# Some old and basic facts about random walks on groups

Wolfgang Woess Institut für Diskrete Mathematik, Technische Universität Graz,
Steyrergasse 30, A-8010 Graz, Austria<EMAIL_ADDRESS>
###### Abstract.

This note contains old instead of new results about random walks on groups,
which may serve as a supplement to the author’s monograph [15]. First, we
exhibit a basic exercise on the periodicity classes of random walk. The second
topic concerns some basics on ratio limits for random walks, which had been
published “only” in German in the 1970ies.

##  1\. Introduction

In all that follows, $\Gamma$ will be a countable, discrete group written
multiplicatively, with elements typically denoted by $x,y$, etc., unit element
$e$, and $\mu$ a probability measure on $\Gamma$. The support of $\mu$ is

$S_{\mu}=\\{x\in\Gamma:\mu(x)>0\\}\,.$

Recall that the random walk on $\Gamma$ with law $\mu$ is the time-homogeneous
Markov chain with state space $\Gamma$ and transition probabilities
$p_{\mu}(x,y)=\mu(x^{-1}y)$. The $n$-step transition probabilities are then
$p_{\mu}^{(n)}(x,y)=\mu^{(n)}(x^{-1}y)$, where $\mu^{(n)}$ is the
$n^{\textrm{th}}$ convolution power of $\mu$. Its support is $S_{\mu}^{n}$,
the set of all products $x_{1}\cdots x_{n}$ of group elements. Thus, the
random walk is _irreducible_ in the sense of Markov chains (resp., non-
negative matrics) if and only if

(1) $\bigcup_{n=1}^{\infty}S_{\mu}^{n}=\Gamma\,,$

i.e., the semigroup genereated by the support is all of $\Gamma$. (In general,
for two subsets $A,B$ of $\Gamma$, their product is $AB=\\{xy:x\in A\,,\;y\in
B\\}$, and $A^{-1}=\\{x^{-1}:x\in A\\}$.)

$\bullet\;$ In this note, we always assume irreducibility.

In §2, we recall a basic fact about the period and aperiodicity of the random
walk, and in §3, we recall some ratio limit theorems.

##  2\. Aperiodicity

As an irreducible Markov chain, the random walk has a _period_

$d=d(\mu)=\gcd\\{n\in\mathbb{N}:\mu^{(n)}(e)>0\\},$

compare with [15, page 3]. The random walk is called _aperiodic,_ if
$d(\mu)=1$. For general $d$, as for any irreducible Markov chain, there is a
partition in subsets

$\Gamma=C_{0}\,\dot{\cup}\,C_{1}\,\dot{\cup}\,\dots\,\dot{\cup}\,C_{d-1}\,,$

with $e\in C_{0}\,$, such that the random walk wanders through the seta
$C_{j}$ cyclically: if $x\in C_{j}$ and $y\in C_{k}$ then
$\mu^{(n)}(x^{-1y})>0$ only when $n\equiv k-j(\mod d)$, and then this holds
fal all but finitely many such $n$. We now recall a group-theoretical fact
which was considered a basic exercise in earlier days but now is sometimes
being “discovered” by younger researchers. The following has also has a
generalisation to random walks on locally compact groups, which was studied by
Woess [14] – which however is written in French and not available online,
apart from the author’s webpage.

######  Proposition.

$\displaystyle\Gamma_{0}=\bigcup_{k=1}^{\infty}S_{\mu}^{-k}S_{\mu}^{k}$
is a normal subgroup of $\Gamma$ with index $d$. The factor group
$\Gamma/\Gamma_{0}$ is cyclic of order $d$. One has $C_{0}=\Gamma_{0}\,$, and
the sets $C_{j}$ are the cosets of $\Gamma_{0}\,$.

The probability measure $\mu^{(d)}$ is supported by $\Gamma_{0}$ and
irreducible on that subgroup, with period $1$.

###### Proof.

We first show that by (1), one has that $\Gamma_{0}$ is a normal subgroup;
this can be found, e.g., in the lecture notes by Mukherjea and Tserpes [7, p.
97].
(a) It is clear that $e\in\Gamma_{0}$ and that $x^{-1}\in\Gamma_{0}$ whenever
$x\in\Gamma_{0}\,$.
(b) If $x\in\Gamma$ then there is $n$ such that $x\in S_{\mu}^{n}$. Hence
$x^{-1}S_{\mu}^{-k}S_{\mu}^{k}x\subset S_{\mu}^{-n-k}S_{\mu}^{k+n}$, so that
$x^{-1}\Gamma_{0}x\subset\Gamma_{0}\,$.
(c) Finally, by (b),

$(S_{\mu}^{-k}S_{\mu}^{k})(S_{\mu}^{-n}S_{\mu}^{n})\subset(\Gamma_{0}S_{\mu}^{-n})S_{\mu}^{n}=S_{\mu}^{-n}\Gamma_{0}S_{\mu}^{n}\subset\Gamma_{0}\,,$

so that $\Gamma_{0}$ is closed with respect to the group product.

Next, we observe that $d(\mu)=\gcd N_{\mu}$, where
$N_{\mu}=\\{n\in\mathbb{N}:e\in S_{\mu}^{n}\\}$, and that the latter set is
closed under addition. This implies via elementary number theory that there
are $n_{1},n_{2}$ such that $e\in S_{\mu}^{n_{1}}\cap S_{\mu}^{n_{2}}$ and
$d(\mu)=n_{2}-n_{1}\,$. Therefore

$e\in
S_{\mu}^{-n_{1}}S_{\mu}^{n_{1}}S_{\mu}^{d}\subset\Gamma_{0}S_{\mu}^{d}\,,\quad\text{and}\quad
S_{\mu}^{-d}\subset S_{\mu}^{-d}\Gamma_{0}S_{\mu}^{d}\subset\Gamma_{0}\,.$

Therefore also $S_{\mu}^{nd}\subset\Gamma_{0}$ for all $n\in\mathbb{N}$.

Now suppose that $S_{\mu}^{k}\cap S_{\mu}^{m}\neq\emptyset$ for $k\neq m$, and
let $x$ be an element of that set. By (1), there is $n$ such that $x^{-1}\in
S_{\mu}^{n}$. Then $e\in S_{\mu}^{k+n}\cap S_{\mu}^{m+n}$, whence $k+n$ and
$m+n$ belong to $N_{\mu}\,$. We conclude that $d$ divides $m-k$.

In particular, let $x_{0}\in S_{\mu}$. If $x_{0}^{k}\in\Gamma_{0}$ then
$x_{0}^{k}\in S_{\mu}^{-n}S_{\mu}^{-n}$ for some $n$, so that
$S_{\mu}^{k+n}\cap S_{\mu}^{n}\neq\emptyset$ and $d$ divedes $k$. Thus, the
cosets $x_{0}^{k}\Gamma_{0}$, $k=0\,,\dots,d-1$ are all disctinct, and
$x_{0}^{d}\Gamma_{0}=\Gamma_{0}$. This holds for any $x_{0}\in S_{\mu}\,$, and
it shows that $\Gamma/\Gamma_{0}$ is cyclic of order $d$.

Turning to the random walk, if $x\in x_{0}^{k}\Gamma^{0}$ for
$k\in\\{0,\dots,d-1\\}$ and $p(x,y)=\mu(x^{-1}y)>0$ then $y\in S_{\mu}^{k+1}$
and $yx_{0}^{d-1-k}\in S_{\mu}^{d}\subset\Gamma_{0}\,$. Consequently, $y\in
x_{0}^{k+1}\Gamma_{0}\,$.

By all that we have proved so far, we have that
$S_{\mu}^{n}\cap\Gamma_{0}=\emptyset$ when $d$ does not divide $n$, so that

$\bigcup_{n=1}^{\infty}S_{\mu}^{nd}=\Gamma_{0}\,,$

which means that $\mu^{(d)}$ is irreducible on $\Gamma_{0}\,$. Since $N_{\mu}$
is closed under addition, there is $n_{0}$ such that $nd\in\mathbb{N}_{\mu}$
for all $n\geq n_{0}\,$. Therefore $\mu^{(d)}$ has period 1 on $\Gamma_{0}\,$.
∎

If $\mu$ has period 1, then it is called aperiodic. By the Proposition, we may
restrict attention to that case.

##  3\. Spectral radius and convergence

For irreducible $\mu$ as in the preceding section, the number

$\rho(\mu)=\limsup_{n\to\infty}\mu^{(n)}(x)^{1/n}$

is independent of $x\in\Gamma$. It is called the _spectral radius_ in a
variety of references, including [15]. We stick to this terminology, but one
should be careful: in general, this is not the spectral radius of an operator;
it _is_ the spectral radius (and norm) of $\mu$ as a convolution operator on
$\ell^{2}(\Gamma)$ when $\mu$ is symmetric, i.e., $\mu(x^{-1})=\mu(x)$ for all
$x$, and on a weighted $\ell^{2}$-space when $\mu$ is reversible, see [15,
§10]. The study of this number, for general irreducible Markov chains, goes
back to Pruitt [9] and Vere-Jones [13]; see also the monograph by Seneta [11].

######  Lemma.

For every $x\in\Gamma$, one has convergence:

$\lim_{n\to\infty}\mu^{(n)}(x)^{1/n}=\rho(\mu)\,,$

and the sequence $\mu^{(n)}(e)^{1/n}$ converges to its limit from below.
Furthermore, there is $k_{0}$ such that

$\mu^{(n)}(e)<\rho(\mu)^{n}\quad\text{strictly for all }\;n\geq k_{0}\,.$

###### Proof.

The convergence result is the multiplicative version of Fekete’s Lemma [2].
Let $a_{n}=\mu^{(n)}(e)$. Then $a_{n}>0$ for all $n\geq n_{0}$ and
$a_{m}a_{n}\leq a_{m+n}\leq 1$. For fixed $m\in n$, let $n=qm+r$ with
$n_{0}\leq r=r_{n}<n_{0}+m$. Then

$a_{n}\geq a_{r}a_{m}^{q}\geq c_{m}a_{m}^{q}\,,\quad\text{where}\quad
c_{m}=\min\\{a_{r}:n_{0}\leq r<n_{0}+m\\}>0.$

Therefore $a_{n}^{1/n}\geq c_{m}^{1/n}a_{m}^{q/n}$, and letting $n\to\infty$,

$\rho(\mu)\geq\liminf_{n\to\infty}a_{n}^{1/n}\geq a_{m}^{1/m}\,.$

This holds for every $m$, and letting $m\to\infty$, we get that $a_{m}^{1/m}$
tends to $\rho(\mu)$ from below. For general $x$, there is $k$ such that
$\mu^{(k)}(x)>0$, and $\mu^{(n)}(x)\geq\mu^{(k)}(x)\mu^{(n-k)}(e)$. Taking
$n^{\text{th}}$ roots on both sides, we get that also $\mu^{(n)}(x)^{1/n}$
converges.

The last observation is due to Gerl [4]: fix $x_{0}\in
S_{\mu}\setminus\\{e\\}$. Then $x_{0}^{-1}\in S_{\mu}^{r_{0}}$ for some
$r_{0}\in\mathbb{N}$, and by the above,

$\mu^{(n)}(x_{0})\mu^{(n)}(x_{0}^{-1})>0\quad\text{for all }\;n\geq
n_{0}+r_{0}=:k_{0}$

Now suppose that for some $n\geq k_{0}$, $\mu^{(n)}(e)=\rho(\mu)^{n}.$ Then

$\rho(\mu)^{2n}\geq\mu^{(2n)}(e)=\sum_{x\in\Gamma}\mu^{(n)}(x)\mu^{(n)}(x^{-1})=\rho(\mu)^{2n}+\sum_{x\neq
e}\mu^{(n)}(x)\mu^{(n)}(x^{-1}).$

But then $\mu^{(n)}(x_{0})\mu^{(n)}(x_{0}^{-1})=0$, a contradiction. ∎

The following result of Gerl [4] is less straightforward, and was the primary
motivation for writing this little note. It is based on an argument in [3],
compare also with Guivarc’h [5, pp. 18-19]. According to Le Page [6], the
argument can be traced back to Orey and Kingman [8].

######  Theorem 1.

If $\mu$ is irreducible and aperiodic on $\Gamma$, then

$\lim_{n\to\infty}\frac{\mu^{(n+1)}(x)}{\mu^{(n)}(x)}=\rho(\mu)\quad\text{for
every }\;x\in\Gamma\,.$

###### Proof.

Write $\rho=\rho(\mu)$. Recall the transition probabilities
$p(x,y)=\mu(x^{-1}y)$ of the random walk. Irreducibility yields that there is
a positive $\rho$-subharmonic function $h:\Gamma\to(0\,,\,\infty)$, that is,

$\sum_{y\in\Gamma}\mu(x^{-1}y)h(y)\leq\rho\,h(x)\quad\text{for every
}\;x\in\Gamma\,.$

See [9], [11] or [15, Lemma 7.2]. In many cases, there even is such a function
which is $\rho$-harmonic, i.e., equality holds at every $x$.

We now define a new Markov chain on $\Gamma$, the _$h$ -process,_ with
transition probabilities

$p_{h}(x,y)=\frac{\mu(x^{-1}y)h(y)}{\rho\,h(x)}.$

This is in general not a group-invariant random walk. The transition matrix
(denoted $Q$ in [4])

$P_{h}=\bigl{(}p_{h}(x,y)\bigr{)}_{x,y\in\Gamma}$

is substochastic, i.e., $\sum_{y}p_{h}(x,y)\leq 1$ for all $x$, so that there
may be a positive probability that the Markov chain “dies” at $x$.
Furthermore, along with $\mu$ it is irreducible and aperiodic: for all $x,y$,
there is $n_{x,y}$ such that $p_{h}^{(n)}(x,y)>0$ for all $n\geq n_{x,y}\,$,
where $p_{h}^{(n)}(x,y)$ is the $(x,y)$-entry of the matrix power
$P_{h}^{n}\,$. We have by Lemma 3

$p_{h}^{(n)}(x,y)=\frac{\mu^{(n)}(x^{-1}y)h(y)}{\rho^{n}\,h(x)}\quad\text{and}\quad\lim_{n\to\infty}p_{h}^{(n)}(x,y)^{1/n}=1\quad\text{for
all }\;x,y\in\Gamma\,.$

In particular, for all $x$,

$0<p_{h}^{(n)}(x,x)=\frac{\mu^{(n)}(e)}{\rho^{n}}<1\quad\text{for all $n\geq
k_{0}\,$.}$

We now fix $m\geq k_{0}$ and set $a=a_{m}=1-p_{h}^{(m)}(x,x)$, so that
$0<a<1$, as well as

$Q=\frac{1}{a}\bigl{(}P_{h}^{m}-(1-a)I\bigr{)}\,,$

where $I$ is the identity matrix over $\Gamma$. (The matrix $Q$ is denoted $R$
in [4].) We shall also need the matrix $E$ over $\Gamma$ with all entries
$=1$. For the next lines of the proof, we just write $P$ for $P_{h}^{m}\,$.
Then $Q$ is also non-negative, substochastic and irreducible, and
$P=a\,Q+(1-a)I$. Note that $Q$ commutes with $P_{h}\,$, and that $P_{h}E\leq
E$. Then

$P^{n}=\sum_{k=0}^{n}p_{a}(n,k)\,Q^{k}\,,\quad\text{where}\quad
p_{a}(n,k)={n\choose k}a^{k}(1-a)^{n-k}\,.$

The latter is the probability that the sum $S_{n}=X_{1}+\dots+X_{n}$ of i.i.d.
Bernoulli random variables with $\mathsf{Pr}[X_{k}=1]=a$ has value $k$. For
$\varepsilon>0$, consider the set

$C_{n}(\varepsilon)=\bigl{\\{}k\in\\{0\,,\dots,n\\}:p_{a}(n,k)\leq(1+\varepsilon)\,p_{a}(n+1,k+1)\bigr{\\}}$

and its complement $C_{n}(\varepsilon)^{c}$ in $\\{0\,,\dots,n\\}$. Then

$\sum_{k\in
C_{n}^{c}}p_{a}(n,k)=\mathsf{Pr}\biggl{[}\frac{S_{n}+1}{n+1}-a>\varepsilon\,a\biggr{]}\,.$

This is a large deviation probability, which is well known to decay
exponentially, i.e., there is $\delta>0$ such that it is $\leq e^{-n\delta}$.
In our specific case, this can also be verified by combinatorial computations.
See e.g. Rényi [10, p. 324]. Then, using matrix products and elementwise
inquality between matrices,

$\displaystyle\frac{1}{a}P^{n+1}-\frac{1-a}{a}P^{n}=QP^{n}$
$\displaystyle=\sum_{k=0}^{n}p_{a}(n,k)Q^{k+1}$ $\displaystyle\leq e^{-\delta
n}E+(1+\varepsilon)\sum_{k\in C_{n}(\varepsilon)}p_{a}(n+1,k+1)Q^{k+1}$
$\displaystyle\leq e^{-\delta n}E+(1+\varepsilon)P^{n+1}\,.$

Reassembling the terms,

$\Bigl{(}1-\frac{a}{1-a}\,\varepsilon\Bigr{)}P^{n+1}\leq\frac{a}{1-a}\,e^{-\delta
n}E+P^{n}\,.$

We multiply from the left with $P_{h}^{r}$, where $r\in\mathbb{N}_{0}$ is
arbitrary, and get for the matrix elements

$\left(1-\frac{a}{1-a}\varepsilon\right)\frac{p_{h}^{(mn+m+r)}(x,y)}{p_{h}^{(mn+r)}(x,y)}\leq\frac{a}{1-a}\,\frac{e^{-\delta
n}}{p_{h}^{(mn+r)}(x,y)}+1\,.$

We are not dividing by $0$ if $n$ is sufficiently large, and since
$p_{h}^{(mn+r)}(x,y)^{1/n}\to 1$, the right hand side tends to 1 as
$n\to\infty$. Since we can choose $\varepsilon$ arbitrarily small, we get

(5) $\limsup_{n\to\infty}\frac{p_{h}^{(mn+m+r)}(x,y)}{p_{h}^{(mn+r)}(x,y)}\leq
1\,,$

and this holds for every $m\geq k_{0}$ and every $r\geq 0$.

For an analogous lower bound, we use the set

$D_{n}(\varepsilon)=\bigl{\\{}k\in\\{0\,,\dots,n\\}:p_{a}(n+1,k+1)\leq(1+\varepsilon)p_{a}(n,k)\bigr{\\}}$

and observe that

$\sum_{k\in
C_{n}^{c}}p_{a}(n+1,k+1)=\mathsf{Pr}\biggl{[}\frac{S_{n+1}+1}{n+1}-a<-\frac{\varepsilon}{1+\varepsilon}\,a\biggr{]}\,.$

also decays exponentially, and is bounded by $e^{-\delta n}$ for some
$\delta>0$. Then

$\displaystyle P^{n+1}$ $\displaystyle\leq e^{-\delta
n}E+(1+\varepsilon)\sum_{k\in D_{n}(\varepsilon)}p_{a}(n,k)Q^{k+1}$
$\displaystyle\leq e^{-\delta n}E+(1+\varepsilon)P^{n}Q=e^{-\delta
n}E+(1+\varepsilon)\left(\frac{1}{a}P^{n+1}-\frac{1-a}{a}P^{n}\right)\,.$

Reassembling the terms,

$(1+\varepsilon)P^{n}\leq\frac{a}{1-a}\,e^{-\delta
n}E+\Bigl{(}1+\frac{\varepsilon}{1-a}\Bigr{)}P^{n+1}\,.$

Proceeding as above, we get for all $m\geq k_{0}$ and all $r\geq 0$

(6) $\liminf_{n\to\infty}\frac{p_{h}^{(mn+m+r)}(x,y)}{p_{h}^{(mn+r)}(x,y)}\geq
1\,,$

for every $m\geq k_{0}$ and every $r\geq 0$. Since these two numbers chan be
chosen arbitrarily, it is an easy exercise to deduce from (5) and (6) that

$\lim_{n\to\infty}\frac{p_{h}^{(n+1)}(x,y)}{p_{h}^{(n)}(x,y)}=1\,,$

and the stated result follows. ∎

######  Remark.

The last theorem is not restricted to random walks on groups. If $P$ is the
transition matrix of an irreducible Markov chain on a countable set – say –
$\Gamma$, and it is _strongly aperiodic,_ that is,

$\gcd\bigl{\\{}n\in\mathbb{N}:\inf_{x}p^{(n)}(x,x)>0\bigr{\\}}=1\,,$

then the same proof applies to show that

$None$
$\lim_{n\to\infty}\frac{p^{(n+1)}(x,y)}{p^{(n)}(x,y)}=\rho(P)\quad\text{for
all }\;x,y\in\Gamma\,.$

In the situation of Theorem 1, assume for simplicity that $S_{\mu}$ is finite.
Then it is well known and easy to decuce that the sequence of measures

$\Bigl{(}\frac{\mu^{(n)}}{\mu^{(n)}(e)}\Bigr{)}_{n\in\mathbb{N}}$

is relatively compact in the topology of pointwise convergence, and every
limit measure $\nu$ satisfies the convolution equation

(8) $\mu*\nu=\rho(\mu)\cdot\nu\,,$

or in other words, the function $h(x)=\nu(x^{-1})$ is $\rho(\mu)$-harmonic,
that is

$\sum_{y}\mu(x^{-1}y)h(y)=\rho(\mu)\,h(x)\quad\text{for all }\;x\in\Gamma\,.$

[4] and (under slightly different conditions, where the state space is not
necessarily discrete) [5] prove the following ratio limit theorem.

######  Theorem 2.

Assume that $\mu$ is irreducible and aperiodic, and that $S_{\mu}$ is finite.
Suppose that (P) is a certain property of positive measures $\nu$ on $\Gamma$
such that

  * •

every limit along a pointwise convergent subsequence

$\Bigl{(}\frac{\mu^{(n_{k})}}{\mu^{(n_{k})}(e)}\Bigr{)}_{k\in\mathbb{N}}$

must have property (P), and

  * •

there is a unique positive measure $\nu$ with $\nu(e)=1$ that satisfies (8).

Then

$\lim_{n\to\infty}\frac{\mu^{(n)}(x)}{\mu^{(n)}(e)}=\nu(x)\quad\text{for all
}\;x\in\Gamma.$

The proof is clear in view of the observations preceding the statement of
Theorem 2, i.e., relative compactness and (8), which are left to the reader as
exercises or to be looked up in old references. Finiteness of $S_{\mu}$ can be
relaxed. Typical examples of application are (virtually) Abelian groups, where
property (P) is empty, because there is a unique solution $\nu$ of (8): there
is a good amount of literature from the 1960ies on this. The most significant
reference for Abelian groups is Stone [12].

Other typical examples are isotropic random walks on free groups, and property
(P) is that $\nu$ is also isotropic; see [4]. It should be noted that in those
cases, one also has stronger results, namely local limit theorems; see [15,
Chapter III].

Author’s final, personal remarks. When I wrote the monograph [15] in the
$2^{\text{nd}}$ half of the 1990ies, ratio limit theorems were not an active
topic, but replaced by the study of the asymptotic type of random walk
transition probabilities and the sharper local limit theorems. Therefore,
having to keep the book size under control, I had “sacrificed” the material of
ratio limit theorems. In the meantime, the subject has “woken up” again, e.g.
in a recent paper of Dor-On [1] and current work of Dougall and Sharp.

## References

  * [1] Dor-On, A.: _Toeplitz quotient $C^{*}$-algebras and ratio limits for random walks._ Documenta Math. 26 (2021) 1529–1556.
  * [2] Fekete, M.: _Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten._ Math. Zeitsch. 17, no. 1 (1923) 228–249.
  * [3] Gerl, P.: _Diskrete, mittelbare Gruppen._ Monatsh. Math. 77 (1973) 307–318.
  * [4] Gerl, P.: _Wahrscheinlichkeitsmaße auf diskreten Gruppen._ Arch. Math. (Basel) 31, no. 6 (1978/79) 611–619.
  * [5] Guivarc’h, Y: _Théorèmes quotients pour les marches aléatoires._ In _Conference on Random Walks_ (Kleebach, 1979), pp. 15–28, 3, Astérisque 74 (1980) 15–28.
  * [6] Le Page, É.: _Thórèmes quotients pour certaines marches aléatoires._ C. R. Acad. Sci. Paris Sér. A 279 (1974) 69–72.
  * [7] Mukherjea, A., and Tserpes, N. A.: Measures on Topological Semigroups: Convolution Products and Random Walks. Lecture Notes in Mathematics 547. Springer, Berlin–New York, 1976.
  * [8] Kingman, J. F. C., and Orey, St.: _Ratio limit theorems for Markov chains._ Proc. Amer. Math. Soc. 15 (1964) 907–910.
  * [9] Pruitt, W. E.: _Eigenvalues of non-negative matrices._ Ann. Math. Statist. 35 (1964) 1797–1800.
  * [10] Rényi, A.: _Wahrseheinlichkeitsrechnung._ Deut. Verl. d. Wissenschaften, Berlin, 1962.
  * [11] Seneta, E.: _Nonnegative matrices and Markov chains._ Second edition. Springer Series in Statistics, New York, 1981.
  * [12] Stone, Ch.: _Ratio limit theorems for random walks on groups._ Trans. Amer. Math. Soc. 125 (1966) 86–100.
  * [13] Vere-Jones, D.: _Ergodic properties of nonnegative matrices. I._ Pacific J. Math. 22 (1967) 361–386.
  * [14] Woess, W.: Périodicité de mesures de probabilité sur les groupes topologiques. In _Random Walks and Stochastic Processes on Lie Groups_ (Nancy, 1981), Inst. Élie Cartan 7 170–180 (1983).
  * [15] Woess, W.: _Random Walks on Infinite Graphs and Groups_ , Cambridge Tracts in Mathematics 138, Cambridge University Press, Cambridge, 2000.


11institutetext: Department of Computer Science, Vanderbilt University,
Nashville, TN, 3723522institutetext: Department of Electrical and Computer
Engineering, Vanderbilt University, Nashville, TN, 37235
22email<EMAIL_ADDRESS>
# Unsupervised Domain Adaptation for Vestibular Schwannoma and Cochlea
Segmentation via Semi-supervised Learning and Label Fusion

Han Liu equal contribution11 Yubo Fan* 11 Can Cui 11 Dingjie Su 11 Andrew
McNeil 22 Benoit M. Dawant 22

###### Abstract

Automatic methods to segment the vestibular schwannoma (VS) tumors and the
cochlea from magnetic resonance imaging (MRI) are critical to VS treatment
planning. Although supervised methods have achieved satisfactory performance
in VS segmentation, they require full annotations by experts, which is
laborious and time-consuming. In this work, we aim to tackle the VS and
cochlea segmentation problem in an unsupervised domain adaptation setting. Our
proposed method leverages both the image-level domain alignment to minimize
the domain divergence and semi-supervised training to further boost the
performance. Furthermore, we propose to fuse the labels predicted from
multiple models via noisy label correction. In the MICCAI 2021 crossMoDA
challenge111https://crossmoda.grand-challenge.org/, our results on the final
evaluation leaderboard showed that our proposed method has achieved promising
segmentation performance with mean dice score of 79.9% and 82.5% and ASSD of
1.29 mm and 0.18 mm for VS tumor and cochlea, respectively. The cochlea ASSD
achieved by our method has outperformed all other competing methods as well as
the supervised nnU-Net.

###### Keywords:

Vestibular schwannoma Cochlea Unsupervised domain adaptation Semi-supervised
learning Label fusion

##  1 Introduction

Vestibular schwannoma (VS) is a benign tumor that arises from the Schwann
cells of the vestibular nerve, which connects the brain and the inner ear. To
facilitate the follow-up and treatment planning of VS, automatic methods to
segment the VS tumors and the cochlea from magnetic resonance imaging (MRI)
have been proposed [1]. While the most commonly used modality for VS
segmentation is contrast-enhanced T1 (ceT1), high-resolution T2 (hrT2) imaging
has been demonstrated to be a possible alternative with less risk and lower
cost[2].

Supervised segmentation methods have shown to be effective for VS segmentation
[3], but they require to fully annotate image data which may not be an option
in practice. Weakly-supervised methods require less annotation efforts, such
as scribbles and bounding boxes, and sometimes they even achieve a level of
performance comparable to the supervised ones [4]. In this work, we aim at
segmenting the VS tumor and the cochlea in hrT2 without any hrT2 annotations
during training. We consider the problem as an unsupervised domain adaptation
(UDA) problem. Specifically, we are provided with a dataset consisting of ceT1
images and hrT2 images, but only the ceT1 images have the segmentation labels.

There are mainly two types of methods to tackle the UDA problem, domain
alignment and techniques based on semi-supervised learning (SSL). Domain
alignment focuses on reducing the distribution discrepancy by optimizing some
divergence metric [5, 6] or via adversarial learning [7, 8]. Domain gaps can
be bridged by image-level alignment [20, 21], feature-level alignment [22,
23], or the combination of the two [24]. On the other hand, due to the lack of
labels in the target domain, techniques originating from SSL can be utilized
to improve the performance. Zou et al. [9] and Zhang et al. [26] use self-
training based methods which iteratively generate pseudo labels and use them
to retrain the network. To alleviate the negative impact from the noisy pseudo
labels, learning from noisy labels has also received increasing interest.
Motivated by [15], Zhang et al. [27] use a confident learning module to
characterize the label errors and correct them to achieve a more robust
training. Mean teacher [29], as another SSL-based technique, can be also used
in UDA to provide competitive performance [10, 28]. Inspired by previous
works, we focus on exploring methods that combine image-level domain alignment
and SSL for UDA.

##  2 Methods

###  2.1 Problem Formulation

For an unsupervised domain adaptation problem, we have access to a source
domain $D^{S}=\\{(x^{s}_{i},y^{s}_{i})|i=1,2,\cdots,n_{s}\\}$, and a target
domain $D^{T}=\\{x^{t}_{j}|j=1,2,\cdots,n_{t}\\}$, where $Y^{S}$ and $Y^{T}$
share the same $K$ classes. In our case, source and target domains correspond
to ceT1 and hrT2 respectively and $K=3$ representing background, VS and
cochlea. We aim to train a segmentation network $F_{t}$ that learns from the
source domain and is capable to achieve robust and accurate segmentation
performance on the target domain, without accessing the target domain labels
$Y^{T}$.

###  2.2 Image-level Domain Alignment

Image-level domain alignment is a simple but effective method to tackle UDA
problem by reducing the distribution mismatch at the image-level, i.e., pseudo
image synthesis. Here, we propose to train the segmentation model $F_{t}$ with
the pseudo target domain images $\tilde{X}^{T}$, which are generated by
unpaired image-to-image translation. We explore both end-to-end training and
two-stage training. For end-to-end training, we rely on the Contrastive
Unpaired Translation (CUT) [11] as the backbone for image synthesis and add an
extra segmentation module $F_{t}$ on top of the synthesized images. This
method is referred as CUTSeg. We select CUT for unpaired image-to-image
translation because it can be trained faster and is less memory-intensive than
the CycleGAN [12], allowing more flexibility when adding the 3D CNN-based
segmentation module. During training, we first train the CUT model alone till
it achieves reasonable synthesis performance. Then we train the CUTSeg end-to-
end with the CUT subnetwork initialized with the pre-trained weights and the
segmentation module trained from scratch. For two-stage training, we use the
CycleGAN to generate pseudo hrT2 images $\tilde{X}^{T}$. To improve the data
diversity, we train both 2D and 3D CycleGANs and collect pseudo images from
different epochs. Lastly, we train a segmentation module $F_{t}$ using
$\tilde{X}^{T}$.

###  2.3 Semi-supervised Training

Though image-level domain alignment can minimize the domain divergence, the
unlabeled target domain images $X^{T}$ are not directly involved in training
the segmentation model $F_{t}$. To overcome this limitation, we propose to
adapt a semi-supervised learning method named Mean Teacher [13] to make better
use of $X^{T}$. Specifically, a student model along with a teacher model with
the same network architecture are created and both models are initialized with
the best model weights obtained from Section 2.2. In our semi-supervised
setting, the labeled images are the pseudo hrT2 images while the unlabeled
images are the real hrT2 images. During training, the labeled pseudo images
are fed to the student model and the segmentation loss $L_{seg}$ is computed
in a supervised manner. For unlabeled images, we first augment the same image
twice with different intensity transformation parameters. The augmented images
are then fed to the student model and the teacher model separately and a
consistency loss $L_{con}$ is computed. As described in Equations 1-4, we use
Dice loss [14] and Cross-Entropy (CE) loss as $L_{seg}$ and Mean Squared Error
(MSE) loss as $L_{con}$, where $p_{ik}$ is the predicted probability of the
$i$th voxel at the $k$th output channel. Note that both $L_{seg}$ and
$L_{con}$ are used to update the weights of the student model. The weights of
the teacher model are updated as an exponential moving average (EMA) of the
student weights, where the EMA decay coefficient is set as 0.99. As suggested
in [13], the teacher prediction is more likely to be correct at the end of the
training and thus the teacher model is taken as our final $F_{t}$.

$L_{seg}=L_{Dice}+L_{CE}$ (1)

$L_{Dice}=1-\frac{2\sum_{k}^{K}\sum_{i}^{N}p_{ik}y_{ik}}{\sum_{k}^{K}\sum_{i}^{N}p_{ik}^{2}+\sum_{k}^{K}\sum_{i}^{N}y_{ik}^{2}}$
(2)

$L_{CE}=-\frac{1}{N}\sum_{k}^{K}\sum_{i}^{N}y_{ik}\log p_{ik}$ (3)

$L_{con}=\frac{1}{N}\sum_{k}^{K}\sum_{i}^{N}(p_{ik}^{Teacher}-p_{ik}^{Student})^{2}$
(4)

###  2.4 Noisy Label Correction as Label Fusion

In this challenge, we have obtained three models (as shown in Figure 1) that
were trained with different strategies and each model alone has achieved
satisfactory result on the validation leaderboard. Specifically, the first
model is obtained by two-stage training using the pseudo images from 2D
CycleGAN, followed by a semi-supervised learning method, i.e., Mean Teacher.
The second model is initialized with the teacher model and fine-tuned using
the pseudo images from 3D CycleGAN. The third model is a CUTSeg model.
Training details can be found in Section 3.1.

Empirically, ensembles tend to yield better predictive performance when there
is a significant diversity among the models. Here, we propose to fuse the
labels from different models by treating the label fusion task as a noisy
label correction problem. We adapt a confident learning method called CleanLab
[15] which provides exact noise estimation and label error finding. Note that
we use CleanLab to directly fuse labels at the inference phase rather than
update the pseudo labels iteratively during training. Specifically, we first
obtain the softmax outputs of two models and convert one output to a one-hot
encoded label mask. The one-hot encoded mask is considered as a ’noisy label’
and corrected by the softmax outputs from the other model. Once the labels
from the first two models are fused, the fused labels are treated as noisy
labels and fused again with the softmax outputs from the remaining model. The
labels fused from the three models are used as our final predictions.

Figure 1: The schematic diagram of our proposed method

##  3 Experiments and Results

###  3.1 Experimental Design

In this section, we describe different methods in our experiments and
summarize the training details in Table 1.

####  3.1.1 Methods #1-3

First, we compare different segmentation backbones including 2D nnU-Net, 3D
nnU-Net, and the 2.5D U-Net proposed in [3], which are referred to as method
#1-3.

####  3.1.2 Method #4

We utilize a semi-supervised learning method named Mean Teacher to leverage
the unlabeled real T2 images. Details are described in Section 2.3.

####  3.1.3 Method #5

Here, we explore the feasibility of using self-training to improve the
segmentation performance of Mean Teacher. Specifically, we use the mean
teacher model to obtain the pseudo labels on the real T2 images. Then we fine-
tune the teacher model obtained from method #4 with the pseudo-labeled real T2
images. The pseudo labels are iteratively updated at the end of each epoch by
a confident learning method [15]. Note that methods #1-4 are trained using
pseudo images generated from 2D CycleGAN.

####  3.1.4 Method #6

We incorporate more training data to further boost the performance.
Specifically, we fine-tune the teacher model from method #4 on additional
pseudo images generated from 3D CycleGAN.

####  3.1.5 Method #7

Here, we train an end-to-end CUTSeg model and the details can be found in
Section 2.2.

####  3.1.6 Methods #8-9

Lastly, we fuse the predictions from different models by CleanLab as described
in section 2.4.

Table 1: Experimental design

# |  Method | Training image |  Training label | Model init.
---|---|---|---|---
1 |  nnU-Net (2D) | S-T2 (2D) |  T1 | scratch
2 |  nnU-Net (3D) | S-T2 (2D) |  T1 | scratch
3 |  U-Net (2.5D) | S-T2 (2D) |  T1 | scratch
4 |  Mean teacher | S-T2 (2D) + R-T2 |  T1 | #3
5 |  #4 + self-training | R-T2 |  #4 (pseudo label) | #4 (teacher)
6 |  #4 + fine-tuning | S-T2 (3D) |  T1 | #4 (teacher)
7 |  CUTSeg | R-T1 + R-T2 |  T1 | scratch
8 |  #4 $\circ$ #6 | - |  - | -
9 |  #4 $\circ$ #6 $\circ$ #7 | - |  - | -

  * •

In the method column, $\circ$ represents label fusion using CleanLab. In the
training data column, S and R represent synthetic data from CycleGAN and real
data, respectively. 2D and 3D represent synthetic data generated from 2D
CycleGAN and 3D CycleGAN, respectively.

###  3.2 Data and Implementation

The dataset was released by the MICCAI challenge crossMoDA 2021 [16]. All
images were obtained on a 32-channel Siemens Avanto 1.5T scanner using a
Siemens single-channel head coil [17]. ceT1 images have in-plane resolution of
0.41×0.41 mm and slice thickness of 1.0 or 1.5 mm. For hrT2 images, the in-
plane resolution varies from 0.47×0.47 mm to 0.55×0.55 mm and slice thickness
is 1.0 or 1.5 mm. The VS and cochleae were manually segmented in consensus by
the treating neurosurgeon and physicist using both the ceT1 and hrT2 images.
We randomly split the images into 185 and 25 for training and validation
respectively. Since the Field of View (FoV) of the source and target domain
images varies significantly, we crop each image into a cubic box, or ROI,
using single-atlas registration [18]. As shown in Figure 2, the ROI on the
atlas image is manually cropped around the right side of the brain. To obtain
the ROI on the left side, we flip the volume left-to-right before performing
registration.

For preprocessing, in two-stage training (methods #1-6), we resample the
images to the most common spacing in the target domain, i.e., (0.46875,
0.468975, 1.5) and normalize the intensity to [0, 1]. In end-to-end training
(method #7), we first train a CUTSeg model for 82 epochs with an auxiliary
consistency loss, which is a Mean Absolute Error (MAE) loss between the
segmentation result of the real hrT2 image and the prediction from the two-
stage training model. This model is used to make inferences for the testing
images with in-plane resolution less than 0.5 mm. We use another CUTSeg model
which is fine-tuned on the hrT2 images with in-plane resolution more than 0.5
mm to make inference for the testing images with such resolution. For all our
segmentors, we use the 2.5D U-Net architecture proposed in [3]. For post-
processing, we first reduce the false positive VS prediction by removing the
isolated components whose center is more than 15 voxels along the z-axis from
the adjacent cochlea center. Then we take the largest connected components for
both VS and cochlea within each ROI.

Figure 2: An illustration of our cropped ROI on the target domain atlas image.

For training, we use Adam optimizer with weight decay $10^{-4}$ and batch size
$1$. The learning rates are initialized to $5\times 10^{-4}$, $5\times
10^{-5}$ and $2\times 10^{-4}$ for two-stage training, Mean Teacher, and
CUTSeg, respectively. The hyperparameters are determined by grid-search within
the range of $10^{-2}$ to $10^{-6}$. The best hyperparameters are selected
based on the segmentation performance on our own validation set. The CNNs are
implemented in PyTorch [19] and MONAI on a Ubuntu desktop with an NVIDIA RTX
2080 Ti GPU. For quantitative evaluation, we measure the Dice score and
Average Symmetric Surface Distance (ASSD) between the segmentation results and
the ground truth.

Table 2: Quantitative results on validation leaderboard |  | Dice$\uparrow$ (%) | ASSD$\downarrow$ (mm)
---|---|---|---
# | Method | VS | Cochlea | VS | Cochlea
1 | nnU-Net (2D) | 72.90±22.77 | 68.14±12.62 | 1.32±2.83 | 1.27±3.65
2 | nnU-Net (3D) | 71.77±27.29 | 79.95±4.12 | 2.71±5.09 | 0.20±0.07
3 | U-Net (2.5D) | 74.81±22.14 | 80.39±3.18 | 1.37±2.63 | 0.22±0.05
4 | Mean teacher | 80.81±9.09 | 80.50±6.36 | 0.63±0.30 | 0.20±0.06
5 | #4 + self-training | 80.73±5.21 | 81.16±4.25 | 0.69±0.35 | 0.19±0.04
6 | #4 + fine-tuning | 81.66±10.5 | 81.24±3.52 | 0.61±0.37 | 0.20±0.06
7 | CUTSeg | 81.00±7.50 | 81.45±3.53 | 0.66±0.29 | 0.19±0.06
8 | #4 $\circ$ #6 | 82.08±4.77 | 81.46±3.51 | 0.58±0.31 | 0.19±0.06
9 | #4 $\circ$ #6 $\circ$ #7 | 83.02±7.72 | 82.20±3.10 | 0.57±0.27 | 0.18±0.05

###  3.3 Experimental Results

Table 2 shows the evaluation metrics on the validation leaderboard of our
developed methods. By comparing method #1-3, we notice that the 2.5D U-Net
architecture inspired by [3] outperforms the 2D and 3D nnU-Nets and thus is
used as the segmentation backbone in all our experiments. We also find that by
incorporating the real unlabeled T2 images, our Mean Teacher model (method #4)
is able to increase the VS dice score from 74.81% to 80.81%, demonstrating the
effectiveness of our semi-supervised learning strategy. We observe that a
smaller learning rate and appropriate model initialization, i.e., the pre-
trained segmentation model weights from method #3, are critical for the
effectiveness of MT. Since the Mean Teacher achieved the best performance
among method #1-4 on leaderboard during validation phase, we use the best
weights obtained by method #4 for model initialization for other methods,
i.e., methods #5-6. By comparing method #4 and #5, we observe that self-
training slightly outperforms the mean teacher model on cochlea but
underperforms on VS tumor. By fine-tuning with more training data, method #6
shows slight improvements on both VS and cochlea compared to method #4.
Furthermore, we find that the CUTSeg model (method #7) achieves comparable
segmentation performance to method #4. Lastly, in method #8 and #9, we show
that though CleanLab is unable to improve performance by iteratively updating
labels during training (method #5), it is however an effective method to fuse
the predictions from different models during the inference phase. In the
evaluation phase, our proposed method (method #9) achieved dice scores of
79.9% and 82.5% and ASSD of 1.29 mm and 0.18 mm for VS tumor and cochlea,
respectively. We note that the ASSD of cochlea achieved by our method (0.18
mm) is the lowest among all unsupervised methods and even lower than that
achieved by the supervised nnU-Net (0.22 mm).

Figure 3: Qualitative results on the validation set. A to C and D to F display
the best and worst VS segmentation results. The image ID and the corresponding
dice score are also shown. Our experiments show that VS tumors with higher
inhomogeneity, e.g., D, are more difficult to segment.

##  4 Discussion

###  4.1 ROI Selection Strategy

In this challenge, we observe that the FoV of source and target domain images
varies significantly. To minimize the impact of FoV difference, we manually
determine a cubic box as ROI and use rigid registration to obtain the ROI from
each volume as described in section 3.1. Note that our ROI covers either the
left or the right side region around the cochlea while most other teams use
the ROIs that cover both cochleae. Arguably, our ROI selection strategy has
both a positive and a negative impact on the downstream segmentation tasks.
First, this strategy helps to increase the amount of training data since two
ROIs can be extracted from one volume. Second, by flipping the right ROI to
the left, the orientation and relative position of the cochlea within each ROI
remain almost the same, which is greatly beneficial for the accuracy and
robustness of cochlea segmentation. Based on the results from the validation
leaderboard, all our attempted methods have achieved around 80% dice scores,
which are very competitive against other teams’ results. Moreover, in the
evaluation leaderboard, our method has achieved an ASSD of 0.18 mm which is
lower than the supervised nnU-Net of 0.22 mm, suggesting the benefits of our
ROI selection strategy on cochlea segmentation. However, this strategy
inevitably introduces challenges for VS segmentation. Because VS tumor can be
either fully/partially included in both left and right ROIs, the merging
scheme for combining the VS segmentation results from both ROIs needs to be
carefully designed. In the future, it may be interesting to explore whether
using ROIs with different FoVs for VS tumor and cochlea can help improve the
results.

###  4.2 Self-training Strategy

Our experiments show that our self-training strategy cannot help to improve
the segmentation performance. In contrast, both the top-two teams on the
leaderboard found the self-training strategy very beneficial for their
performance. Specifically, the 1st-place team updated the noisy labels by
manually inspecting their qualities and filtered out the unreliable labels. On
the other hand, the 2nd-place team considered the voxels with the higher
probabilities as confident labels to train the next epoch. Hence, we speculate
that the effectiveness of self-training may heavily depend on the filtering
strategy for noisy pseudo labels. Besides, in our self-training experiments,
we only used the real unlabeled data while the aforementioned teams used the
combined data, i.e., pseudo T2 and real T2 images. The combined data might
also be a key factor for the effectiveness of self-training. In the future, an
exciting direction to improve the performance would be to develop an automatic
and smart filtering strategy for self-training.

##  5 Conclusion

In this work, we exploited the image-level domain alignment and semi-
supervised training to tackle the unsupervised domain adaptation segmentation
problem. The results on the validation and evaluation leaderboard of crossMoDA
challenge show that our proposed method can yield promising segmentation
performance on VS tumor and cochlea on hrT2 MRI images without labels.

## References

  * [1] Vokurka, Elizabeth A., et al. ”Using Bayesian tissue classification to improve the accuracy of vestibular schwannoma volume and growth measurement.” American journal of neuroradiology 23.3 (2002): 459-467.
  * [2] Coelho, Daniel H., et al. ”MRI surveillance of vestibular schwannomas without contrast enhancement: clinical and economic evaluation.” The Laryngoscope 128.1 (2018): 202-209.
  * [3] Wang, Guotai, et al. ”Automatic segmentation of vestibular schwannoma from T2-weighted MRI by deep spatial attention with hardness-weighted loss.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2019.
  * [4] Dorent, Reuben, et al. ”Scribble-based Domain Adaptation via Co-segmentation.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2020.
  * [5] Lee, Chen-Yu, et al. ”Sliced wasserstein discrepancy for unsupervised domain adaptation.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2019.
  * [6] Long, Mingsheng, et al. ”Learning transferable features with deep adaptation networks.” International conference on machine learning. PMLR, 2015.
  * [7] Hoffman, Judy, et al. ”Cycada: Cycle-consistent adversarial domain adaptation.” International conference on machine learning. PMLR, 2018.
  * [8] Ganin, Yaroslav, et al. ”Domain-adversarial training of neural networks.” The journal of machine learning research 17.1 (2016): 2096-2030.
  * [9] Zou, Yang, et al. ”Unsupervised domain adaptation for semantic segmentation via class-balanced self-training.” Proceedings of the European conference on computer vision (ECCV). 2018.
  * [10] Perone, Christian S., et al. ”Unsupervised domain adaptation for medical imaging segmentation with self-ensembling.” NeuroImage 194 (2019): 1-11.
  * [11] Park, Taesung, et al. ”Contrastive learning for unpaired image-to-image translation.” European Conference on Computer Vision. Springer, Cham, 2020.
  * [12] Zhu, Jun-Yan, et al. ”Unpaired image-to-image translation using cycle-consistent adversarial networks.” Proceedings of the IEEE international conference on computer vision. 2017.
  * [13] Tarvainen, Antti, and Harri Valpola. ”Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results.” arXiv preprint arXiv:1703.01780 (2017).
  * [14] Milletari, F., Navab, N., & Ahmadi, S. A. (2016, October). V-net: Fully convolutional neural networks for volumetric medical image segmentation. In 2016 fourth international conference on 3D vision (3DV) (pp. 565-571). IEEE.
  * [15] Northcutt, Curtis, Lu Jiang, and Isaac Chuang. ”Confident learning: Estimating uncertainty in dataset labels.” Journal of Artificial Intelligence Research 70 (2021): 1373-1411.
  * [16] Dorent, R., Kujawa, A., Ivory, M., Bakas, S., Rieke, N., Joutard, S., … & Vercauteren, T. (2022). CrossMoDA 2021 challenge: Benchmark of Cross-Modality Domain Adaptation techniques for Vestibular Schwnannoma and Cochlea Segmentation. arXiv preprint arXiv:2201.02831.
  * [17] Shapey, J., Kujawa, A., Dorent, R., Wang, G., Dimitriadis, A., Grishchuk, D., Pad-dick, I., Kitchen, N., Bradford, R., Saeed, S.R., Bisdas, S., Ourselin, S., Vercauteren,T.: Segmentation of vestibular schwannoma from mri — an open annotated datasetand baseline algorithm. Scientific Data (2021), in press. Preprint available athttps://doi.org/10.1101/2021.08.04.21261588medRXiv:10.1101/2021.08.04.21261588
  * [18] Avants, Brian B., et al. ”A reproducible evaluation of ANTs similarity metric performance in brain image registration.” Neuroimage 54.3 (2011): 2033-2044.
  * [19] Paszke, Adam, et al. ”Pytorch: An imperative style, high-performance deep learning library.” Advances in neural information processing systems 32 (2019): 8026-8037.
  * [20] Zhang, Yue, et al. ”Task driven generative modeling for unsupervised domain adaptation: Application to x-ray image segmentation.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2018.
  * [21] Bousmalis, Konstantinos, et al. ”Unsupervised pixel-level domain adaptation with generative adversarial networks.” Proceedings of the IEEE conference on computer vision and pattern recognition. 2017.
  * [22] Zhang, Yifan, et al. ”Collaborative unsupervised domain adaptation for medical image diagnosis.” IEEE Transactions on Image Processing 29 (2020): 7834-7844.
  * [23] Chang, Wei-Lun, et al. ”All about structure: Adapting structural information across domains for boosting semantic segmentation.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2019.
  * [24] Chen, Cheng, et al. ”Unsupervised bidirectional cross-modality adaptation via deeply synergistic image and feature alignment for medical image segmentation.” IEEE transactions on medical imaging 39.7 (2020): 2494-2505.
  * [25] Zhang, Pan, et al. ”Prototypical pseudo label denoising and target structure learning for domain adaptive semantic segmentation.” Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2021.
  * [26] Zhang, Qiming, et al. ”Category Anchor-Guided Unsupervised Domain Adaptation for Semantic Segmentation.” Advances in Neural Information Processing Systems 32 (2019): 435-445.
  * [27] Zhang, Minqing, et al. ”Characterizing label errors: Confident learning for noisy-labeled image segmentation.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2020.
  * [28] Zhao, Ziyuan, et al. ”MT-UDA: Towards Unsupervised Cross-modality Medical Image Segmentation with Limited Source Labels.” International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Cham, 2021.
  * [29] Tarvainen, Antti, and Harri Valpola. ”Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results.” Advances in Neural Information Processing Systems 30 (2017).