved
excellent performances, in particular on the $5$-way $1$-shot case with the
inductive setting. MHFC has significant improvements of at least 7.7%, 10.0%,
6.8% and 4.7% on mini-ImageNet, tiered-ImageNet, CIFAR-FS, FC100 datasets.
Notably, the performance of MHCF with the inductive setting has exceeded many
other methods with transductive or semi-supervised settings. Besides, for the
methods based on semi-supervised setting, the final results are influenced by
the number of employed unlabeled samples. Thus, we observe the impact and list
the results in Appendix B.1. With the increase of unlabeled samples, the
proposed method has become more effective. And the results start to saturate
after $100$ unlabeled samples.

Figure 3. Ablation studies to show the performances of employing different
kinds of features with the inductive setting.

###  5.4. Ablation Studies

In this paper, we propose MHFC for few-shot learning. There exist three
factors influence the classification performance, e.g., (1) introducing multi-
head features; (2) transforming the multi-head features into a unified space;
(3) designing attention mechanism to weight features. This section conducts
ablation studies on $5$-way $1$-shot case with the inductive setting to
evaluate the efficiency of the three blocks.

####  5.4.1. Influence of Multi-Head Feature

The results listed in Table 1, 2 employ two heads of features (SS-R-Fea and
SS-M-Fea). Here, we compare the results of employing the raw single-head
feature with the collaborative feature (e.g., multi-head-fea). The results of
four datasets on $5$-way $1$-shot case with inductive setting are listed in
Figure 3. Obviously, employing the collaborative feature improves
significantly compared with exploiting the raw single-head feature. It
demonstrates the efficiency of our ”multi-head feature” to some extent.
Moreover, the proposed model can integrate more than two heads of features. We
list the comparison results in Appendix B.2.

Figure 4. The comparison results of adding transformation or not on $5$-way
$1$-shot case with the non-standardized inductive setting.

####  5.4.2. Influence of Subspace Transformation

For all the results listed in Table 1, 2 with inductive setting, we select LE
(Belkin and Niyogi, 2002) to transform the multi-head features. While, besides
the LE, there also exist several other choices to pull multi-head features to
a unified space, such as PCA (Tipping and Bishop, 1999), LLE (Roweis and Saul,
2000). We illustrate Figure 4 to show the comparison results. We can see that
all kinds of subspace learning methods are helpful to our MHFC, and LE is the
most.

Table 3. Comparison results with fixed weights on $5$-way few-shot case. (a, b) denotes that the SS-R-Fea’s weight is ”a”, and SS-M-Fea’s weight is ”b”. MHFC employs our designed attention block to update the weights automatically for each episode. The top two results are shown in red and blue. Weight | mini-ImageNet | tiered-ImageNet
---|---|---
$1$-shot |  $5$-shot |  $1$-shot |  $5$-shot
(0.1, 0.9) | $70.85$ | $80.09$ | $79.66$ | $87.12$
(0.3, 0.7) | $71.32$ | 80.72 | $80.25$ | $87.14$
(0.5, 0.5) | 71.88 | $80.15$ | 80.86 | 87.38
(0.7, 0.3) | $70.56$ | $80.48$ | $80.19$ | $87.19$
(0.9, 0.1) | $70.36$ | $79.97$ | $79.69$ | $86.93$
MHFC⋆⋆ | 73.10 | 81.75 | 82.10 | 87.99

####  5.4.3. Influence of Attention Block

We can use multi-head features to describe a category of samples, but the
degrees of importance are different on each head. To this end, it is crucial
to design the attention block to calculate weights for different categories
automatically. Here, we compare the results with fixed weights to our MHFC,
which is listed in Table 3. The results show that the updated weights are more
reasonable for our method. Besides, from Equation 6, we know that $\eta$ is a
parameter to influence the to-be-learned weights. For fairness and
convenience, we have fixed $\eta$ to $1.4$ for all the experiments. We list
the other comparison results in Appendix B.3.

###  5.5. Cross-Domain Few-Shot Learning

After introducing multi-head features from different views, we believe that
the MHFC is an extremely robust method in practical scenarios. Therefore, we
evaluate the proposed method with transductive setting on a cross-domain
dataset: e.g., mini-ImageNet $\longrightarrow$ CUB. In pre-train stage, we use
mini-ImageNet to train the FEM, and in meta-test stage, we classify the CUB
dataset. The results are reported in Table 4. Compared to the state-of-the-
arts, we have significant improvements at least 13.3% on $1$-shot case and
7.8% on $5$-shot case. Thus, the performance on the cross-domain few-shot
learning task demonstrates that the MHFC can solve the DSP better, and the
proposed MHFC would be powerful in real practice.

Table 4. Comparison in cross-domain dataset scenario. Our MHFC is in transductive setting. $(\cdot)^{\flat}$ and $(\cdot)^{\sharp}$ indicate the reported results come from (Boudiaf et al., 2020) and (Mangla et al., 2020), respectively. The top two results are shown in red and blue. Method | mini-ImageNet $\longrightarrow$ CUB
---|---
$5$-way $1$-shot | 5-way 5-shot
Baseline♭ (Chen et al., 2019) | - | $53.1$
ProtoNet♭ (Snell et al., 2017) | - | $62.0$
RelationNet♭ (Sung et al., 2018) | - | $57.7$
GNN♭ (Tseng et al., 2020) | - | $66.9$
Neg-Cosine♭ (Liu et al., 2020) | - | $67.0$
LaplacianShot♭ (Ziko et al., 2020) | - | $66.3$
TIM-GD♭ (Boudiaf et al., 2020) | - | 71.0
MetaOpt♯ (Bertinetto et al., 2019) | $44.79\pm 0.75$ | $64.98\pm 0.68$
Manifold Mixup♯ (Verma et al., 2019) | $46.21\pm 0.77$ | $66.03\pm 0.71$
S2M2♯ (Mangla et al., 2020) | $\textbf{48.24}\pm 0.84$ | $70.44\pm 0.75$
MHFC† | $\textbf{61.57}\pm 1.28$ | $\textbf{78.80}\pm 0.78$

##  6\. Conclusion

Few-shot learning (FSL) based tasks have a fundamental problem, e.g.,
distribution-shift-problem (DSP). To address this challenge, we propose Multi-
Head Feature Collaboration (MHFC), which attempts to collaboratively represent
samples by fusing multi-head features. It is helpful to strengthen the FSL
based model’s efficacy and robustness. MHFC is a simple non-parametric method
that can directly employ the existing FEMs. Experimental results have
demonstrated the effectiveness of MHFC.

## Acknowledgment

The paper was supported by the National Natural Science Foundation of China
(Grant No. 62072468), the Natural Science Foundation of Shandong Province,
China (Grant No. ZR2019MF073), the Fundamental Research Funds for the Central
Universities, China University of Petroleum (East China) (Grant No.
20CX05001A), the Graduate Innovation Project of China University of Petroleum
(East China) (YCX2021117, YCX2021123).

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