# How highly connected can an orbifold be?

Christian Lange  Christian Lange, Ludwig-Maximilians-Universität München,
Mathematisches Institut
Theresienstraße 39, 80333 München, Germany  Ruhr-Universität Bochum,
Mathematisches Institut
Universitätsstr. 150, 44780 Bochum, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Marco Radeschi  Marco Radeschi,
University of Notre Dame,
Department of Mathematics, 255 Hurley, Notre Dame, IN 46556<EMAIL_ADDRESS>
###### Abstract.

On the one hand, we provide the first examples of arbitrarily highly connected
(compact) bad orbifolds. On the other hand, we show that $n$-connected
$n$-orbifolds are manifolds. The latter improves the best previously known
bound of Lytchak by roughly a factor of $2$. For compact orbifolds and in most
dimensions we prove slightly better bounds. We obtain sharp results up to
dimension $5$.

##  1\. Introduction

Orbifolds are a generalization of manifolds that incorporate local symmetries.
They were introduced by Satake under the name of V-manifolds [Sat56, Sat57]
and later rediscovered by Thurston. Orbifolds for instance occur as quotient
spaces of Lie group actions or foliations [LW16], as collapsed limits of
manifolds under Gromov-Hausdorff convergence [Fuk90] and as moduli spaces,
e.g. in Teichmüller theory.

Thurston defined the notion of orbifold coverings and showed the existence of
an orbifold version of coverings and fundamental groups analogous to the
classical one [Thu79]. More generally, every orbifold can be thought as the
quotient of an almost free action of a compact Lie group on a manifold, and
this allows to define algebraic invariants, like _orbifold homotopy_ \- and
_(co)homology groups_ , simply as the usual homotopy and (co)homology groups
of the associated Borel construction. In the special case of the orbifold
fundamental group one indeed recovers Thurston’s definition, because orbifold
coverings are in one-to-one correspondence with coverings of a model for the
Borel construction.

An orbifold is called _good_ if it is covered by a manifold and _bad_
otherwise. It is a natural question to ask which measurable parameters tell
good and bad orbifolds apartor identify manifolds among orbifolds. Such
identifiers can have also practical applications. The fact that an orbifold
which admits a Riemannian metric of constant curvature is always good [MM91]
was for instance applied in [Le15] in the context of Alexandrov geometry. In
[LW16] Lytchak and Wilking completed the program initiated by Ghys [Gh84] and
Gromoll–Grove [GG88] of classifying Riemannian foliations of spheres, and a
large part of the work amounted to showing that a compact $7$-connected
$8$-orbifold is in fact a manifold. In the context of Riemannian orbifolds all
of whose geodesics are closed [ALR21] the authors applied a cohomology
characterization of manifolds among orbifolds, see Proposition 2.1, in order
to show that odd-dimensional orbifolds all of whose geodesics are closed are
covered by spheres.

In terms of topological parameters, Davis asked whether a contractible
orbifold is a manifold [Da11, p. 28]. A positive answer to this question was
provided by Lytchak, who actually showed that a $(2n-2)$-connected
$n$-dimensional orbifold is a manifold [Ly13], see also Section 2.2. In the
same paper, [Ly13] Lytchak moreover asked whether arbitrarily highly connected
bad orbifolds exist.

In this paper we answer Lytchak’s question in the positive, and we improve his
bound by roughly a factor of $2$.

On improving the bound we have the following result:

######  Theorem A.

A $n$-connected $n$-orbifold is a manifold for any $n\geq 1$. A $2n$-connected
$(2n+1)$-orbifold is a manifold for any $n\geq 1$.

Note that the even-dimensional part of this theorem is only a statement about
non-compact orbifolds: A compact $n$-orbifold has a fundamental class and can
thus not be $n$-connected. In the compact case, we obtain a better bound by
additionally exploiting Lefschetz duality and the classification of homotopy
spheres that are covered by spheres [Sje73].

######  Theorem B.

A compact $(2n-2)$-connected $2n$-orbifold is a manifold for any $n\geq 3$.
For $n\geq 3$ not a power of $2$ a compact $(2n-2)$-connected
$(2n+1)$-orbifold is a manifold.

In particular, our argument simplifies the tricky part (i.e. Section 4) in the
work [LW16] of Lytchak and Wilking concerned with showing that a certain
compact $7$-connected $8$-orbifold is a manifold. Moreover, our proofs of
Theorem A and B show that for orbifolds with nonempty strata only in
sufficiently low codimensions their conclusions still hold when assuming less
connectivity.

On the constructive side we answer Lytchak’s question about the existence of
highly connected bad orbifolds in the following way.

######  Theorem C.

For any $n\geq 4$, there exist compact $\lfloor n/2-1\rfloor$-connected bad
orbifolds in dimension $n$.

Moreover, in low dimensions we present some specific constructions that yield
higher connectedness than provided by Theorem C. For instance, there exists a
compact, $3$-connected, bad $4$-orbifold. All our results are summarized in
the following section.

###  1.1. Summary of the results

Given a dimension $n$, let $\kappa(n)$ be the maximum $k$ such that there
exists a bad, $k$-connected, $n$-orbifold, and let $\kappa_{c}(n)$ be the
maximum $k$ such that there exists a bad, $k$-connected, compact $n$-orbifold.
Then the statements we prove can be written as follows:

######  Theorem D (Bounds for $\kappa(n)$).

The following hold:

  1. (i)

$\left\lfloor{n\over 2}\right\rfloor-1\leq\kappa(n)<2\left\lfloor{n\over
2}\right\rfloor$

  2. (ii)

$\kappa(3)=1$, $\kappa(4)=\kappa(5)=3$.

######  Theorem E (Bounds for $\kappa_{c}(n)$).

The following hold:

  1. (i)

$\left\lfloor{n\over 2}\right\rfloor-1\leq\kappa_{c}(n)<2\left\lfloor{n-1\over
2}\right\rfloor$

  2. (ii)

If $n\neq 2^{k}$, $\kappa_{c}(2n+1)<2n-2$.

  3. (iii)

$\kappa_{c}(3)=1$ and $\kappa_{c}(4)=3$.

###  1.2. Structure of the paper

In Section 2 we recall some background about orbifolds. In particular, we
recall a cohomological characterization of manifolds among orbifolds, see
Proposition 2.1. In Section 3 we provide specific examples of highly connected
orbifolds in low dimensions. Moreover, we prove Theorem C about the existence
of arbitrarily highly connected orbifolds. Finally, in Section 4 we prove our
Theorems A and B.

Let us close the introduction with the following questions:

######  Question 1.1.

What are the precise values of $\kappa(n)$ and $\kappa_{c}(n)$?

######  Question 1.2.

(When) does $\kappa=\kappa_{c}$ hold?

Acknowledgements. We are grateful to Alexander Lytchak for comments on a draft
version of this paper.

##  2\. Preliminaries

###  2.1. Orbifolds

An _$n$ -dimensional Riemannian orbifold_ is a metric length space
$\mathcal{O}$ such that each point in $\mathcal{O}$ has a neighborhood that is
isometric to the quotient of an $n$-dimensional Riemannian manifold $M$ by an
isometric action of a finite group $\Gamma$. Every such Riemannian orbifold
has a canonical smooth orbifold structure in the usual sense [La20].
Conversely, every smooth (effective) orbifold can be endowed with a Riemannian
metric, and then the induced length metric turns it into a Riemannian orbifold
in the above sense. For a point $p$ in $\mathcal{O}$ the isotropy group of a
preimage of $p$ in a Riemannian manifold chart is uniquely determined up to
conjugation in $\mathrm{O}(n)$. Its conjugacy class is called the _local
group_ of $\mathcal{O}$ at $p$ and we also denote it as $\Gamma_{p}$. The
point $p$ is called _regular_ if this group is trivial and _singular_
otherwise. More precisely, an orbifold admits a stratification into manifolds,
where the stratum of codimension $k$ is given by

$\Sigma_{k}=\\{p\in\mathcal{O}\mid\mathrm{codim}\mathrm{Fix}(\Gamma_{p})=k\\}.$

In particular, $\Sigma_{0}$ is the set of regular points.

Riemannian orbifolds for instance arise as a quotient of a Riemannian manifold
$M$ by an effective, isometric and almost free (i.e. isotropy groups are
finite) action of a compact Lie group $G$. In fact, every Riemannian orbifold
arises in this way. Namely, it can be obtained as the quotient of its
(orthonormal) frame bundle by the natural $\mathrm{O}(n)$-action on it. The
homotopy type of the corresponding Borel construction
$B\mathcal{O}:=M\times_{G}EG$, which was first considered by Haefliger
[Hae84], depends only on the orbifold $\mathcal{O}$ and not on the specific
representation of $\mathcal{O}$ as a quotient, see [ALR07, Proposition 1.51].
Hence, if $\mathcal{O}$ is a manifold, we can take $M=\mathcal{O}$,
$G=\\{1\\}$ and see that $B\mathcal{O}\simeq M$. The orbifold homotopy groups,
cohomology groups, etc. of an orbifold $\mathcal{O}$ are defined as the
corresponding invariants of $B\mathcal{O}$. In particular, an orbifold
$\mathcal{O}$ is by definition $k$-connected, if $B\mathcal{O}$ is so.

The notion of an orbifold covering is defined in [Thu79]. An alternative
metric characterization of orbifold coverings is provided in [La20].

In dimension $1$ every orbifold is covered by $\mathbb{R}$. In dimension $2$ a
so-called _football_ orbifold (also known as spindle or complex weighted
projective line) provide examples of simply connected, bad, compact
$2$-orbifold, see Figure 1. In fact, any bad, simply connected $2$-orbifold is
of this type [Da11].

$\mathbb{Z}_{p}$

$\mathbb{Z}_{q}$

Figure 1. A football orbifold is homeomorphic to $S^{2}$ and has two cyclic
singularities of order $p$ and $q$. Its fundamental group is isomorphic to
$\mathbb{Z}_{\mathrm{gcd}(p,q)}$.

An orbifold (resp. Riemannian orbifold) is diffeomorphic to a manifold (resp.
isometric to a Riemannian manifold) if and only if its cohomology groups are
non-trivial in only finitely many degrees. This statement appears in a more
general form in the work of Quillen [Qu71, Corollary 7.8]. In particular, it
yields an alternative proof for Davis’ question whether a contractible
orbifold is a manifold [Da11, p. 28]. In the special case of an orientable
orbifold an easier argument is provided in [ALR21, Proposition 3.3]. For the
convenience of the reader we present this argument here in a condensed form.

######  Proposition 2.1.

An orientable $n$-orbifold $\mathcal{O}$ is a manifold, i.e. all its local
groups are trivial, if and only if
$H_{{\mathrm{orb}}}^{i}(\mathcal{O})=H^{i}(B\mathcal{O};\mathbb{Z})=0$ for all
$i>n$. This is the case if and only if the cohomology is nontrivial only in
finitely many degrees.

###### Proof.

We only need to show the if part. If $\mathcal{O}$ is not a manifold, then
there exists some $\mathbb{Z}_{p}\subset\mathrm{SO}(n)$ which fixes some
$x\in\mathrm{Fr}(\mathcal{O})$. In this case the projection
$\mathrm{Fr}(\mathcal{O})_{\mathbb{Z}_{p}}:=\mathrm{Fr}(\mathcal{O})\times_{\mathbb{Z}_{p}}E\mathrm{SO}(n)\rightarrow
B\mathbb{Z}_{p}$ admits the section $s(b)=[x,b]$. Hence we obtain an inclusion
$H^{*}(B\mathbb{Z}_{p})\hookrightarrow
H^{*}(\mathrm{Fr}(\mathcal{O})_{\mathbb{Z}_{p}})$. Now consider the fibration

$\mathrm{SO}(n)/\mathbb{Z}_{p}\rightarrow\mathrm{Fr}(\mathcal{O})_{\mathbb{Z}_{p}}\rightarrow
B\mathcal{O}:=\mathrm{Fr}(\mathcal{O})_{\mathrm{SO}(n)}$

Since $\mathcal{O}$ is orientable, $\pi_{1}(B\mathcal{O})$ acts trivially on
$H^{*}(\mathrm{SO}(n)/\mathbb{Z}_{p})$ [ALR21, Lemma 3.1] and thus the
Leray–Serre spectral sequence satisfies

$E_{2}^{p,q}=H^{p}(B\mathcal{O};H^{q}(\mathrm{SO}(n)/\mathbb{Z}_{p}))$

and converges to $H^{p+q}(\mathrm{Fr}(\mathcal{O})_{\mathbb{Z}_{p}})$. Since
$\mathrm{SO}(n)/\mathbb{Z}_{p}$ is a finite-dimensional manifold and
$H^{*}(\mathrm{Fr}(\mathcal{O})_{\mathbb{Z}_{p}})$ is nontrivial in infinitely
many degrees as we have seen above, the same must be true for
$H^{*}(B\mathcal{O})$. ∎

######  Remark 2.2.

An orbifold can be homeomorphic to a manifold even if some of its local groups
are not trivial. The question when this happens is completely answered in
[La19].

###  2.2. Lytchak’s bound

We recall the following argument by Lytchak [Ly13]. Suppose an $n$-orbifold
$\mathcal{O}$ is $k$-connected for some $k\geq 1$. Then $\mathcal{O}$ is in
particular orientable and so one can realize it as an almost free
$G=\mathrm{SO}(n)$ quotient, of a manifold $M$. The orbit map
$o_{p}:G\rightarrow M$ is then $k$-connected as well. Thus for any $l<k$ the
map $H^{l}(M)\rightarrow H^{l}(G)$ is surjective. Since the orbit map
factorizes through $\pi_{p}:G\rightarrow G/G_{p}$, the map
$H^{l}(G/G_{p};\mathbb{Z})\rightarrow H^{l}(G;\mathbb{Z})$ is surjective for
$l<k$ as well. If $\mathcal{O}$ is not a manifold, i.e. if $G_{p}$ is
nontrivial for some $p$, then $G\rightarrow G/G_{p}$ is a nontrivial covering
map for this $p$, in which case the image of $H^{m}(G/G_{p};\mathbb{Z})$ in
$H^{m}(G;\mathbb{Z})=\mathbb{Z}$ is a subgroup of index $|G_{p}|$ for $m=\dim
G=n(n-1)/2$. Since the free part of $H^{*}(G;\mathbb{Z})$ is generated in
degree $2n-3$, it follows that $\mathcal{O}$ is a manifold if $k\geq 2n-2$.

We remark that this argument can be improved for $4$-orbifolds. Indeed, in
this case $H^{6}(G;\mathbb{Z})$ is generated in degree $3$, not just in degree
$\leq 2n-3=5$. Therefore, the above argument shows that a $4$-orbifold
$\mathcal{O}$ is a manifold, if it is $4$-connected. In Section 4 we will see
an alternative proof of this conclusion that works in any dimension.

##  3\. Examples & Constructions

In this section we provide specific examples and general constructions for
highly connected orbifolds.

###  3.1. Low dimensional examples

An example of a compact, simply connected, bad $2$-orbifold is shown in Figure
1 when $p$ and $q$ are coprime. We will in particular see that compact, simply
connected, bad orbifolds exist in all dimensions. An example in dimension $3$
is provided by the following proposition.

######  Proposition 3.1.

There exists a compact simply connected bad $3$-orbifold.

###### Proof.

There exists a $3$-orbifold with underlying space $S^{3}$ whose singular set
is the trivalent graph shown in Figure 2 [Thu79]. The edge weights specify the
orders of the corresponding cyclic local groups. The local groups at the two
upper vertices are the orientation preserving symmetry group $\mathrm{I}$ of
an icosahedron. The local groups at the two lower vertices are the orientation
preserving symmetry group $\mathrm{O}$ of a cube. We claim that this orbifold
is simply connected as an orbifold. To see this we first observe that it is
the double of the dashed $3$-ball. By Seifert–Van Kampen it suffices to show
that this $3$-ball is simply connected as an orbifold. The latter is covered
by two good orbifolds with fundamental group $\mathrm{I}$ and $\mathrm{O}$,
respectively. Their intersection is homotopy equivalent to
$\mathbb{R}^{2}(2,3)$, an open disk with two cyclic orbifold singularities of
order $2$ and $3$, respectively. The fundamental group of this disk has two
generators $x$ and $y$ represented respectively by a loop around the cyclic
singularity of order $2$, and a loop around the cyclic singularity of order
$3$. A presentation of this fundamental group is given by $\left\langle
x,y\mid x^{2},y^{3}\right\rangle$. The fundamental groups of the two good
orbifold balls isomorphic to $\mathrm{O}$ and $\mathrm{I}$ are generated by
$x$ and $y$ as well and have presentations $\left\langle x,y\mid
x^{2},y^{3},(xy)^{4}\right\rangle$ and $\left\langle x,y\mid
x^{2},y^{3},(xy)^{5}\right\rangle$, respectively. Applying Seifert–Van Kampen
again now proves the claim. ∎

$5$

$4$

$3$

$2$

$3$

$2$

Figure 2. Compact simply connected bad $3$-orbifold

In dimension $4$ higher connectedness can be achieved.

######  Proposition 3.2.

There exists a compact $3$-connected bad $4$-orbifold.

###### Proof.

If $\rho$ is an irreducible representation of $\mathrm{SU}(2)$ on a complex
even-dimensional vector space $\mathbb{C}^{n}$, then the induced action of
$\mathrm{SU}(2)$ on the unit sphere in $\mathbb{C}^{n}$ is almost free but not
free, and so the corresponding quotient, which is also referred to as a
_weighted quaternionic projective space_ , is an orbifold [GG88, p. 154]. In
particular, for $n=4$ the quotient is a compact $4$-orbifold that is not a
manifold. Moreover, since $S^{7}$ is $3$-connected and $\mathrm{SU}(2)\cong
S^{3}$ is $2$-connected, the long exact sequence in homotopy implies trivially
that this $4$-orbifold is $3$-connected. ∎

Note that by taking products with $\mathbb{R}^{n}$ Proposition 3.2 provides
examples of (noncompact) $3$-connected $n$-orbifolds for any $n\geq 5$. In
dimension $5$ we are only aware of $2$-connected compact examples. In fact,
the example in the following proposition can be shown to be not $3$-connected.

######  Proposition 3.3.

There exists a compact $2$-connected bad $5$-orbifold.

###### Proof.

Consider the maps $\rho_{1}:\mathrm{SU}(2)\to\mathrm{SU}(3)$ given by
inclusion, and the map $\rho_{2}:\mathrm{SU}(2)\to\mathrm{SU}(3)$ given by the
composition $\mathrm{SU}(2)\to\mathrm{SO}(3)\to\mathrm{SU}(3)$. Finally,
consider the action of $\mathrm{SU}(2)$ on $\mathrm{SU}(3)$ given by $g\cdot
h=\rho_{1}(g)h(\rho_{2}(g))^{-1}$, which was proved to be almost free but not
free by Yeroshkin [Ye15]. In particular the biquotient
$\mathcal{\mathcal{O}}:=\mathrm{SU}(3)/\\!/\mathrm{SU}(2)$ is a 5-dimensional
orbifold. Since both $\mathrm{SU}(3)$ and $\mathrm{SU}(2)$ are 2-connected,
the long exact sequence in homotopy implies as above that
$\mathcal{\mathcal{O}}$ is $2$-connected as well. ∎

###  3.2. Arbitrarily highly connected bad orbifolds.

Here we present a construction of arbitrarily highly connected orbifolds. It
is based on the existence of certain parallelizable lens spaces as well as a
surgery construction by Milnor that allows to kill homotopy groups.

######  Proposition 3.4.

For any $n\geq 4$, there exist compact $\lfloor n/2-1\rfloor$-connected bad
orbifolds in dimension $n$.

###### Proof.

The proof is constructive. Let $M$ be either a stably parallelizable lens
space $M:=S^{n-1}/\mathbb{Z}_{p}$, whose existence is guaranteed by [EMSS], or
the parallelizable product $M:=S^{n-2}/\mathbb{Z}_{p}\times S^{1}$ of such a
lens space with $S^{1}$, depending on whether $n$ is even or odd.

Take $W:=M\times[0,1]$, a compact parallelizable $n$-manifold with $\partial
W=M\cup M$. By the corollary of [Mil61, Theorem 2] it is possible to perform
surgery and turn it into a $\lfloor n/2-1\rfloor$-connected manifold
$W^{\prime}$, with $\partial W^{\prime}=M\cup M$.

Finally, let $\mathcal{O}$ be the orbifold obtained by gluing two copies of
$X:=\mathbb{D}^{n}/\mathbb{Z}_{p}$ to $W^{\prime}$ along their boundary. We
claim that $\mathcal{O}$ is $\lfloor n/2-1\rfloor$-connected as an orbifold.
Since $W^{\prime}$ is simply connected, it follows that
$\pi_{1}^{{\mathrm{orb}}}(\mathcal{O})=1$ by Seifert–Van Kampen. Moreover,
since for $i=1,\ldots\lfloor n/2-1\rfloor$ we have that
$H_{i}^{\mathrm{orb}}(W^{\prime})=H_{i}(W^{\prime})=0$ and that
${i_{1}}_{*}:H_{i}^{\mathrm{orb}}(M\cup M)\to H_{i}^{\mathrm{orb}}(X\cup X)$
is an isomorphism

for $i<n-1$, cf. [La19, Proposition 3.5.], Mayer-Vietoris implies that
$H_{i}^{{\mathrm{orb}}}(\mathcal{O})=0$ for $i=1,\ldots,\lfloor n/2-1\rfloor$.
Now Hurewicz implies that $\mathcal{O}$ is $\lfloor n/2-1\rfloor$-connected. ∎

######  Remark 3.5.

Killing homotopy groups via surgery becomes more delicate past degree
$\left\lfloor{n/2}-1\right\rfloor$ since new surgery might reintroduce
homology in lower degrees, and it involves analyzing the intersection form in
$H_{\left\lfloor{n/2}\right\rfloor}(W)$ (when $\dim W$ is even) or the linking
form in $\mathrm{Tor}(H_{\left\lfloor{n/2}\right\rfloor}(W))$ (when $\dim W$
is odd), see Milnor [Mil61] and Wall [Wal62]. In these cases, however, one
also needs vanishing conditions on $\partial W$ such as
$H_{\left\lfloor{n/2}\right\rfloor}(\partial
W)=H_{\left\lfloor{n/2}\right\rfloor+1}(\partial W)$ which are in general not
satisfied in the examples above.

##  4\. Stratification & Obstructions

The first step in proving our obstruction results is to rule out the presence
of $2$-torsion in sufficiently highly connected orbifolds. The following lemma
generalizes a conclusion in the proof of [LW16, Lemma 3.2].

######  Lemma 4.1.

Let $\mathcal{O}$ be a $2^{a-1}$-connected $n$-orbifold for some $a\geq 1$.
Then the local groups of $\mathcal{O}$ contained in strata of codimension
$<2^{a}$ have no $2$-torsion.

###### Proof.

In any case $\mathcal{O}$ is simply connected and so we can realize it as an
$\mathrm{SO}(n)$ quotient of its frame bundle. Suppose a local group
$\Gamma_{p}$ of $\mathcal{O}$ contains a subgroup $\Gamma_{0}$ of order $2$
and a fixed point subspace of codimension $<2^{a}$. The image of
$\mathrm{SO}(n)p\times E\mathrm{SO}(n)\subset\mathrm{Fr}(\mathcal{O})\times
E\mathrm{SO}(n)$ under the projection $\mathrm{Fr}(\mathcal{O})\times
E\mathrm{SO}(n)\rightarrow B\mathcal{O}$ is a model for the classifying space
of $\Gamma_{p}$. If we pull back the tangent bundle $T\mathcal{O}$ via the map
$B\Gamma_{0}\rightarrow B\Gamma_{p}\hookrightarrow B\mathcal{O}$ we get a
bundle isomorphic to $V=E\Gamma_{0}\times_{\Gamma_{0}}\mathbb{R}^{n}$.

Suppose that the non-zero element $\iota\in\Gamma_{0}\subset\mathrm{SO}(n)$
has $2m<2^{a}$ times the eigenvalue $-1$. Then we can write $2m$ “in binary”
as $2m=\sum_{j=1}^{N}2^{e_{j}}$, where $0<e_{1}<e_{2}<\ldots e_{N}<a$.
Furthermore, $V$ is a bundle over $\mathbb{R}\mathbb{P}^{\infty}\cong
B\Gamma_{0}$ that decomposes as the sum of $2m$ canonical line bundles and
$n-2m$ trivial line bundles. Thus the total Stiefel–Whitney class is given by

$(1+w)^{2m}=\prod_{j=1}^{m}(1+w)^{2^{e_{j}}}=\prod_{j=1}^{m}(1+w^{2^{e_{j}}})=1+w^{2^{e_{1}}}+R$

where $1$ is the generator of
$H^{0}(\mathbb{R}\mathbb{P}^{\infty};\mathbb{Z}_{2})$, $w$ is the generator of
$H^{1}(\mathbb{R}\mathbb{P}^{\infty};\mathbb{Z}_{2})\cong\mathbb{Z}_{2}$, and
$R$ is a multiple of $w^{2^{e_{1}}+1}$. This implies that
$w_{2^{e_{1}}}(V)\neq 0$. However, since $V$ is a pullback of $T\mathcal{O}$,
and $\mathcal{O}$ is $2^{a-1}$-connected, we have
$w_{1}(V)=\ldots=w_{2^{a-1}}(V)=0$. Moreover, since the Stiefel-Whitney
classes $w_{j}(V)$, $j<2^{a}$ are generated by
$w_{1}(V),\ldots,w_{2^{a-1}}(V)$ via Steenrod powers [MS74, p. 94], it follows
that $w_{j}(V)=0$ and thus for all $j<2^{a}$. This is a contradiction with
$w_{2^{e_{1}}}(V)\neq 0$, since by assumption $2^{e_{1}}<2m<2^{a}$. ∎

It turns out that the absence of $2$-torsion has strong representation-
theoretical implications as the following lemma shows. It for instance
reflects the fact that any irreducible subgroup of $\mathrm{SO}(3)$ contains
elements of order $2$.

######  Lemma 4.2.

A real representation without trivial components of a finite group $G$ of odd
order has even dimension.

###### Proof.

Assume that the statement is false. Then there exists a nontrivial odd-
dimensional irreducible real representation of $G$. Since it has odd
dimension, its complexification $\rho:G\to U(V)$ (which we can turn to be
unitary as $G$ is finite) is irreducible as well. Let
$\chi_{\rho}:G\to\mathbb{R}$ be the character of this representation:
$\chi_{\rho}(g)=\operatorname{tr}(\rho(g))$. The fact that $G$ has odd order
implies that the map $g\mapsto g^{2}$ is a bijection of $G$. This allows us to
compute the Schur-indicator of $\rho$:

$\iota_{\rho}:={1\over|G|}\sum_{g\in
G}\chi_{\rho}(g^{2})={1\over|G|}\sum_{g\in
G}\chi_{\rho}(g)=\langle\chi_{\rho},\chi_{\rho_{0}}\rangle=0.$

Here $\rho_{0}$ denotes the trivial character and the last relation holds by
the Schur orthogonality relation since $\rho$ nontrivial. Now the claim
follows by contradiction since an irreducible complex representation with
trivial Schur indicator cannot be realized over the real numbers [Wol84,
Theorem 4.5.6]. ∎

The conclusion of Lemma 4.2 will be essential in our proof as it allows us to
skip every second dimension. This will be necessary because later on we want
to apply the following proposition to turn a highly connected orbifold into a
$k$-connected orbifold without singular strata in codimension $\geq k+1$.

Recall that given an orbifold $\mathcal{O}$, a _strong suborbifold_ is a
subset $\Sigma\subset\mathcal{O}$ such that for every $p\in\Sigma$ and every
chart $\phi:U\to V\subset\mathcal{O}$ around $p$ (i.e. $U$ is a Riemannian
manifold with an isometric action of a finite group $\Gamma$, $\phi$ is
$\Gamma$-invariant and induces an isometry $U/\Gamma\simeq V$), the preimage
$\phi^{-1}(\Sigma\cap V)$ is a submanifold fixed set-wise by the group
$\Gamma$.

######  Proposition 4.3.

Let $\mathcal{O}$ be a simply connected $n$-orbifold with a closed strong
suborbifold ${\Sigma}\subset\mathcal{O}$ of codimension $k$. Then the pair
$(\mathcal{O},\mathcal{O}\setminus\Sigma)$ is $(k-1)$-connected in the
orbifold topology.

###### Proof.

Let $D\Sigma$ denote a tubular neighbourhood of $\Sigma$, with smooth boundary
$S\Sigma:=\partial D\Sigma$. At any point $p\in\Sigma$, consider a local chart
$\pi_{p}:\tilde{U}_{p}\to U_{p}$, where $\tilde{U}_{p}$ is an open set in
$\mathbb{R}^{n}$ on which the local group $\Gamma_{p}$ acts by fixing the
(singleton) preimage $\tilde{p}$ of $p$. We can furthermore introduce an
orbifold Riemannian metric such that the local group $\Gamma_{p}$ acts by
isometries.

By definition of strong suborbifold, the preimage
$\tilde{\Sigma}_{p}:=\pi_{p}^{-1}(\Sigma\cap U_{p})$ is a submanifold of
$\tilde{U}_{p}$, fixed (as a set) by the local group $\Gamma_{p}$. In
particular, $\pi_{p}^{-1}(D\Sigma)=D(\tilde{\Sigma}_{p})$ is preserved by the
action of $\Gamma_{p}$, as well as $\partial
D(\tilde{\Sigma}_{p})=\pi_{p}^{-1}(S\Sigma)$.

Consider now the frame bundle $\pi:\mathrm{Fr}(\mathcal{O})\to\mathcal{O}$,
and recall that on each local chart $U_{p}$, one has
$\pi^{-1}(U_{p})\simeq\tilde{U}_{p}\times_{\Gamma_{p}}\mathrm{SO}(n)$. In
particular we have:

$\displaystyle\pi^{-1}(\Sigma\cap U_{p})=$
$\displaystyle\tilde{\Sigma}_{p}\times_{\Gamma_{p}}\mathrm{SO}(n)$
$\displaystyle\pi^{-1}(D\Sigma\cap U_{p})=$ $\displaystyle
D(\tilde{\Sigma}_{p})\times_{\Gamma_{p}}\mathrm{SO}(n)$
$\displaystyle\pi^{-1}(S\Sigma\cap U_{p})=$ $\displaystyle(\partial
D(\tilde{\Sigma}_{p}))\times_{\Gamma_{p}}\mathrm{SO}(n)=\partial(D(\tilde{\Sigma}_{p})\times_{\Gamma_{p}}\mathrm{SO}(n))$

Since the projection $D(\tilde{\Sigma}_{p})\to\tilde{\Sigma}_{p}$ is a
$\Gamma_{p}$-equivariant trivial disc bundle, with sphere bundle $\partial
D(\tilde{\Sigma}_{p})\to\tilde{\Sigma}_{p}$, there is an induced map

$\pi^{-1}(D\Sigma\cap
U_{p})=D(\tilde{\Sigma}_{p})\times_{\Gamma_{p}}\mathrm{SO}(n)\to\tilde{\Sigma}_{p}\times_{\Gamma_{p}}\mathrm{SO}(n)=\pi^{-1}(\Sigma\cap
U_{p})$

which is again a disc bundle, with sphere bundle given by
$\pi^{-1}(S\Sigma\cap U_{p})$.

In particular, $\pi^{-1}(D\Sigma)$ is a disk bundle over
$\tilde{\Sigma}=\pi^{-1}(\Sigma)$, with sphere bundle and $\pi^{-1}(S\Sigma)$.
Consider the map of fibrations:

$\textstyle{S^{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{D}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{-1}(S\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{-1}(D\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{\Sigma}}$

since the inclusion $S^{k-1}\to\mathbb{D}^{k}$ is $(k-1)$-connected, it
follows by the 5-lemma that the inclusion
$\pi^{-1}(S\Sigma)\to\pi^{-1}(D\Sigma)$ is $(k-1)$-connected as well.
Furthermore, since the inclusion $\pi^{-1}(S\Sigma)\to\pi^{-1}(D\Sigma)$ is
$\mathrm{SO}(n)$-equivariant, it induces fibrations

$\textstyle{\pi^{-1}(S\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{-1}(D\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{S\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{-1}(S\Sigma)\times_{\mathrm{SO}(n)}E\mathrm{SO}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{-1}(D\Sigma)\times_{\mathrm{SO}(n)}E\mathrm{SO}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{D\Sigma}}$$\textstyle{B\mathrm{SO}(n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\mathrm{SO}(n)}$

where $\widehat{D\Sigma},\widehat{S\Sigma}$ are the classifying spaces of
$D\Sigma$ and $S\Sigma$, respectively. Applying the 5-lemma again to the long
exact sequence in homotopy induced by the fibrations above, it follows that
the map $\widehat{S\Sigma}\to\widehat{D\Sigma}$ induced by the inclusion
${S\Sigma}\subset{D\Sigma}$ is $(k-1)$-connected. In particular,
$\pi_{m}^{{\mathrm{orb}}}(S\Sigma)\to\pi_{m}^{\mathrm{orb}}(D\Sigma)$ is an
isomorphism for $m\leq k-2$, and surjective for $m=k-1$.

Apply now Seifert–Van Kampen theorem to the partition $\mathcal{O}=U\cup V$
where $U=D\Sigma$ and $V=\mathcal{O}\setminus\Sigma$ (if $\Sigma$ is
disconnected then we can apply Seifert–Van Kampen to each connected component
of $\Sigma$ separately, or apply the groupoid version of Seifert–Van Kampen).
Let us name the inclusions

$i_{U}:U\cap V\to U,\quad i_{V}:U\cap V\to V,\quad j_{U}:U\to\mathcal{O},\quad
j_{V}:V\to\mathcal{O}.$

Then $\pi_{*}^{\mathrm{orb}}(U\cap V)\simeq\pi_{*}^{\mathrm{orb}}(S\Sigma)$,
and $\pi_{1}^{\mathrm{orb}}(\mathcal{O})$ is the pushout of

$\pi_{1}^{\mathrm{orb}}(U)\stackrel{{\scriptstyle{i_{U}}_{*}}}{{\longleftarrow}}\pi_{1}^{\mathrm{orb}}(U\cap
V)\stackrel{{\scriptstyle{i_{V}}_{*}}}{{\longrightarrow}}\pi_{1}^{\mathrm{orb}}(V)$

but since the map ${i_{U}}_{*}$ is an isomorphism by the discussion above, it
follows that the pushout of the diagram is isomorphic to
$\pi_{1}^{\mathrm{orb}}(V)=\pi_{1}^{\mathrm{orb}}(\mathcal{O}\setminus\Sigma)$.
Therefore,
$\pi_{1}^{\mathrm{orb}}(\mathcal{O}\setminus\Sigma)\simeq\pi_{1}^{\mathrm{orb}}(\mathcal{O})\simeq
0$.

Finally, by excision we have
$H_{m}^{\mathrm{orb}}(\mathcal{O},\mathcal{O}\setminus\Sigma)\simeq
H_{m}^{\mathrm{orb}}(D\Sigma,S\Sigma)\simeq 0$ for $m\leq k-1$. By the
relative Hurewicz isomorphism, this implies
$\pi_{m}(\mathcal{O},\mathcal{O}\setminus\Sigma)=0$ for all $m\leq k-1$, thus
finishing the proof. ∎

Since the orbifold strata of $\mathcal{O}$ are special examples of strong
suborbifolds, the previous proposition implies the following.

######  Corollary 4.4.

Let $\mathcal{O}$ be a simply connected $n$-orbifold. Then the pair
$(\mathcal{O},\mathcal{O}\setminus\Sigma_{\geq k})$ is $(k-1)$-connected in
the orbifold topology, where $\Sigma_{\geq k}$ denotes the union of all strata
of codimension at least $k$.

Now we can prove our first obstruction theorem.

###### Proof of Theorem A.

In dimension $1$ no bad orbifolds exist. In dimension $2$ any bad orbifold is
compact. Hence, we can assume that the dimension is at least $3$.

By Lemma 4.1 all local groups have odd order. Lemma 4.2 then implies that all
singular strata have even codimension.

Assume by contradiction that $\mathcal{O}$ is not a manifold, and let $2k$
denote the lowest codimension of a singular stratum. Let $\Sigma_{\geq 2k+2}$
be the union of strata of codimension $\geq 2k+2$. By Corollary 4.4 and the
fact that $\mathcal{O}$ is $2k$-connected, the complement
$\mathcal{O}^{\prime}:=\mathcal{O}\setminus\Sigma_{\geq 2k+2}$ is
$2k$-connected as well, and it contains only singular strata of codimension
$2k$, which we call $\Sigma$.

By abuse of notation we denote by $\Sigma\subseteq\mathcal{O}^{\prime}$ the
connected component of $\Sigma$, and let $D\Sigma$ be a tubular neighbourhood
of $\Sigma$ and $S\Sigma:=\partial D\Sigma$. By the proof of Proposition 4.3,
the sets $D\Sigma$ and $\Sigma$ are in fact suborbifolds of $\mathcal{O}$, and
the map $\widehat{S\Sigma}\to\widehat{D\Sigma}$ between their classifying
spaces induced by the inclusion $S\Sigma\to D\Sigma$ is an $S^{2k-1}$-bundle.
By the assumption on the structure of the singular set, $S\Sigma$ is a
manifold and in fact $\widehat{S\Sigma}$($\simeq S\Sigma$) has finite
dimensional cohomology. On the other hand, letting $\Gamma$ be the local group
at $\Sigma$ one has that
$H^{*}(\widehat{D\Sigma})=H^{*}_{\mathrm{orb}}(D\Sigma)$ is nonzero in
infinitely many degrees.

From the Gysin sequence of $S\Sigma\to D\Sigma$ there exists an element $e\in
H^{2k}_{\mathrm{orb}}(D\Sigma)$ with a long exact sequence

$\ldots H^{j}_{\mathrm{orb}}(S\Sigma)\to
H^{j-(2k-1)}_{\mathrm{orb}}(D\Sigma)\stackrel{{\scriptstyle\cup
e}}{{\longrightarrow}}H^{j+1}_{\mathrm{orb}}(D\Sigma)\to
H^{j+1}_{\mathrm{orb}}(S\Sigma)\to\ldots$

By choosing an index $j$ such that $H^{\geq j}_{\mathrm{orb}}(S\Sigma)=0$ and
$H^{j-(2k-1)}_{\mathrm{orb}}(D\Sigma)\neq 0$, it follows that $e\neq 0$. By
then choosing $j=2k-1$ one gets

$H^{2k-1}_{\mathrm{orb}}(S\Sigma)\to
H^{0}_{\mathrm{orb}}(D\Sigma)\stackrel{{\scriptstyle\cup
e}}{{\longrightarrow}}H^{2k}_{\mathrm{orb}}(D\Sigma)\to
H^{2k}_{\mathrm{orb}}(S\Sigma)\to\ldots$

i.e. $0\neq e\in\ker(j_{1}^{*}:H^{2k}_{\mathrm{orb}}(D\Sigma)\to
H^{2k}_{\mathrm{orb}}(S\Sigma))$. Apply now the Mayer-Vietoris sequence to the
partition $(D\Sigma,\mathcal{O}^{\prime}\setminus\Sigma)$ of
$\mathcal{O}^{\prime}$:

$0=H^{2k}_{\mathrm{orb}}(\mathcal{O}^{\prime})\to
H^{2k}_{\mathrm{orb}}(D\Sigma)\oplus
H^{2k}_{\mathrm{orb}}(\mathcal{O}^{\prime}\setminus\Sigma)\stackrel{{\scriptstyle
j_{1}^{*}-j_{2}^{*}}}{{\longrightarrow}}H^{2k}_{\mathrm{orb}}(S\Sigma)\to\ldots$

But this is a contradiction, because of
$0\neq(e,0)\in\ker(j_{1}^{*}-j_{2}^{*})$. ∎

In the compact case we can show more by exploiting Lefschetz duality.

######  Proposition 4.5.

Any compact $(2n-2)$-connected $2n$-orbifold $\mathcal{O}$, $n\geq 3$, is
actually a manifold.

###### Proof.

By the preceding proof we can assume that every singular point is isolated. We
denote the union of these singular points by $\Sigma$. Since $\Sigma$ is
finite, for sufficiently small $\epsilon>0$ the $\epsilon$-balls around these
singular points are disjoint. For such an $\epsilon$ we set
$U=B_{\epsilon}(\Sigma)$. By Proposition 4.3 the complement
$V=\mathcal{O}\setminus U$ is a compact $(2n-2)$-connected manifold with
boundary, and by Lefschetz duality
$H^{i}_{{\mathrm{orb}}}(V)=H_{2n-i}^{{\mathrm{orb}}}(V,V\cap U)$. In
particular $H_{i}^{\mathrm{orb}}(V,V\cap U)=0$ for $i=2,\ldots,2n-1$. Via the
long exact sequence in homology for the pair $(V,V\cap U)$ one gets for
$i=2,\ldots,2n-1$ that

$0=H_{i}^{\mathrm{orb}}(V,V\cap U)\to H_{i-1}^{{\mathrm{orb}}}(V\cap U)\to
H_{i-1}^{{\mathrm{orb}}}(V)=0.$

Hence, each component of $V\cap U$ is homotopy equivalent to a homology
$(2n-1)$-sphere. On the other hand, $U\cap V\simeq S^{2n-1}/\Gamma_{p}$, where
the local group $\Gamma_{p}$ acts freely on $S^{2n-1}$. Based on work by
Zassenhaus [Zas35] it was shown in [Sje73] that there are no homology spheres
in dimensions higher than $3$ which are covered by a genuine sphere. Because
of $(2n-1)>3$ this implies that $\Gamma_{p}$ is trivial and so the claim
follows. ∎

A slight modification of the proof of Proposition 4.5 also shows the following
statement.

######  Proposition 4.6.

Let $n>3$ be an integer that is not a power of $2$. Any compact
$(2n-2)$-connected $(2n+1)$-orbifold $\mathcal{O}$ is actually a manifold.

###### Proof.

By the proof of Theorem A we can assume that the singular set is a disjoint
union of embedded circles. By Proposition 4.3 the complement
$V=\mathcal{O}\setminus B_{\epsilon}(\Sigma)$ is a compact $(2n-2)$-connected
manifold with boundary, and by Lefschetz duality
$H^{i}_{{\mathrm{orb}}}(V)=H_{2n+1-i}^{{\mathrm{orb}}}(V,V\cap U)$. In
particular $H_{i}^{\mathrm{orb}}(V,V\cap U)=0$ for $i=3,\ldots,2n$. Via the
long exact sequence in homology for the pair $(V,V\cap U)$ one gets for
$i=3,\ldots,2n-1$ that

$0=H_{i}^{\mathrm{orb}}(V,V\cap U)\to H_{i-1}^{{\mathrm{orb}}}(V\cap U)\to
H_{i-1}^{{\mathrm{orb}}}(V)=0,$

i.e. $H_{i}^{{\mathrm{orb}}}(V\cap U)=0$ for $i=2,\ldots,2n-2$. On the other
hand, we know that each component of $V\cap U$ is homotopy equivalent to
$S^{1}\times S^{2n-1}/\Gamma_{p}$, where the local group $\Gamma_{p}$ acts
freely on $S^{2n-1}$. The homological computation above together with the
Künneth formula implies that $S^{2n-1}/\Gamma_{p}$ is a homology
$(2n-1)$-sphere. Because of $(2n-1)>3$ this again implies that $\Gamma_{p}$ is
trivial [Sje73]. ∎

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# Dense FixMatch: a simple semi-supervised learning method
for pixel-wise prediction tasks

Miquel Martí i Rabadán Corresponding Author<EMAIL_ADDRESS>KTH Royal
Institute of Technology, Stockholm, Sweden  Univrses AB, Stockholm, Sweden
Alessandro Pieropan  Univrses AB, Stockholm, Sweden
Hossein Azizpour  KTH Royal Institute of Technology, Stockholm, Sweden  Atsuto
Maki  KTH Royal Institute of Technology, Stockholm, Sweden

###### Abstract

We propose Dense FixMatch, a simple method for online semi-supervised learning
of dense and structured prediction tasks combining pseudo-labeling and
consistency regularization via strong data augmentation. We enable the
application of FixMatch in semi-supervised learning problems beyond image
classification by adding a matching operation on the pseudo-labels. This
allows us to still use the full strength of data augmentation pipelines,
including geometric transformations.

We evaluate it on semi-supervised semantic segmentation on Cityscapes and
Pascal VOC with different percentages of labeled data and ablate design
choices and hyper-parameters. Dense FixMatch significantly improves results
compared to supervised learning using only labeled data, approaching its
performance with 1/4 of the labeled samples.

##  1 Introduction

Semi-supervised learning (SSL) has shown great potential to reduce the
annotation costs of training deep learning models. Modern methods achieve
competitive results at a fraction of the amount of annotated samples required
for standard supervised learning [1, 2, 26]. The potential cost savings are
even larger for structured or dense prediction tasks, such as object
detection, instance or semantic segmentation since the annotation cost for
such tasks is much larger than for image classification.

Figure 1: Dense FixMatch (blue) on unlabeled data improves the performance of
semi-supervised semantic segmentation on Cityscapes val set using DeepLabv3+
with ResNet-101 backbone over supervised baselines (red) across different
amounts of labeled samples. $\star$ represents the mean over four different
runs with random labeled data splits. Results for individual runs are shown
with circles. Figure 2: Dense FixMatch diagram for semantic segmentation. From
an input image (top-left), two different views are created via $\alpha$ and
$\mathcal{A}$, the weak and strong augmentation pipelines respectively. The
squares represent the crops used for obtaining both views. Top: the first view
is used by the teacher in the Mean Teacher [28] framework to generate pseudo-
labels. These are matched to the second view after applying the inverse of the
weak augmentation $\alpha^{-1}$, and then the strong one $\mathcal{A}$.
Bottom: the second view is passed to the student model to obtain predictions
to train against the pseudo-labels via the consistency loss, possible to
define thanks to the shared structure and reference frame between both.

However, SSL methods have been mainly developed and studied with image-level
classification in mind [28, 21, 31, 1, 26]. Only more recently, methods have
appeared adapting or proposing solutions to structured or dense tasks such as
object detection [38, 15, 27, 18, 32] or semantic segmentation [11, 22, 16,
39, 5, 14]. Still, most works have focused on improving performance on
specific tasks and not aimed at finding methods that could be applied to
different tasks. Only a handful of methods are generic enough to be used for
multiple tasks with no or few changes [11, 31, 38, 5, 27]. Designing task-
generic methods is important for ease of portability to new tasks and the goal
of our work, as well as a must in multi-task learning scenarios.

To this end, we perform simple but effective modifications to FixMatch [26] to
adapt it for a larger class of dense or structured task, staying as close as
possible to the original formulation. We call our approach Dense FixMatch and
summarise it in Figure 2. We align the reference frame of the pseudo-labels
obtained from the weakly-augmented view with that of the predictions obtained
from the strongly-augmented view. This way, we can define a consistency loss
at each output location while still using the full set of possible
augmentations. Using strong and varied augmentations has been identified as a
key component of self-training with input-consistency [30, 11] since they
allow exploring larger neighbourhoods of the training data points in the input
data manifold as well as in different directions. In addition, we incorporate
the Mean Teacher (MT) framework so that learning is more robust to noisy
pseudo-labels and imbalanced class size [18, 33].

We evaluate our approach on semantic segmentation with Cityscapes and Pascal
VOC 2012 datasets and show the results in Figure 1 and Table 1, outperforming
supervised baselines across different labeled data regimes by a large margin
and achieving comparable results to other works in the literature. We also
compare different mini-batch sampling approaches to assess whether it is
feasible to use our method for semi-supervised multi-task learning where
separate labeled and unlabeled data sampling is not possible [19].

Our contributions are as follows:

  * •

We propose Dense FixMatch, a simple method that adds a matching operation
between pseudo-labels and predictions to FixMatch thereby enabling its use on
semi-supervised learning for any dense or structured task.

  * •

We study its performance on semi-supervised semantic segmentation on
Cityscapes and Pascal VOC 2012, showing improvements across multiple labeled
data regimes over supervised baselines. For Cityscapes, we get improvements of
up to +0.1 mIoU for 93 and 186 labeled samples reaching 0.6697 mIoU and 0.7110
mIoU respectively, and +0.04 mIoU when using all labeled samples and extra
unlabeled data reaching 0.8082 mIoU.

  * •

We ablate our design choices and hyper-parameters to give practitioners
insights on how to tune it for new tasks and datasets.

##  2 Related work

The success of semi-supervised learning [35] has come mainly from its
application to image classification with deep learning. Wei et al. [30] proved
that SSL methods based on (a) self-training and (b) consistency regularization
will achieve high accuracy with respect to ground-truth labels with the key to
their success being to explore large enough neighbourhoods of the pseudo-
labeled examples in the input data manifold, for example via aggressive data
augmentation. Self-training or pseudo-labeling methods rely on bootstrapping
current model predictions on unlabeled data and using them as labels.
Consistency regularization relies on the assumption that small perturbations
of the data points in either input or latent space should not change the
output.

FixMatch [26] and Noisy Student self-training [31] are two methods combining
such building blocks: the former follows an online approach where pseudo-
labels are generated during training, and the latter has subsequent pseudo-
labeling and re-training phases. Both use strong data augmentation to train
against the pseudo-labels. Other works have also used other kinds of
perturbations for the consistency objective, such as adversarial examples
[21], network perturbations [17, 28, 31] or MixUp [37, 1] as well as other
techniques to tackle distribution misalignment between true- and pseudo-labels
[2].

SSL applied to tasks other than image classification has also seen significant
developments in recent years. For semi-supervised object detection, multiple
works have used consistency regularization and perturbations via data
augmentation [15, 27, 18, 32]. For semi-supervised semantic segmentation, the
work in [11] found that strong and varied perturbations are required and
proposed CutMix [36] as the strong augmentation. CCT [22] enforces consistency
between predictions perturbing latent features. GCT [16] uses two differently
initialized networks for co-training and a flaw detection module. CPS [5]
instead enforces consistency against hard pseudo-labels. Pseudoseg [39] uses
strong augmentation and fuses pseudo-labels from decoder predictions with ones
from GradCAM [25]. ST++ [34] does self-training with strong data augmentation
in the re-training phase while selecting and prioritizing reliable images. AEL
[14] focuses on balancing the performance between classes via different task-
specific strategies. U2PL [29] uses unreliable pseudo-labels for negative
learning.

In contrast, we follow FixMatch as close as possible to keep the benefits of
using online pseudo-labeling and consistency regularization between
predictions on weakly and strongly augmented images. We add only a spatial
matching operation to enable its use in dense and structured tasks and the MT
framework for improving pseudo-label quality.

##  3 Dense FixMatch

We adapt FixMatch [26] for its use in structured and dense prediction tasks in
the semi-supervised setting.

Our method assumes the standard framework of semi-supervised learning where
labeled samples $X_{L}$ contribute to the supervised objective
$\mathcal{L}_{s}$ and unlabeled samples $X_{U}$ are used in an unsupervised
objective $\mathcal{L}_{u}$, with the option to use the labeled samples also
for the latter. The unsupervised loss weight $\lambda$ trades off the
contribution of both objectives to the final loss.

$\mathcal{L}=\mathcal{L}_{s}(\mathbf{x},\mathbf{y},\theta)+\lambda\mathcal{L}_{u}(\mathbf{x},\theta)$
(1)

To define the unsupervised or consistency objective, FixMatch uses image-level
pseudo-labels obtained from a weakly-augmented version of the unlabeled images
(via augmentation pipeline $\alpha$) to supervise learning on the strongly-
augmented version of the same images (via $\mathcal{A}$). For image
classification, the output is expected to be invariant to the applied
transformations and so the obtained pseudo-label can be directly used for this
purpose. In contrast, this is not possible when the output of the task at hand
has a spatial structure related to that of the input and thus will vary
depending on the applied augmentations. This is the case for dense or
structured tasks such as semantic segmentation or object detection, among
others. For those tasks, any geometric transformation of the input
equivariantly transforms its corresponding output. Therefore, when using
geometric transformations as part of the weak and strong augmentation
pipelines, the obtained pseudo-labels will not generally match pixel-to-pixel
or at each location.

We adopt a simple approach to align the predictions of one view (e.g. weak
augmentation $\alpha$) to the reference frame of the other view (e.g. strong
augmentation $\mathcal{A}$). Specifically, we first apply the inverse
geometric transformation of the first view to the predictions obtained on it
and then apply the geometric transformation of the second view so that
predictions on both views end up in the same reference frame. This simple
mechanism enables to define a consistency objective between the two views for
any dense or structured task, including semantic segmentation, object
detection, and instance segmentation, while still being able to use different
geometric transformations in both augmentation pipelines. Figure 2 illustrates
our approach for the case of semantic segmentation.

We define the supervised objective $\mathcal{L}_{s}$ by applying a per-sample,
location-wise objective $l_{s}$ between a prediction
$\mathbf{\hat{y}}_{i}=f(\alpha(\mathbf{x}_{i});\theta)$ by the model $f$ with
weights $\theta$ on a weakly-augmented view of the labeled sample
$\alpha(\mathbf{x}_{i})$ and its corresponding (augmented) ground-truth label
$\alpha(\mathbf{y}_{i})$, and averaging over all valid locations $j$:

$\mathcal{L}_{s}=\frac{1}{B_{L}}\sum_{\mathbf{x}_{i}\in\mathcal{B}^{l}}\frac{1}{S_{i}}\sum_{j=1}^{S_{i}}l_{s}(\mathbf{\hat{y}}_{i,j},\alpha(\mathbf{y}_{i})_{j}),$
(2)

where $\mathcal{B}^{l}$ is the part of the mini-batch corresponding to labeled
samples with size $B_{L}$, and $S_{i}$ is the number of valid locations in the
augmented ground-truth label $\alpha(\mathbf{y}_{i})$, e.g. for semantic
segmentation the total number of valid pixels, width $\times$ height. The
label corresponding to location $j$ is denoted by $\mathbf{y}_{i,j}$.

The unsupervised or consistency objective $\mathcal{L}_{u}$ is then defined by
applying the same or a different location-wise objective $l_{u}$ between the
predictions of a strongly-augmented view
$\mathbf{\tilde{y}}_{i}=f(\mathcal{A}(\mathbf{x}_{i});\theta)$ and the pseudo-
label $\mathbf{\bar{y}}_{i}$ obtained from a weakly-augmented view of the same
sample after spatially matching them:

$\mathcal{L}_{u}=\frac{1}{B_{U}}\sum_{\mathbf{x}_{i}\in\mathcal{B}^{u}}\frac{1}{S_{i}}\sum_{j=1}^{S_{i}}l_{u}(\mathbf{\tilde{y}}_{i,j},\operatorname*{match}(\mathbf{\bar{y}}_{i};\alpha,\mathcal{A})_{j}),$
(3)

where,
$\mathbf{\bar{y}}_{i}=\operatorname*{pseudolabel}(\mathbf{\hat{y}}_{i})$ is a
pseudo-label created from prediction $\mathbf{\hat{y}}_{i}$,
$\operatorname*{match}()$ is the matching operation applied to spatially align
pseudo-label $\mathbf{\bar{y}}_{i}$ with prediction $\mathbf{\tilde{y}}_{i}$,
$\mathcal{B}^{u}$ is the part of the mini-batch corresponding to unlabeled
samples with size $B_{U}$, and $S_{i}$ is the number of valid locations in the
matched pseudo-labels.

For our experiments on semantic segmentation, we adopt the cross-entropy loss
for both $l_{s}$ and $l_{u}$. We describe next the details of the
$\operatorname*{match}$ and $\operatorname*{pseudolabel}$ operations, the
latter being dependent on the task. Note that, for the unsupervised loss
$\mathcal{L}_{u}$, gradients are back-propagated only through the predictions
on strongly-augmented samples $\mathbf{\tilde{y}}$, and not through the
pseudo-labels $\mathbf{\bar{y}}$.

From prediction to pseudo-label. Depending on the task at hand, obtaining
pseudo-labels from the model predictions requires some post-processing that is
usually done outside of the model. For our experiments in semantic
segmentation, the model outputs normalized classification probabilities, using
$\operatorname*{softmax}$ operation, for each possible class at each output
location. Obtaining a “hard” pseudo-label means retaining only the most likely
class at each location, which is achieved by applying the
$\operatorname*{argmax}$ operation. In addition, it is common to use only
high-confidence predictions as pseudo-labels [31, 26]. To do so we define a
confidence threshold $\tau$ above which to retain the labels. Locations with
predictions of confidence below the threshold will not have valid
corresponding pseudo-labels and thus will not contribute to the loss:

$\begin{split}\operatorname*{pseudolabel}(\mathbf{\hat{y}}_{i,j};\tau)=\\\
\mathbbm{1}(\max(\mathbf{\hat{y}}_{i,j})\geq\tau)\operatorname*{argmax}(\mathbf{\hat{y}}_{i,j};\theta)).\end{split}$
(4)

Matching operation. There are multiple options to spatially align or match
predictions and pseudo-labels. One could achieve so by (a) bringing both to
the original image reference frame inverting the transforms applied to both
views; (b) bringing the prediction to the pseudo-label reference frame by
applying to the prediction the inverse of the geometric transform in the
strong augmentation pipeline and the geometric transform in the weak
augmentation pipeline; or (c) bringing the pseudo-label to the prediction
reference frame by first applying the inverse of the geometric transforms in
the weak augmentation pipeline, $\alpha^{-1}$, and then the transforms of the
strong augmentation pipeline, $\mathcal{A}$, to the pseudo-label
$\mathbf{\bar{y}}_{i}$. We choose the latter since it importantly does not
operate on the predictions, only on the pseudo-labels, and thus does not
require to back-propagate gradients through the matching operation:

$\operatorname*{match}(\mathbf{\bar{y}}_{i};\alpha,\mathcal{A})=\mathcal{A}(\alpha^{-1}(\mathbf{\bar{y}}_{i})).$
(5)

It is important to note that applying both $\alpha^{-1}$ and $\mathcal{A}$
might produce invalid results for some locations of the pseudo-labels, e.g.
the inverse of a random crop will place the pseudo-labels on the original
location of the original sample view but areas outside the crop will not be
defined. Such areas with invalid pseudo-labels must be tracked so that
predictions on them do not contribute to the loss.

Augmentation pipelines. Dense FixMatch, like FixMatch, uses two different
augmentations pipelines for generating the pseudo-labels and for training
against them. Pseudo-labels are generated via the weak augmentation pipeline
$\alpha$, while predictions to train on the pseudo-labels are obtained via the
strong augmentation pipeline $\mathcal{A}$. FixMatch [26] applied to image
classification uses a standard flip-and-shift augmentation strategy as
$\alpha$ and either RandAugment [7] or CTAugment [2] as $\mathcal{A}$, always
applying Cutout [8] last. We adopt similar choices for semantic segmentation,
choosing RandAugment [7] for the strong augmentation $\mathcal{A}$ for
simplicity, including both geometric and color transforms. We use random crops
for both pipelines before applying the rest of transforms to ensure the same
input size and instead of the shift operation in $\alpha$ since both achieve a
similar effect. RandAugment randomly samples two transformations from a list
of augmentations and a random magnitude for each within a range. The pool of
augmentations includes both geometric and color transforms and we study the
importance of using both types in Section 4.3. Instead, other works focusing
on dense tasks and inspired by FixMatch avoid the misalignment by dropping the
geometric transforms [39, 18], applying the same to both views [20], or
applying them only as part of $\mathcal{A}$ [32].

Mean Teacher. In order to obtain cleaner and more stable pseudo-labels [33,
18], we use the teacher in MT [28] instead of the same model to generate
pseudo-labels on the weakly-augmented views:
$\mathbf{\tilde{y}}_{i}=f(\alpha(\mathbf{x}_{i});\bar{\theta})$. The teacher
weights $\bar{\theta}$ are updated after each training step with an
exponential moving average (EMA) of the student weights according to Eq. 6,
with decay rate $m$. We ablate this choice in Section 4.3.

$\bar{\theta}\xleftarrow{}m\bar{\theta}+(1-m)\theta.$ (6)

##  4 Experiments

Datasets. We use Cityscapes [6] and Pascal VOC 2012 [10] datasets for
evaluating our method on semi-supervised semantic segmentation. Cityscapes
consists of 2975 samples for training, with fine annotations for 19 classes,
and 500 samples for evaluation. In addition, further 20000 samples are
available in the extra set with coarse annotations, but we will use them later
as unlabeled samples only. Pascal VOC consists of 1464 samples for training in
the original set with annotations for 21 classes including background, and
1449 samples for evaluation in the validation set. Moreover, there are further
9118 labeled samples in the augmented set from SBD [12]. As is common in the
literature, we simulate the semi-supervised setting with labeled and unlabeled
splits of the training set with different labeled data regimes or ratios of
labeled samples. For each split, we generate four different random splits with
no guarantees of class balance. For Pascal VOC, we split the original set and
use the augmented set as unlabeled data.

Method Backbone Sampling 93(1/32) 186(1/16) 372(1/8) 744(1/4) 1488(1/2)
2975All Supervised RN-50 $.5579_{\pm.0091}$ $.6004_{\pm.0012}$
$.6550_{\pm.0051}$ $.6943_{\pm.0065}$ $.7332_{\pm.0095}$ $.7608_{\pm.0054}$
RN-101 $.5686_{\pm.0080}$ $.6122_{\pm.0025}$ $.6628_{\pm.0081}$
$.6979_{\pm.0049}$ $.7421_{\pm.0081}$ $.7652_{\pm.0023}$ Dense FixMatch RN-50
Explicit $.6581_{\pm.0202}$ $.7013_{\pm.0079}$ $.7243_{\pm.0049}$
$.7504_{\pm.0063}$ $.7599_{\pm.0063}$ Implicit $.6554_{\pm.0158}$
$.7065_{\pm.0065}$ $.7339_{\pm.0055}$ $.7547_{\pm.0070}$ $.7637_{\pm.0079}$
RN-101 Explicit $.6697_{\pm.0119}$ $.7110_{\pm.0061}$ $.7283_{\pm.0069}$
$.7572_{\pm.0015}$ $.7666_{\pm.0050}$ Implicit $.6481_{\pm.0178}$
$.7083_{\pm.0098}$ $.7391_{\pm.0028}$ $.7565_{\pm.0058}$ $.7635_{\pm.0088}$

Table 1: Results of Dense FixMatch on Cityscapes val set with few labeled
samples on different amounts of labeled data, ResNet-50/101 backbones and
DeepLabv3+, and either explicit or implicit mini-batch sampling settings.
Dense FixMatch significantly improves over the baselines for both settings.

Method Backbone Sampling 92(1/16) 183(1/8) 366(1/4) 732(1/2) 1464Original
10582Augmented Supervised RN-50 $.4075_{\pm.0114}$ $.5361_{\pm.0257}$
$.6183_{\pm.0153}$ $.6788_{\pm.0056}$ $.7214_{\pm.0029}$ $.7522_{\pm.0038}$
RN-101 $.4482_{\pm.0256}$ $.5771_{\pm.0247}$ $.6534_{\pm.0081}$
$.7059_{\pm.0041}$ $.7454_{\pm.0023}$ $.7722_{\pm.0017}$ Dense FixMatch RN-50
Explicit $.5215_{\pm.0246}$ $.6249_{\pm.0374}$ $.6902_{\pm.0045}$
$.7169_{\pm.0010}$ $.7391_{\pm.0012}$ Implicit $.4928_{\pm.0284}$
$.5892_{\pm.0334}$ $.6729_{\pm.0076}$ $.7031_{\pm.0028}$ $.7432_{\pm.0045}$
RN-101 Explicit $.5485_{\pm.0501}$ $.6582_{\pm.0334}$ $.7204_{\pm.0059}$
$.7473_{\pm.0008}$ $.7716_{\pm.0020}$ Implicit $.4984_{\pm.0300}$
$.6133_{\pm.0313}$ $.7047_{\pm.0053}$ $.7414_{\pm.0044}$ $.7710_{\pm.0046}$

Table 2: Results of Dense FixMatch on Pascal VOC 2012 val set with few labeled
samples on different amounts of labeled data, ResNet-50/101 backbones and
DeepLabv3+, and explicit or implicit settings.

Model. Following the literature, we use DeepLabv3+ [4] on ResNet-50 or
ResNet-101 backbones [13] to define our semantic segmentation model, giving
predictions at the same spatial resolution as the input. The model is
initialized with ImageNet [24] pre-trained weights.

Implementation details. We implement and train our models using PyTorch [23]
with distributed training and mixed precision on up to four NVIDIA A100 GPUs,
depending on the total batch size and the backbone used. We use the computer
vision library Kornia [9] for implementing the data augmentation pipelines for
its support of invertible geometric transforms and differentiable
augmentations for multiple tasks, including semantic segmentation. We follow
EMAN [3] and use the exponential moving average of the Batch Normalization
statistics of the student to update the teacher.

Evaluation. We evaluate the performance of our method using as the main metric
the mean Intersection-over-Union (mIOU) over all classes. For simplicity, we
use full-resolution, single-scale, and single-pass evaluation in contrast to
the sliding evaluation approach used in other works [5, 14, 29]. For
stability, the model used for evaluation has weights following the EMA of the
weights obtained during training. For most experiments, this means exactly
using the teacher in the MT framework for evaluation. For each labeled data
regime, we train our model on each of the four random data splits using also a
different random seed for each training run. We take the best checkpoint of
each run according to the mIOU and compute the mean and standard deviation
over the four runs.

Training details. We follow the training details of [29]. We employ two
different mini-batch sampling strategies for SSL: (a) the common explicit
setting in which labeled and unlabeled data are sampled separately, and (b)
the alternative implicit setting in which all data is sampled uniformly
regardless of labels [19]. We train each setting for the equivalent of 80 or
240 epochs on the full train set for the supervised baseline, i.e. 52910 or
89250 updates, for Pascal VOC and Cityscapes, respectively.

###  4.1 Results on few labeled samples

In Figure 1 and Tables 1 and 2, we show results for the common splits of 1/32
to 1/2 of all the full train set for Cityscapes and 1/16 to the full original
train set for Pascal VOC 2012, respectively. We compare few-supervision
baselines using only the labeled data with Dense FixMatch, which also uses the
unlabeled samples in either the explicit or implicit settings. Our method
performs better than the baselines for both mini-batch sampling approaches.
The explicit setting is slightly better for more label-scarce regimes but both
give similar results with more labeled data.

Method Backbone mIoU Supervised RN-50 $.7608_{\pm.0054}$ RN-101
$.7652_{\pm.0023}$ train extra Explicit Implicit Dense FixMatch RN-50 ✓
$.7869_{\pm.0018}$ ✓ $.7916_{\pm.0026}$ $.7935_{\pm.0017}^{\dagger}$ ✓ ✓
$\mathbf{.7998_{\pm.0020}}$ $.7948_{\pm.0016}$ RN-101 ✓ $.7907_{\pm.0020}$ ✓
$.8005_{\pm.0010}$ $.7974_{\pm.0020}^{\dagger}$ ✓ ✓
$\mathbf{.8082_{\pm.0024}}$ $.8012_{\pm.0013}$

Table 3: Results on Cityscapes full labeled set and extra unlabeled samples
for the supervised baselines and semi-supervised Dense FixMatch in both the
explicit and implicit mini-batch sampling settings for SSL. We also compare
using our method as a regularization on the labeled data only and using it on
both labeled and unlabeled data. †is our setting for the main experiments.

$\tau$ $\lambda$ mIoU 0 0.2 0.6566 0.5 0.6464 1 0.6594 2 0.6581 5 0.5699 0.5
0.1 0.6447 0.2 0.6609 0.5 0.6495 1 0.6501† 2 0.6554 5 0.6090 10 0.5276 0.8 1
0.6388 0.95 1 0.6148

Table 4: Ablation study on the pseudo-label confidence threshold $\tau$ and
the consistency loss weight $\lambda$. †values used in the main experiments.

Method mIoU Supervised 0.5409 Mean Teacher∗ 0.5820 Crop relation Augmentation
MT Dense FixMatch Same Crop+color $\checkmark$ 0.5794 Same Crop+color+cutout
$\checkmark$ 0.6531 Min. overlap Crop+color+cutout $\checkmark$ 0.6542 Min.
overlap Crop+geom. $\checkmark$ 0.6539 Min. overlap Crop+geom.+cutout
$\checkmark$ 0.6517 Min. overlap Crop+color+geom. $\checkmark$ 0.6579 Same
Crop+color+geom. $\checkmark$ 0.6157 Any Crop+color+geom.+cut. $\checkmark$
0.6300 Same Crop+color+geom.+cut. $\checkmark$ 0.6660 Min. overlap
Crop+color+geom.+cut. $\checkmark$ 0.6594† Min. overlap Crop+color+geom.+cut.
0.6275

Table 5: Ablation on the use of Mean Teacher (MT) framework, the relation
between crops in the two views of Dense FixMatch and the use of the geometric,
color or all augmentations in RandAugment. ∗Using MT on its own with the
augmentation pipeline of the supervised baseline. We use a logistic warm-up
schedule for the consistency weight during the first 60 epochs. †is our
setting for the main experiments.

###  4.2 Results on full labeled set and extra unlabeled samples

In addition, we evaluate Dense FixMatch in the more realistic setting where
all labeled samples are used and extra unlabeled data is available in
Cityscapes. We use all samples from the extra set but discard the coarse
annotations and just treat them as unlabeled samples. We also compare to the
fully-supervised baseline using only the labeled data and using the
consistency loss as a regularization term only, i.e. computed on the same
labeled data as the supervised loss. Results are shown in Table 4. Computing
the loss on both labeled and unlabeled samples gives the best results.

###  4.3 Ablations

In Table 4, we ablate the main hyper-parameters of our method using the 93
labeled samples regime of semi-supervised semantic segmentation with
Cityscapes, with a single data split and seed, and the explicit setting. For
dense tasks such as semantic segmentation, other works have reported that
using a low $\tau$ or removing it altogether gives better final results [39],
hypothesizing that discarding predictions of lower confidence makes the loss
dominated by easy classes [20, 14]. Moreover, low-confidence predictions in
semantic segmentation tend to concentrate in truly ambiguous regions such as
the boundaries between objects of different classes [29] so discarding them
means removing supervision mainly from boundary pixels which are in fact the
most informative. Values for $\lambda$ between 0.2 and 2 give comparable
results.

In Table 5 we ablate our design choices: the relation between crops in both
views, the choice of augmentations, and the use of MT. The best results are
obtained when using MT and all possible augmentations, although using the same
crop instead of overlapping crops between views improves results. We
hypothesize this is due to a larger number of valid locations in the matched
pseudo-labels.

Method Road Side. Build. Wall Fence Pole T.light T.sign Veg. Terr. Sky Person
Rider Car Truck Bus Train Motor. Bic. Mean % pixels 36.02 7.06 25.61 0.41 0.81
1.16 0.18 0.59 14.08 1.22 3.80 1.08 0.09 6.73 0.25 0.23 0.21 0.14 0.34
Supervised .962 .722 .873 .293 .296 .500 .497 .563 .896 .469 .914 .734 .336
.888 .114 .276 .318 .273 .677 $.558_{\pm.262}$ Dense FixMatch (E) .976 .810
.902 .414 .424 .574 .619 .710 .906 .575 .935 .778 .544 .924 .560 .699 .295
.488 .715 $.676_{\pm.195}$ Dense FixMatch (I) .974 .820 .900 .467 .432 .578
.603 .688 .913 .587 .931 .769 .448 .924 .495 .667 .471 .475 .661
$.674_{\pm.184}$

Method Bg. Plane Bicy. Bird Boat Bottle Bus Car Cat Chair Cow Table Dog Horse
Motor. Person Pott. Sheep Sofa Train Tv Mean % pixels 72.45 1.09 0.83 0.15
0.94 1.57 3.46 2.60 0.58 0.65 1.25 2.08 1.13 0.58 0.90 5.27 0.56 0.34 0.49
1.72 1.35 Supervised .883 .702 .379 .227 .574 .443 .764 .639 .336 .114 .429
.215 .309 .332 .542 .647 .199 .511 .202 .610 .307 $.446_{\pm.206}$ Dense
FixMatch (E) .890 .784 .523 .639 .500 .555 .843 .684 .776 .188 .514 .439 .384
.388 .752 .622 .356 .576 .352 .743 .553 $.574_{\pm.180}$ Dense FixMatch (I)
.899 .740 .459 .010 .559 .583 .833 .721 .431 .192 .469 .347 .418 .559 .693
.672 .364 .356 .333 .747 .545 $.520_{\pm.213}$

Table 6: Class-wise IoU on val set of Cityscapes (top) when training on a 93 labeled samples data split and on val set of Pascal VOC 2012 (bottom) when training on a 92 labeled samples data split. We show in bold the best result for each class and in red the classes that perform worse than the supervised baseline for both the explicit (E) and implicit (I) mini-batch sampling settings. Input | Supervised | Dense FixMatch(E) | Dense FixMatch(I) | Ground truth
---|---|---|---|---
|  |  |  |
|  |  |  |
Figure 3: Qualitative results for SSL training on Cityscapes with 93 labeled
samples for the supervised baseline and Dense FixMatch in the explicit (E) and
implicit (I) mini-batch sampling settings, shown on samples in the validation
set.

###  4.4 Class-wise analysis

Semantic segmentation is known for its large class imbalance. In both
Cityscapes and Pascal VOC the predominant classes appear in a few orders of
magnitude more pixels than the least frequent ones [6, 10]. In Table 6, we
give per-class results comparing the supervised baseline to Dense FixMatch on
a single data split of 93 or 92 labeled samples for Cityscapes and Pascal VOC
2012 respectively. We also report the percentage of pixels for each class. For
Cityscapes, our method improves significantly on average over the baseline,
and does so while improving for all but one class. Importantly, it also
reduces the effect of class-imbalance since the gap between the best
performing classes and the worst performing ones is reduced significantly, as
it can be seen by the lower standard deviation across classes. In Pascal VOC
there is a very large imbalance between the background class and the rest.
Dense FixMatch improves the results on average, but up to 3 classes get lower
IoU. For the implicit setting, the least-frequent class Bird is never learnt.

###  4.5 Qualitative results

In Figure 3, we give some examples for the 93 labeled samples experiments with
Cityscapes comparing the baseline and Dense FixMatch for both the explicit and
implicit settings. We see how both settings give similar results and
outperform the supervised baseline, as seen in the more defined boundaries
between road and side-walk and poles.

##  5 Conclusions

We proposed Dense FixMatch, a simple method that puts together the most
important components in modern deep semi-supervised learning and adds a
matching operation on the pseudo-labels. In this way, it can be used for
multiple dense or structured prediction tasks with the full strength of data
augmentation pipelines, including strong geometric transformations. We
evaluated it on semi-supervised semantic segmentation on Cityscapes and Pascal
VOC and ablate design choices as well as hyper-parameters. This gives future
practitioners insights on how to tune the proposed method for other datasets
and tasks.

## Acknowledgments

This work was partially supported by the Wallenberg AI, Autonomous Systems and
Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.

The computations were partially enabled by resources provided by the Swedish
National Infrastructure for Computing (SNIC) at Chalmers Centre for
Computational Science and Engineering (C3SE) partially funded by the Swedish
Research Council through grant agreement no. 2018-05973.

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