Higher Obstructions for Complex Supermanifolds

Higher Obstructions of Complex Supermanifolds

Kowshik BETTADAPURA

K. Bettadapura

Yau Mathematical Sciences Center, Tsinghua University, Haidian, Beijing,
100084, China<EMAIL_ADDRESS>
Received April 29, 2018, in final form August 30, 2018; Published online
September 07, 2018

In this article we introduce the notion of a ‘good model’ in order to study
the higher obstructions of complex supermanifolds. We identify necessary and
sufficient conditions for such models to exist. Illustrations over Riemann
surfaces are provided.

complex supergeometry; supermanifolds; obstruction theory 32C11; 58A50

##  1 Introduction

Complex supermanifolds are spaces modelled on the data of a complex manifold
$X$ and holomorphic vector bundle $E\rightarrow X$. Accordingly, we refer to
the pair $(X,E)$ as a ‘model’. Supermanifolds can be either split or non-
split. A splitting is a global isomorphism to some fixed model space. Berezin
in [2, pp. 163–169] observed that to any complex supermanifold, there will be
associated a hierarchy of inductively defined cohomology classes representing
obstructions to the existence of a splitting.111In the terminology of Berezin,
a splitting is referred to as a ‘retraction’ and a split supermanifold is
referred to as ‘simple’. These classes are called _obstruction classes to
splitting_ , or simply ‘obstructions’. To any supermanifold one can always
define its _primary_ obstruction. If this vanishes, another will appear in its
place, sitting at a higher degree (in a suitable sense). Accordingly, these
classes are referred to as ‘higher obstructions’.

Donagi and Witten in [7] observed that the higher obstructions to splitting a
supermanifold $\mathfrak{X}$ might fail to represent a genuine obstruction to
splitting $\mathfrak{X}$. That is, it might well be that $\mathfrak{X}$ is
split even if it supports an atlas in which a higher obstruction to the
existence of a splitting does not vanish. Such a phenomenon seems to be
difficult to illustrate in practice. For instance, it was shown by the author
in [3] that certain deformations of super Riemann surfaces will _not_ support
such atlases.

In this article we are motivated by the following question:

When does a higher obstruction to splitting $\mathfrak{X}$ represent a genuine
obstruction? (1.1)

Following the terminology of Donagi and Witten, if a higher obstruction is not
genuine it is referred to as _exotic_.

We summarise the main ideas and results in this article below.

###  1.1 Article summary

Obstruction classes are associated to supermanifolds, while obstruction spaces
are associated to models. In our attempt to address the question in (1.1), we
will look to classify models $(X,E)$. In Definition 4.1 we introduce the
notion of a ‘good model’ $(X,E)$. This is a model on which every higher
obstruction for any supermanifold modelled on $(X,E)$ will be genuine. Lemma
4.2 clarifies the relation to the splitting problem. Our main result is
Theorem 4.3 where we obtain necessary and sufficient conditions for $(X,E)$ to
be a good model.

Section 5 is devoted to illustrations of models which are good and otherwise.
We restrict our attention to holomorphic vector bundles on Riemann surfaces.
Over the projective line $\mathbb{P}^{1}_{\mathbb{C}}$, when the bundle has
rank $3$, a general characterisation of good models is derived in Theorem 5.3.
It is based essentially on an example by Donagi and Witten in [7]. This is
then extended to higher rank for a particular class of bundles on
$\mathbb{P}^{1}_{\mathbb{C}}$ in Theorem 5.5.

The relation of higher obstructions to the splitting problem is subtle. In
Appendix A we submit a proof of the assertion in Theorem 2.12, being: if the
primary obstruction of a supermanifold $\mathfrak{X}$ does not vanish, then
$\mathfrak{X}$ is non-split. There is no reason for the analogous statement
for higher obstructions to hold. It is for this reason that we introduce
variants of splitting such as ‘strong splitting’ and ‘weak non-splitting’ in
Definition 3.6.

##  2 Complex supermanifolds

There are a number of different categories of supermanifolds and each comes
with its own level of subtlety as detailed in [1]. Those considered in this
article fall under the umbrella of the ‘algebro-geometric’ supermanifolds.
These are defined as locally ringed spaces which are modelled on a given
space. Standard reference include [5, 10]. One can however bypass that
language and give an equivalent definition in terms of Čech cohomology sets.
This is the approach we consider here since it is more convenient for the
purposes of this article.

###  2.1 Green’s automorphism groups

Let $X$ be a complex manifold and $E\rightarrow X$ a holomorphic vector
bundle. Throughout this article we will refer to the pair $(X,E)$ as a
_model_. Denote by $\mathcal{E}$ the sheaf of local sections of $E$. We can
then form the sheaf of exterior algebras $\wedge^{\bullet}\mathcal{E}$ on $X$.

######  Remark 2.1.

Note that $\wedge^{\bullet}\mathcal{E}$ is a sheaf of $\mathbb{Z}_{2}$-graded,
supercommutative algebras. It also admits a compatible $\mathbb{Z}$-grading.
We will however only consider automorphisms of $\wedge^{\bullet}\mathcal{E}$
which preserve the $\mathbb{Z}_{2}$-grading. The sheaf of such automorphisms
will be denoted
$\operatorname{{\mathcal{A}}ut}_{\mathbb{Z}_{2}}(\wedge^{\bullet}\mathcal{E})$.
However we will suppress the subscript ‘$\mathbb{Z}_{2}$’ so as to avoid
cumbersome notation and simply write
$\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}$.

Let $\mathcal{J}_{(X,E)}\subset\wedge^{\bullet}\mathcal{E}$ be the ideal
generated by local sections of $\mathcal{E}$. As an $\mathcal{O}_{X}$-module,
$\mathcal{J}_{(X,E)}=\oplus_{j>0}\wedge^{j}\mathcal{E}$. The following sheaf
of non-abelian groups were defined by Green in [8]:

$\displaystyle\mathcal{G}_{(X,E)}^{(k)}:=\big{\\{}\alpha\in\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\,|\,\alpha(u)-u\in\mathcal{J}_{(X,E)}^{k}\big{\\}}.$
(2.1)

The Čech cohomology of $X$ valued in $\mathcal{G}_{(X,E)}^{(k)}$ is of
particular importance. We present now our working definition of a
supermanifold.

######  Definition 2.2.

A _supermanifold modelled on $(X,E)$_ is an element of the Čech cohomology set
$\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(2)}\big{)}$.

Generally $\check{H}^{1}(X,G)$, for $G$ a sheaf of non-abelian groups on $X$,
will be a pointed set. The basepoint corresponds to the identity section of
$G$. That is, under the inclusion $\\{e\\}\rightarrow G$, for $\\{e\\}$ the
trivial group, one obtains a map
$\check{H}^{1}(X,\\{e\\})\rightarrow\check{H}^{1}(X,G)$ of pointed sets. Now
$\check{H}^{1}(X,\\{e\\})$ is a one-point set. The basepoint in
$\check{H}^{1}(X,G)$ is the image of this one-point set.

######  Definition 2.3.

The basepoint in $\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(2)}\big{)}$ is
referred to as _the split supermanifold_ modelled on $(X,E)$, or simply the
_split model_.

The 1-cohomology set $\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(2)}\big{)}$
will be referred to as the _set of supermanifolds modelled on $(X,E)$_.

###  2.2 Isomorphisms of supermanifolds

We will make use of an early observation of Grothendieck in [9], discussed
also in [4, p. 160], regarding sheaves of non-abelian groups.

######  Theorem 2.4.

Let $X$ be a $($paracompact$)$ topological space and suppose the following is
a short exact sequence of sheaves of $($not necessarily abelian$)$ groups on
$X$

$\displaystyle e\longrightarrow A\longrightarrow B\longrightarrow
C\longrightarrow e.$

Then there exists a long exact sequence

$\displaystyle e$ $\displaystyle\longrightarrow H^{0}(X,A)\longrightarrow
H^{0}(X,B)\longrightarrow H^{0}(X,C)$ (2.2)
$\displaystyle\longrightarrow\check{H}^{1}(X,A)\longrightarrow\check{H}^{1}(X,B)\longrightarrow\check{H}^{1}(X,C),$
(2.3)

where the sequence in (2.2) is as groups whereas that in (2.3) is as pointed
sets. Furthermore, there exists an action of $H^{0}(X,C)$ on the set
$\check{H}^{1}(X,A)$ such that the following diagram commutes

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\check{H}^{1}(X,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{H}^{1}(X,B)}$$\textstyle{\check{H}^{1}(X,A)/H^{0}(X,C).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We have so far defined supermanifolds as elements of a certain Čech cohomology
set. As for when two supermanifolds are isomorphic, this is defined as
follows. Firstly observe for $k=2$ the following short exact sequence of
sheaves of groups

$\displaystyle\\{e\\}\longrightarrow\mathcal{G}^{(2)}_{(X,E)}\longrightarrow\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\longrightarrow\operatorname{{\mathcal{A}}ut}\mathcal{E}\longrightarrow\\{e\\}.$
(2.4)

Applying Theorem 2.4 gives a long exact sequence on cohomology. Consider the
following piece

$\displaystyle\cdots\longrightarrow
H^{0}\big{(}X,\operatorname{{\mathcal{A}}ut}\mathcal{E}\big{)}\longrightarrow\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}\longrightarrow\check{H}^{1}\big{(}X,\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\big{)}\longrightarrow\cdots.$
(2.5)

Isomorphisms are defined as follows.

######  Definition 2.5.

Two supermanifolds modelled on $(X,E)$ are _isomorphic_ if and only if their
image in
$\check{H}^{1}\big{(}X,\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\big{)}$
coincide.

Note, $H^{0}(X,\operatorname{{\mathcal{A}}ut}\mathcal{E})$ coincides with the
group $\operatorname{Aut}(E)$ of global automorphisms. From the latter part of
Theorem 2.4 we have an action of the group $\operatorname{Aut}(E)$ on
$\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$ and a well-defined
map from the orbits to
$\check{H}^{1}\big{(}X,\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\big{)}$.
This leads to the following result, first described by Green in [8], on the
general classification of supermanifolds. As per the definitions we have made
it follows from Theorem 2.4.

######  Theorem 2.6.

There exists a bijective correspondence

$\displaystyle\frac{\left\\{\text{\rm supermanifolds modelled on
$(X,E)$}\right\\}}{\text{\rm up to
isomorphism}}\cong\frac{\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}}{\operatorname{Aut}(E)}.$

###  2.3 Obstruction theory

Recall that in Definition 2.3 we termed the basepoint in
$\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$ the _split model_.
With the notion of isomorphism in Definition 2.5 we have:

######  Definition 2.7.

A supermanifold if said to be _split_ if and only if it is isomorphic to the
split model.

Obstruction theory for supermanifolds is concerned with the following question
which is referred to as _the splitting problem_ :

######  Question 2.8.

Given a supermanifold, how can one tell if it is split?

The splitting problem has a long history dating back to its original
formulation by Berezin in [2], framed using the term ‘retraction’. Examples of
non-split supermanifolds were provided by Berezin and contemporaries including
Palamodov in [13], Green in [8] and Manin in [10]. Outside of mathematical
curiosity however, the relevance of the splitting problem to theoretical
physics is not so clear. Recent advances by Donagi and Witten in [6, 7] serve
to show that non-splitting is a phenomenon that, at least in superstring
theory, cannot be ignored. As our starting point in addressing the splitting
problem, we have the following result by Green in [8].

######  Proposition 2.9.

For each $k\geq 2$ there exists a short exact sequence of sheaves of groups

$\displaystyle\\{e\\}\longrightarrow\mathcal{G}^{(k+1)}_{(X,E)}\longrightarrow\mathcal{G}^{(k)}_{(X,E)}\longrightarrow\mathcal{Q}^{(k)}_{(X,E)}\longrightarrow\\{e\\},$

where the inclusion
$\mathcal{G}^{(k+1)}_{(X,E)}\rightarrow\mathcal{G}^{(k)}_{(X,E)}$ is normal
and the factor $\mathcal{Q}^{(k)}_{(X,E)}$ is abelian.

An important utility of Proposition 2.9 is in relating non-abelian cohomology
to the cohomology of abelian sheaves. By Theorem 2.4 the short exact sequence
in Proposition 2.9 will induce, for each $k$, a long exact sequence on
cohomology. The piece of most relevance for our present purposes is

$\displaystyle\cdots\longrightarrow\check{H}^{1}\big{(}X,\mathcal{G}^{(k+1)}_{(X,E)}\big{)}\longrightarrow\check{H}^{1}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}\stackrel{{\scriptstyle\omega_{*}}}{{\longrightarrow}}H^{1}\big{(}X,\mathcal{Q}^{(k)}_{(X,E)}\big{)},$
(2.6)

where in the latter we have used the identification of Čech cohomology and
sheaf cohomology for abelian sheaves.

######  Definition 2.10.

To any model $(X,E)$ we term the following important constructs for each
$k\geq 2$:

  * •

the sheaf $\mathcal{Q}^{(k)}_{(X,E)}$ in Proposition 2.9 is referred to as the
$k$-th _obstruction sheaf_ ;

  * •

the cohomology group $H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}$ is
referred to as the _$k$ -th obstruction space_.

Inspecting (2.6) shows: there will exist a class in the second obstruction
space associated to _any_ supermanifold $\mathfrak{X}$ modelled on $(X,E)$. We
denote this class by $\omega_{*}(\mathfrak{X})$.

######  Definition 2.11.

Let $\mathfrak{X}$ be a supermanifold modelled on $(X,E)$. The class
$\omega_{*}(\mathfrak{X})$ in the second obstruction space will be referred to
as the _primary obstruction_ to splitting $\mathfrak{X}$.

The terminology in the above definition is justified in the following
classical result.

######  Theorem 2.12.

Let $\mathfrak{X}$ be a supermanifold. If its primary obstruction is non-
vanishing, then $\mathfrak{X}$ is non-split.

###### Proof.

A proof of this theorem, stated using the notion of ‘retractibility,’ can be
found in [2, Theorem 4.6.2, p. 158]. For completeness, we present a proof
based on our formulation of isomorphisms of supermanifolds in Definition 2.5.
As the proof is a little involved, we defer it to Appendix A. ∎

Note that Theorem 2.12 above cleanly addresses the splitting problem, as posed
in Question 2.8. If the primary obstruction to splitting vanishes however, it
far more subtle to adequately address this problem.

##  3 Higher obstructions

###  3.1 Higher atlases

We are interested in the splitting problem in instances where the primary
obstruction vanishes. To that extent we begin with the following string of
definitions.

######  Definition 3.1.

An element in $\check{H}^{1}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}$ is
referred to as a _$(k-1)$ -split atlas_ for a supermanifold modelled on
$(X,E)$.

######  Definition 3.2.

If a supermanifold $\mathfrak{X}$ lies in the image of a $(k-1)$-split atlas
$\mathfrak{X}^{(k)}$, then $\mathfrak{X}$ is said to be _$(k-1)$ -split_.
Conversely, the image of $\mathfrak{X}^{(k)}$ in
$\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$ will be referred to
as the _supermanifold associated to $\mathfrak{X}^{(k)}$_.

Associated to any $(k-1)$-split atlas will be a class in the $k$-th
obstruction space, which is the cohomology group
$H^{1}\big{(}X,\mathcal{Q}^{(k)}_{(X,E)}\big{)}$. Following Definition 2.11 we
have:

######  Definition 3.3.

The class in the $k$-th obstruction space associated to any $(k-1)$-split
atlas will be referred to as the _primary obstruction of the $(k-1)$-split
atlas_.

######  Definition 3.4.

If the primary obstruction of a $(k-1)$-split atlas $\mathfrak{X}^{(k)}$ is
non-vanishing, then $\mathfrak{X}^{(k)}$ is said to be _obstructed_.

The primary obstructions associated to $(k-1)$-split atlases are precisely the
‘higher’ obstruction classes associated to supermanifolds, as described by
Berezin (see [2, p. 164]) and more recently by Donagi and Witten in [7, p.
15]. In these texts the higher obstructions were defined inductively as
follows. Starting with a $(k-1)$-split atlas $\mathfrak{X}^{(k)}$ for a
supermanifold $\mathfrak{X}$, one wants to lift this to a $k$-split atlas for
$\mathfrak{X}$. The obstruction to doing so is precisely the primary
obstruction of $\mathfrak{X}^{(k)}$. The primary obstruction of
$\mathfrak{X}^{(k)}$ _is_ the $(k-1)$-th obstruction to splitting
$\mathfrak{X}$. Hence, the $k$-th obstruction to splitting $\mathfrak{X}$ is
defined if and only if the $(k-1)$-th obstruction to splitting $\mathfrak{X}$
vanishes.

###  3.2 The weak splitting problem

To see the relation of higher obstructions to the splitting problem, note that
the groups $\mathcal{G}^{(k)}_{(X,E)}$, defined in (2.1), will be trivial for
$k$ sufficiently large. Indeed, if $q$ denotes the rank of the holomorphic
vector bundle $E\rightarrow X$, then $\mathcal{J}_{(X,E)}^{k}=(0)$ for all
$k>q$. Hence $\mathcal{G}^{(k)}_{(X,E)}=\\{e\\}$ for all $k>q$. This leads to
the following result, which can also be found (albeit phrased differently) in
[2, 13]:

######  Theorem 3.5.

A supermanifold $\mathfrak{X}$ is split if and only if it admits a
$(k-1)$-split atlas for $k>q$.

We present now a weaker notion of splitting for higher atlases suited to the
purposes of this article.

######  Definition 3.6.

A $(k-1)$-split atlas $\mathfrak{X}^{(k)}$, $k>2$, is said to be _weakly non-
split_ if its associated supermanifold does not coincide with the basepoint.
Otherwise, it is said to be _strongly split_. We say $\mathfrak{X}^{(k)}$ is
split or non-split if its associated supermanifold is split or non-split.

Note that non-split will imply _weakly_ non-split but not necessarily
conversely. Similarly, strongly split will imply split, but not necessarily
conversely. These notions lead to the following variant of the splitting
problem.

######  Question 3.7.

When will a given $(k-1)$-split atlas be weakly non-split?

We refer to our proof of Theorem 2.12 in Appendix A for an illustration of the
subtleties involved in resolving the splitting problem as outlined in Question
2.8.

###  3.3 Exotic atlases

In [7] it was observed that the map
$\check{H}^{1}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}\rightarrow\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$
will generally fail to be either injective or surjective. This means the
existence of an obstructed, $(k-1)$-split atlas for a supermanifold
$\mathfrak{X}$ does _not_ imply $\mathfrak{X}$ will be non-split – in contrast
with the $k=2$ case in Theorem 2.12. The following definition captures
precisely those obstructed atlases which represent split supermanifolds.

######  Definition 3.8.

A $(k-1)$-split atlas $\mathfrak{X}^{(k)}$, for $k>2$, is said to be _exotic_
if it entertains the following two properties:

  1. (i)

$\mathfrak{X}^{(k)}$ is obstructed;

  2. (ii)

$\mathfrak{X}^{(k)}$ is strongly split.

With the sequence in Proposition 2.9 we can describe an exotic atlas more
concretely. Inspecting the cohomology sequence in (2.6) one step to the left
and for the sequence of groups
$\mathcal{G}^{(k-1)}_{(X,E)}\rightarrow\mathcal{G}^{(k)}_{(X,E)}\rightarrow\mathcal{Q}^{(k-1)}_{(X,E)}$
gives

$\displaystyle\cdots$ $\displaystyle\longrightarrow
H^{0}\big{(}X,\mathcal{Q}^{(k-1)}_{(X,E)}\big{)}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\check{H}^{1}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}$
(3.1)
$\displaystyle\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\check{H}^{1}\big{(}X,\mathcal{G}^{(k-1)}_{(X,E)}\big{)}\longrightarrow
H^{1}\big{(}X,\mathcal{Q}^{(k-1)}_{(X,E)}\big{)}.$ (3.2)

We prove:

######  Lemma 3.9.

Let $\mathfrak{X}^{(k)}$ be an obstructed, $(k-1)$-split atlas for $k>2$. If
there exists a non-zero $\phi\in
H^{0}\big{(}X,\mathcal{Q}^{(k-1)}_{(X,E)}\big{)}$ such that
$\alpha(\phi)=\mathfrak{X}^{(k)}$, then $\mathfrak{X}^{(k)}$ will be exotic.

###### Proof.

Let $\mathfrak{X}^{(k)}$ be a $(k-1)$-split atlas and suppose
$\alpha(\phi)=\mathfrak{X}^{(k)}$, for some non-zero $\phi\in
H^{0}\big{(}X,\mathcal{Q}^{(k-1)}_{(X,E)}\big{)}$. In assuming
$\mathfrak{X}^{(k)}$ is obstructed, it cannot represent the basepoint in
$\check{H}^{1}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}$. Now by exactness of
the sequence in (3.1)–(3.2) note, under $\beta$, that $\mathfrak{X}^{(k)}$
will map to the basepoint in
$\check{H}^{1}\big{(}X,\mathcal{G}^{(k-1)}_{(X,E)}\big{)}$. Hence it will map
to the basepoint in $\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$.
Thus $\mathfrak{X}^{(k)}$ is obstructed and strongly split. Hence
$\mathfrak{X}^{(k)}$ is exotic. ∎

Further assumptions on the global sections of the obstruction sheaves yield
the following sufficient conditions forbidding the existence of exotic
atlases.

######  Proposition 3.10.

Suppose that $H^{0}\big{(}X,\mathcal{Q}^{(k)}_{(X,E)}\big{)}=(0)$ for all
$k>2$. Then there do not exist exotic atlases for any supermanifold modelled
on $(X,E)$.

###### Proof.

Suppose $H^{0}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}=(0)$. Then from (3.1)
we have the exact sequence of pointed sets

$\displaystyle\\{e\\}\longrightarrow\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(k+1)}\big{)}\stackrel{{\scriptstyle\iota_{*}^{k+1}}}{{\longrightarrow}}\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(k)}\big{)}.$
(3.3)

Unlike for rings or modules, an exact sequence as in (3.3) above need not
imply $\iota_{*}^{k+1}$ will be injective as a map of sets. But for
$\mathfrak{X}^{(k+1)}\in\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(k+1)}\big{)}$
we can nevertheless conclude the following implication from exactness of
(3.3):

$\displaystyle\big{(}\iota_{*}^{k+1}\big{(}\mathfrak{X}^{(k+1)}\big{)}=\\{e\\}\big{)}\Longrightarrow\big{(}\mathfrak{X}^{(k+1)}=\\{e\\}\big{)}.$
(3.4)

Assume now that $H^{0}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}=(0)$ for all
$k$. Then the implication (3.4) will hold for all $k$. Let
$\mathfrak{X}^{(k+1)}\in\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(k+1)}\big{)}$
and suppose it maps to the basepoint in
$\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(2)}\big{)}$. It will then follow
from (3.4) that $\mathfrak{X}^{(k+1)}=\\{e\\}$. Hence $\mathfrak{X}^{(k+1)}$
cannot be obstructed and so, by definition, cannot be exotic. ∎

######  Remark 3.11.

For $E\rightarrow X$ a holomorphic, rank $q$ vector bundle it is superfluous
to assume $H^{0}\big{(}X,\mathcal{Q}^{(q)}_{(X,E)}\big{)}=(0)$ in Proposition
3.10. This is because $\mathcal{G}^{(k)}_{(X,E)}=\\{e\\}$ for all $k>q$,
meaning we already have
$\\{e\\}\rightarrow\check{H}^{1}\big{(}X,\mathcal{G}^{(q)}_{(X,E)}\big{)}\rightarrow\check{H}^{1}\big{(}X,\mathcal{G}^{(q-1)}_{(X,E)}\big{)}$.
That is, Proposition 3.10 will be valid under the slightly weakened assumption
that $H^{0}\big{(}X,\mathcal{Q}^{(k)}_{(X,E)}\big{)}=(0)$ for all $2<k<q$.

##  4 Good models

The existence of an exotic atlas depends essentially on the model $(X,E)$.
This leads to the following definition.

######  Definition 4.1.

A model $(X,E)$ is said to be _good_ if there do not exist any exotic atlases
for any supermanifold modelled on $(X,E)$. Otherwise, it is said to support
exotic atlases.

The prime motivation for introducing good models lies in the following result
which follows from the definitions.

######  Lemma 4.2.

Let $(X,E)$ be a good model. Then if a $(k-1)$-split atlas modelled on $(X,E)$
is obstructed, it will be weakly non-split.

It is a meaningful endeavour to understand when a given model $(X,E)$ will be
good. In Proposition 3.10 we found sufficient conditions in this vein. We
arrive now at our main result in this article, concerning both necessary and
sufficient conditions for a model to be good.

######  Theorem 4.3.

A model $(X,E)$ will be a good if and only if taking global sections preserves
exactness of the sequence

$\displaystyle\\{e\\}\longrightarrow\mathcal{G}^{(k+1)}_{(X,E)}\longrightarrow\mathcal{G}^{(k)}_{(X,E)}\longrightarrow\mathcal{Q}^{(k)}_{(X,E)}\longrightarrow\\{e\\}$

for all $k\geq 2$.

###### Proof.

From the exact sequence of sheaves of groups
$\mathcal{G}^{(k+1)}_{(X,E)}\rightarrow\mathcal{G}^{(k)}_{(X,E)}\rightarrow\mathcal{Q}^{(k)}_{(X,E)}$
we have the following piece of the long exact sequence on cohomology

$\displaystyle\\{e\\}$ $\displaystyle\longrightarrow
H^{0}\big{(}X,\mathcal{G}^{(k+1)}_{(X,E)}\big{)}\longrightarrow
H^{0}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}\longrightarrow
H^{0}\big{(}X,\mathcal{Q}^{(k)}_{(X,E)}\big{)}$
$\displaystyle\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\check{H}^{1}\big{(}X,\mathcal{G}^{(k+1)}_{(X,E)}\big{)}\longrightarrow\cdots.$
(4.1)

In the direction $(\Leftarrow)$ in the statement of this theorem, assume (4.1)
is exact for all $k\geq 2$. Then either
$H^{0}\big{(}X,\mathcal{Q}^{(k)}_{(X,E)}\big{)}=(0)$ for all $k\geq 2$ or the
map $\alpha$ is constant, sending every $\phi\in
H^{0}\big{(}X,\mathcal{Q}^{(k-1)}_{(X,E)}\big{)}$ to the basepoint $\\{e\\}$.
In either case, the proof of Proposition 3.10 will apply since we will obtain
exact sequences in (3.3) for all $k$. In the direction $(\Rightarrow)$,
suppose $(X,E)$ is a good model. We will show that $\alpha$ is constant.
Firstly, by definition we know that there will not exist any exotic atlases.
Therefore, for any $\phi\in H^{0}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}$,
the atlas $\alpha(\phi)$ must be an unobstructed, $k$-split atlas, for
otherwise it would be exotic by Lemma 3.9. Observe that as a result we have
the following lifting $\widetilde{\alpha}$ of $\alpha$, represented by the
dashed arrow

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
30.73166pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 58.14832pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\check{H}^{1}\big{(}X,\mathcal{G}^{(k+2)}_{(X,E)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
86.58583pt\raise-30.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
144.5185pt\raise-30.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{0}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
34.67137pt\raise-14.63889pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.61111pt\hbox{$\scriptstyle{\widetilde{\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 73.34462pt\raise-6.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.05396pt\raise-37.99306pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 58.14832pt\raise-42.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
58.14832pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\check{H}^{1}\big{(}X,\mathcal{G}^{(k+1)}_{(X,E)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
86.58583pt\raise-63.75pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.04167pt\hbox{$\scriptstyle{\omega_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 86.58583pt\raise-73.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
142.44pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
142.44pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\check{H}^{1}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}}$}}}}}}}{\hbox{\kern-3.0pt\raise-85.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
54.73166pt\raise-85.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H^{1}\big{(}X,\mathcal{Q}^{(k+1)}_{(X,E)}\big{)}.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(4.2)

The lifting $\widetilde{\alpha}$ exists since the vertical maps in (4.2) are
exact and unobstructedness of $\alpha(\phi)$ means
$\omega_{*}\big{(}\alpha(\phi)\big{)}=0$. Hence from $\phi$ we obtain a
$(k+1)$-split atlas $\widetilde{\alpha}(\phi)$. We claim that
$\widetilde{\alpha}(\phi)$ is also unobstructed. To see this note that (4.2)
will commute and so $\widetilde{\alpha}(\phi)$ will map to the basepoint in
$\check{H}^{1}\big{(}X,\mathcal{G}^{(k)}_{(X,E)}\big{)}$. As such it will map
to the basepoint in $\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$
and so be split. This means, if $\widetilde{\alpha}(\phi)$ were obstructed, it
would be exotic, contradicting that $(X,E)$ is a good model. Hence
$\widetilde{\alpha}(\phi)$ must be unobstructed. Applying this argument again
to $\widetilde{\alpha}(\phi)$ will show that $\widetilde{\alpha}$ will itself
lift to some $\widetilde{\alpha}^{\prime}$ valued in
$\check{H}^{1}\big{(}X,\mathcal{G}^{(k+3)}_{(X,E)}\big{)}$ and that to $\phi$
will be associated an unobstructed, $(k+2)$-split atlas. We can proceed
inductively now and keep lifting the map $\alpha$ to get unobstructed,
$k^{\prime}$-split atlases for any $k^{\prime}>k$, resulting in the
commutative diagram

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
30.73166pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 59.25835pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\check{H}^{1}\big{(}X,\mathcal{G}^{(k^{\prime})}_{(X,E)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
84.55807pt\raise-30.6111pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{0}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.26753pt\raise-13.77777pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-4.52779pt\hbox{$\scriptstyle{\widetilde{\alpha}^{k^{\prime}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 71.65964pt\raise-6.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
37.04008pt\raise-38.10416pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
54.73166pt\raise-42.6111pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
54.73166pt\raise-42.6111pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\check{H}^{1}\big{(}X,\mathcal{G}^{(k+1)}_{(X,E)}\big{)}.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(4.3)

Since $\mathcal{G}^{(k^{\prime})}_{(X,E)}$ is trivial for $k^{\prime}$
sufficiently large, commutativity of (4.3) requires $\alpha$ map $\phi$ to the
basepoint $\\{e\\}$ for all $\phi$. Thus $\alpha$ is constant. This argument
is independent of $k\geq 2$ and depends only on $(X,E)$ being a good model.
The theorem now follows. ∎

######  Remark 4.4.

Concerning the diagram (4.2), it is argued in [2, Theorem 4.7.1, pp. 163–164]
and in [7, p. 16] that there will always exist a lift $\widetilde{\alpha}$ of
$\alpha$ when $k$ is odd.

##  5 Illustrations

To convince the reader that exotic atlases and good models exist we present
some illustrations. It will be convenient to firstly present the following
more explicit characterisation of the obstruction sheaves which we will make
use of in our applications. A justification can be found in [8].

######  Lemma 5.1.

To any model $(X,E)$ there exists an isomorphism of sheaves

$\displaystyle\mathcal{Q}^{(k)}_{(X,E)}=\begin{cases}\wedge^{k}\mathcal{E}\otimes
T_{X},&\text{$k$ is even},\\\
\mathcal{E}^{*}\otimes\wedge^{k}\mathcal{E},&\text{$k$ is odd}.\end{cases}$
(5.1)

###  5.1 On Riemann surfaces

Let $X$ be a complex manifold and $E\rightarrow X$ a holomorphic vector bundle
of rank $3$. Then $\mathcal{G}_{(X,E)}^{(k)}=\\{e\\}$ for all $k>3$. With the
identifications of the obstruction sheaves in (5.1) we obtain the following
long exact sequence, corresponding to the more general case in (4.1)

$\displaystyle\\{e\\}$ $\displaystyle\longrightarrow
H^{0}\big{(}X,\mathcal{G}_{(X,E)}^{(3)}\big{)}\longrightarrow
H^{0}\big{(}X,\mathcal{G}_{(X,E)}^{(2)}\big{)}\longrightarrow
H^{0}\big{(}X,\wedge^{2}\mathcal{E}\otimes T_{X}\big{)}$
$\displaystyle\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(3)}\big{)}\longrightarrow\cdots.$
(5.2)

Theorem 4.3 says $(X,E)$ will be a good model if and only if $\alpha$ in (5.2)
vanishes. This is generally a non-trivial condition to check. It will be true
however if $h^{0}\big{(}X,\wedge^{2}\mathcal{E}\otimes T_{X}\big{)}=0$.222For
any abelian sheaf $\mathcal{A}$ on $X$ we denote $h^{i}(X,\mathcal{A}):=\dim
H^{i}(X,\mathcal{A})$.

######  Proposition 5.2.

Let $X$ be a Riemann surface of genus $g$ and $E\rightarrow X$ a rank $3$,
holomorphic vector bundle. Suppose it splits into a sum of non-negative,
holomorphic line bundles and $\deg E<3g-3$. Then $(X,E)$ will be a good model.

###### Proof.

It suffices to show $h^{0}\big{(}X,\wedge^{2}\mathcal{E}\otimes
T_{X}\big{)}=0$. Let $X$ be a genus $g$ Riemann surface. Assuming $E$ is
holomorphically split we can write $E=L_{d_{1}}\oplus L_{d_{2}}\oplus
L_{d_{3}}$, for $L_{d_{i}}\rightarrow X$ a line bundle on $X$ of degree
$d_{i}$. If $L_{d_{i}}$ is non-negative, then $d_{i}\geq 0$. Note that
$\wedge^{2}E=L_{d_{1}+d_{2}}\oplus L_{d_{1}+d_{3}}\oplus L_{d_{2}+d_{3}}$; and
$\deg\wedge^{2}E=2\deg E=2(d_{1}+d_{2}+d_{3})$. With $\deg T_{X}=2-2g$ and the
above description of $\wedge^{2}E$ we see that
$h^{0}\big{(}\wedge^{2}\mathcal{E}\otimes T_{X}\big{)}=0$ when

$\displaystyle d_{1}+d_{2}<2g-2,\qquad d_{1}+d_{3}<2g-2\qquad\text{and}\qquad
d_{2}+d_{3}<2g-2.$

These conditions are equivalent to $\deg E<3g-3$ when $d_{i}\geq 0$ for each
$i$. ∎

###  5.2 In genus zero

We can be considerably more specific on the projective line.

######  Theorem 5.3.

Let $E=L_{d_{1}}\oplus L_{d_{2}}\oplus L_{d_{3}}$ be a rank $3$, holomorphic
vector bundle on $\mathbb{P}^{1}_{\mathbb{C}}$, where $L_{d_{i}}$ is a line
bundle on $\mathbb{P}^{1}_{\mathbb{C}}$ of degree $d_{i}$. Then
$(\mathbb{P}_{\mathbb{C}}^{1},E)$ will be a good model if and only if at least
one of $d_{i}\neq-1$.

Before giving a proof of Theorem 5.3 we provide some preliminary remarks about
exotic atlases. Regarding their existence, this was addressed in [7, pp.
15–16] for the present situation, i.e., when $E$ has rank $3$. We comment on
this here, but in a little more generality. Let $E\rightarrow X$ be a
holomorphic vector bundle of rank $q$. Note that
$\mathcal{G}_{(X,E)}^{(k)}=\\{e\\}$ for all $k>q$. Hence
$\mathcal{G}^{(q)}_{(X,E)}\cong\mathcal{Q}_{(X,E)}^{(q)}$ which implies
$\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(q)}\big{)}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(q)}\big{)}$.
With the exact sequence

$\displaystyle\cdots\longrightarrow
H^{0}\big{(}X,\mathcal{Q}^{(q-1)}_{(X,E)}\big{)}\longrightarrow\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(q)}\big{)}\longrightarrow\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(q-1)}\big{)}\longrightarrow\cdots$

we deduce:

######  Lemma 5.4.

Let $E\rightarrow X$ be a rank $q$, holomorphic vector bundle. Any
$(q-1)$-split atlas in
$\check{H}^{1}\big{(}X,\mathcal{G}_{(X,E)}^{(q)}\big{)}$ in the image of a
non-trivial section in $H^{0}\big{(}X,\mathcal{Q}^{(q-1)}_{(X,E)}\big{)}$ will
be obstructed and hence, by Lemma 3.9, exotic.

We now present a proof of Theorem 5.3.

###### Proof of Theorem 5.3.

Write
$\mathcal{E}=\mathcal{O}(d_{1})\oplus\mathcal{O}(d_{2})\oplus\mathcal{O}(d_{3})$,
where $\mathcal{E}$ denotes the sheaf of sections of $E$. Then

$\displaystyle\mathcal{Q}^{(2)}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}=\mathcal{O}(d_{1}+d_{2}+2)\oplus\mathcal{O}(d_{1}+d_{3}+2)\oplus\mathcal{O}(d_{2}+d_{3}+2)\qquad\text{and}$
$\displaystyle\mathcal{Q}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}^{(3)}=\mathcal{O}(d_{1}+d_{2})\oplus\mathcal{O}(d_{1}+d_{3})\oplus\mathcal{O}(d_{2}+d_{3}).$

In order to construct an exotic atlas for a supermanifold modelled on
$\big{(}\mathbb{P}^{1}_{\mathbb{C}},E\big{)}$ it will be necessary for

  1. (i)

$h^{0}\big{(}\mathcal{Q}^{(2)}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}\big{)}\neq 0$
(by Proposition 3.10 and Remark 3.11) and

  2. (ii)

$h^{1}\big{(}\mathcal{Q}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}^{(3)}\big{)}\neq 0$
(since
$\mathcal{G}^{(3)}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}\cong\mathcal{Q}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}^{(3)}$
here).

From Bott’s formula on the dimension of the cohomology of line bundles on
projective space,333See, e.g., [11, p. 4]. both (i) and (ii) will be satisfied
if and only if $(d_{1},d_{2},d_{3})=(-1,-1,-1)$. Now in this case, where
$(d_{1},d_{2},d_{3})=(-1,-1,-1)$, we have by Serre duality

$\displaystyle
H^{0}\big{(}\mathbb{P}_{\mathbb{C}}^{1},\mathcal{Q}^{(2)}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}\big{)}\cong
H^{1}\big{(}\mathbb{P}_{\mathbb{C}}^{1},\mathcal{Q}^{(3)}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}\big{)}.$

Upon observing
$H^{1}\big{(}\mathbb{P}_{\mathbb{C}}^{1},\mathcal{Q}^{(3)}_{(\mathbb{P}^{1}_{\mathbb{C}},E)}\big{)}\cong\check{H}^{1}\big{(}\mathbb{P}^{1}_{\mathbb{C}},\mathcal{G}^{(3)}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}\big{)}$,
the theorem will then follow from Lemma 5.4 applied to $q=3$. ∎

We will now consider those bundles of higher rank which decompose into copies
of a single line bundle, i.e., $E=\oplus^{q}L_{d}$. As we will see, the case
$d=-1$ will be similar to that in Theorem 5.3.

######  Theorem 5.5.

Let $E\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ be a rank $q$, holomorphic
vector bundle. Suppose $E=\oplus^{q}L_{d}$ where
$L_{d}\rightarrow\mathbb{P}_{\mathbb{C}}^{1}$ is a line bundle of degree $d$.
Then

  1. $(i)$

if $d=-1$ then $\big{(}\mathbb{P}^{1}_{\mathbb{C}},E\big{)}$ will support
exotic atlases;

  2. $(ii)$

if $d<-1$, then $\big{(}\mathbb{P}^{1}_{\mathbb{C}},E\big{)}$ will be a good
model.

###### Proof.

Let $\mathcal{O}(d)$ denote the sheaf of sections of $L_{d}$ so that
$\mathcal{E}=\oplus^{q}\mathcal{O}(d)$. We begin by proving (ii). We have
generally $\wedge^{k}\mathcal{E}=\oplus^{\binom{q}{k}}\mathcal{O}(kd)$. As
such the obstruction sheaves are given by

$\displaystyle\mathcal{Q}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(k)}=\begin{cases}\oplus^{\binom{q}{k}}\mathcal{O}(kd+2),&\text{$k$
is even},\\\ \oplus^{k\binom{q}{k}}\mathcal{O}\big{(}(k-1)d\big{)},&\text{$k$
is odd}.\end{cases}$ (5.3)

Note that $2\leq k\leq q$ here. Evidently, if $d<-1$ the obstruction sheaves
will be sums of line bundles of negative degree. Hence
$H^{0}\big{(}\mathbb{P}_{\mathbb{C}}^{1},\mathcal{Q}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(k)}\big{)}=(0)$.
Part (ii) then follows from Proposition 3.10. As for part (i), firstly note
that

$\displaystyle\wedge^{\bullet}\mathcal{E}$
$\displaystyle=\wedge^{0}\mathcal{E}\oplus\wedge^{1}\mathcal{E}\oplus\wedge^{2}\mathcal{E}\oplus\cdots\oplus\wedge^{q}\mathcal{E}$
$\displaystyle=\mathcal{O}(0)\oplus\big{[}\oplus^{q}\mathcal{O}(d)\big{]}\oplus\big{[}\oplus^{\binom{q}{2}}\mathcal{O}(2d)\big{]}\oplus\cdots\oplus\mathcal{O}(qd).$

We have
$\operatorname{{\mathcal{A}}ut}\mathcal{E}\\!\subset\\!\oplus^{q^{2}}\mathcal{O}(0)$
and
$\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\\!\subset\\!\oplus_{0\leq
a\leq b\leq q}\mathcal{O}((b-a)d)$. Note in particular that $b-a\geq 0$.
Regarding the sheaf $\mathcal{G}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(k)}$
observe that it can be realised as a subsheaf of $\oplus_{0\leq a<b\leq
q}\mathcal{O}((b-a)d)$ where now $b-a\geq k-1$. Therefore if $d<0$:

$\displaystyle
H^{0}\big{(}\mathbb{P}_{\mathbb{C}}^{1},\mathcal{G}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(k)}\big{)}=\\{1\\},$
(5.4)

for all $k\geq 2$. Now if $d=-1$ then
$h^{0}\big{(}\mathbb{P}^{1}_{\mathbb{C}},\mathcal{Q}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(2)}\big{)}=\binom{q}{2}\neq
0$ from (5.3). Since
$H^{0}\big{(}\mathbb{P}_{\mathbb{C}}^{1},\mathcal{G}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(2)}\big{)}=\\{e\\}$
from (5.4) we have the exact sequence

$\displaystyle\\{1\\}\longrightarrow
H^{0}\big{(}\mathbb{P}^{1}_{\mathbb{C}},\mathcal{Q}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(2)}\big{)}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\check{H}^{1}\big{(}\mathbb{P}^{1}_{\mathbb{C}},\mathcal{G}_{(\mathbb{P}_{\mathbb{C}}^{1},E)}^{(3)}\big{)},$

where $\alpha$ is the boundary map from (5.2). In particular $\alpha$ does not
vanish and any atlas in its image will be exotic. ∎

###  5.3 In genus one

Here we can relax some of conditions in Proposition 5.2, although we cannot be
as general as in the genus zero case.

######  Proposition 5.6.

Let $C$ be a Riemann surface of genus one and $E\rightarrow C$ a holomorphic,
rank $3$ vector bundle. Then $(C,E)$ will be a good model when either:

  1. $(i)$

$\deg\mathcal{Q}_{(C,E)}^{(2)}=h^{0}\big{(}\mathcal{Q}_{(C,E)}^{(2)}\big{)}$
or

  2. $(ii)$

$\deg\mathcal{Q}_{(C,E)}^{(2)}=-h^{1}\big{(}\mathcal{Q}_{(C,E)}^{(2)}\big{)}$.

###### Proof.

In genus one note that $T_{C}=\mathcal{O}_{C}$ is a line bundle of degree
zero. Hence we have a natural isomorphism
$\mathcal{Q}_{(C,E)}^{(2)}\cong\mathcal{Q}_{(C,E)}^{(3)}$. The proposition now
follows from the Riemann–Roch theorem. ∎

##  Appendix A Proof of Theorem 2.12

We will firstly present an erroneous proof as it will be instructive in
illustrating the subtlety involved in the splitting problem.

###### False proof.

Let $\mathfrak{X}$ be a supermanifold modelled on $(X,E)$ and suppose its
primary obstruction $\omega_{*}(\mathfrak{X})$ does not vanish. The map
$\omega_{*}\colon\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}\rightarrow
H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(2)}\big{)}$ is a map of pointed sets,
sending the basepoint
$\\{e\\}\in\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$ to the
basepoint $0\in H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(2)}\big{)}$. Since
$\omega_{*}(\mathfrak{X})\neq 0$, $\mathfrak{X}$ cannot be split. ∎

The fault in the above reasoning lies in our failure to consider isomorphisms
induced by the global symmetries $\operatorname{Aut}(E)$ acting on
$\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$ (cf. Definition 2.5
and Theorem 2.6). If $\operatorname{Aut}(E)$ fixes the basepoint $\\{e\\}$
then the above argument would be valid, but there is of course no reason for
it to fix the basepoint in general. Consider instead the following

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Aut}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
83.26518pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\delta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 100.83725pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
100.83725pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
113.19397pt\raise-21.25pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.04167pt\hbox{$\scriptstyle{\omega_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 125.55115pt\raise-30.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
181.67169pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
181.67169pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\check{H}^{1}\big{(}X,\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
273.31345pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
273.31345pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}{\hbox{\kern-3.0pt\raise-42.5pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
47.0903pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
93.4306pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(2)}\big{)}.}$}}}}}}}{\hbox{\kern
212.49257pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
277.06345pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(A.1)

Since (A.1) is exact we have: if a supermanifold $\mathfrak{X}$ is split, then
$\mathfrak{X}=\delta(\phi)$ for some $\phi\in\operatorname{Aut}(E)$. Observe
that Theorem 2.12 will then follow from:

######  Proposition A.1.

The composition of maps $\omega_{*}\delta$ in (A.1) vanishes.

The proof we submit of Proposition A.1 is based on the discussion in [7, p.
16]. We will cite the following result whose proof can be found in [12].

######  Lemma A.2.

Let $(X,E)$ be a model. The subgroup $\mathbb{C}^{\times}\cdot
1_{E}<\operatorname{Aut}(E)$ acts on the $k$-th obstruction space
$H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}$ by sending
$v\mapsto\lambda^{k}v$ for any vector $v\in
H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(k)}\big{)}$.

###### Proof of Proposition A.1.

From Theorem 2.4 we know that $\operatorname{Aut}(E)$ will act on
$\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$. Denote this action
by $\star$. Our first claim is that the subgroup $\mathbb{C}^{\times}\cdot
1_{E}$ will act trivially on the image of $\operatorname{Aut}(E)$. To see
this, let $(U_{i}\rightarrow X)$ be an open cover and
$\phi\in\operatorname{Aut}(E)$. If
$\widetilde{\phi}_{i}\in\big{(}\operatorname{{\mathcal{A}}ut}\wedge^{\bullet}\mathcal{E}\big{)}(U_{i})$
are local lifts of $\phi_{i}$, then

$\displaystyle\delta(\phi)_{ij}=\widetilde{\phi}_{i}\widetilde{\phi}_{j}^{-1}.$

Now clearly any $\lambda\cdot 1_{E}\in\mathbb{C}^{\times}\cdot
1_{E}<\operatorname{Aut}(E)$ will act by sending
$\phi_{i}\mapsto\lambda\phi_{i}$. Hence,

$\displaystyle\big{(}\lambda\cdot
1_{E}\big{)}\star\delta(\phi)_{ij}=\big{(}\lambda\widetilde{\phi}_{i}\big{)}\big{(}\lambda^{-1}\widetilde{\phi}_{j}^{-1}\big{)}=\widetilde{\phi}_{i}\widetilde{\phi}_{j}^{-1}=\delta(\phi)_{ij}.$
(A.2)

The group $\operatorname{Aut}(E)$ will act on
$\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$ and this action will
induce an action on $H^{1}\big{(}X,\mathcal{Q}_{(X,E)}^{(2)}\big{)}$. The
action of $\mathbb{C}^{\times}\cdot 1_{E}$ mentioned in Lemma A.2 is
compatible with this induced $\operatorname{Aut}(E)$ action. In particular we
can conclude: for a supermanifold
$\mathfrak{X}\in\check{H}^{1}\big{(}X,\mathcal{G}^{(2)}_{(X,E)}\big{)}$ and
any $\lambda\cdot 1_{E}\in\mathbb{C}^{\times}\cdot 1_{E}$,

$\displaystyle\omega_{*}\big{(}(\lambda\cdot
1_{E})\star\mathfrak{X}\big{)}=\lambda^{2}\omega_{*}\big{(}\mathfrak{X}\big{)}.$
(A.3)

In comparing (A.2) and (A.3) we see that if $\mathfrak{X}=\delta(\phi)$, we
must have $\omega_{*}\big{(}\mathfrak{X}\big{)}=0$. This proves Proposition
A.1 from whence Theorem 2.12 follows. ∎

### Acknowledgements

The author would like to acknowledge the helpful feedback of the anonymous
referees.

## References

  * [1] Bartocci C., Bruzzo U., Hernández Ruipérez D., The geometry of supermanifolds, Mathematics and its Applications, Vol. 71, Kluwer Academic Publishers Group, Dordrecht, 1991.
  * [2] Berezin F.A., Introduction to superanalysis, Mathematical Physics and Applied Mathematics, Vol. 9, D. Reidel Publishing Co., Dordrecht, 1987.
  * [3] Bettadapura K., On the problem of splitting deformations of super Riemann surfaces, Lett. Math. Phys., to appear, arXiv:1610.07541.
  * [4] Brylinski J.-L., Loop spaces, characteristic classes and geometric quantization, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008.
  * [5] Deligne P., Morgan J.W., Notes on supersymmetry (following Joseph Bernstein), in Quantum Fields and Strings: a Course for Mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, 41–97.
  * [6] Donagi R., Witten E., Super Atiyah classes and obstructions to splitting of supermoduli space, Pure Appl. Math. Q. 9 (2013), 739–788, arXiv:1404.6257.
  * [7] Donagi R., Witten E., Supermoduli space is not projected, in String-Math 2012, Proc. Sympos. Pure Math., Vol. 90, Amer. Math. Soc., Providence, RI, 2015, 19–71, arXiv:1304.7798.
  * [8] Green P., On holomorphic graded manifolds, Proc. Amer. Math. Soc. 85 (1982), 587–590.
  * [9] Grothendieck A., A general theory of fibre spaces with structure sheaf, University of Kansas, 1995.
  * [10] Manin Yu.I., Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, Vol. 289, Springer-Verlag, Berlin, 1988.
  * [11] Okonek C., Schneider M., Spindler H., Vector bundles on complex projective spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2011\.
  * [12] Onishchik A.L., On the classification of complex analytic supermanifolds, Lobachevskii J. Math. 4 (1999), 47–70.
  * [13] Palamodov V.P., Invariants of analytic $Z_{2}$-manifolds, Funct. Anal. Appl. 17 (1983), 68–69.


# Zero-shot Classification using Hyperdimensional Computing

Authors removed for the double-blind review  Samuele Ruffino12, Geethan
Karunaratne1, Michael Hersche12,
Luca Benini2, Abu Sebastian1, and Abbas Rahimi1
Corresponding author: Geethan Karunaratne (email<EMAIL_ADDRESS>1IBM
Research – Zurich, Switzerland 3ETH Zurich, Switzerland

###### Abstract

Classification based on Zero-shot Learning (ZSL) is the ability of a model to
classify inputs into novel classes on which the model has not previously seen
any training examples. Providing an auxiliary descriptor in the form of a set
of attributes describing the new classes involved in the ZSL-based
classification is one of the favored approaches to solving this challenging
task. In this work, inspired by Hyperdimensional Computing (_HDC_), we propose
the use of stationary binary codebooks of symbol-like distributed
representations inside an attribute encoder to compactly represent a
computationally simple end-to-end trainable model, which we name
Hyperdimensional Computing Zero-shot Classifier (_HDC-ZSC_). It consists of a
trainable image encoder, an attribute encoder based on HDC, and a similarity
kernel. We show that _HDC-ZSC_ can be used to first perform zero-shot
attribute extraction tasks and, can later be repurposed for Zero-shot
Classification tasks with minimal architectural changes and minimal model
retraining. _HDC-ZSC_ achieves Pareto optimal results with a 63.8% top-1
classification accuracy on the CUB-200 dataset by having only 26.6 million
trainable parameters. Compared to two other state-of-the-art non-generative
approaches, _HDC-ZSC_ achieves 4.3% and 9.9% better accuracy, while they
require more than 1.85$\times$ and 1.72$\times$ parameters compared to _HDC-
ZSC_ , respectively.

###### Index Terms:

Zero-shot Learning, Hyperdimensional Computing, Fine-grained Classification

##  I Introduction

Object recognition using classical machine learning approaches heavily depends
on the availability of a sufficiently large labeled training dataset.
Particularly for fine-grained classification tasks, creating high-quality
annotations is expensive and time-consuming; therefore, models with
generalization capability to classify images from unseen (or partially seen)
classes have been studied over the past few years. Few-shot Learning has
recently emerged as a popular topic of research in this regard, which attempts
to rapidly train the existing models by admitting only one or very few
instances of training data per class [1, 2, 3].

In an extreme case, Zero-shot Learning (ZSL) aims at classifying an object
without having seen any instances from the same class during training [4, 5,
6, 7, 8, 9, 10]. To enable the recognition of previously unseen classes, in
ZSL, the learner is provided with a unique descriptor that distinguishes the
unseen class with respect to the classes the learner was trained on. The
descriptor is typically a collection of attributes, each of which can take a
finite range of values. Attributes generated per instance can be human-
annotated or the captions accompanying the images [11]. A class-level
descriptor can be generated by observing a sample statistic from all instance-
level attribute value distributions of the class concerned. The goal of ZSL
is, having seen certain combinations of values for the set of attributes, to
successfully interpolate to other value combinations corresponding to the
unseen classes and thereby inferring the classes of the input examples
provided.

Figure 1: The general model structure employed for Zero-shot Classification in
this work. It comprises two main modules: a pre-trained foundation model image
encoder and an HDC-based attribute encoder. The attribute encoder consists of
weights with fixed binary vectors and binary vector operations, providing
opportunities for implementation in resource-constrained edge devices. Modules
filled with gray color remain stationary during inference.

A ZSC model generally consists of an image encoder and an attribute encoder as
shown in Fig. 1. At inference time, the image encoder accepts an image from an
_unseen_ class on which the model was not trained. In this example, a duck
image is presented to the model, but no images from the duck class were
provided during training. Therefore it happens to be an input from a class
that the model has neither seen nor tuned to before. However, the model gets
access to auxiliary information of all classes including the unseen ones via
the attribute encoder. In this example, the attributes unique to the class of
duck such as bill color, head color, wing shape, size, etc. are fed in as an
auxiliary descriptor for the duck class. Having seen the attributes in
different combinations for other classes during training, a ZSL model infers
that the attributes present in the given image belong to the description of
the duck class.

There are several approaches for solving ZSL problems, which can be broadly
divided into two categories: non-generative and generative. In non-generative
approaches, mapping functions are explored for image and attribute
descriptions such that their alignments can be generalized by applying batch-
level contrastive loss functions [11, 4]. The generative approaches on the
other hand rely on larger models capable of artificially manufacturing
instances for a given unseen class descriptor thereby helping effectively turn
the problem into a few-shot learning problem [12].

This work presents a new non-generative end-to-end training method for Zero-
shot Classification (ZSC) featuring attribute encoders containing fixed and
compact dictionaries. Thanks to the theory of Hyperdimensional Computing
(_HDC_) [13], these compact codebooks, representing values and attribute
groups, can be initialized randomly with binary codevectors when sufficiently
high dimensionality is given. _HDC_ has recently been combined with neural
networks which not only achieved the state-of-the-art accuracy in image-based
few-shot learning [14] and few-shot continual learning [15, 16] but also led
to energy-efficient hardware implementations. Here, we further expand this
combination to the ZSC tasks.

The contributions of this work are listed below:

  * •

Introducing a novel hybrid architecture for ZSC named _HDC-ZSC_. It comprises
a trainable foundation model image encoder that extracts features/embeddings
from images, an _HDC_ -based fixed (stationary) attribute encoder that
transforms auxiliary attributes into high-dimensional binary/bipolar vector
embeddings, and finally, a kernel that measures the similarity between the two
embeddings, as shown in Fig. 1.

  * •

Proposing a training methodology to boost ZSC accuracy beyond previous state-
of-the-art non-generative methods. It consists of pre-training the model on a
standard image classification task at first, followed by a domain-specific
attribute extraction task. Here we predict the attributes present in an image
to match the ground-truth attributes of the target. In the final phase of
training, the matured model is exposed to the ZSC task by retraining to
discriminate attribute vectors representing the classes.

  * •

Evaluating our model on the CUB-200 dataset on the attribute extraction task
first, followed by the ZSC. For ZSC, the final _HDC-ZSC_ achieves an average
top-1 classification accuracy of 63.8% on the CUB-200 dataset. Compared to two
other non-generative state-of-the-art approaches [4, 17], this is a 4.3% and
9.9% improvement in accuracy, yet our model is made up of 1.85$\times$ and
1.72$\times$ fewer parameters compared to the same reference models,
respectively.

To put these results into perspective, more computationally intensive
prominent generative approaches require 1.75$\times$–2.58$\times$ more
parameters while their accuracies reach only a meager 3.9% increase compared
to _HDC-ZSC_. When compared with both state-of-the-art generative and non-
generative approaches, _HDC-ZSC_ , and its variants, are in the Pareto front
with respect to accuracy and model parameter count (see Fig. 4). This could
potentially pave the way for its implementation as an application target in
low-power embedded platforms [18].

##  II Background

#### Zero-shot Learning

The human brain is able to learn new concepts by encountering a few examples
or applying deductive, inductive, or abductive reasoning. However, the
training of classical machine learning models, deep learning models, in
particular, rely on a large dataset with an adequate number of training
examples for each class. The idea of ZSL stems from the quest for general
artificial intelligence models capable of recognizing classes having seen zero
or a few instances thereof similar to the way the brain functions. In fact,
some argue that one-shot learning is a specific case of ZSL called
_transductive ZSL_ , in which the attribute descriptors are provided in an
unstructured pictorial form [19]. In ZSL the learner is first exposed to a
training dataset $\mathcal{D}_{r}$ composing of: (i) the input object set
$\mathcal{X}_{r}=\\{\mathbf{x}_{1_{r}},\mathbf{x}_{2_{r}},...,\mathbf{x}_{N_{r}}\\}$
which are three-dimensional matrices in the case of images, and $N_{r}$ stands
for the number of training samples, (ii) the set of class labels
$\mathcal{Y}_{r}$ associated with the input objects, (iii) the class attribute
matrix $\mathbf{A}_{r}\in\mathbb{R}^{C_{r}\,\times\,\alpha}$, which consists
of an attribute vector of length $\alpha$, per unique class. $C_{r}$
represents the total number of unique classes in the training dataset and,
$\alpha$ refers to the number of attributes that are identifiable across
training and testing datasets.

Similarly, a testing dataset $\mathcal{D}_{e}$ can be defined with
$\mathcal{X}_{e}$, $N_{e}$, $\mathcal{Y}_{e}$, $\mathbf{A}_{e}$, $C_{e}$
referring to the testing dataset’s input object set, number of samples, class
label set, class attribute matrix, and the number of unique classes,
respectively. In ZSL, the set of class labels in the training and testing
datasets are mutually exclusive
($\mathcal{Y}_{r}\bigcap\mathcal{Y}_{e}=\emptyset$).

All ZSL methods associate seen and unseen classes through some form of
_auxiliary descriptors_ which encodes distinguishing properties of objects
(e.g., language description, attributes, etc.), and establish links between
previously seen data and new unseen data. Different ZSL methods have been
developed in the past years. The notable types include (i)  _Learning Linear
Compatibility_ [20], [4], [7], in which a linear connection is learned between
image and attribute encodings, (ii)  _Learning Non-Linear Compatibility_ [21],
[22], in which the learned connection is non-linear, (iii)  _Learning
Intermediate Attribute Classifiers_ [23], in which an attribute predictor’s
scores are used for class prediction, (iv)  _Hybrid Models_ [6], [8], [24],
[25], which measures semantic similarity between images and attributes.
Additionally, there exist approaches based on Generative Adversarial Network
[26], [12], [27], [28], which employ supplementary networks to generate unseen
class examples during training.

This work follows the linear compatibility approach similar to _ESZSL_ [4] and
_CLIP_ [11]. ESZSL uses the squared loss to learn the bilinear compatibility
and regularizes the objective w.r.t Frobenius norm, whereas CLIP tries to
maximize the cosine similarity of correct image-attribute pairs while
minimizing the same for incorrect pairs.

#### Hyperdimensional Computing (_HDC_)

_HDC_ is a vector space model that encodes symbols starting from randomly
initialized high-dimensional vectors called _atomic hypervectors_. It has been
shown that as the dimensionality of the vector space grows, randomly
initialized vectors tend to become _quasi-orthogonal_ to each other [13]. As
one use-case of _HDC_ , the quasi-orthogonal atomic hypervectors representing
a key vector can be combined with the corresponding value vector using a
_variable binding_ operation ($\odot$) to generate a hypervector that
represents the compound symbol within the same dimensional space. In addition
to the binding operation, _HDC_ is defined by several other elementary
algebraic operations such as bundling ($+$), permutation ($\rho$), and
unbinding ($\oslash$). Hardware prototype of HDC showed substantial energy
savings and extremely robust operation [29].

##  III Methods

In this work, an end-to-end approach for ZSL is proposed, exploiting class-
attribute association and _HDC_. The concept is illustrated in Fig. 1. The
inputs to the model are provided in two modalities: (i) a batch of images
$\mathbf{X}=(\mathbf{x}_{1},\mathbf{x}_{2},...,\mathbf{x}_{B})$ in matrix form
with batch size $B$, and (ii) a class attributes matrix
$\mathbf{A}=(\mathbf{a}_{1},...,\mathbf{a}_{C})\in\mathbb{R}^{C\times\alpha}$,
where each row is an attribute vector of $\alpha$ dimension with $C$ vectors
in total representing all classes. The images of dimensions $h,w,l$ are passed
through an image encoder $\gamma(\cdot):\mathbb{R}^{h\times w\times
l}\rightarrow\mathbb{R}^{d}$ and the attributes are passed through an
attribute encoder $\phi(\cdot):\mathbb{R}^{\alpha}\rightarrow\mathbb{R}^{d}$,
both of which generate embeddings of the same dimensionality ($d$). The
embeddings are then compared via a bi-similarity kernel _cossim_ , that
relates image and class embeddings [4], [11], as follows:

$cossim(\gamma(\mathbf{X}),\phi(\mathbf{A}))=\frac{1}{K}\,\frac{\gamma(\mathbf{X})^{T}\cdot\phi(\mathbf{A})}{||\gamma(\mathbf{X})||\,\,||\phi(\mathbf{A})||},$

where $K$ is the learnable temperature scaling parameter. The resulting
similarity matrix of dimension $B\times C$ is used for error computation with
respect to the batch of ground-truth labels and updating the image encoder’s
weights over a series of epochs. Finally, the predicted class for every unseen
queried image $\mathbf{x}$ is computed as

$\hat{\mathbf{y}}=\operatorname*{arg\,max}_{i\in\\{1,...,C\\}}\,cossim(\gamma(\mathbf{x}),\phi(\mathbf{a}_{i})).$

In summary, our approach employs three main computational modules:

  * •

Image Encoder $\gamma(\cdot)$: extracts embeddings from images;

  * •

Attribute Encoder $\phi(\cdot)$: extract embeddings from class attributes;

  * •

Similarity Kernel: measure the similarity (e.g., dot-product, cosine
similarity) between image and attribute embeddings.

###  III-A _HDC_ -based Attribute Extraction

Figure 2: Different phases of Zero-shot Classification training. (a) The
backbone network is pre-trained on ImageNet data (b) The backbone network and
the projection FC are pre-trained on attribute extraction (c) The backbone
network and the projection FC are further fine-tuned with training on Zero-
shot Classification task with images of classes in the training dataset.
Modules filled with gray color are stationary.

The attribute encoder first assigns randomly initialized $d$-dimensional
binary/bipolar dense random hypervectors to all atomic level symbols. Dense
random hypervector generation can be easily realized by sampling from the
Rademacher distribution. With a sufficiently high dimension, the atomic
vectors tend to be quasi-orthogonal to each other.

In this task, we identify atomic hypervectors from two codebooks. They are the
_attribute groups_ codebook
($\mathbf{g}_{1},\mathbf{g}_{2},...,\mathbf{g}_{G}$) and the _attribute
values_ codebook ($\mathbf{v}_{1},\mathbf{v}_{2},...,\mathbf{v}_{V}$). By
assigning an atomic hypervector for the two sources separately instead of
assigning one for each appropriate attribute group/value combination
($\mathbf{b}_{1},\mathbf{b}_{2},...,\mathbf{b}_{\alpha}$), we reduce the
memory requirement For example, in the CUB-200 dataset, there are $\alpha=312$
attribute group/value combinations; however, there are only $G=28$ groups
(e.g., crown colors, leg colors, sizes, etc.) and $V=61$ unique values (e.g.,
blue, brown, large, etc.). This leads to a 71% reduction in memory requirement
for storing atomic hypervectors. This results in just 17 KB of memory for
storing the atomic hypervectors which is a negligible amount compared to the
image encoder memory requirement which is typically several hundreds of MB.

The next step involves binding appropriate group and value hypervectors
together using _variable binding_ to materialize the attribute level
codevectors on the fly. As described in [30], for dense bipolar hypervectors,
this is simply an elementwise multiplication operation (for binary
hypervectors the operation becomes elementwise XOR operation) ($\odot$) as in
$\mathbf{b}_{x}=\mathbf{g}_{y}\odot\mathbf{v}_{z}$. This shows how
$x^{th}\in\\{1,...,\alpha\\}$ attribute vector $\mathbf{b}_{x}$ in the
attribute dictionary is generated implicitly by binding appropriate
$y^{th}\in\\{1,...,G\\}$ _group_ and $z^{th}\in\\{1,...,V\\}$ _value_
hypervectors from the corresponding codebooks. According to the theory of
_HDC_ , the binding operation produces vectors quasi-orthogonal to the operand
vectors. Hence it allows for preserving quasi-orthogonality at the attribute
level.

During the attribute prediction, the matrix containing the attribute vectors
$\mathbf{B}=~{}(\mathbf{b}_{1},\mathbf{b}_{2},...,\mathbf{b}_{\alpha})\in\\{-1,+1\\}^{\alpha\times
d}$ is fed to the similarity kernel which additionally takes the output
embedding by the image encoder $\gamma(\mathbf{x})\in\mathbb{R}^{d}$ and
performs cosine similarity calculation as
$\mathbf{q}=cossim(\gamma(\mathbf{x}),\mathbf{B})$.

For the image encoder, we use ResNet50 as the backbone network followed by a
single fully connected (FC) layer as the image encoder. As shown in Fig. 2(a)
the backbone network is pre-trained with ImageNet1K data by connecting the
backbone network to the 1000-dimensional softmax layer via a temporary
$FC^{\prime}$ layer in phase I. Once this pre-training stage is complete the
backbone network is retained with the updated weights for phase II while the
$FC^{\prime}$ layer is replaced with a $FC$ layer which is used to project the
backbone network embeddings to a dimension ($d$) preferred by ZSC tasks. The
process is pictorially illustrated in Fig. 2(b). The image encoder after pre-
training for the attribute extraction task has matured weights in both the
backbone network and the $FC$ layer so that it can be used as the starting
point for ZSC in phase III.

There is a large class imbalance in the attribute extraction task due to the
dominating number of inactive attributes (e.g., only one color attribute per
color group is typically activated at a time among all 15 possible values).
For this reason, a weighted Binary Cross Entropy Loss (BCEL) is employed
between the similarity vector $\mathbf{q}$ and the ground-truth attributes.
The loss is used to update the weights of both the $FC$ layer and the ResNet50
using backpropagation while keeping the attribute encoder designed with
codevectors fixed.

###  III-B Zero-shot Classification (ZSC)

Figure 3: _HDC-ZSC_ for Inference in Zero-shot Classification. Weights in both
image and attribute encoders are stationary.

Fig. 2(c) shows the training procedure of _HDC-ZSC_ for ZSC. The image encoder
$\gamma(\cdot)$ is pre-trained on ImageNet1K data as well as on the attribute
extraction task (see Section III-A). As the first step, the attribute encoder
$\phi(\cdot)$ is defined as the product between the class attributes matrix
$\mathbf{A}\in\mathbb{R}^{C\times\alpha}$ and the attribute vector matrix
($\mathbf{B}\in\\{-1,+1\\}^{\alpha\times d}$), that was populated during
attribute extraction task as given in $\phi=\mathbf{A}\times\mathbf{B}$.

The result of the attribute and image encoders are fed to the similarity
kernel, which computes class logits via
$\mathbf{p}=cossim(\gamma(\mathbf{x}),\phi)$. Cross entropy loss (CEL) between
$p$ and the ground-truth class is used to update the $FC$ layer weights in
$\gamma$ over several epochs so that the image embeddings are aligned with the
attribute embeddings. Having been pre-trained on ImageNet1K and attribute
extraction, the backbone network remains stationary during this fine-tuned
retraining step.

Finally, the trained model is deployed for ZSC inference as shown in Fig. 3.
All weights of the model remain stationary at this stage.

##  IV Experiments

###  IV-A Experimental Setup

#### Dataset

We demonstrate the effectiveness of our framework on Caltech-UCSD
Birds-200-2011 (CUB-200-2011) [31], a fine-grained dataset of 200 bird species
containing 11,788 images in total. The dataset contains continuous class
attributes (312-dimensional visual attributes) which are used as the class
attribute matrix $\mathbf{A}$. We evaluate our model on two commonly used
train-test splits, available in the literature, namely: (i) _noZS_ where
samples of 100 classes are available in each of train and test datasets, (ii)
_ZS_ where 150 classes are available for training and the remaining 50 classes
for testing.

#### Evaluation Metrics

We use the top-1% accuracy as a metric for the attribute extraction task. To
address the class imbalance issue, we use the _Weighted Mean Average
Precision_ (WMAP) as another metric. WMAP is a modified version of Average
Precision designed to compensate for attributes that are less frequent in the
dataset. For ZSC, a commonly used top-1 and top-5 accuracy is employed.

#### Implementation Details

We resize input images to 256$\times$256 and augment the training dataset with
random rotation in the range of $[-45^{\circ},+45^{\circ}]$, center cropping,
and random horizontal flip, before feeding them to the image encoder for both
attribute and class extraction task. We implemented our framework in Pytorch
and optimized it using _AdamW_ optimizer [32] with default settings and
_cosine annealing learning rate scheduler_ [33]. Finally, the results are
obtained on the aforementioned metrics by running five trials with different
seeds, and average and standard deviation results are reported as:
$\mu\pm\sigma$.

###  IV-B Experimental Results

#### Attribute extraction

Table I summarizes the results of weighted mean average precision (WMAP) and
top-1% accuracy, comparing our approach with Finetag [34] and A3M [35]
respectively. To facilitate a fair comparison, _noZS_ split is used as was the
case in the works compared against. The results show that our _HDC-ZSC_
outperforms the previous approaches across both the metrics, by $+4.14\%$ and
$+36.71\%$ respectively.

TABLE I: Attribute extraction task results comparison with Finetag and A3M using W_MAP and top-1% accuracy as evaluation metric | Attribute
---
Group
| Finetag
---
(W_MAP)
| Ours
---
(W_MAP)
| A3M
---
(top-1% acc)
| Ours
---
(top-1% acc)
bill shape | 54 | 58 | 60 | 90
wing color | 57 | 60 | 45 | 90
upperpart color | 55 | 57 | 43 | 90
underpart color | 59 | 62 | 58 | 93
breast pattern | 15 | 61 | 58 | 81
back color | 50 | 53 | 45 | 91
tail shape | 25 | 25 | 34 | 84
uppertail color | 40 | 42 | 43 | 93
head pattern | 30 | 33 | 35 | 89
breast color | 58 | 61 | 57 | 92
throat color | 57 | 61 | 60 | 93
eye color | 76 | 76 | 81 | 98
bill length | 73 | 76 | 72 | 80
forehead color | 56 | 59 | 51 | 92
tail collor | 42 | 44 | 38 | 90
nape color | 55 | 58 | 49 | 92
belly color | 58 | 61 | 59 | 93
wing shape | 24 | 25 | 32 | 80
size | 55 | 56 | 58 | 81
shape | 47 | 49 | 57 | 94
back pattern | 44 | 45 | 46 | 77
tail pattern | 41 | 43 | 43 | 77
belly pattern | 60 | 62 | 62 | 81
primary color | 62 | 66 | 51 | 90
leg color | 32 | 37 | 46 | 92
bill color | 42 | 47 | 47 | 91
crown color | 56 | 60 | 53 | 93
wing pattern | 48 | 50 | 48 | 72
average | 48.96 | 53.11 | 51.11 | 87.82

#### Zero-shot Classification

Fig. 4 compares _HDC-ZSC_ accuracy performance and number of parameters with
other state-of-the-art approaches, both generative and non-generative methods.
Additionally, results from an alternative _Trainable-MLP model_ , in which
fixed _HDC_ codebooks-based attribute encoders are replaced by 2-layer
trainable MLP, are included in the table for reference. When compared to
generative methods, our models reduce the number of total network parameters
by more than 1.75$\times$–2.58$\times$ while maintaining a competitive top-1%
accuracy. Specifically, comparing _HDC-ZSC_ with ESZSL [4], as a non-
generative model similar to our approach, we obtain a $+9.9\%$ accuracy
improvement while improving the parameter efficiency by more than
1.72$\times$. As it is shown later in Section IV-C, our models train with a
minimal number of epochs which enlarges computational cost savings even
further compared to the above parameter savings.

Figure 4: Comparison of Zero-shot Classification accuracy vs model parameter
count. Our models _HDC-ZSC_ and _Trainable-MLP model_ are both in the Pareto
front.

###  IV-C Ablation study and hyperparameter tuning

We present results from training image and attribute encoders of different
complexity and maturity in Table II. This result is obtained with a common
hyperparameter set without model-wise tuning. From this, we choose the best
performing $ResNet50$ with an $FC$ projection layer of $d=1536$ as the
preferred configuration of the image encoder for our models. Note that this
preferred configuration outperforms the relatively larger $ResNet101$-based
image encoder in terms of Top-1% accuracy. Further investigations could be
carried out as a part of future work to find configurations that suit even
more lightweight image encoders.

TABLE II: Training of different image and attribute encoders. Pre-training stage II is skipped when the projection $FC$ layer is absent. The figure of merit is Top-1% accuracy Image Encoder | Pre-train | $d$ |  | Attribute
---
Encoder
| HDC
---
ZSC
| MLP
---
Trainable
ResNet50 | I,III | 2048 | 55 | 60
ResNet50+FC | I,II,III | 1536 | 58 | 61
I,II,III | 2048 | 50 | 57
ResNet101 | I,III | 2048 | 53 | 56
Figure 5: Hyperparameter tuning for _HDC-ZSC_ on validation split (50 disjoint
classes).

Fig. 5 provides results from the hyper-parameter exploration for _HDC-ZSC_.
The hyperparameters explored include number of epochs, batch size, learning
rate, temperature scale, and weight decay. One interesting observation is that
the number of epochs required to achieve the best accuracy is around 10 which
enables ZSC training to be realized in a reasonable time in low-power hardware
systems. The final accuracies are reported with the best-performing hyper-
parameter combination is adopted.

##  V Conclusions and outlook

In this work, we present the concept of integrating lightweight _HDC_
-enhanced attribute encoders in the backend of non-generative Zero-shot
Learning models. This approach allows for reducing the model parameter count
by representing attributes in compact and stationary codebooks leading to over
1.72$\times$ model parameter storage efficiency while improving top-1
classification accuracy by over 4.3% compared to other non-generative
approaches.

This outcome opens avenues to explore hardware implementations of the model
for both inference and minimal training in low-power heterogeneous embedded
systems such as the one described in [36]. For example, the convolutional
image encoder can be processed in a general-purpose processor cluster, or any
convolutional accelerator, whereas the stationary binary attribute encoder
weights and similarity kernel operations can be offloaded to analog [37] or
digital [38] non-von Neumann accelerators.

## Acknowledgments

We would like to acknowledge the early technical investigations by Junyuan
Cui.

## References

  * [1] O. Vinyals, C. Blundell, T. Lillicrap _et al._ , “Matching networks for one shot learning,” _Advances in Neural Information Processing Systems (NeurIPS)_ , vol. 29, 2016.
  * [2] J. Snell, K. Swersky, and R. Zemel, “Prototypical networks for few-shot learning,” _Advances in Neural Information Processing Systems (NeurIPS)_ , vol. 30, 2017.
  * [3] V. Sunder, A. Srinivasan, L. Vig _et al._ , “One-shot information extraction from document images using neuro-deductive program synthesis,” _International Workshop on Neural-Symbolic Learning and Reasoning (NeSy)_ , 2019.
  * [4] B. Romera-Paredes and P. Torr, “An embarrassingly simple approach to zero-shot learning,” in _Proceedings of the 32nd International Conference on Machine Learning (ICML)_ , vol. 37. PMLR, 2015, pp. 2152–2161.
  * [5] E. Altszyler, P. Brusco, N. Basiou _et al._ , “Zero-shot multi-domain dialog state tracking using descriptive rules,” _International Workshop on Neural-Symbolic Learning and Reasoning (NeSy)_ , 2020.
  * [6] M. Norouzi, T. Mikolov, S. Bengio _et al._ , “Zero-shot learning by convex combination of semantic embeddings,” _arXiv preprint arXiv:1312.5650_ , 2013\.
  * [7] E. Kodirov, T. Xiang, and S. Gong, “Semantic autoencoder for zero-shot learning,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2017, pp. 3174–3183.
  * [8] Z. Zhang and V. Saligrama, “Zero-shot learning via semantic similarity embedding,” in _Proceedings of the IEEE International Conference on Computer Vision (ICCV)_ , December 2015.
  * [9] D. Huynh and E. Elhamifar, “Compositional fine-grained low-shot learning,” _arXiv preprint arXiv:2105.10438_ , 2021.
  * [10] S. Narayan, A. Gupta, F. S. Khan _et al._ , “Latent embedding feedback and discriminative features for zero-shot classification,” in _European Conference on Computer Vision (ECCV)_. Springer, 2020, pp. 479–495.
  * [11] A. Radford, J. W. Kim, C. Hallacy _et al._ , “Learning transferable visual models from natural language supervision,” in _International Conference on Machine Learning (ICML)_. PMLR, 2021, pp. 8748–8763.
  * [12] J. Li, M. Jing, K. Lu _et al._ , “Leveraging the invariant side of generative zero-shot learning,” in _2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2019, pp. 7394–7403.
  * [13] P. Kanerva, “Hyperdimensional computing: An introduction to computing in distributed representation with high-dimensional random vectors,” _Cognitive Computation_ , vol. 1, no. 2, pp. 139–159, 2009.
  * [14] G. Karunaratne, M. Schmuck, M. Le Gallo _et al._ , “Robust high-dimensional memory-augmented neural networks,” _Nature Communications_ , vol. 12, no. 1, pp. 1–12, 2021.
  * [15] M. Hersche, G. Karunaratne, G. Cherubini _et al._ , “Constrained Few-shot Class-incremental Learning,” in _Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2022.
  * [16] G. Karunaratne, M. Hersche, J. Langeneager _et al._ , “In-memory realization of in-situ few-shot continual learning with a dynamically evolving explicit memory,” in _IEEE 48th European Solid State Circuits Conference (ESSCIRC)_ , 2022, pp. 105–108.
  * [17] H. Jiang, R. Wang, S. Shan _et al._ , “Transferable contrastive network for generalized zero-shot learning,” _CoRR_ , vol. abs/1908.05832, 2019.
  * [18] P. D. Schiavone, D. Rossi, A. Pullini _et al._ , “Quentin: an ultra-low-power pulpissimo soc in 22nm fdx,” in _2018 IEEE SOI-3D-Subthreshold Microelectronics Technology Unified Conference (S3S)_ , 2018, pp. 1–3.
  * [19] Y. Xian, C. H. Lampert, B. Schiele _et al._ , “Zero-shot learning—a comprehensive evaluation of the good, the bad and the ugly,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 41, no. 9, pp. 2251–2265, 2018.
  * [20] Z. Akata, F. Perronnin, Z. Harchaoui _et al._ , “Label-embedding for image classification,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 38, no. 7, pp. 1425–1438, 2015.
  * [21] Y. Xian, Z. Akata, G. Sharma _et al._ , “Latent embeddings for zero-shot classification,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2016, pp. 69–77.
  * [22] R. Socher, M. Ganjoo, C. D. Manning _et al._ , “Zero-shot learning through cross-modal transfer,” _Advances in Neural Information Processing Systems (NeurIPS)_ , vol. 26, 2013.
  * [23] C. H. Lampert, H. Nickisch, and S. Harmeling, “Attribute-based classification for zero-shot visual object categorization,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 36, pp. 453–465, 2014.
  * [24] S. Changpinyo, W.-L. Chao, B. Gong _et al._ , “Synthesized classifiers for zero-shot learning,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2016, pp. 5327–5336.
  * [25] V. K. Verma and P. Rai, “A simple exponential family framework for zero-shot learning,” in _Machine Learning and Knowledge Discovery in Databases_. Springer International Publishing, 2017, pp. 792–808.
  * [26] Y. Xian, S. Sharma, B. Schiele _et al._ , “F-VAEGAN-D2: A feature generating framework for any-shot learning,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2019.
  * [27] R. Felix, I. Reid, G. Carneiro _et al._ , “Multi-modal cycle-consistent generalized zero-shot learning,” in _Proceedings of the European conference on computer vision (ECCV)_ , 2018, pp. 21–37.
  * [28] Y. Xian, T. Lorenz, B. Schiele _et al._ , “Feature generating networks for zero-shot learning,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2018, pp. 5542–5551.
  * [29] A. Rahimi, P. Kanerva, and J. M. Rabaey, “A robust and energy-efficient classifier using brain-inspired hyperdimensional computing,” in _Proceedings of the 2016 International Symposium on Low Power Electronics and Design (ISLPED)_ , 2016.
  * [30] M. Schmuck, L. Benini, and A. Rahimi, “Hardware optimizations of dense binary hyperdimensional computing: Rematerialization of hypervectors, binarized bundling, and combinational associative memory,” _ACM Journal on Emerging Technologies in Computing Systems (JETC)_ , vol. 15, no. 4, pp. 1–25, 2019.
  * [31] C. Wah, S. Branson, P. Welinder _et al._ , “Cub-200 2011 dataset,” California Institute of Technology, Tech. Rep. CNS-TR-2011-001, 2011.
  * [32] I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,” in _International Conference on Learning Representations (ICLR)_ , 2019.
  * [33] ——, “SGDR: Stochastic gradient descent with warm restarts,” _arXiv preprint arXiv:1608.03983_ , 2016.
  * [34] R. Zakizadeh, M. Sasdelli, Y. Qian _et al._ , “Finetag: Multi-attribute classification at fine-grained level in images,” _arXiv preprint arXiv:1806.07124_ , 2018.
  * [35] K. Han, J. Guo, C. Zhang _et al._ , “Attribute-aware attention model for fine-grained representation learning,” in _Proceedings of the 26th ACM international conference on Multimedia_ , 2018, pp. 2040–2048.
  * [36] A. Garofalo, G. Ottavi, F. Conti _et al._ , “A heterogeneous in-memory computing cluster for flexible end-to-end inference of real-world deep neural networks,” _IEEE Journal on Emerging and Selected Topics in Circuits and Systems (JETCAS)_ , vol. 12, no. 2, pp. 422–435, 2022.
  * [37] R. Khaddam-Aljameh, M. Stanisavljevic, J. F. Mas _et al._ , “Hermes-core—a 1.59-tops/mm 2 pcm on 14-nm cmos in-memory compute core using 300-ps/lsb linearized cco-based adcs,” _IEEE Journal of Solid-State Circuits_ , vol. 57, no. 4, pp. 1027–1038, 2022.
  * [38] M. Eggimann, A. Rahimi, and L. Benini, “A 5 $\mu$w standard cell memory-based configurable hyperdimensional computing accelerator for always-on smart sensing,” _IEEE Transactions on Circuits and Systems I: Regular Papers_ , vol. 68, no. 10, pp. 4116–4128, 2021.