# Random iteration on hyperbolic Riemann surfaces

Marco Abate Partially supported by 2017 PRIN grant “Real and Complex
Manifolds: Topology, Geometry and Holomorphic Dynamics”, Ministry of
University and Research, Italy, and by 2020 PRA grant “Sistemi dinamici in
logica, geometria, fisica matematica e scienza delle costruzioni”, University
of Pisa, Italy. Dipartimento di Matematica, Università di Pisa, Largo
Pontecorvo 5, I-56127 Pisa, Italy. E-mail<EMAIL_ADDRESS>Argyrios
Christodoulou  School of Mathematical Sciences, Queen Mary University of
London, Mile End Road, London E1 4NS, United Kingdom. E-mail:
<EMAIL_ADDRESS>
(September 19, 2021)

###### Abstract

Let $\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(X,X)$ be a sequence of
holomorphic self-maps of a hyperbolic Riemann surface $X$. In this paper we
shall study the asymptotic behavior of the sequences obtained by iteratively
left-composing or right-composing the maps $\\{f_{\nu}\\}$; the sequences of
self-maps of $X$ so obtained are called left (respectively, right) iterated
function systems. We shall obtain the analogue for left iterated function
systems of the theorems proved by Beardon, Carne, Minda and Ng for right
iterated function systems with value in a Bloch domain; and we shall extend to
the setting of general hyperbolic Riemann surfaces results obtained by Short
and the second author in the unit disk $\mathbb{D}$ for iterated function
systems generated by maps close enough to a given self-map.

##  1 Introduction

A classical result in one variable holomorphic dynamics is the _Wolff-Denjoy
theorem:_

######  Theorem 1.1 (Wolff [Wolff1926a, Wolff1926b, Wolff1926c], Denjoy
[Denjoy1926]; 1926).

Let $f\in\mathop{\mathrm{Hol}}\nolimits(\mathbb{D},\mathbb{D})$ be a
holomorphic self-map of the unit disk $\mathbb{D}\subset\mathbb{C}$, not an
elliptic automorphism. Then there exists a point
$\tau\in\overline{\mathbb{D}}$ such that the sequence $\\{f^{k}\\}$ of
iterates of $f$ converges, uniformly on compact subsets, to the constant
$\tau$.

In this statement, an _elliptic automorphism_ is an automorphism of
$\mathbb{D}$ with a fixed point in $\mathbb{D}$. Up to a conjugation an
elliptic automorphism is a rotation, and so its dynamics is well understood;
Theorem 1.1 describes completely the dynamics of all other holomorphic self-
maps of $\mathbb{D}$. The point $\tau$ in the statement is the _Wolff point_
of the function $f$.

This result has been generalised by Heins in 1941 to hyperbolic Riemann
surfaces as follows:

######  Theorem 1.2 (Heins [Heins1941a]; 1941).

Let $X$ be a hyperbolic Riemann surface, and let
$f\in\mathop{\mathrm{Hol}}\nolimits(X,X)$. Then either:

  1. (i)

$f$ has an attracting fixed point in $X$, or

  2. (ii)

$f$ is a periodic automorphism, or

  3. (iii)

$f$ is a pseudoperiodic automorphism, or

  4. (iv)

the sequence $\\{f^{k}\\}$ is compactly divergent.

Furthermore, the case (iii) can occur only if $X$ is either simply connected
(and $f$ has a fixed point) or doubly connected (and $f$ has no fixed points).

An _attracting fixed point_ $z_{0}$ of $f$ is a fixed point (that is,
$f(z_{0})=z_{0}$) where the derivative of $f$ (which is well-defined because
$z_{0}$ is a fixed point) has modulus less than 1; in particular, it follows
that the sequence of iterates $\\{f^{k}\\}$ converges, uniformly on compact
subsets, to the constant function $z_{0}$. A _periodic automorphism_ is an
automorphism $f\in\mathop{\mathrm{Aut}}\nolimits(X)$ such that
$f^{q}=\mathop{\mathrm{id}}\nolimits_{X}$ for some $q\geq 1$ — and then the
dynamics of $f$ is trivial. A _pseudoperiodic automorphism_ is an automorphism
$f\in\mathop{\mathrm{Aut}}\nolimits(X)$ such that there exists a subsequence
of iterates converging to the identity $\mathop{\mathrm{id}}\nolimits_{X}$.
Elliptic automorphisms of $\mathbb{D}$ are always periodic or pseudoperiodic.
A doubly connected hyperbolic Riemann surface is necessarily biholomorphic to
a pointed disk or to an annulus; furthermore, pseudoperiodic automorphisms are
conjugated to rotations, either of the disk or of the pointed disk or of an
annulus, and so their dynamics is well known.

Finally, a sequence $\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(X,Y)$
of holomorphic maps between two Riemann surfaces is _compactly divergent_ if
for every compact $K\subseteq X$ and every compact $L\subseteq Y$ there is a
$\nu_{0}\in\mathbb{N}$ such that $f_{\nu}(K)\cap
L=\mathord{\mathchoice{\ooalign{$\hfil\displaystyle/\hfil$\crcr$\displaystyle\mathrel{\hbox{\kern
0.0pt\raise
0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}{\ooalign{$\hfil\textstyle/\hfil$\crcr$\textstyle\mathrel{\hbox{\kern
0.0pt\raise
0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}{\ooalign{$\hfil\scriptstyle/\hfil$\crcr$\scriptstyle\mathrel{\hbox{\kern
0.0pt\raise
0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}{\ooalign{$\hfil\scriptscriptstyle/\hfil$\crcr$\scriptscriptstyle\mathrel{\hbox{\kern
0.0pt\raise 0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}}$ for all
$\nu\geq\nu_{0}$. Roughly speaking, a compactly diverging sequence is
diverging to infinity (or, more precisely, is converging to the infinity point
of the Alexandroff compactification of $Y$). Restricting to the case of self-
maps for simplicity, when $X=Y=D$ is a domain in a larger compact Riemann
surface $\widehat{X}$ then a compactly divergent sequence of iterates is
converging toward the boundary, in the sense that all accumulation points of
the sequence are constant maps contained in the boundary, and the set of
accumulation points is closed and connected. Furthermore, when the boundary of
$D$ is sufficiently nice then Heins has obtained a complete analogue of the
Wolff-Denjoy theorem:

######  Theorem 1.3 (Heins [Heins1988]; 1988).

Let $D\subset\widehat{X}$ be a hyperbolic domain in a compact Riemann surface
$\widehat{X}$. Assume that $\partial D$ consists of a finite number of
isolated points or disjoint Jordan curves. Let
$f\in\mathop{\mathrm{Hol}}\nolimits(D,D)$ be such that the sequence of
iterates $\\{f^{k}\\}$ is compactly divergent in $D$. Then the sequence
$\\{f^{k}\\}$ converges, uniformly on compact subsets, to a point
$\tau\in\partial D$.

See also [Abate] for proofs and other related results.

A sequence of iterates is obtained by composing the same map over and over.
Stimulated from problems coming from continued fractions theory, computer
simulations of dynamical systems and other sources, in the ’80s mathematicians
have started to study properties of sequences obtained by composing different
self-maps. This area of exploration is often called “random dynamics” because
the self-maps to consider can be chosen at random according to a suitable
probability distribution, for instance with mass concentrated around a fixed
self-map, thus obtaining random perturbations of the sequence of iterates. We
shall not pursue a probabilistic approach in this paper, preferring a more
topological approach, but we kept the term “random dynamics” in the title as
done, for instance, in [BeardonCarneMindaNg2004].

Let us define the sequences of self-maps appearing in random dynamics.

######  Definition 1.4.

Let $\\{f_{\nu}\\}$ be a sequence of self-maps of a Riemann surface $X$, all
different from the identity. The _left_ (or _direct_ or _forward_) _iterated
function system_ (or _composition system_) generated by $\\{f_{\nu}\\}$ is the
sequence of self-maps $\\{L_{\nu}\\}$ given by

$L_{\nu}=f_{\nu}\circ f_{\nu-1}\circ\cdots\circ f_{0}\;.$

The _right_ (or _reverse_ or _backward_) _iterated function system_ generated
by $\\{f_{\nu}\\}$ is instead the sequence of self-maps $\\{R_{\nu}\\}$ given
by

$R_{\nu}=f_{0}\circ f_{1}\circ\cdots\circ f_{\nu}\;.$

When all the maps $f_{\nu}$ belong to a given family $\cal F$ of self-maps of
$X$ we shall say that the corresponding left or right iterated function system
is _in_ $\cal F$.

Clearly, general left or right iterated function systems can have very erratic
behaviours. Moreover, left and right iterated function systems generated by
the same sequence can behave very differently; consider for instance the case
when all $f_{\nu}$’s are constant. So to get meaningful theorems one has
somehow to restrict the class $\mathcal{F}$ of functions used to generate the
iterated function systems.

In this paper we shall consider iterated function systems generated by two
classes of self-maps: maps belonging to
$\mathop{\mathrm{Hol}}\nolimits(X,\Omega)$, where $\Omega\subset X$ is a Bloch
subdomain of $X$; and maps sufficiently close to a given self-map
$F\in\mathop{\mathrm{Hol}}\nolimits(X,X)$.

Roughly speaking (see Section 2 for a precise definition) a Bloch subdomain
$\Omega$ of a hyperbolic Riemann surface is a domain $\Omega\subset X$ such
that all holomorphic maps from $X$ into $\Omega$ are strict contraction with
respect to the Poincaré distance of $X$. Right iterated function systems in
$\mathop{\mathrm{Hol}}\nolimits(X,\Omega)$ have been thoroughly studied by
Beardon, Carne, Minda and Ng [BeardonCarneMindaNg2004] in 2004. Left iterated
function systems, on the other hand, are not so well studied, with the
exception of a few results due to Gill in the unit disk (see [Gill1991a,
Gill2012, Gill2017]) and to Keen and Lakic on plane domains (see
[KeenLakic2007, Section 11.2]).

In this paper, along the lines followed in [BeardonCarneMindaNg2004] for right
iterated function systems, we shall complete the study of left iterated
function systems in Bloch domains proving the following result:

######  Theorem 1.5.

Let $\Omega\subset X$ be a Bloch domain in a hyperbolic Riemann surface $X$,
and let $\\{f_{\nu}\\}$ be a sequence of holomorphic self-maps of $X$ such
that $f_{\nu}(X)\subseteq\Omega$ for all $\nu\in\mathbb{N}$, that is
$\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(X,\Omega)$. Then:

  1. (i)

all limit points of the left iterated function systems $\\{L_{\nu}\\}$
generated by $\\{f_{\nu}\\}$ are constant;

  2. (ii)

let $z_{\nu}\in\Omega$ be the unique fixed point of $f_{\nu}$. Then
$\\{L_{\nu}\\}$ converges to a constant $z_{\infty}\in X$ if and only if
$z_{\nu}\to z_{\infty}$.

Random iteration for holomorphic self-maps ot the unit disk $\mathbb{D}$
sufficiently close to a given map have been thoroughly studied by Short and
the second author in 2019 (see [ChristodoulouShort2019]). In this paper we
shall extend their results to the setting of generic hyperbolic Riemann
surfaces in the spirit of Heins theorems. According to Theorem 1.3 we need to
distinguish three cases: when $F\in\mathop{\mathrm{Hol}}\nolimits(X,X)$ has an
attracting fixed point; when the sequence $\\{F^{\nu}\\}$ of iterates of $F$
is compactly divergent; and when $F$ is a periodic or pseudoperiodic
automorphism of $X$.

When $F$ has an attracting fixed point we shall prove a fairly complete
result:

######  Theorem 1.6.

Let $X$ be a hyperbolic Riemann surface and let
$F\in\mathop{\mathrm{Hol}}\nolimits(X,X)$ be with an attracting fixed point
$z_{0}\in X$. Then:

  1. (i)

there exists a neighbourhood $\mathcal{U}$ of $F$ in
$\mathop{\mathrm{Hol}}\nolimits(X,X)$ such that every right iterated function
system generated by $\\{f_{\nu}\\}\subset\mathcal{U}$ converges to a constant
in $X$;

  2. (ii)

if $\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(X,X)$ is a sequence
converging to $F$ then the left iterated function system generated by
$\\{f_{\nu}\\}$ converges to $z_{0}$.

When the sequence of iterates of $F$ is compactly divergent, simple examples
(see, e.g., Example 4.6) show that in general we cannot deduce much on the
behavior of right iterated function systems generated by functions close to
$F$. On the other hand, if $f_{\nu}$ converges to $F$ fast enough we shall
prove that the behavior of $\\{L_{\nu}\\}$ is dictated by the dynamical
behavior of $F$:

######  Theorem 1.7.

Let $X$ be a hyperbolic Riemann surface, and let
$F\in\mathop{\mathrm{Hol}}\nolimits(X,X)$ be such that the sequence of
iterates $\\{F^{\nu}\\}$ is compactly divergent. Then we can find a sequence
of neighbourhoods
$\mathcal{U}_{\nu}\subset\mathop{\mathrm{Hol}}\nolimits(X,X)$ of $F$ such that
if $f_{\nu}\in\mathcal{U}_{\nu}$ for all $\nu\in\mathbb{N}$ then the left
iterated function system $\\{L_{\nu}\\}$ generated by $\\{f_{\nu}\\}$ is
compactly divergent. Furthermore, if $X\subset\widehat{X}$ is a hyperbolic
domain in a compact Riemann surface and $\\{F^{\nu}\\}$ converges to a point
$\tau\in\partial X$ then $\\{L_{\nu}\\}$ converges to $\tau$.

Surprisingly enough, if the convergence of $f_{\nu}$ to $F$ is too slow then
the left iterated function system might be oscillating even when the sequence
of iterates of $F$ converges to a point in the boundary; see Example 4.5.

Finally, if $F$ is a periodic or pseudoperiodic automorphism of $X$, in
Section 4 we shall see that the only interesting case is $X=\mathbb{D}$, that
has already been studied in [ChristodoulouShort2019].

##  2 Preliminaries

In this section we collect known definitions and results that we shall use in
the sequel.

Let $X$ be a hyperbolic Riemann surface. We shall denote by $\omega_{X}$ the
Poincaré distance on $X$, which is a complete distance whose main property is
the classical Schwarz-Pick lemma:

######  Theorem 2.1.

Let $X$ and $Y$ be two hyperbolic Riemann surfaces, and $f\colon X\to Y$ a
holomorphic function. Then

$\hbox{}\hfilneg\hbox to0.0pt{\hss$\forall z_{1},\,z_{2}\in
X\quad\qquad$}\hfil\omega_{Y}\bigl{(}f(z_{1}),f(z_{2})\bigr{)}\leq\omega_{X}(z_{1},z_{2})\;.$

Furthermore, equality at some $z_{1}\neq z_{2}$ implies that $f$ is a covering
map; conversely, if $f$ is a covering map then for every $w_{1}$, $w_{2}\in Y$
and $z_{1}\in X$ with $f(z_{1})=w_{1}$ we can find $z_{2}\in X$ so that
$f(z_{2})=w_{2}$ and $\omega_{Y}(w_{1},w_{2})=\omega_{X}(z_{1},z_{2})$.

For a proof see, e.g., [BeardonMinda2007].

For each $z\in X$ and $R>0$ we shall denote by $B_{X}(z,R)$ the ball with
respect to $\omega_{X}$ centred in $z$ and with radius $R$. The completeness
of $\omega_{X}$ implies that the closed balls $\overline{B_{X}(z,R)}$ are
compact in $X$.

######  Definition 2.2.

If $\Omega\subset X$ is a domain in a hyperbolic Riemann surface $X$ and
$z\in\Omega$ put

$R(z;\Omega,X)=\sup\\{r>0\mid B_{X}(z,r)\subseteq\Omega\\}$

and

$R(\Omega,X)=\sup_{z\in\Omega}R(z;\Omega,X)=\sup\\{r>0\mid\exists
z\in\Omega:B_{X}(z,r)\subseteq\Omega\\}\;.$

We say that $\Omega$ is a _Bloch domain_ of $X$ if $R(\Omega,X)<+\infty$.

It is easy to see that relatively compact subdomains of a hyperbolic Riemann
surface are Bloch domains; but it is not too difficult to find examples of
Bloch domains which are not relatively compact (see, e.g.,
[BeardonCarneMindaNg2004]).

The main property of Bloch domains is the following:

######  Proposition 2.3 (Beardon, Carne, Minda, Ng [BeardonCarneMindaNg2004]).

Let $\Omega\subset X$ be a Bloch domain in a hyperbolic Riemann surface $X$.
Then there exists $0<\ell<1$ such that

$\omega_{X}\bigl{(}f(z),f(w)\bigr{)}\leq\ell\,\omega_{X}(z,w)$

for all $z$, $w\in X$ and all
$f\in\mathop{\mathrm{Hol}}\nolimits(X,\Omega)\subset\mathop{\mathrm{Hol}}\nolimits(X,X)$.

A _hyperbolic domain_ $D$ in a compact Riemann surface $\widehat{X}$ is a
domain $D\subset\widehat{X}$ which is a hyperbolic Riemann surface on its own.
A useful property of the Poincaré distance of hyperbolic domains is the
following:

######  Proposition 2.4.

Let $D\subset\widehat{X}$ be a hyperbolic domain in a compact Riemann surface.
Take $\tau_{0}\in\partial D$, and a sequence $\\{z_{\nu}\\}\subset D$
converging to $\tau_{0}$. Let $\\{w_{\nu}\\}\subset D$ be another sequence,
and assume there is $M>0$ so that

$\hbox{}\hfilneg\hbox
to0.0pt{\hss$\forall\nu\in\mathbb{N}\quad\qquad$}\hfil\omega_{D}(z_{\nu},w_{\nu})<M\;.$

Then $w_{\nu}\to\tau_{0}$ as $\nu\to+\infty$.

For a proof see, e.g., [Milnor2000].

We shall need a few properties of the spaces of holomorphic maps between
hyperbolic Riemann surfaces, that we shall always consider endowed with the
compact-open topology (which is equivalent to the topology of uniform
convergence on compact sets). The first one is the classical Vitali theorem
(for a proof see, e.g., [Abate]*Theorem 1.1.45):

######  Theorem 2.5 (Vitali).

Let $X$ and $Y$ be hyperbolic Riemann surfaces, and let $\\{f_{\nu}\\}$ be a
sequence of functions in $\mathop{\mathrm{Hol}}\nolimits(X,Y)$. Assume there
is a set $A\subset X$ with at least one accumulation point such that
$\\{f_{\nu}(z)\\}$ converges for every $z\in A$. Then $\\{f_{\nu}\\}$
converges uniformly on compact subsets of $X$ to a function
$f\in\mathop{\mathrm{Hol}}\nolimits(X,Y)$.

The next two results are again due to Heins [Heins1941b]:

######  Theorem 2.6.

Let $X$ be a hyperbolic Riemann surface with non-abelian fundamental group.
Then $\mathop{\mathrm{id}}\nolimits_{X}$ is isolated in
$\mathop{\mathrm{Hol}}\nolimits(X,X)$. In particular,
$\mathop{\mathrm{Aut}}\nolimits(X)$ is discrete, and each
$f\in\mathop{\mathrm{Aut}}\nolimits(X)$ is isolated in
$\mathop{\mathrm{Hol}}\nolimits(X,X)$.

######  Proposition 2.7.

Let $X$ be a hyperbolic Riemann surface not biholomorphic to $\mathbb{D}$ or
$\mathbb{D}^{*}=\mathbb{D}\setminus\\{0\\}$. Then
$\mathop{\mathrm{Aut}}\nolimits(X)$ is open and closed in
$\mathop{\mathrm{Hol}}\nolimits(X,X)$.

For a modern proof see, e.g., [Abate]*Theorem 1.2.19, Corollary 1.2.24.

The next result is another consequence of Heins’ works; we include a proof for
completeness.

######  Corollary 2.8.

Let $X$ be a hyperbolic Riemann surface and
$f\in\mathop{\mathrm{Hol}}\nolimits(X,X)$ a self-covering of $X$. Assume that
$f$ has a fixed point $z_{0}\in X$. Then $f$ is a periodic or pseudoperiodic
automorphisms of $X$, and $z_{0}$ is not attracting.

###### Proof.

Let $\pi_{X}\colon\mathbb{D}\to X$ the universal covering map. Given
$\tilde{z}_{0}\in\pi_{X}^{-1}(z_{0})$ we can choose a lifting
$\tilde{f}\colon\mathbb{D}\to\mathbb{D}$ of $f$ such that
$\tilde{f}(\tilde{z}_{0})=\tilde{z}_{0}$. Since (see, e.g., [Lee2000]*Lemma
12.1) $\tilde{f}$ is a self-covering of $\mathbb{D}$, it must be an
automorphism; therefore $|\tilde{f}^{\prime}(\tilde{z}_{0})|=1$, by the
classical Schwarz-Pick lemma. This clearly implies that
$|f^{\prime}(z_{0})|=1$, and the assertion follows from Theorem 1.2. ∎

Finally, we recall a few standard facts about doubly connected hyperbolic
Riemann surfaces; for proofs see, e.g., [Abate].

A Riemann surface is _doubly connected_ if and only if its fundamental group
is isomorphic to $\mathbb{Z}$. A doubly connected hyperbolic Riemann surface
is necessarily biholomorphic either to the pointed disk
$\mathbb{D}^{*}=\mathbb{D}\setminus\\{0\\}$ or to an annulus

$A(r,1)=\\{z\in\mathbb{C}\mid r<|z|<1\\}$

for some $0<r<1$. A hyperbolic Riemann surface has abelian fundamental group
if and only if it is simply connected (and hence is biholomorphic to
$\mathbb{D}$) or doubly connected.

Finally, we can completely describe the automorphisms of doubly connected
hyperbolic Riemann surfaces. Indeed, every automorphism of $\mathbb{D}^{*}$ is
of the form $\gamma(z)=e^{i\theta}z$ for some $\theta\in\mathbb{R}$; and every
automorphism of $A(r,1)$ is either of the form $\gamma(z)=e^{i\theta}z$ or of
the form $\gamma(z)=e^{i\theta}rz^{-1}$ for some $\theta\in\mathbb{R}$.

##  3 Random iteration on Bloch domains

In this section we shall study left iterated function systems in
$\mathop{\mathrm{Hol}}\nolimits(X,\Omega)$, where $\Omega\subset X$ is a Bloch
domain in a hyperbolic Riemann surface $X$. As in [BeardonCarneMindaNg2004]
the main result is of topological nature:

######  Theorem 3.1.

Let $\\{f_{\nu}\\}\subset C^{0}(X,X)$ be a sequence of continuous self-maps of
a metric space $(X,d)$, and assume that there exists $0<\ell<1$ such that

$\hbox{}\hfilneg\hbox to0.0pt{\hss$\forall x,y\in X\
\forall\nu\in\mathbb{N}\quad\qquad$}\hfil
d\bigl{(}f_{\nu}(x),f_{\nu}(y)\bigr{)}\leq\ell d(x,y)\;.$ (1)

Let $\\{L_{\nu}\\}$ be the left iterated function system generated by
$\\{f_{\nu}\\}$. Then

  1. (i)

every (pointwise) limit point of $\\{L_{\nu}\\}$ is constant;

  2. (ii)

if a subsequence $\\{L_{\nu_{k}}\\}$ converges pointwise to a constant
$x_{0}\in X$ then it converges to $x_{0}$ uniformly on compact subsets;

  3. (iii)

assume that every $f_{\nu}$ has a (necessarily unique) fixed point $x_{\nu}\in
X$. Then $\\{L_{\nu}\\}$ converges uniformly on compact subsets to a constant
function if and only if the sequence $\\{x_{\nu}\\}$ converges in $X$.

###### Proof.

(i) Assume that a subsequence $\\{L_{\nu_{k}}\\}$ converges pointwise to a map
$g\in C^{0}(X,X)$. Then for every $x$, $y\in X$ we have

$d\bigl{(}g(x),g(y)\bigr{)}=\\!\lim_{k\to+\infty}d\bigl{(}f_{\nu_{k}}\circ\cdots\circ
f_{0}(x),f_{\nu_{k}}\circ\cdots\circ
f_{0}(y)\bigr{)}\leq\lim_{k\to+\infty}\ell^{\nu_{k}+1}d(x,y)=0\;,$

and thus $g$ is constant.

(ii) Assume that $L_{\nu_{k}}(x)\to x_{0}$ for each $x\in X$. Let $K\subseteq
X$ be compact, and let $d_{0}=\max_{x\in K}\\{d(x_{0},x)\\}<+\infty$. Then

$\displaystyle d\bigl{(}L_{\nu_{k}}(x),x_{0})$ $\displaystyle\leq
d\bigl{(}L_{\nu_{k}}(x),L_{\nu_{k}}(x_{0})\bigr{)}+d\bigl{(}L_{\nu_{k}}(x_{0}),x_{0}\bigr{)}$
$\displaystyle\leq\ell^{\nu_{k}+1}d(x,x_{0})+d\bigl{(}L_{\nu}(x_{0}),x_{0})\leq\ell^{\nu_{k}+1}d_{0}+d\bigl{(}L_{\nu_{k}}(x_{0}),x_{0})\;,$

for all $x\in K$, and thus $L_{\nu_{k}}\to x_{0}$ uniformly on $K$.

(iii) Assume first that $x_{\nu}\to x_{\infty}\in X$. Then for all $x\in X$ we
have

$\displaystyle d\bigl{(}L_{\nu}(x),\,x_{\infty}\bigr{)}$ $\displaystyle\leq
d\bigl{(}f_{\nu}\circ\cdots\circ
f_{0}(x),f_{\nu}(x_{\nu})\bigr{)}+d(x_{\nu},x_{\infty})$
$\displaystyle\leq\ell d\bigl{(}f_{\nu-1}\circ\cdots\circ
f_{0}(x),x_{\nu}\bigr{)}+d(x_{\nu},x_{\infty})$ $\displaystyle\leq\cdots$
$\displaystyle\leq\ell^{\nu+1}d(x,x_{0})+\sum_{j=0}^{\nu-1}\ell^{\nu-j}d(x_{j},x_{j+1})+d(x_{\nu},x_{\infty})\;.$

The first and third addends in the last line clearly goes to zero as
$\nu\to+\infty$. To prove that the second addend goes to zero too, let
$M=\max_{j}d(x_{j},x_{j+1})<+\infty$, and fix $\varepsilon>0$. Choose
$\nu_{0}\in\mathbb{N}$ so that $d(x_{j},x_{j+1})<\varepsilon(1-\ell)/2$ as
soon as $j\geq\nu_{0}$ and $\nu_{1}\in\mathbb{N}$ such that
$\ell^{j}<\varepsilon(1-\ell)/(2M)$ as soon as $j\geq\nu_{1}$. Then if
$\nu\geq\nu_{0}+\nu_{1}$ we have

$\displaystyle\sum_{j=0}^{\nu-1}\ell^{\nu-j}d(x_{j},x_{j+1})$
$\displaystyle=\sum_{j=0}^{\nu-\nu_{1}}\ell^{\nu-j}d(x_{j},x_{j+1})+\sum_{j=\nu-\nu_{1}+1}^{\nu-1}\ell^{\nu-j}d(x_{j},x_{j+1})$
$\displaystyle\leq
M\sum_{j=0}^{\nu-\nu_{1}}\ell^{\nu-j}+\frac{\varepsilon(1-\ell)}{2}\sum_{j=\nu-\nu_{1}+1}^{\nu-1}\ell^{\nu-j}$
$\displaystyle=M\sum_{j=\nu_{1}}^{\nu}\ell^{j}+\frac{\varepsilon(1-\ell)}{2}\sum_{j=1}^{\nu_{1}-1}\ell^{j}$
$\displaystyle\leq\frac{M}{1-\ell}\ell^{\nu_{1}}+\frac{\varepsilon(1-\ell)}{2}\frac{1-\ell^{\nu_{1}}}{1-\ell}\leq\varepsilon\;.$

In this way we have proved that $L_{\nu}\to x_{\infty}$ pointwise — and hence
uniformly on compact subsets by (ii) — as claimed.

Conversely, assume that $L_{\nu}\to x_{\infty}$ and, by contradiction, that
there exist $r>0$ and a subsequence $\\{x_{\nu_{k}}\\}$ such that
$d(x_{\nu_{k}},x_{\infty})>r$ for all $k$. Fix $x\in X$. Since
$L_{\nu}(x)\to\infty$, we know that

$d\bigl{(}L_{\nu}(x),x_{\infty}\bigr{)}<\frac{1-\ell}{1+\ell}r$

for $\nu$ large enough. Then for $k$ large enough we have

$\displaystyle d(x_{\nu_{k}},x_{\infty})-\frac{1-\ell}{1+\ell}r$
$\displaystyle\leq
d(x_{\nu_{k}},x_{\infty})-d\bigl{(}L_{\nu_{k}}(x),x_{\infty})$
$\displaystyle\leq d\bigl{(}L_{\nu_{k}}(x),x_{\nu_{k}}\bigr{)}$
$\displaystyle\leq\ell d\bigl{(}L_{\nu_{k}-1}(x),x_{\nu_{k}}\bigr{)}\leq\ell
d\bigl{(}L_{\nu_{k}-1}(x),x_{\infty}\bigr{)}+\ell d(x_{\nu_{k}},x_{\infty})$
$\displaystyle\leq\frac{1-\ell}{1+\ell}\ell r+\ell
d(x_{\nu_{k}},x_{\infty})\;,$

and then $d(x_{\nu_{k}},x_{\infty})\leq r$, contradiction. ∎

######  Remark 3.2.

If $(X,d)$ is complete the Banach fixed point theorem implies that every
$f\in\mathop{\mathrm{Hol}}\nolimits(X,X)$ satisfying (1) has a fixed point in
$X$, and thus the hypothesis in (iii) is automatically satisfied.

As recalled in Section 2, if $\Omega\subset X$ is a Bloch domain then there
exists $0<\ell<1$ such that

$\omega_{X}\bigl{(}f(z),f(w)\bigr{)}\leq\ell\omega_{X}(z,w)$

for every $z$, $w\in X$ and every
$f\in\mathop{\mathrm{Hol}}\nolimits(X,\Omega)$. Therefore Theorem 1.5 is a
consequence of Theorem 3.1, as shown in the following result, where for the
sake of completeness we report also what has been proved in
[BeardonCarneMindaNg2004] for right iterated function systems:

######  Corollary 3.3.

Let $\Omega\subset X$ be a Bloch domain in a hyperbolic Riemann surface $X$,
and let $\\{f_{\nu}\\}$ be a sequence of holomorphic self-maps of $X$ such
that $f_{\nu}(X)\subseteq\Omega$ for all $\nu\in\mathbb{N}$, that is
$\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(X,\Omega)$. Then:

  1. (i)

all limit points of the left and right iterated function systems,
$\\{L_{\nu}\\}$ and $\\{R_{\nu}\\}$, generated by $\\{f_{\nu}\\}$ are
constant;

  2. (ii)

let $z_{\nu}\in\Omega$ be the unique fixed point of $f_{\nu}$. Then
$\\{L_{\nu}\\}$ converges to a constant $z_{\infty}\in X$ if and only if
$z_{\nu}\to z_{\infty}$;

  3. (iii)

if there is $z_{0}\in X$ such that the set $\\{f_{\nu}(z_{0})\\}$ is
relatively compact in $X$ then $\\{R_{\nu}\\}$ converges to a constant
function.

###### Proof.

The statements for $\\{R_{\nu}\\}$ are in [BeardonCarneMindaNg2004]. For
$\\{L_{\nu}\\}$, since the Poincaré distance of $X$ is complete, by
Proposition 2.3 and Remark 3.2 we can apply Theorem 3.1, and we are done. ∎

##  4 Random iteration of small perturbations

In this section we shall discuss the behaviour of iterated function systems
generated by functions close enough to a given self-map $F$; in particular we
would like to understand whether the dynamics of the iterated function systems
mimics the dynamics of the sequence of iterates of $F$.

Recalling Theorem 1.2, we see that we have three cases to consider: when $F$
has an attracting fixed point, when $F$ is a periodic or pseudoperiodic
automorphism and when the sequence $\\{F^{k}\\}$ is compactly divergent.

In the first case we have a fairly complete result.

######  Theorem 4.1.

Let $X$ be a hyperbolic Riemann surface and let
$F\in\mathop{\mathrm{Hol}}\nolimits(X,X)$ be with an attracting fixed point
$z_{0}\in X$. Then:

  1. (i)

there exists a neighbourhood $\mathcal{U}$ of $F$ in
$\mathop{\mathrm{Hol}}\nolimits(X,X)$ such that every right iterated function
system generated by $\\{f_{\nu}\\}\subset\mathcal{U}$ converges to a constant
in $X$;

  2. (ii)

if $\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(X,X)$ is a sequence
converging to $F$ then the left iterated function system generated by
$\\{f_{\nu}\\}$ converges to $z_{0}$.

###### Proof.

Fix $r>0$ and let $D=B_{X}(z_{0},r)$. Since $\overline{D}$ is compact and $F$
is not a self-covering of $X$ (see Corollary 2.8), by the general Schwarz-Pick
lemma Theorem 2.1 there is $0<k<1$ such that
$\omega_{X}\bigl{(}F(z),F(w)\bigr{)}\leq k\omega_{X}(z,w)$ for all $z$,
$w\in\overline{D}$. In particular, since $F(z_{0})=z_{0}$, we have
$F(\overline{D})\subseteq\overline{B_{X}(z_{0},kr)}\subset D$. Choose a real
number $t$ such that $kr<t<r$ and put

$\mathcal{U}=\\{h\in\mathop{\mathrm{Hol}}\nolimits(X,X)\mid
h(\overline{D})\subset B_{X}(z_{0},t)\\}\;;$

clearly $\cal U$ is a neighbourhood of $F$. Notice that $D$ is a hyperbolic
Riemann surface, and $B_{X}(z_{0},t)$ is a Bloch domain, because it is
relatively compact in $D$. If $\\{f_{\nu}\\}\subset\cal U$ then, by Corollary
3.3.(iii), the corresponding right iterated function system converges in
$D$—and hence, by Vitali theorem (Theorem 2.5), in $X$—to a constant contained
in $\overline{B_{X}(z_{0},t)}$, and thus (i) is proved.

For (ii), since $f_{\nu}\to F$ we have $f_{\nu}\in\cal U$ for all $\nu$ large
enough; truncating $L_{\nu}$ by finitely many terms on the right and
relabelling we can assume without loss of generality that
$f_{\nu}(\overline{D})\subset D$ for all $\nu\in\mathbb{N}$. Choose now
$z\in\overline{D}$. We have $F^{\nu}(z)$, $L_{\nu}(z)\in\overline{D}$ for all
$\nu\in\mathbb{N}$; hence

$\displaystyle\omega_{X}\bigl{(}L_{\nu}(z),F^{\nu}(z)\bigr{)}$
$\displaystyle\leq\omega_{X}\bigl{(}L_{\nu}(z),F\bigr{(}L_{\nu-1}(z)\bigr{)}\bigr{)}+\omega_{X}\bigl{(}F\bigr{(}L_{\nu-1}(z)\bigr{)},F^{\nu}(z)\bigr{)}$
$\displaystyle\leq\sup_{w\in\overline{D}}\omega_{X}\bigl{(}f_{\nu}(w),F(w)\bigr{)}+k\omega_{X}\bigl{(}L_{\nu-1}(z),F^{\nu-1}(z)\bigr{)}\;.$

Repeating this argument we get by induction

$\omega_{X}\bigl{(}L_{\nu}(z),F^{\nu}(z)\bigr{)}\leq\sum_{j=0}^{\nu}k^{j}\sup_{w\in\overline{D}}\omega_{X}\bigl{(}f_{\nu-j}(w),F(w)\bigr{)}\;.$

Fix $\varepsilon>0$. Since $0<k<1$ and, by assumption, $f_{\nu}\to F$
uniformly on compact subsets, we can find $\nu_{0}$ large enough so that
$k^{j}<\frac{(1-k)\varepsilon}{4r}$ and
$\sup\limits_{w\in\overline{D}}\omega_{X}\bigl{(}f_{j}(w),F(w)\bigr{)}<\frac{(1-k)\varepsilon}{2}$
as soon as $j\geq\nu_{0}/2$. Therefore if $\nu\geq\nu_{0}$ we have

$\displaystyle\omega_{X}\bigl{(}L_{\nu}(z),F^{\nu}(z)\bigr{)}$
$\displaystyle\leq\sum_{j=0}^{\nu/2}k^{j}\sup_{w\in\overline{D}}\omega_{X}\bigl{(}f_{\nu-j}(w),F(w)\bigr{)}$
$\displaystyle\quad+\sum_{j=\nu/2}^{\nu}k^{j}\sup_{w\in\overline{D}}\omega_{X}\bigl{(}f_{\nu-j}(w),F(w)\bigr{)}$
$\displaystyle<\left[\frac{(1-k)\varepsilon}{2}+2rk^{\nu/2}\right]\sum_{j=0}^{\nu/2}k^{j}<2\frac{(1-k)\varepsilon}{2}\frac{1}{1-k}=\varepsilon\;,$

and so $\omega_{X}\bigl{(}L_{\nu}(z),F^{\nu}(z)\bigr{)}\to 0$ uniformly on
$\overline{D}$ as $\nu\to+\infty$. Since Theorem 1.2 implies $F^{\nu}\to
z_{0}$, it follows that $L_{\nu}\to z_{0}$ on $\overline{D}$ and hence, again
by Vitali theorem, on $X$, as claimed. ∎

######  Remark 4.2.

In Theorem 4.1.(i) by changing $f_{0}$ one can change the constant limit of
the right iterated function system $\\{R_{\nu}\\}$.

The next example, taken from [ChristodoulouShort2019], shows that in Theorem
4.1.(ii) we cannot replace the hypothesis $f_{\nu}\to F$ by the hypothesis
$\\{f_{\nu}\\}\subset\mathcal{U}$ for any neighbourhood $\mathcal{U}$ of $F$.

######  Example 4.3.

Let $F\in\mathop{\mathrm{Hol}}\nolimits(\mathbb{D},\mathbb{D})$ be given by
$F(z)=\frac{1}{2}z$, and let
$\mathcal{U}\subset\mathop{\mathrm{Hol}}\nolimits(\mathbb{D},\mathbb{D})$ be a
neighbourhood of $F$. Given $\delta>0$ and $\theta\in\mathbb{R}$, put
$f_{\delta,\theta}(z)=\frac{1}{2}z+\delta e^{i\theta}$. Clearly
$f_{\delta,\theta}\in\mathop{\mathrm{Hol}}\nolimits(\mathbb{D},\mathbb{D})$ as
soon as $\delta<1/2$.

We claim that we can choose $\delta$ small enough so that
$f_{\delta,\theta}\in\mathcal{U}$ for all $\theta\in\mathbb{R}$. Indeed, fix a
compact subset $K\subset\mathbb{D}$ and let $V\subset\mathbb{D}$ be an open
neighbourhood of $F(K)$, so that
$F\in\mathcal{U}(K,V)=\\{h\in\mathop{\mathrm{Hol}}\nolimits(\mathbb{D},\mathbb{D})\mid
h(K)\subset V\\}$. Since $F(K)\cap\partial
V=\mathord{\mathchoice{\ooalign{$\hfil\displaystyle/\hfil$\crcr$\displaystyle\mathrel{\hbox{\kern
0.0pt\raise
0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}{\ooalign{$\hfil\textstyle/\hfil$\crcr$\textstyle\mathrel{\hbox{\kern
0.0pt\raise
0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}{\ooalign{$\hfil\scriptstyle/\hfil$\crcr$\scriptstyle\mathrel{\hbox{\kern
0.0pt\raise
0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}{\ooalign{$\hfil\scriptscriptstyle/\hfil$\crcr$\scriptscriptstyle\mathrel{\hbox{\kern
0.0pt\raise 0.8pt\hbox{$\scriptstyle\bigcirc$}}}$}}}$, there is a
$\delta_{0}>0$ such that $d\bigl{(}w,F(K)\bigr{)}<\delta_{0}$ implies $w\in
V$, where $d$ denotes the Euclidean distance. As a consequence, if
$\delta\leq\delta_{0}$ we have $f_{\delta,\theta}(K)\subset V$ for all
$\theta\in\mathbb{R}$. Since $\mathcal{U}$ contains a finite intersection of
sets of the form $\mathcal{U}(K,V)$ the claim follows.

Given $\\{\theta_{\nu}\\}\subset\mathbb{R}$ it is easy to check by induction
that the left iterated function system $\\{L_{\nu}\\}$ generated by
$\\{f_{\delta,\theta_{\nu}}\\}$ is given by

$L_{\nu}(z)=\frac{1}{2^{\nu+1}}z+\delta\sum_{j=0}^{\nu}\frac{1}{2^{j}}e^{i\theta_{\nu-j}}\;.$

For instance, taking $e^{i\theta_{j}}=(-1)^{j}$ we get

$L_{\nu}(0)=\delta(-1)^{\nu}\sum_{j=0}^{\nu}\left(-\frac{1}{2}\right)^{j}$

that does not converge when $\nu\to+\infty$, and thus $\\{L_{\nu}\\}$ cannot
be convergent.

The next case is when $\\{F^{\nu}\\}$ is compactly divergent. If $f_{\nu}$
converges to $F$ fast enough then the dynamics of the left iterated function
system generated by $\\{f_{\nu}\\}$ is dictated by the dynamics of
$\\{F^{\nu}\\}$:

######  Theorem 4.4.

Let $X$ be a hyperbolic Riemann surface, and let
$F\in\mathop{\mathrm{Hol}}\nolimits(X,X)$ be such that the sequence of
iterates $\\{F^{\nu}\\}$ is compactly divergent. Then we can find a sequence
of neighbourhoods
$\mathcal{U}_{\nu}\subset\mathop{\mathrm{Hol}}\nolimits(X,X)$ of $F$ such that
if $f_{\nu}\in\mathcal{U}_{\nu}$ for all $\nu\in\mathbb{N}$ then the left
iterated function system $\\{L_{\nu}\\}$ generated by $\\{f_{\nu}\\}$ is
compactly divergent. Furthermore, if $X\subset\widehat{X}$ is a hyperbolic
domain in a compact Riemann surface $\widehat{X}$ and $\\{F^{\nu}\\}$
converges to a point $\tau\in\partial X$ then $\\{L_{\nu}\\}$ converges to
$\tau$.

###### Proof.

Fix a reference point $z_{0}\in X$. For $\nu\in\mathbb{N}$ set

$D_{\nu}=\overline{B_{X}\bigl{(}z_{0},1+\omega_{X}(F^{\nu}(z_{0}),z_{0})\bigr{)}}\;.$

Since $\omega_{X}\bigl{(}F^{\nu}(z_{0}),z_{0}\bigr{)}\to+\infty$ as
$\nu\to+\infty$ we have $X=\bigcup\limits_{\nu\in\mathbb{N}}D_{\nu}$. Given
$\nu\in\mathbb{N}$, choose $z_{1},\ldots,z_{r}\in D_{\nu}$ such that
$F(D_{\nu})\subseteq\bigcup\limits_{j=1}^{r}B_{X}\bigl{(}F(z_{j}),(3\cdot
2^{\nu+1})^{-1}\bigr{)}$. Put $B_{j}=B_{X}\bigl{(}F(z_{j}),(3\cdot
2^{\nu+1})^{-1}\bigr{)}$, $K_{j}=D_{\nu}\cap F^{-1}(\overline{B_{j}})$ and
$\widetilde{B_{j}}=B_{X}\bigl{(}F(z_{j}),2^{-\nu-2}\bigr{)}$; in particular,
$D_{\nu}=\bigcup\limits_{j=1}^{r}K_{j}$ and if $z\in K_{j}$ then
$\omega_{X}\bigl{(}F(z),F(z_{j})\bigr{)}\leq(3\cdot 2^{\nu+1})^{-1}$. Finally,
put

$\mathcal{U}_{\nu}=\bigcap_{j=1}^{r}\bigl{\\{}h\in\mathop{\mathrm{Hol}}\nolimits(X,X)\bigm{|}h(K_{j})\subset\widetilde{B_{j}}\bigr{\\}}\;;$

this is a neighbourhood of $F$ because
$F(K_{j})\subseteq\overline{B_{j}}\subset\widetilde{B_{j}}$ for all
$j=1,\ldots,r$ by construction.

Take $h\in\mathcal{U}_{\nu}$. If $z\in D_{\nu}$ we must have $z\in K_{j}$ for
some $j=1,\ldots,r$; then

$\displaystyle\omega_{X}\bigl{(}h(z),F(z)\bigr{)}$
$\displaystyle\leq\omega_{X}\bigl{(}h(z),F(z_{j})\bigr{)}+\omega_{X}\bigl{(}F(z_{j}),F(z)\bigr{)}$
(2) $\displaystyle<\frac{1}{2^{\nu+2}}+\frac{1}{3\cdot
2^{\nu+1}}<\frac{1}{2^{\nu+1}}\;.$

Now take $\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(X,X)$ with
$f_{\nu}\in\mathcal{U}_{\nu}$ for all $\nu\in\mathbb{N}$; notice that (2)
implies that $f_{\nu}\to F$. We will prove, by induction, that

$\omega_{X}\bigl{(}L_{\nu}(z_{0}),F^{\nu+1}(z_{0})\bigr{)}<1-\frac{1}{2^{\nu+1}}$

for all $\nu\in\mathbb{N}$. For $\nu=0$ it follows immediately from (2).
Assume it holds for $\nu-1$; then

$\displaystyle\omega_{X}\bigl{(}L_{\nu}(z_{0}),$
$\displaystyle\,F^{\nu+1}(z_{0})\bigr{)}$
$\displaystyle\leq\omega_{X}\bigl{(}L_{\nu}(z_{0}),F\bigl{(}L_{\nu-1}(z_{0})\bigr{)}\bigr{)}+\omega_{X}\bigl{(}F\bigl{(}L_{\nu-1}(z_{0})\bigr{)},F^{\nu+1}(z_{0})\bigr{)}$
$\displaystyle\leq\omega_{X}\bigl{(}L_{\nu}(z_{0}),F\bigl{(}L_{\nu-1}(z_{0})\bigr{)}\bigr{)}+\omega_{X}\bigl{(}L_{\nu-1}(z_{0}),F^{\nu}(z_{0})\bigr{)}$
$\displaystyle<\omega_{X}\bigl{(}L_{\nu}(z_{0}),F\bigl{(}L_{\nu-1}(z_{0})\bigr{)}\bigr{)}+1-\frac{1}{2^{\nu}}\;.$

Now, since

$\displaystyle\omega_{X}\bigl{(}L_{\nu-1}(z_{0}),z_{0}\bigr{)}$
$\displaystyle\leq\omega_{X}\bigl{(}L_{\nu-1}(z_{0}),F^{\nu}(z_{0})\bigr{)}+\omega_{X}\bigl{(}F^{\nu}(z_{0}),z_{0}\bigr{)}$
$\displaystyle<1+\omega_{X}\bigl{(}F^{\nu}(z_{0}),z_{0}\bigr{)}\;,$

we have $L_{\nu-1}(z_{0})\in D_{\nu}$. Therefore since
$f_{\nu}\in\mathcal{U}_{\nu}$ we can use (2) to get

$\omega_{X}\bigl{(}L_{\nu}(z_{0}),F\bigl{(}L_{\nu-1}(z_{0})\bigr{)}\bigr{)}=\omega_{X}\bigl{(}f_{\nu}\bigl{(}L_{\nu-1}(z_{0})\bigr{)},F\bigl{(}L_{\nu-1}(z_{0})\bigr{)}\bigr{)}<\frac{1}{2^{\nu+1}}$

and thus

$\omega_{X}\bigl{(}L_{\nu}(z_{0}),\,F^{\nu+1}(z_{0})\bigr{)}<\frac{1}{2^{\nu+1}}+1-\frac{1}{2^{\nu}}=1-\frac{1}{2^{\nu+1}}$

as claimed.

As a consequence for any $z\in X$ we have

$\displaystyle\omega_{X}\bigl{(}L_{\nu}(z),$ $\displaystyle\,z_{0}\bigr{)}$
$\displaystyle\geq\omega_{X}\bigl{(}F^{\nu+1}(z_{0}),z_{0}\bigr{)}-\omega_{X}\bigl{(}F^{\nu+1}(z_{0}),L_{\nu}(z_{0})\bigr{)}-\omega_{X}\bigl{(}L_{\nu}(z_{0}),L_{\nu}(z)\bigr{)}$
$\displaystyle\geq\omega_{X}\bigl{(}F^{\nu+1}(z_{0}),z_{0}\bigr{)}-1-\omega_{X}(x_{0},z)\to+\infty$

as $\nu\to+\infty$, and thus $\\{L_{\nu}\\}$ is compactly divergent.

Finally, assume that $X\subset\widehat{X}$ is a hyperbolic domain and that
$F^{\nu}\to\tau\in\partial X$. Then since
$\omega_{X}\bigl{(}L_{\nu}(z_{0}),\,F^{\nu+1}(z_{0})\bigr{)}<1$ for all
$\nu\in\mathbb{N}$ we can apply Proposition 2.4 to get
$L_{\nu}(z_{0})\to\tau$. Moreover, for any $z\in D$ we also have
$\omega_{X}\bigl{(}L_{\nu}(z),L_{\nu}(z_{0})\bigr{)}\leq\omega_{X}(z,z_{0})$;
so another application of Proposition 2.4 yields $L_{\nu}(z)\to\tau$ for all
$z\in X$, and we are done. ∎

Surprisingly enough, if the convergence of $f_{\nu}$ to $F$ is too slow then
the left iterated function system might not be convergent, as the next example
shows.

######  Example 4.5.

Let $F\in\mathop{\mathrm{Hol}}\nolimits(\mathbb{H}^{+},\mathbb{H}^{+})$ be
given by $F(w)=w-1$. Moreover, for $\nu\in\mathbb{N}$ put

$\varphi_{\nu}(w)=\frac{\nu
w-1}{w+\nu}=\frac{w\cos\theta_{\nu}-\sin\theta_{\nu}}{w\sin\theta_{\nu}+\cos\theta_{\nu}},$

where $\theta_{0}=\pi/2$ and $\theta_{\nu}=\arctan\frac{1}{\nu}$ for $\nu\geq
1$. It is easy to check that each $\varphi_{\nu}$ is an elliptic automorphism
of $\mathbb{H}^{+}$ fixing $i$; furthermore
$\varphi_{\nu}\to\mathop{\mathrm{id}}\nolimits_{\mathbb{H}^{+}}$ as
$\nu\to+\infty$.

Now let $g_{\nu}=\varphi_{\nu}\circ F\circ\varphi^{-1}_{\nu}$. By
construction, each $g_{\nu}$ is a parabolic automorphism of $\mathbb{H}^{+}$;
furthermore we have

$g_{\nu}(w)=\frac{(\nu^{2}+\nu+1)w-\nu^{2}}{w+\nu^{2}-\nu+1}\;,$

and a quick computation shows that $g_{\nu}(\nu)=\nu$. In particular, for each
$w\in\mathbb{H}^{+}$ as soon as $k$ is sufficiently large we have
$g_{\nu}^{k}(w)$ arbitrarily close to $\nu$. Moreover, $g_{\nu}\to F$ as
$\nu\to+\infty$.

We shall now build by induction a sequence
$\\{f_{\nu}\\}\subset\mathop{\mathrm{Aut}}\nolimits(\mathbb{H}^{+})$
generating a left iterated function system $\\{L_{\nu}\\}$ and an increasing
sequence $\\{\nu_{j}\\}\subset\mathbb{N}$ with the following properties:

  * (a)

$\nu_{0}=0$ and $\nu_{1}=1$;

  * (b)

$f_{0}=g_{0}$ and $f_{1}=F$;

  * (c)

$\nu_{2j}\geq\nu_{2j-1}+j$ for all $j\geq 1$;

  * (d)

$f_{\nu_{2j-1}+1}=\cdots=f_{\nu_{2j}-j}=g_{j}$ and
$f_{\nu_{2j}-j+1}=\cdots=f_{\nu_{2j+1}}=F$ for all $j\geq 1$;

  * (e)

$|L_{\nu_{2j}}(i)|<1/2^{j}$ and $|L_{\nu_{2j+1}}(i)|>j$ for all
$j\in\mathbb{N}$.

Since $|g_{0}(i)|<1$ and $|F\circ g_{0}(i)|>0$ condition (e) is satisfied for
$j=0$. Now choose $n_{1}\in\mathbb{N}$ such that
$|g_{1}^{n_{1}}\bigl{(}L_{\nu_{1}}(i)\bigr{)}-1|<1/2$; in particular,
$\bigl{|}(F\circ g_{1}^{n_{1}})\bigl{(}L_{\nu_{1}}(i)\bigr{)}\bigr{|}<1/2$.
Choose now $m_{1}\in\mathbb{N}$ so that $\bigl{|}(F^{m_{1}+1}\circ
g_{1}^{n_{1}})\bigl{(}L_{\nu_{1}}(i)\bigr{)}\bigr{|}>1$; putting
$\nu_{2}=\nu_{1}+n_{1}+1$, $\nu_{3}=\nu_{2}+m_{1}$,
$f_{\nu_{1}+1}=\cdots=f_{\nu_{2}-1}=g_{1}$ and
$f_{\nu_{2}}=\cdots=f_{\nu_{3}}=F$ we get $|L_{\nu_{2}}(i)|<1/2$ and
$|L_{\nu_{3}}(i)|>2$, that is conditions (c)–(e) are satisfied for $j=1$.

Now given $j\geq 1$ assume by induction that we have found
$\nu_{0}<\cdots<\nu_{2j-1}$ and
$f_{0},\ldots,f_{\nu_{2j-1}}\in\mathop{\mathrm{Aut}}\nolimits(\mathbb{H}^{+})$
satisfying (a)–(e). Choose $n_{j}\in\mathbb{N}$ such that

$|g_{j}^{n_{j}}\bigl{(}L_{\nu_{2j-1}}(i)\bigr{)}-j|<\frac{1}{2^{j}}\;;$

in particular, $\bigl{|}(F^{j}\circ
g_{1}^{n_{j}})\bigl{(}L_{\nu_{2j-1}}(i)\bigr{)}\bigr{|}<1/2^{j}$. Choose now
$m_{j}\in\mathbb{N}$ so that $\bigl{|}(F^{m_{j}+j}\circ
g_{j}^{n_{j}})\bigl{(}L_{\nu_{2j-1}}(i)\bigr{)}\bigr{|}>j$; putting
$\nu_{2j}=\nu_{2j-1}+n_{j}+j$ and $\nu_{2j+1}=\nu_{2j}+m_{j}$ and choosing
$f_{\nu_{2j-1}+1},\ldots,f_{\nu_{2j+1}}$ as in (d) we get
$|L_{\nu_{2j}}(i)|<1/2^{j}$ and $|L_{\nu_{2j+1}}(i)|>j$, as required.

In this way we have constructed a sequence
$\\{f_{\nu}\\}\subset\mathop{\mathrm{Aut}}\nolimits(\mathbb{H}^{+})$
converging (very slowly) to $F$ generating a left iterated function system
with $L_{\nu_{2j}}(i)\to 0$ and $L_{\nu_{2j+1}}(i)\to\infty$ as $j\to+\infty$;
in particular $\\{L_{\nu}\\}$ does not converge.

The left iterated function system we constructed in this example, though not
converging, it is still compactly divergent. It would be interesting to find
an example (or to prove that it does not exists) of a self-map
$F\in\mathop{\mathrm{Hol}}\nolimits(\mathbb{D},\mathbb{D})$ such that
$\\{F^{\nu}\\}$ is compactly divergent and of a sequence $\\{f_{\nu}\\}$
converging to $F$ so that $\\{L_{\nu}\\}$ is not compactly divergent.

Finally, there is no hope to get a version of Theorem 4.4 for right iterated
function systems:

######  Example 4.6.

Let $F\in\mathop{\mathrm{Hol}}\nolimits(\mathbb{H}^{+},\mathbb{H}^{+})$ be
given by $F(w)=w+1$, and define
$\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(\mathbb{H}^{+},\mathbb{H}^{+})$
by setting $f_{0}(w)=i+e^{2\pi iz}$ and $f_{\nu}=F$ for $\nu\geq 1$. Then
$f_{\nu}\to F$ in the fastest possible way but $R_{\nu}=f_{0}$ for all
$\nu\in\mathbb{N}$, and thus $\\{R_{\nu}\\}$ is not even compactly divergent.

We are left with the case when $F$ is a periodic or pseudoperiodic
automorphism of $X$. If the fundamental group of $X$ is not abelian, Theorem
2.6 implies that $F$ is isolated in $\mathop{\mathrm{Hol}}\nolimits(X,X)$;
therefore any map close enough to $F$ coincides with $F$ and the study of
random iteration of functions sufficiently close to $F$ reduces to the study
of the dynamics of $F$, which is trivial.

If the fundamental group of $X$ is abelian, we know that $X$ is biholomorphic
either to $\mathbb{D}$ or to $\mathbb{D}^{*}$ or to an annulus $A(r,1)$ with
$0<r<1$. If $X$ is biholomorphic to $\mathbb{D}^{*}$ then every holomorphic
self-map of $\mathbb{D}^{*}$ extends to a holomorphic self-map of
$\mathbb{D}$, and thus random iteration on $\mathbb{D}^{*}$ reduces to random
iteration on $\mathbb{D}$.

If $X$ is biholomorphic to an annulus $A(r,1)$ then Proposition 2.7 says that
$\mathop{\mathrm{Aut}}\nolimits(X)$ is open in
$\mathop{\mathrm{Hol}}\nolimits(X,X)$; in particular, maps sufficiently close
to $F$ are automorphisms of $X$. The group
$\mathop{\mathrm{Aut}}\nolimits\bigl{(}A(r,1)\bigr{)}$ has two connected
components, $A_{1}=\\{\varphi_{\theta,1}\mid\theta\in\mathbb{R}\\}$ and
$A_{-1}=\\{\varphi_{\theta,-1}\mid\theta\in\mathbb{R}\\}$, where
$\varphi_{\theta,1}(z)=e^{i\theta}z$ and
$\varphi_{\theta,-1}(z)=e^{i\theta}rz^{-1}$. If $F\in A_{1}$ then every
holomorphic self-map of $A(r.1)$ sufficiently close to $F$ belongs to $A_{1}$;
since $A_{1}\subset\mathop{\mathrm{Aut}}\nolimits(\mathbb{D})$, in this case
random iteration of holomorphic self-maps close to $F$ is reduced to random
iteration on $\mathbb{D}$. If instead $F\in A_{-1}$ then every holomorphic
self-map of $A(r.1)$ sufficiently close to $F$ belongs to $A_{-1}$. Since it
is easy to check that

$\varphi_{\theta,-1}\circ\varphi_{\eta,-1}=\varphi_{\theta-\eta,1}\;,\
\varphi_{\theta,1}\circ\varphi_{\eta,-1}=\varphi_{\theta+\eta,-1}\ \hbox{and}\
\varphi_{\theta,-1}\circ\varphi_{\eta,1}=\varphi_{\theta-\eta,-1}\;,$

we see that every iterated function system generated by self-maps close enough
to $F$ splits in the union of an iterated function system contained in
$A_{1}$, obtained considering an even number of maps, and of the composition
of $\varphi_{0,-1}$ with an iterated function system again contained in
$A_{1}$, obtained considering an odd number of maps. Thus in this case too we
are led to the study of random iteration in $\mathbb{D}$.

Summing up, when $F$ is a periodic or pseudoperiodic automorphism of $X$ for
our aims we can safely assume that $X=\mathbb{D}$. This situation has been
studied in [ChristodoulouShort2019]; for the sake of completeness we conclude
this paper by reporting the main result in this case, referring to
[ChristodoulouShort2019] for proofs, examples and comments.

######  Theorem 4.7.

Let $F\in\mathop{\mathrm{Aut}}\nolimits(\mathbb{D})$ be a periodic or
pseudoperiodic (and hence elliptic) automorphism of $\mathbb{D}$; the case
$F=\mathop{\mathrm{id}}\nolimits_{\mathbb{D}}$ is allowed. Let
$\\{f_{\nu}\\}\subset\mathop{\mathrm{Hol}}\nolimits(\mathbb{D},\mathbb{D})$ be
a sequence of non-constant holomorphic self-maps of $\mathbb{D}$ for which

$\sum_{\nu=0}^{\infty}\omega\bigl{(}f_{\nu}(a),F(a)\bigr{)}<+\infty\quad\hbox{and}\quad\sum_{\nu=0}^{\infty}\omega\bigl{(}f_{\nu}(b),F(b)\bigr{)}<+\infty$
(3)

for two distinct points $a$, $b\in\mathbb{D}$. Then $f_{\nu}\to F$ as
$\nu\to+\infty$, and the sequences $\\{F^{-\nu}\circ L_{\nu}\\}$ and
$\\{R_{\nu}\circ F^{-\nu}\\}$ converge to non-constant holomorphic self-maps
of $\mathbb{D}$.

2020 Mathematics Subject Classification: 37H12 (primary); 37F99, 30D05
(secondary).

Keywords: Wolff-Denjoy theorem; random iteration; iterated function system;
Bloch domain.

## References


# Robust Saliency Guidance for Data-free Class Incremental Learning

Xialei Liu 1,∗, Jiang-Tian Zhai 1, Andrew D. Bagdanov 2 Ke Li 3 Ming-Ming
Cheng 1
1 TMCC, CS, Nankai University 2 MICC, University of Florence 3 Tencent Youtu
Lab Corresponding author (xialei@nankai.edu.cn)The first two authors
contribute equally.

###### Abstract

Data-Free Class Incremental Learning (DFCIL) aims to sequentially learn tasks
with access only to data from the current one. DFCIL is of interest because it
mitigates concerns about privacy and long-term storage of data, while at the
same time alleviating the problem of catastrophic forgetting in incremental
learning. In this work, we introduce robust saliency guidance for DFCIL and
propose a new framework, which we call RObust Saliency Supervision (ROSS), for
mitigating the negative effect of saliency drift. Firstly, we use a teacher-
student architecture leveraging low-level tasks to supervise the model with
global saliency. We also apply boundary-guided saliency to protect it from
drifting across object boundaries at intermediate layers. Finally, we
introduce a module for injecting and recovering saliency noise to increase
robustness of saliency preservation. Our experiments demonstrate that our
method can retain better saliency maps across tasks and achieve state-of-the-
art results on the CIFAR-100, Tiny-ImageNet and ImageNet-Subset DFCIL
benchmarks. Code will be made publicly available.

##  1 Introduction

Deep neural networks achieve state-of-the-art performance on many computer
vision tasks. However, most of these tasks consider a static world in which
tasks are well-defined, stationary, and all training data is available in a
single training session. The real world consists of dynamically changing
environments and data distributions, which – especially given the
computational burden of training large CNNs – has led to renewed interest
learning new tasks incrementally while avoiding catastrophic forgetting of old
ones [27, 11].

Class Incremental Learning (CIL) [26, 1] is a scenario that considers the
possibility of adding new classes to already-trained models. Most CIL methods
rely on a memory buffer that stores data from past tasks [29, 2, 9, 33]. In
this paper, we consider Data-Free Class Incremental Learning (DFCIL), which is
a more challenging scenario in which _no data from previous tasks is
retained_. This is a realistic scenario and of great interest due to privacy
concerns or restrictions on long-term storage of data. The inability to retain
examples from past tasks, however, significantly exacerbates the problem of
catastrophic forgetting.

Figure 1: Illustration of saliency drift across 5 tasks on ImageNet-Subset.
SSRE [43] is a representative method without saliency supervision which
results in random saliency drift toward the background. RRR [10] vanilla
saliency supervision method which however fails to avoid saliency drift across
boundary regions. These phenomena lead to incorrect predictions: Orange to
Bowl, Dog to Shoe, Bird to Boat. In comparison, ROSS maintains robust saliency
and preventing saliency drift across boundaries.

There are several recent works that consider the DFCIL problem. DeepInversion
[37] inverts trained networks from random noise to generate images as
exemplars mixed with current samples for training. SDC [38] updates previous
prototypes of each learned class by hypothesizing that semantic drift of
previous classes can be approximated and estimated using new data. Other
previous works propose representation learning methods for overcoming
catastrophic forgetting [41, 42]. As pointed out in IL2A [41], learning better
representations can reduce representation bias when transferred to new tasks.
Incorporating self-supervised learning tasks, such as Barlow Twins [28] and
rotation prediction [42], has also been proposed to achieve more stable
representations and alleviate forgetting.

CNNs naturally learn to _attend_ to features that are discriminative for the
tasks they are trained to solve. Catastrophic forgetting also occurs in DFCIL
due to that model’s attention to _salient_ features drifting to features
specific to the new task. Standard regularization approaches do little to
prevent this saliency drift when learning new tasks. One direct method of
regularizing saliency is to apply distillation on saliency maps of old samples
[10]. However, this is complicated by the inability to save samples from
previous tasks in the DFCIL setting. Another method is to apply saliency
distillation between current task samples and previous task attention [8].
This method however suffers from the semantic gap between current and old
classes when enforcing saliency consistency. A lack of robust saliency
regularization may also lead to attention drifting toward the background in
future tasks. As demonstrated in Table 1, a conventional baseline (e.g. SSRE
[42]) results in random saliency drift toward the background, and vanilla
saliency supervision (e.g. RRR [10]) fails to avoid saliency drift across
boundary regions in future tasks. Our Robust Saliency Supervision (ROSS)
approach, however, keeps saliency focused on the foreground across
incrementally learned tasks (for more details, see Section 4).

Motivated by these observations, we propose the Robust Saliency Supervision
(ROSS) approach which incorporates three components to address this problem.
ROSS first uses a teacher-student architecture in which the teacher model
provides low-level supervision of salient regions and salient region boundary
maps. This serves as a stationary supervision signal over the incremental
model. Additionally, we apply dilated boundary maps to avoid saliency drift
across object boundaries at intermediate layers in the CNNs. Since saliency
drift usually happens across tasks, encouraging the model to focus on
important foreground regions with dilated boundary supervision reduces the
likelihood of saliency shifting toward the background. Finally, inspired by
SDC [38], we propose a module to inject saliency noise into some feature
channels and train the network to denoise them. This helps the network further
resist saliency drift across tasks.

The main contributions of this work are: (i) We provide new insight into
robust saliency supervision under DFCIL settings; we also show the negative
effect of methods with no or trivial saliency supervision, which illustrates
the superiority of our method and motivates the need for saliency drift
mitigation in DFCIL. (ii) We propose the Robust Saliency Supervision (ROSS)
framework with three components that combine to mitigate the saliency drift
problem. (iii) We show that ROSS can be easily integrated it into other state-
of-the-art methods, such as MUC [23], IL2A [41], PASS [42], SSRE [43], leading
to significant performance gains. (iv) Our experiments demonstrate that ROSS
outperforms all existing DFCIL methods and even several exemplar-based methods
on the CIFAR-100, Tiny-ImageNet, and ImageNet-Subset DFCIL benchmarks.

##  2 Related Work

We first discuss previous work on incremental learning from the recent
literature and then describe work on DFCIL.

###  2.1 Incremental Learning

A variety of methods have been proposed for incremental learning in the past
few years [6, 1]. Recent works can be coarsely grouped into three categories:
replay-based, regularization-based, and parameter-isolation methods. Replay-
based methods mitigate the task-recency bias by retaining training samples
from previous tasks. In addition to replaying samples, BiC [33], PODNet [9],
and iCaRL [29] apply a distillation loss to prevent forgetting and enhance
model stability. GEM [24], AGEM [3], and MER [30] exploit past-task exemplars
by modifying gradients on current training samples to match old samples.
Rehearsal-based methods may cause models to overfit to stored samples.
Regularization-based approaches such as LwF [20], EWC [15], and DMC [39] offer
ways to learn better representations while leaving enough plasticity for
adaptation to new tasks. Parameter-isolation methods [25, 34] use models with
different computational graphs for each task. With the help of growing models,
new model branches mitigate catastrophic forgetting at the cost of more
parameters and computational cost.

As for saliency-guided incremental learning, LwM [8] regularizes the saliency
activation of previous classes on current task data. However, there exist
semantic gaps between current classes and old ones, which results in an
inaccurate distillation target for preserving saliency activations on old
samples. RRR [10] directly saves the old saliency activation of each sample
with Grad-CAM in a replay buffer and applies distillation to memorize this old
knowledge, which requires storeing additional samples during incremental
learning. Despite these initial works on saliency-guided incremental learning,
saliency drift still remains problematic and leads to catastrophic forgetting.

Figure 2: Overall framework of our Robust Saliency Supervision (ROSS). We
apply a teacher model to generate saliency and boundary maps. The boundary map
is dilated and downsampled to provide supervision at different stages of the
encoder. A decoder is attached after the encoder for low-level distillation,
which serves as the robust teacher saliency guidance. To prevent saliency
drift in later training phases, we introduce saliency noise into each encoder
stage. The model is trained to denoise and reduce the saliency drift on
testing current data in future phases. ROSS can be integrated into a DFCIL
approach to provide robust saliency guidance across incremental tasks.

###  2.2 Data-free Class Incremental Learning

Compared to conventional class incremental learning, data-free class
incremental learning is more appropriate for applications where training data
is sensitive and may not be stored in perpetuity. DAFL [4] uses a GAN to
generate synthetic samples from past tasks as an alternative to storing actual
data. DeepInversion, which inverts trained networks using random noise to
generate images, is another popular DFCIL method [37]. Always Be Dreaming
further improves on DeepInversion for DFCIL [32]. SDC attempts to overcome the
problems caused by semantic drift when training new tasks on old class samples
[38]. It directly estimates prototypes of each learned class to use in a
nearest class mean classifier. PASS [42] and IL2A [41] are prototype-based
replay methods for efficient and effective DFCIL. Both introduce an efficient
way of generating prototypes of old classes. Since these prototypes are
features computed from past training samples, original images need not be
retained. SSRE [43] introduces a re-parameterization method to trade-off
between old and new knowledge. Our Robust Saliency Supervision (ROSS) method
uses three three new components to reduce saliency drift in DFCIL and is
complementary to several of the approaches mentioned above.

##  3 Robust Saliency Supervision for DFCIL

We first define the Data-free Class Incremental Learning (DFCIL) scenario and
our framework for low-level saliency supervision. Then we describe our
approach to dilated boundary supervision and saliency noise injection that
further mitigate saliency drift in DFCIL. Our overall framework is illustrated
in Figure 2.

###  3.1 Data-Free Class-Incremental Learning

Class-incremental learning aims to sequentially learn tasks consisting of
disjoint classes of samples. Let $t\in\\{1,2,..T\\}$ denote the incremental
learning tasks. The training data $D_{t}$ for each task contains classes
$C_{t}$ with $N_{t}$ training samples
$\\{(x_{t}^{i},y_{t}^{i})\\}_{i=1}^{N_{t}}$, where $x^{i}_{t}$ are images and
$y_{t}^{i}\in C_{t}$ are their labels.

Most deep networks applied to class-incremental learning can be split into two
components: a feature extractor $F_{\theta}$ and a common classifier
$G_{\phi}$ which grows with each new task $t+1$ to include classes $C_{t+1}$.
The feature extractor $F_{\theta}$ first maps the input $x$ to a deep feature
vector $z=F_{\theta}(x)\in\mathbb{R}^{d}$, and then the unified classifier
$G_{\phi}(z)\in\mathbb{R}^{|C_{t}|}$ is a probability distribution over
classes $C_{t}$ that is used to make predictions on input $x$.

Class-incremental learning requires that the model be capable of correctly
classifying all samples from previous tasks at any training task – that is,
when learning task $t$, the model must not forget how to classify samples from
classes from tasks $t^{\prime}<t$. Data-free class-incremental learning
additionally restricts models to learn each new task without access to samples
from previous ones. This typically leads to learning objectives that minimize
a loss function $\mathcal{L}$ defined on current training data $D_{t}$:

$\displaystyle\mathcal{L}_{t}^{\text{CIL}}(x,y)$
$\displaystyle=\mathcal{L}_{\text{ce}}(G_{\phi_{t}}(F_{\theta_{t}}(x)),y)+\mathcal{L}^{\text{method}}_{t},$
(1)

where $\mathcal{L}_{\text{ce}}$ is the standard cross-entropy classification
loss and $L^{\text{method}}_{t}$ is a method-specific loss that mitigates
forgetting of past tasks when learning current task $t$.

###  3.2 Auxiliary Low-level Supervisions for DFCIL

We propose to learn stable features from low-level stationary tasks shared
across all incremental tasks during class-incremental learning. Low-level
vision tasks like salient object detection require useful representations of
input images. By learning these feature representations across tasks, the
model can focus on key area of input images and exploit learned, stable
features with less representation drift since the low-level features change
very little between tasks.

Saliency map prediction is relevant to image classification since the
foreground largely determines the results, while the background is
comparatively less important. When learning new tasks with new classes, the
background of images of new classes may contain new visual concepts that
introduce undesirable noise and lead to forgetting essential previous
knowledge. The effectiveness of saliency features for learning classification
tasks was demonstrated by Saliency Guided Training [14]. Additional
supervision of salient region boundaries can aid salient object detection
tasks for both segmentation and localization [40]. The positive interaction
between these two tasks brings richer attention to features relevant to the
main classification task. It can provide positive guidance in the form of
stationary knowledge across class-incremental tasks. Some examples are
illustrated in Figure 5.

We incorporate low-level vision tasks into the network using a teacher-student
model. A teacher model generates low-level representations of each input image
$x$ (saliency and boundary maps in our experiments). We use CSNet [5] to
generate saliency and boundary maps, as it is lightweight and efficient, but
any low-level model producing saliency maps can be used in our framework. The
boundary map is computed with a Laplacian filter over the estimated saliency
map. We add a decoder $D_{\psi}$ [18] after the backbone $F_{\theta}$ to
predict low-level saliency and boundary maps for input images. The average L2
distance between the student and teacher maps is used as a low-level
distillation loss:

$\displaystyle\mathcal{L}^{\text{lm}}_{t}(x)$
$\displaystyle=\frac{||D_{\psi}(F_{\theta}(x))-A(x)||_{2}}{\sqrt{N}},$ (2)

where $A(x)$ denotes the output of the teacher network on input $x$,
consisting of a saliency map $A_{s}(x)$ and a boundary map $A_{b}(x)$.
$D_{\psi}((F_{\theta}(x))$ are combined saliency and boundary maps produced by
the decoder, and $N$ is the number of pixels in the student and teacher
saliency maps.

###  3.3 Boundary-guided Mid-level Saliency Drift Regularization

Figure 3:  We dilate the teacher boundary map and apply a binary cross entropy
loss at three stages in the CNN backbone to prevent mid-level student
attention from drifting into boundary regions.

The multi-task supervision of salient regions and boundaries described in the
previous section encourages the network to learn representations sufficient to
reconstruct the teacher outputs. However, it does little to guide attention at
intermediate layers in the CNN. To guard against saliency drift at these
intermediate layers in the backbone, we use the generated boundary maps as a
type of adaptive supervision as shown in Figure 3. When applying dilated
boundary supervision, we add a penalty term on object boundaries to avoid
drift into the background. We first use 0.5 as the threshold to binarize the
teacher boundary map and then obtain dilated teacher boundary maps by:

$\displaystyle B_{d}(x)$ $\displaystyle=\text{Dilate}(A_{b}(x),d)$ (3)

where $A_{b}(x)$ is the original teacher boundary map of image $x$, which is
generated from the saliency map a Laplacian filter, and $d$ denotes the
dilation radius applied on the teacher boundary map for controlling the
strictness of boundary-guided saliency.

Rather than use a decoder at each layer as described above, the student
saliency map is generated using Grad-CAM [31] at three stages of the CNN
backbone (see Figure 2). We also experiment with several other methods for
generating student saliency maps and report these results in the Supplementary
Material. The dilated teacher boundary map $B_{d}(x)$ is downsampled to match
the feature map dimensions at these three stages in order to compare the Grad-
CAM generated saliency boundary maps with the teacher. We use the binary cross
entropy loss for supervision on dilated boundary regions. The loss is defined
as:

$\displaystyle\mathcal{L}^{\text{dbs}}_{t}(x)$
$\displaystyle=-\frac{\sum_{j=1}^{N}B_{d}(x,j)\log(1-S(x,j))}{\sum_{j=1}^{N}B_{d}(x,j)},$
(4)

where $S(x,j)$ denotes the student saliency map of image $x$ at pixel $j$,
$B_{d}(x,j)$ is the dilated teacher boundary map at pixel $j$, and $N$ is the
number of pixels in $x$. We compute this loss only within dilated boundary
regions, that is where $B_{d}(x,j)=1$. This loss helps the student saliency
map have no intersection with the dilated teacher boundary region.

###  3.4 Saliency Noise Injection

Although we apply low-level teacher-student distillation and dilated boundary
supervision to maintain robust saliency representations across tasks, the
model can still forget saliency on samples from previous tasks. To address
this, we force the model to recover the correct saliency estimation from
injected saliency noise.

Algorithm 1 Pseudo Code of our ROSS

1:The number of tasks $T$, training samples
$D_{t}=\\{(x_{i},y_{i})\\}_{i=1}^{N}$ of task $t$, initial parameters
$\Theta^{0}=\\{\theta_{0},\phi_{0},\psi_{0}\\}$ containing parameters of
feature extractor $F_{\theta}$, classifier $G_{\phi}$, and low-level decoder
$D_{\psi}$.

2:Model $\Theta^{T}$

3:for $t\in$ $\\{1,2,...,T\\}$ do

4: $\Theta^{t}$ ← $\Theta^{t-1}$

5: while not converged do

6: Sample $(x,y)\mbox{ from }D_{t}$

7: $\mathcal{L}^{\text{CIL}}_{t}$ ← SaliencyNoiseInjection$(x,y)$

8: $\mathcal{L}^{\text{lm}}_{t}$ ← LowLevelMultitask$(x,A(x))$

9: $S$ ← ComputeStudentSaliency($x,y$)

10: $\mathcal{L}^{\text{dbs}}_{t}$ ← DilatedBoundarySupervision$(S,A(x))$

11: update $\Theta^{t}$ by minimizing $\mathcal{L}^{\mathrm{all}}_{t}$ from
Eq. 5

12: end while

13:end for

At each task there is no available training data from previous or future
tasks, and therefore we cannot directly know saliency drift on these samples.
Instead of supervising the model with ground-truth saliency drift signals, we
introduce saliency noise on random feature channels. We use a random ellipse
to approximate the potential saliency drift in future tasks and the model is
trained to denoise within each stage. Therefore the model can learn to
effectively reduce real saliency drift.

We generate elliptical noise using a very simple approach. There are six
parameter dimensions: the center coordinate $(x,y)$, the major and minor axis
lengths $(a,b)$, the rotation angle $\alpha$, and the mask weight $w$. A
detailed explanation of this process is given in the Supplementary Material.
With the help of dilated boundary supervision, each stage learns to eliminate
this additional saliency noise and this aids generalization for future tasks
and mitigates saliency forgetting in previous ones.

###  3.5 Learning Objective and Train Algorithm

The overall learning objective combines the low-level multi-task learning,
dilated boundary supervision, and random saliency noise injection modules:

$\displaystyle\mathcal{L}^{\text{all}}_{t}$
$\displaystyle=\mathcal{L}^{\mathrm{CIL}}_{t}+\mathcal{L}^{\text{lm}}_{t}+\mathcal{L}^{\text{dbs}}_{t}.$
(5)

For training with saliency-aware supervision, ROSS processes each sample with
three proposed losses and updates its parameters for preserving robust
saliency, as shown in Algorithm 1.

Metric: |  Accuracy $\boldsymbol{\uparrow}$ |  Average Forgetting $\boldsymbol{\downarrow}$
---|---|---
| Method | 5 tasks | 10 tasks | 20 tasks | 5 tasks | 10 tasks | 20 tasks
Exemplar-based | iCaRL-CNN† |  40.12$\pm$1.0 |  39.65$\pm$0.8 |  35.47$\pm$0.8 |  42.13$\pm$0.8 |  45.69$\pm$0.8 |  43.54$\pm$0.7
iCaRL-NCM† |  49.74$\pm$0.8 |  45.13$\pm$0.7 |  40.68$\pm$0.6 |  24.90$\pm$0.9 |  28.32$\pm$0.7 |  35.53$\pm$0.7
LUCIR† |  55.06$\pm$1.0 |  50.14$\pm$0.9 |  48.78$\pm$0.9 |  21.00$\pm$1.5 |  25.12$\pm$1.3 |  28.65$\pm$1.3
EEIL† |  52.35$\pm$0.6 |  47.67$\pm$0.5 |  41.59$\pm$0.5 |  23.36$\pm$0.8 |  26.65$\pm$0.9 |  32.40$\pm$0.7
RRR† |  57.22$\pm$0.8 |  55.74$\pm$0.8 |  51.35$\pm$0.7 |  18.05$\pm$0.8 |  18.59$\pm$0.8 |  18.40$\pm$0.7
Data-free | LwF_MC |  36.17$\pm$0.9 |  17.04$\pm$0.9 |  15.88$\pm$0.8 |  44.23$\pm$1.2 |  50.47$\pm$1.0 |  55.46$\pm$1.0
EWC |  09.32$\pm$0.7 |  08.47$\pm$0.5 |  08.23$\pm$0.5 |  60.17$\pm$0.8 |  62.53$\pm$0.7 |  63.89$\pm$0.5
MUC |  38.45$\pm$0.9 |  19.57$\pm$0.8 |  15.65$\pm$0.8 |  40.28$\pm$1.3 |  47.56$\pm$1.1 |  52.65$\pm$1.0
IL2A |  55.13$\pm$0.7 |  45.32$\pm$0.7 |  45.24$\pm$0.6 |  23.78$\pm$1.1 |  30.41$\pm$1.0 |  30.84$\pm$0.7
PASS |  55.67$\pm$1.2 |  49.03$\pm$0.9 |  48.48$\pm$0.7 |  25.20$\pm$0.8 |  30.25$\pm$0.7 |  30.61$\pm$0.7
SSRE |  56.33$\pm$0.9 |  55.01$\pm$0.7 |  50.47$\pm$0.6 |  18.37$\pm$1.1 |  19.48$\pm$1.0 |  19.00$\pm$1.0
|  ROSS (Ours) | 59.26$\pm$0.5 | 57.93$\pm$0.4 | 53.78$\pm$0.4 | 16.42$\pm$0.7 | 17.66$\pm$0.8 | 17.78$\pm$0.6
Table 1: Average top-1 accuracy and forgetting on CIFAR-100 for different
numbers of tasks. Replay-based methods storing 20 exemplars from each previous
class are denoted by ${\dagger}$. The best overall results are in bold. We run
all experiments three times and report average accuracy and standard
deviations.

##  4 Experimental Results

In this section we first describe the experimental setup first and then we
compare ROSS to other state-of-the-art methods on several DFCIL benchmarks. In
Section 4.3 we give further analysis over the different components of our
proposed approach.

Dataset | CIFAR-100 | Tiny-ImageNet
---|---|---
Method | 5 tasks | 10 tasks | 20 tasks | 5 tasks | 10 tasks | 20 tasks
MUC | 38.45 | 19.57 | 15.65 | 18.95 | 15.47 |  09.14
+ROSS |  49.17 (+10.72) |  40.34 (+20.77) |  37.86 (+22.21) |  32.47 (+13.46) |  30.13 (+14.66) |  27.70 +18.56
IL2A | 55.13 | 45.32 | 45.24 | 36.77 | 34.53 | 28.68
+ROSS |  58.74 (+3.61) |  53.24 (+7.92) |  53.07 (+7.83) |  42.49 (+5.72) |  41.34 (+6.81) |  40.59 (+11.91)
PASS | 55.67 | 49.03 | 48.48 | 41.58 | 39.28 | 32.78
+ROSS |  59.10 (+3.43) |  54.45 (+5.42) |  52.37 (+3.89) |  44.05 (+2.47) |  43.06 (+3.78) |  42.57 (+9.79)
SSRE | 56.33 | 55.01 | 50.47 | 41.45 | 40.07 | 39.25
+ROSS |  59.26 (+2.93) |  57.93 (+2.92) |  53.78 (+3.31) |  44.13 (+2.68) |  43.86 (+3.79) |  43.55 (+4.30)

Table 2: We report the performance gain of top-1 accuracy by applying ROSS to
other DFCIL methods in a plug-and-play way. Absolute gains are marked in
(red).

###  4.1 Experimental Setup

We follow standard experimental protocols for DFCIL on three benchmark
datasets.

Datasets. We perform experiments on CIFAR-100 [16], Tiny-ImageNet [17], and
ImageNet-Subset [7]. For most experiments, we train the model on half of the
classes for the first task, and then equally distribute the remaining classes
across each of the subsequent tasks. The convention we use is: $F+C\times T$
means that the first task contains $F$ classes, and the next $T$ tasks each
contain $C$ classes. This is a common configuration for DFCIL used in PASS
[42] and SSRE [43]. We consider three configurations for CIFAR-100 and
ImageNet-Subset: 50 + 5 $\times$ 10, 50 + 10 $\times$ 5, 40 + 20 $\times$ 3\.
For Tiny-ImageNet we generate three settings: 100 + 5 $\times$ 20, 100 + 10
$\times$ 10, and 100 + 20 $\times$ 5.

State-of-the-art methods. Since we focus on DFCIL, we mainly compare with
data-free state-of-the-art approaches: SSRE [43], PASS [42], IL2A [41], EWC
[15], LwF-MC [29], and MUC [23]. To demonstrate the effectiveness of our
method, we also compare its performance with several exemplar-based methods
like iCaRL (both nearest-mean and CNN) [29], EEIL [2], and LUCIR [13]. We also
compare with RRR [10] integrated with SSRE, which focuses on preserving
saliency using exemplar replay.

Implementation details. We use ResNet-18 [12] as a feature extraction
backbone. This is the same base network used in SSRE [43] and PASS [42], two
state-of-the-art DFCIL approaches. We use the decoder in [18] to estimation
student low-level saliency maps. All experiments are trained from scratch
using Adam for 100 epochs with an initial learning rate 0.001. The learning
rate is reduced by a factor 10 at epochs 45 and 90. For exemplar-based
approaches, we use herding [29] to select and store 20 samples per class
following common settings [29, 13]. We implement RRR [10] with SSRE to fairly
compare it with ROSS. We report two common metrics for class incremental
learning: top-1 accuracy and average forgetting for all classes learned up to
task $t$. We perform three runs of all experiments and report mean performance
and variance. For dilated boundary supervision, we set $d$ of the three
boundary dilation stages to be 5%, 10% and 15% of the image size.

###  4.2 Comparison with the State-of-the-art

Figure 4:  Results on Tiny-ImageNet and ImageNet-Subset for different numbers
of tasks. Our method outperforms others, especially on longer task sequences
(i.e. more, but smaller, tasks).

Dataset | Tiny-ImageNet | ImageNet-Subset
---|---|---
Method | 5 tasks | 10 tasks | 20 tasks | 5 tasks | 10 tasks | 20 tasks
LwF_MC | 17.12 | 12.33 | 8.75 | 24.10 | 20.01 | 17.42
MUC | 17.98 | 14.54 | 12.70 | 27.89 | 22.65 | 20.12
PASS | 41.58 | 39.28 | 32.78 | 52.61 | 50.44 | 46.07
SSRE | 41.67 | 39.89 | 39.76 | 58.46 | 57.51 | 50.05
ROSS | 44.13 | 43.86 | 43.55 | 63.14 | 60.94 | 57.60

Table 3: Top-1 accuracy on all learned classes after the last task.

We report the comparative performance on CIFAR-100 in Table 1. ROSS
outperforms all data-free approaches. For exemplar-based methods like iCaRL
[29], EEIL [2], and LUCIR [13], our method still has significantly better
performance. On longer sequences (i.e. 10 and 20 tasks), our method
significantly reduces forgetting when learning new classes compared with other
DFCIL methods. ROSS outperforms the best method SSRE by about 3% accuracy on
the last task. The performance improvement can be also observed in terms of
average forgetting.

As we can see in Table 3 and Figure 4 for Tiny-ImageNet and ImageNet-Subset,
although our method has the similar top-1 accuracy on the first task in Figure
4, it has better performance at most intermediate tasks and also the final
one. For longer sequences in Figure 4, the gap between our method and the best
baseline is largely consistent, showing the effectiveness of our method at
relieving forgetting. The performance gain in Table 3 is larger on Tiny-
ImageNet and ImageNet-Subset compared to CIFAR100, and this demonstrates the
generality of our method to datasets with larger images and object scales. It
is worth mentioning that ROSS also produces results with smaller variance. We
believe this to be due to ROSS reducing saliency drift to background regions,
which may include random noise.

Plug-and-play with other DFCIL methods. Some existing DFCIL methods, for
example PASS [42], IL2A [41] and SSRE [43], focus on reducing forgetting via
embedding regularization. Considering the importance of saliency to image
classification, it is natural consider whether ROSS can be integrated into
these methods. The results in Table 2 show the performance gain brought by
integrating ROSS with these methods. Adding ROSS doubles the performance for
MUC in many cases and significantly improves IL2A and PASS. When we
incorporate it into the best baseline SSRE, it yields a consistent gain of
about 3%. These results clearly show that ROSS, by specifically mitigating
saliency drift, is complementary to other methods in relieving forgetting in
DFCIL. They additionally demonstrate the significance of saliency drift as a
cause of catastrophic forgetting in DFCIL.

Figure 5: Visualization of the saliency (a) and boundary (b) maps from our
student encoder-decoder network with original images from different tasks at
different stages of incremental learning. Our method also produces stable low-
level results while reducing forgetting in classification. (c) The MAE loss of
low-level tasks between the student and teacher network across tasks.

Task 1 |  |  |  |  |
---|---|---|---|---|---
Task 3 |  |  |  |  |
Task 5 |  |  |  |  |
| Stage 0 | Stage 1 | Stage 2 | Stage 3 | Stage 4
(a)

Ours |  |
---|---|---
Baseline |  |
| Base task | Last task
(b)

Figure 6: Visualization of saliency across tasks at different stages as shown
in (a). Stage 0 denotes the image-level saliency map. Some saliency drift at
later tasks are recovered through multiple stages. Dilated boundary
supervision also shrinks the saliency map at deeper stages, increasing their
robustness. (b) Visualization of the embedding $F_{\phi}(x)$ with and without
ROSS. Compared to the baseline, our method preserves more discriminative
representations between classes from one task and between different tasks.

###  4.3 Additional Analysis

In this section we take a deeper look at the method we propose. If not
specified, the results are produced using ROSS integrated into SSRE [43].

Ablation Study. To assess each component ROSS, we performed a set of ablations
using the 10-task setting on CIFAR-100 (see Table 4). We do ablations on both
PASS [42] and SSRE [43]. Low-level multi-task supervision is crucial and
improve by 2.2% and 1.2% for PASS and SSRE, respectively. Dilated boundary
supervison further boosts the performance by about 1-2 points. Saliency noise
injection is also helpful for both methods and it improves by 1.5% points for
PASS. In total, ROSS improves baselines by 5.5% and 2.9% points, respectively.
Note that SSRE is the previous state-of-the-art method and our method
outperforms it by a large margin.

Method & Tasks | LM | DBS | SNI | Accuracy
---|---|---|---|---
Baseline (PASS) |  |  |  | 49.0
Variants | ✓ |  |  | 51.2
| ✓ | ✓ |  | 53.0
| ✓ | ✓ | ✓ | 54.5
Baseline (SSRE) |  |  |  | 55.0
Variants | ✓ |  |  | 56.2
| ✓ | ✓ |  | 57.3
| ✓ | ✓ | ✓ | 57.9
Table 4: Ablative experiments on each component of our proposed method.
Experiments are on CIFAR-100 in the 10 task setting and we report the top-1
accuracy in %, demonstrating the effect of integrating ROSS into PASS and
SSRE. We use DBS, LM, SNI to denote the three components of ROSS: dilated
boundary supervision, low-level multi-task supervision, and saliency noise
injection.

Low-level multi-task. To analyze the effect of our proposed low-level saliency
supervision, we performed experiments on ImageNet-Subset in the 5 and 10 task
settings. We first plot the loss across tasks in Figure 5(c). After learning
to predict boundary and saliency maps in the first task, the network maintains
good performance for the rest of the 5 task sequence. This shows that the low-
level tasks are stable during continual learning. Furthermore, we visualize
the results of saliency and boundary map prediction during incremental
learning in Figure 5(a-b). We give some examples of predicted boundary and
saliency maps after learning different tasks. Although CIL involves samples of
different classes, we see that the low-level outputs are relatively stable and
class-agnostic. Since the model is able to stably predict these low-level
features across tasks, it therefore can preserve useful prior knowledge for
continual learning.

Saliency denoising across stages. To show the effect of saliency denoising, we
look at the 5-task setting on ImageNet-Subset. We visualize a sample and its
saliency at different training phases and encoder stages in Figure
LABEL:fig:pa1. With the help of the denoising process, the encoder is able to
mitigate intermediate saliency drift and maintain accurate attention. When
testing on old samples, the network reduces the saliency drift and focuses on
important regions.

Discriminative embeddings over different classes. Since our method helps the
model focus on the foreground, more class-specific pixels are used to
computing embeddings. This makes them more discriminative with less
distracting background information. In Figure LABEL:fig:pc1 we use t-SNE to
visualize embeddings of five initial classes after learning the base task and
the last task in the 10-task setting on ImageNet-Subset in. At the base task,
both Baseline (SSRE) and Ours (SSRE+ROSS) perform well. When visualizing after
the last task, it is clear that ROSS can helps maintain discriminative
features between tasks while the Baseline has overlapping embeddings.

##  5 Conclusions

In this paper we propose robust saliency guidance for DFCIL. The insight
behind ROSS is to consider guiding the model to focus on salient regions with
less drift. We show that robust saliency guidance is crucial to mitigate
forgetting across tasks. Experiments demonstrate that ROSS is effective and
surpasses the state-of-the-art. ROSS can be easily combined with other
methods, leading to large performance gains over baselines. Qualitative
results also show that low-level tasks are stable across different tasks,
resulting in less catastrophic forgetting.

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|  |  |  |
---|---|---|---|---
Figure 7: Visualization of some generated saliency noise maps.

##  Appendix A Further Ablation Studies

#### Ablation on student saliency methods.

To show the generalization of ROSS, we use several methods to compute student
saliency maps and report the results in Table 5. Grad-CAM performs the best,
although other methods yield performance gains, demonstrating the
effectiveness of ROSS.

Method | 5 Tasks | 10 Tasks | 20 Tasks
---|---|---|---
baseline (SSRE) | 40.2 | 40.0 | 39.3
CAM | 41.2 | 40.7 | 40.4
SmoothGrad | 42.1 | 41.0 | 40.4
Grad-CAM | 44.1 | 43.9 | 43.5
Table 5:  Ablation on methods for generating student saliency maps on Tiny-
Imagenet.

#### Ablation on low-level teacher maps.

In the manuscript we use CSNet [5] to compute all the teacher saliency and
boundary maps because it is very lightweight. Compared to our main model, the
teacher model has fewer than 1% parameters and requires 1.5% of the FLOPs (as
shown in Table 6). Note that we compute all low-level maps offline before new
tasks, and so the extra FLOPs should be amortized over the number of epochs.
Therefore, the additional FLOPs required by the teacher model is only about
0.015% of the main model, which is negligible in practice.

To show the effectiveness of ROSS, we perform an ablation on the low-level
teacher maps. We replace them with the Grad-CAM generated from a ResNet-152
network. To avoid information leakage, ResNet-152 was trained from scratch.
Before each new task, we first train it only on task data and use the Grad-CAM
output to supervise saliency in our incremental model. From Table 7 we see
that ROSS still outperforms other methods. Moreover, ROSS is applicable to
other teacher models for predicting saliency maps, (e.g. DFI [21] or PoolNet
[22]) with more parameters, and produces even better performance with larger
networks.

Model | Parameter(M) | FLOPS(G)
---|---|---
Ours | 17.9 | 0.78
Teacher model | 0.0941 | 0.012
Table 6:  Parameters and FLOPs of the teacher model. FLOPs are computed using $3\times 32\times 32$ images.  Low-level source (Method) | Accuracy (%)
---|---
PASS | 39.3
SSRE | 40.0
ResNet152 (Ours) | 42.1
CSNet (Ours) | 43.9
PoolNet (Ours) | 44.2
DFI (Ours) | 44.4
Table 7:  Ablation on low-level teacher saliency maps on Tiny-ImageNet with 10
tasks.

#### Ablation on student architecture and teacher pretraining.

We select PASS [42] as our baseline method to apply ROSS to (as shown in Table
8 and Table 9). Experiments in these two tables are conducted on ImageNet-
Subset with 5 tasks. Since some methods use ImageNet pretrained weights for
better saliency map estimation, we train CSNet [5] from scratch on the dataset
(with and without pretraining) for salient object detection [35, 19, 36]. This
allows us to verify that no information leakage happens due to pretraining the
saliency network on ImageNet. The teacher network without pretraining works
almost as well as pretraining the saliency network on ImageNet. We also
compare the number of parameters of different methods in Table 8. This shows
that adding network capacity for PASS from ResNet-18 to ResNet-32 with more
parameters only improves the performance marginally. Ours with ResNet-18 based
on PASS achieves a significant gain surpassing SSRE which has more parameters.

Method | Parameter (M) | Accuracy (%)
---|---|---
PASS-Res18 | 14.5 | 50.4
PASS-Res32 | 21.7 | 51.2
SSRE-Res18 | 19.4 | 58.7
Ours-Res18 | 17.9 | 61.5
Table 8: Comparison of different student network architectures. Method-Res18 denotes applying Method with ResNet18 as its backbone. Method | Accuracy (%)
---|---
No pretraining | 61.5
Pretrained teacher model | 62.0
Table 9: Ablation on teacher network pretraining.

##  Appendix B More Details on Salience Noise Generation

For each ellipse there are 6 dimensions: the center coordinate $(x,y)$, the
rotation angle $\alpha$, the mask weight $w$, and the major and minor axes
$(a,b)$. $x$, $y$, $\alpha$ and $w$ are sampled from a uniform distribution
over ranges: $x\in[0,H)$, $y\in[0,W)$, $\alpha\in[0,2\pi)$, $w\in[0,1]$. $H$
and $W$ denote the height and width of input images. To generate ellipses of
appropriate size, we draw the major and minor axes from a Gaussian
distribution with $\mu_{a}=\max(H,W)/2$, $\sigma_{a}=\max(H,W)/6$,
$\mu_{b}=\min(H,W)/2$, $\sigma_{b}=\min(H,W)/6$. The sampled $a,b$ is clipped
to$[0,max(H,W)/2]$ and $[0,min(H,W)/2]$, respectively. For each ellipse, we
create a saliency map $S_{i}$. We repeat this random generation process 3-5
times and apply an element-wise max operation on the $S_{i}$ to obtain a
single saliency map $S$. Then we crop and resize $S$ to the original image
size, with crop size sampled from a uniform distribution in
$[\min(H,W)/2,\min(H,W)]$, introducing center-aware saliency noise to the
network for training. Finally, we apply a Gaussian blur on $S$ to better
simulate a realistic saliency map. The kernel size for Gaussian blurring is
the closest odd integer to $\min(H,W)/20$. For each encoder feature map, 10%
of randomly selected channels are directly masked with $S$, where each
selected channel will have an independent $S$. We visualize several generated
samples in Figure 7.