# Geometric properties of some generalized Mathieu power series inside the
unit disk

Stefan Gerhold, Živorad Tomovski, Deepak Bansal and Amit Soni  TU Wien,
Vienna, Austria<EMAIL_ADDRESS>University of Ostrava, Faculty of
Sciences, Department of Mathematics, 30\. Dubna 22701 03 Ostrava, Czech
Republic<EMAIL_ADDRESS>Department of Mathematics, University
College of Engineering and Technology, Bikaner 334004, Rajasthan, India
<EMAIL_ADDRESS>Department of Mathematics, Govt. Engineering
College, Bikaner, 334004, Rajasthan, India<EMAIL_ADDRESS>
###### Abstract.

We consider two parametric families of special functions: One is defined by a
power series generalizing the classical Mathieu series, and the other one is a
generalized Mathieu type power series involving factorials in its
coefficients. Using criteria due to Fejér and Ozaki, we provide sufficient
conditions for these functions to be close-to-convex or starlike inside the
unit disk, and thus univalent.

###### Key words and phrases:

Univalent function, Starlike function, Close-to-convex function, Generalized
Mathieu-type series.

###### 2010 Mathematics Subject Classification:

33E20, 40A10, 30C45

Živorad Tomovski was supported by DAAD, during his visit to the Department of
Physics at the University of Potsdam in Germany from 15 June 2021 to 15
September 2021 to collaborate with Ralf Metzler.

##  1\. Introduction and Preliminaries

Let $\mathbb{U}=\\{z\in\mathbb{C}:|z|<1\\}$ denote the open unit disk and
$\mathscr{A}$ denote the class of all analytic functions inside the unit disk
$\mathbb{U},$ normalized by the conditions $f(0)=0,$ $f^{\prime}(0)=1$. We
denote by $\mathscr{S}$ the class of all functions $f\in\mathscr{A}$ which are
univalent in $\mathbb{U},$ i.e.

$\mathscr{S}=\\{f\in\mathscr{A}|\;f\;\mbox{is one-to-one in}\;\mathbb{U}\\}.$

A set $\Omega\subseteq\mathbb{C}$ containing the origin is called starlike
with respect to the origin if for any point $z\in\Omega$ the line segment from
the origin to $z$ lies in the interior of $\Omega$. A function
$f\in\mathscr{A}$ that maps the unit disk $\mathbb{U}$ onto a starlike domain
is called starlike function, and the class of such functions is denoted by
$\mathscr{S}^{*}$. Analytically, starlike functions are characterized by the
condition

$\Re\left(\frac{zf^{\prime}(z)}{f(z)}\right)>0,\quad z\in\mathbb{U}.$

An analytic function $f\in\mathscr{A}$ is called close-to-convex if
$\Re\left\\{e^{i\theta}zf^{\prime}(z)/g(z)\right\\}>0,\,z\in\mathbb{U},$ for
some $\theta\in\mathbb{R}$ and for some starlike function
$g\in\mathscr{S}^{*}$. Taking $g(z)=z$, it is easy to see that every starlike
function is close-to-convex. However, the converse is not true. The Noshiro-
Warschawski theorem implies that close-to-convex functions are univalent in
$\mathbb{U}$, but the converse is not true in general. Thus, it is convenient
to show that $f$ is close-to-convex in order to check the univalency of $f$.
Geometrically an analytic function $f$ is called close-to-convex in
$\mathbb{U}$, if the complement of $f(\mathbb{U})$ can be written as the union
of non-intersecting half-lines. These classes are studied in detail in the
literature (see the books of Duren [7] and Goodman [8]). Among many papers
dealing with these geometric properties for certain special functions, we
mention [21] and the references therein.

The following infinite series was named after Émile Leonard Mathieu
(1835-1890), who investigated it in his 1890 monograph [15] on elasticity of
solid bodies:

(1.1) $S(r)=\sum\limits_{n=1}^{\infty}{\frac{2n}{(n^{2}+r^{2})^{2}}},\quad
r\in\mathbb{R}_{+}.$

An integral representation of the series $S(r)$ is given by (see [11])

$S(r)=\frac{1}{r}\int_{0}^{\infty}{\frac{t\sin(rt)}{e^{t}-1}dt}.$

The generalized Mathieu type power series or generalized Mathieu type power
series of fractional order $\mu$ is defined by (see [23], [25]):

$F_{\mu}(r;z)=\sum\limits_{n=1}^{\infty}{\frac{2n}{(n^{2}+r^{2})^{\mu+1}}}z^{n},\quad\mu,r\in\mathbb{R}_{+},\;|z|<1.$

In 1998, Alzer et al. [1] obtained the following bounds for Mathieu’s series
(1.1):

$\frac{1}{r^{2}+\frac{1}{2\zeta(3)}}<S(r)<\frac{1}{r^{2}+1/6},$

where $\zeta$ denotes the zeta function. One can refer to [1, 4, 5, 6, 10, 16,
20, 22, 24] about the study of Mathieu’s series, its generalizations and
inequalities. In the present investigation, our aim is to study geometric
properties of generalized Mathieu type power series. It is obvious that
$z\mapsto{F}_{\mu}(r;z)\notin\mathscr{A}$, so we use the following
normalization:

$\displaystyle\mathbb{F}_{\mu}(r;z)$
$\displaystyle=\frac{(r^{2}+1)^{\mu+1}}{2}\sum\limits_{n=1}^{\infty}{\frac{2n}{(n^{2}+r^{2})^{\mu+1}}}z^{n}$
(1.2)
$\displaystyle=z+\sum\limits_{n=2}^{\infty}{\frac{n(r^{2}+1)^{\mu+1}}{(n^{2}+r^{2})^{\mu+1}}}z^{n}.$

For $\mu=1$, $\mathbb{F}_{\mu}(r;z)$ becomes a Mathieu type power series (see
[23]). Geometric properties of the series $\mathbb{F}_{1}(r;z)$ have already
been discussed in [3]. Recently, Gerhold et al. [14] considered the following
generalized Mathieu type power series:

${Q}_{\mu}(r;z)=\sum\limits_{n=1}^{\infty}{\frac{2\;n!}{((n!)^{2}+r^{2})^{\mu+1}}}z^{n},\quad\mu,r\in\mathbb{R}^{+},|z|<1.$

Since ${Q}_{\mu}(r;z)\notin\mathcal{A}$, we define the normalization

$\displaystyle{\mathbb{Q}}_{\mu}(r;z)$
$\displaystyle=\frac{(r^{2}+1)^{\mu+1}}{2}\sum\limits_{n=1}^{\infty}{\frac{2\;n!}{((n!)^{2}+r^{2})^{\mu+1}}}z^{n}$
(1.3)
$\displaystyle=z+\sum\limits_{n=2}^{\infty}{\frac{n!(r^{2}+1)^{\mu+1}}{((n!)^{2}+r^{2})^{\mu+1}}}z^{n}.$

In the present investigation our aim is to find geometric properties of
$\mathbb{F}_{\mu}(r;z)$ and ${\mathbb{Q}}_{\mu}(r;z)$. A sequence of real
numbers $\\{a_{n}\\}_{n\geq 1}$ satisfying the condition

$2a_{n+1}\leq a_{n}+a_{n+2},\quad n\geq 1,$

is called convex sequence. It is clear that if $f(x)$ is a convex function (of
a real variable) for $x\geq 1$, then the sequence $a_{n}=f(n),n=1,2...$ is
convex. We need the following lemmas to prove our main results.

######  Lemma 1.1.

(Ozaki [19], Corollary 7). Let
$f(z)=z+\sum\limits_{n=2}^{\infty}{a_{n}z^{n}}$. Suppose

$1\geq 2a_{2}\geq\cdots\geq(n+1)a_{n+1}\geq\cdots\geq 0$

or

$1\leq 2a_{2}\leq\cdots\leq(n+1)a_{n+1}\leq\cdots\leq 2.$

then $f$ is close-to-convex with respect to the starlike function $z/(1-z).$

######  Lemma 1.2.

(Fejér [13], Satz IX) If $\\{a_{n}\\}$ is a non-negative real sequence with
$a_{1}=1$ and such that $\\{na_{n}\\}_{n\geq 1}$ and
$\\{na_{n}-(n+1)a_{n+1}\\}_{n\geq 1}$ are non-increasing, then the function
$f(z)=z+\sum\limits_{n=2}^{\infty}{a_{n}z^{n}}$ is in $\mathscr{S}^{*}$.

######  Lemma 1.3.

Let $\\{a_{n}\\}_{n\geq 1}$ be a non-increasing sequence of non-negative real
numbers with $a_{1}=1$ which is convex, i.e.

$a_{1}-a_{2}\geq\cdots\geq a_{k}-a_{k+1}\geq\cdots\geq 0.$

Then

$\Re\bigg{(}\sum\limits_{n=1}^{\infty}a_{n}z^{n-1}\bigg{)}>\frac{1}{2},\quad
z\in\mathbb{U}.$

###### Proof.

According to Lemma 3.4 in [2], this is due to Fejér [13]. As we could not find
the result in this reference, we give a proof for the reader’s convienence,
which is a simple variation of a proof found in another paper by Fejér ([12],
Theorem 1), and without claiming originality. Let $z=re^{i\theta}$ with
$r\in(0,1)$ and $\theta\in(0,2\pi).$ (The cases $r=0$ and $\theta=0$ are both
trivial.) Define $\tilde{a}_{n}=r^{n-1}a_{n}.$ As the radius of convergence of
$\sum a_{n}z^{n}$ is at least $1$, we have $\tilde{a}_{n}=o(1).$ The sequence
$\tilde{a}_{n}$ inherits decrease and convexity from $a_{n}$ (see p. 98 in
[13]), and it is easy to see that it is actually _strictly_ convex. We define
$s_{n}=1/2+\sum_{k=1}^{n}\cos k\theta$ and

$\sigma_{n}=\sum_{k=0}^{n}s_{k}=\frac{1}{2}\Big{(}\frac{\sin\big{(}(n+1)\theta/2\big{)}}{\sin(\theta/2)}\Big{)}^{2},$

which satisfies $0\leq\sigma_{n}=O(1).$ Using summation by parts twice, we
find, for $N\geq 3,$

$\displaystyle\Re\bigg{(}\sum_{n=1}^{N+1}$ $\displaystyle
a_{n}z^{n-1}\bigg{)}=1+\sum_{n=1}^{N}\tilde{a}_{n+1}\cos
n\theta=1+\sum_{n=1}^{N}\tilde{a}_{n+1}(s_{n}-s_{n-1})$
$\displaystyle=1+\tilde{a}_{N+1}(\sigma_{N}-\sigma_{N-1})-\tilde{a}_{2}s_{0}-\Big{(}(\tilde{a}_{N+1}-\tilde{a}_{N})\sigma_{N-1}-(\tilde{a}_{3}-\tilde{a}_{2})\sigma_{0}$
$\displaystyle\qquad-\sum_{n=3}^{N}\sigma_{n-2}(\tilde{a}_{n+1}-2\tilde{a}_{n}+\tilde{a}_{n-1})\Big{)}.$

Then, $N\to\infty$ yields

$\displaystyle\Re\bigg{(}\sum_{n=1}^{\infty}a_{n}z^{n-1}\bigg{)}$
$\displaystyle=1-\tfrac{1}{2}\tilde{a}_{2}+\tfrac{1}{2}(\tilde{a}_{3}-\tilde{a}_{2})+\sum_{n=3}^{\infty}\sigma_{n-2}(\tilde{a}_{n+1}-2\tilde{a}_{n}+\tilde{a}_{n-1})$
$\displaystyle>1-\tilde{a}_{2}+\tfrac{1}{2}\tilde{a}_{3}=\frac{1}{2}+\frac{\tilde{a}_{1}-2\tilde{a}_{2}+\tilde{a}_{3}}{2}\geq\frac{1}{2}.\qed$

##  2\. Close-to-convexity and starlikeness of $\mathbb{F}_{\mu}(r;z)$

######  Theorem 2.1.

If $\mu>0\;\mbox{and}\;0<r\leq\sqrt{\mu},$ then $\mathbb{F}_{\mu}(r;z)$ is
close-to-convex with respect to the starlike function $z/(1-z).$

###### Proof.

In view of Lemma 1.1 it is sufficient to prove that the sequence
$\\{na_{n}\\}$ is non-increasing. Here the $a_{n}$ are the coefficients in the
series expansion of $\mathbb{F}_{\mu}(r;z)$ given by (1.2). Let

$f(n)=na_{n}=\frac{n^{2}(r^{2}+1)^{\mu+1}}{(n^{2}+r^{2})^{\mu+1}}.$

Now it is sufficient to show that the function
$f(x)=\frac{x^{2}(r^{2}+1)^{\mu+1}}{(x^{2}+r^{2})^{\mu+1}}$ decreases.
Differentiating $f(x)$, we have

$f^{\prime}(x)=\frac{2x(r^{2}+1)^{\mu+1}}{(x^{2}+r^{2})^{\mu+2}}(r^{2}-x^{2}\mu),\quad
x\geq 1,\mu>0.$

The denominator is positive for all $x\geq 1$ and $r>0,$ therefore
$f^{\prime}(x)<0$ provided that $0<r\leq\sqrt{\mu}.$ This completes the proof.
∎

###### Alternative proof.

Using the Bernoulli inequality we can directly prove that the sequence
$\\{na_{n}\\}$ is non-increasing under the stated condition.

$\displaystyle na_{n}-(n+1)a_{n+1}$
$\displaystyle=\frac{n^{2}(r^{2}+1)^{\mu+1}}{(n^{2}+r^{2})^{\mu+1}}-\frac{(n+1)^{2}(r^{2}+1)^{\mu+1}}{((n+1)^{2}+r^{2})^{\mu+1}}$
$\displaystyle=\left[\frac{(r^{2}+1)}{(n^{2}+r^{2})((n+1)^{2}+r^{2})}\right]^{\mu+1}b_{n},$

where

$\displaystyle b_{n}$
$\displaystyle=n^{2}((n+1)^{2}+r^{2})^{\mu+1}-(n+1)^{2}(n^{2}+r^{2})^{\mu+1}$
$\displaystyle=n^{2}(n^{2}+2n+1+r^{2})^{\mu+1}-(n^{2}+2n+1)(n^{2}+r^{2})^{\mu+1}$
$\displaystyle=n^{2}(n^{2}+r^{2})^{\mu+1}\left(1+\frac{2n+1}{n^{2}+r^{2}}\right)^{\mu+1}-n^{2}(n^{2}+r^{2})^{\mu+1}-(2n+1)(n^{2}+r^{2})^{\mu+1}$
$\displaystyle=n^{2}(n^{2}+r^{2})^{\mu+1}\left[\left(1+\frac{2n+1}{n^{2}+r^{2}}\right)^{\mu+1}-1\right]-(2n+1)(n^{2}+r^{2})^{\mu+1}$
$\displaystyle\geq
n^{2}(n^{2}+r^{2})^{\mu+1}\frac{(2n+1)(\mu+1)}{n^{2}+r^{2}}-(2n+1)(n^{2}+r^{2})^{\mu+1}$
$\displaystyle=(2n+1)(n^{2}+r^{2})^{\mu}[n^{2}(\mu+1)-(n^{2}+r^{2})]$
$\displaystyle=(2n+1)(n^{2}+r^{2})^{\mu}[n^{2}\mu-r^{2}]\geq 0,$

provided $\mu>0\;\mbox{and}\;0<r\leq\sqrt{\mu}$. ∎

######  Theorem 2.2.

If $\mu>0$ and $0<r\leq\sqrt{\frac{{(3+5\mu)-\sqrt{17\mu^{2}+26\mu+9}}}{2}},$
then $\mathbb{F}_{\mu}(r;z)$ is starlike in $\mathbb{U}$.

###### Proof.

We already proved in Theorem 2.1 that $\\{na_{n}\\}$ is non-increasing for all
$0<r\leq\sqrt{\mu}$. To show that $\mathbb{F}_{\mu}(r;z)$ is starlike in
$\mathbb{U}$, using Lemma 1.2, it is sufficient to show that the sequence
$\\{na_{n}-(n+1)a_{n+1}\\}$ is also non-increasing. That is

$\displaystyle na_{n}-2(n+1)a_{n+1}+(n+2)a_{n+2}\geq 0$ $\displaystyle\iff$
$\displaystyle(r^{2}+1)^{\mu+1}\left[\frac{n^{2}}{(n^{2}+r^{2})^{\mu+1}}-2\frac{(n+1)^{2}}{((n+1)^{2}+r^{2})^{\mu+1}}+\frac{(n+2)^{2}}{((n+2)^{2}+r^{2})^{\mu+1}}\right]\geq
0$ $\displaystyle\iff$ $\displaystyle[f(n)-2f(n+1)+f(n+2)]\geq 0,$

where

$f(x)=\frac{x^{2}}{(x^{2}+r^{2})^{\mu+1}},\quad x\geq 1,\mu>0.$

To show $[f(n)-2f(n+1)+f(n+2)]\geq 0$, $n=1,2,3,4,\ldots$, it is sufficient to
prove that $f(x)$ is a convex function in the real sense or that
$f^{\prime\prime}(x)\geq 0,x\geq 1$. Differentiating twice, we have

$f^{\prime\prime}(x)=\frac{2[x^{4}(\mu+2\mu^{2})-x^{2}(3+5\mu)r^{2}+r^{4}]}{(x^{2}+r^{2})^{\mu+3}},\quad
x\geq 1,\mu>0.$

The denominator is positive for all $x\geq 1$ and $r>0$. Let
$\phi(x)=x^{4}(\mu+2\mu^{2})-x^{2}(3+5\mu)r^{2}+r^{4}$. Obviously
$\phi^{\prime}(x)=4x^{3}(2\mu^{2}+\mu)-2(3+5\mu)r^{2}x\geq 0$ for all $x\geq
1,\mu>0$ and $0<r\leq\sqrt{\frac{2(2\mu^{2}+\mu)}{5\mu+3}}$. Thus
$f^{\prime\prime}(x)\geq 0$ provided $\phi(1)\geq 0$, which in turn gives
$0<r\leq\sqrt{\frac{{(5\mu+3)-\sqrt{17\mu^{2}+26\mu+9}}}{2}}$ . This completes
the proof. ∎

######  Theorem 2.3.

For $\mu>0$ and $0<r\leq\sqrt{\frac{2\mu+1}{3}}$,

$\Re\bigg{(}{\frac{\mathbb{F}_{\mu}(r;z)}{z}}\bigg{)}>\frac{1}{2},\quad
z\in\mathbb{U}.$

###### Proof.

First we prove that

$\left\\{{{a_{n}}}\right\\}_{n=1}^{\infty}=\left\\{\frac{n(r^{2}+1)^{\mu+1}}{(n^{2}+r^{2})^{\mu+1}}\right\\}_{n=1}^{\infty}$

is a decreasing sequence, i.e.

$a_{n}-a_{n+1}\geq 0,\quad n\in\mathbb{N}.$

Now

$\displaystyle a_{n}-a_{n+1}\geq 0$ $\displaystyle\iff$
$\displaystyle(r^{2}+1)^{\mu+1}\left[\frac{n}{(n^{2}+r^{2})^{\mu+1}}-\frac{n+1}{((n+1)^{2}+r^{2})^{\mu+1}}\right]\geq
0$ $\displaystyle\iff$
$\displaystyle(r^{2}+1)^{\mu+1}\left[f(n)-f(n+1)\right]\geq 0,$

where

(2.4) $f(x)=\frac{x}{(x^{2}+r^{2})^{\mu+1}},\quad x\geq 1,\mu>0.$

To show $f(n)-f(n+1)\geq 0$, $n=1,2,3,\ldots,$ it is sufficient to prove that
$f(x)$ is a decreasing function in the real sense or that $f^{\prime}(x)<0$,
$x\geq 1$. We have

$f^{\prime}(x)=\frac{r^{2}-(1+2\mu)x^{2}}{(x^{2}+r^{2})^{\mu+2}}\leq 0,\quad
x\geq 1,\mu>0\;{\rm and}\;0<r\leq\sqrt{1+2\mu}.$

Next we prove that $\left\\{{{a_{n}}}\right\\}_{n=1}^{\infty}$ is a convex
decreasing sequence. For this we show

$a_{n+2}-a_{n+1}\geq a_{n+1}-a_{n},\quad n\in\mathbb{N}.$

Now

$\displaystyle a_{n}-2a_{n+1}+a_{n+2}\geq 0$ $\displaystyle\iff$
$\displaystyle(r^{2}+1)^{\mu+1}\left[\frac{n}{(n^{2}+r^{2})^{\mu+1}}-2\frac{n+1}{((n+1)^{2}+r^{2})^{\mu+1}}+\frac{n+2}{((n+2)^{2}+r^{2})^{\mu+1}}\right]\geq
0$ $\displaystyle\iff$
$\displaystyle(r^{2}+1)^{\mu+1}\left[f(n)-2f(n+1)+f(n+2)\right]\geq 0,$

where $f(x)$ is given by (2.4). To show $\left[f(n)+f(n+2)-2f(n+1)\right]\geq
0$ , $n=1,2,3,4,\ldots$, it suffices to prove that $f(x)$ is a convex function
in the real sense or that $f^{\prime\prime}(x)\geq 0$, $x\geq 1$. It can be
easily verified that

$f^{\prime\prime}(x)=\frac{2x[(2\mu^{2}+3\mu+1)x^{2}-3(\mu+1)r^{2}]}{(x^{2}+r^{2})^{\mu+3}}\geq
0$

for all

$x\geq 1,\mu>0\;{\rm and}\;0<r\leq\sqrt{\frac{2\mu+1}{3}}.$

Thus $\left\\{{{a_{n}}}\right\\}_{n=1}^{\infty}$ is a convex decreasing
sequence. Now applying Lemma 1.3 to
$\left\\{{{a_{n}}}\right\\}_{n=1}^{\infty}$, we have

$\Re\bigg{(}\sum\limits_{n=1}^{\infty}{{a_{n}}{z^{n-1}}}\bigg{)}>\frac{1}{2},\quad
z\in\mathbb{U},$

which is equivalent to

$\Re\bigg{(}\frac{\mathbb{F}_{\mu}(r;z)}{z}\bigg{)}>\frac{1}{2},\quad
z\in\mathbb{U}.$

∎

######  Theorem 2.4.

For $\mu>0$ and $0<r\leq\sqrt{\frac{{(5\mu+3)-\sqrt{17\mu^{2}+26\mu+9}}}{2}}$,

$\Re\big{(}{\mathbb{F}^{\prime}_{\mu}(r;z)}\big{)}>\frac{1}{2},\quad
z\in\mathbb{U}.$

###### Proof.

From (1.2),

$\mathbb{F}^{\prime}_{\mu}(r;z)=1+\sum\limits_{n=2}^{\infty}{\frac{n^{2}(r^{2}+1)^{\mu+1}}{(n^{2}+r^{2})^{\mu+1}}z^{n-1}}.$

We have shown in Theorems 2.1 and 2.2 that the sequence

$\frac{n^{2}(r^{2}+1)^{\mu+1}}{(n^{2}+r^{2})^{\mu+1}}$

is convex and non-increasing, and so the result follows from Lemma 1.3. ∎

##  3\. Close-to-convexity and starlikeness of $\mathbb{Q}_{\mu}(r;z)$

######  Theorem 3.1.

If $\mu>0$ and $0<r\leq\sqrt{\mu},$ then $\mathbb{Q}_{\mu}(r;z)$ is close-to-
convex w.r.t. the starlike function $z/(1-z).$

###### Proof.

Again, we apply Lemma 1.1. Let

$C_{n}={\frac{n!(r^{2}+1)^{\mu+1}}{((n!)^{2}+r^{2})^{\mu+1}}}.$

Using the Bernoulli inequality we can directly prove that the sequence
$\\{nC_{n}\\}$ is non-increasing under the stated condition. We have

$\displaystyle nC_{n}-(n+1)C_{n+1}$
$\displaystyle=\frac{n\;n!(r^{2}+1)^{\mu+1}}{{[(n!)^{2}+r^{2}]^{\mu+1}}}-\frac{(n+1)\;(n+1)!(r^{2}+1)^{\mu+1}}{{[((n+1)!)^{2}+r^{2}]^{\mu+1}}}$
$\displaystyle=\left[\frac{(r^{2}+1)}{((n!)^{2}+r^{2})(((n+1)!)^{2}+r^{2})}\right]^{\mu+1}B_{n},$

where

$\displaystyle B_{n}$
$\displaystyle=n\;n![((n+1)!)^{2}+r^{2}]^{\mu+1}-(n+1)(n+1)!((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle=n\;n![((n^{2}+2n+1)(n)!)^{2}+r^{2}]^{\mu+1}-(n^{2}+2n+1)\;n!\;((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle=n\;n![(n^{2}+2n)((n)!)^{2}+((n)!)^{2}+r^{2}]^{\mu+1}-(n^{2}+2n+1)\;n!\;((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle=n\;n![(n!)^{2}+r^{2}]^{\mu+1}\left[1+\frac{(n^{2}+2n)((n)!)^{2}}{(n!)^{2}+r^{2}}\right]^{\mu+1}-(n^{2}+2n+1)\;n!\;((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle\geq
n\;n![(n!)^{2}+r^{2}]^{\mu+1}\left[1+\frac{(\mu+1)(n^{2}+2n)((n)!)^{2}}{(n!)^{2}+r^{2}}\right]-(n^{2}+2n+1)\;n!\;((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle=n\;n![(n!)^{2}+r^{2}]^{\mu+1}+n(n^{2}+2n)(\mu+1)((n)!)^{3}[(n!)^{2}+r^{2}]^{\mu}-(n^{2}+2n+1)\;n!\;((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle\geq\;n![(n!)^{2}+r^{2}]^{\mu+1}+(n^{2}+2n)(\mu+1)((n)!)^{3}[(n!)^{2}+r^{2}]^{\mu}-(n^{2}+2n+1)\;n!\;((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle=(n^{2}+2n)(\mu+1)((n)!)^{3}[(n!)^{2}+r^{2}]^{\mu}-(n^{2}+2n)\;n!\;((n!)^{2}+r^{2})^{\mu+1}$
$\displaystyle=(n)!(n^{2}+2n)((n!)^{2}+r^{2})^{\mu}(\mu(n!)^{2}-r^{2})\geq 0,$

provided that $\mu-r^{2}\geq 0$ or $0<r\leq\sqrt{\mu}$. ∎

######  Theorem 3.2.

If $\mu\geq 2$ and $0<r\leq\sqrt{\mu},$ then $\mathbb{Q}_{\mu}(r;z)$ is
starlike in $\mathbb{U}$.

###### Proof.

We have shown in the preceding proof that $\\{nC_{n}\\}_{n\geq 1}$ is non-
increasing. Now we prove that it is convex, provided that $\mu\geq 2,$ in
order to apply Lemma 1.2. We have $nC_{n}=g(n),$ where

$g(x)=\frac{x\Gamma(x+1)(1+r^{2})^{\mu+1}}{\big{(}\Gamma(x+1)^{2}+r^{2}\big{)}^{\mu+1}},\quad
x\geq 0.$

Define

$h(x)=g(x)-g(x+1).$

Our goal is to show that $\\{h(n)\\}_{n\geq 1}$ is non-increasing. We have

$\displaystyle h(1)-h(2)$
$\displaystyle=1-8\Big{(}\frac{r^{2}+1}{r^{2}+4}\Big{)}^{\mu+1}+18\Big{(}\frac{r^{2}+1}{r^{2}+36}\Big{)}^{\mu+1}$
$\displaystyle>1-8\Big{(}\frac{r^{2}+1}{r^{2}+4}\Big{)}^{\mu+1}$
$\displaystyle\geq 1-8\Big{(}\frac{\mu+1}{\mu+4}\Big{)}^{\mu+1}\geq 0,$

where the last estimate follows from $\mu\geq 2$ and the fact that
$\Big{(}\frac{\mu+1}{\mu+4}\Big{)}^{\mu+1}$ decreases w.r.t. $\mu$. (This is
the only step where $\mu\geq 2$ is used; the rest of the proof works for
$\mu>0$.) Very similarly, it can be shown that

$h(2)\geq h(3)\geq h(4).$

To complete the proof, we show that $g(x)$ is a convex function for $x\geq 4.$
We have

$g^{\prime\prime}(x)=\Gamma(x+1)\big{(}\Gamma(x+1)^{2}+r^{2}\big{)}^{-\mu-3}(1+r^{2})^{\mu+1}A(x),$

where

(3.5)
$A(x)=2(r^{2}+\Gamma(x+1)^{2})\big{[}r^{2}-(2\mu+1)\Gamma(x+1)^{2}\big{]}\psi(x+1)\\\
+x\big{[}r^{4}-2r^{2}(4\mu+3)\Gamma(x+1)^{2}+(2\mu+1)^{2}\Gamma(x+1)^{4}\big{]}\psi(x+1)^{2}\\\
+x\big{(}r^{2}+\Gamma(x+1)^{2}\big{)}\big{(}r^{2}-(2\mu+1)\Gamma(x+1)^{2}\big{)}\psi^{\prime}(x+1),$

and $\psi=\Gamma^{\prime}/\Gamma$ denotes the digamma function. We will apply
the following estimates:

(3.6) $-2r^{2}(4\mu+3)c+(2\mu+1)^{2}c^{2}\geq 0,\quad\mu>0,0\leq
r\leq\sqrt{\mu},c\geq 2,$ (3.7) $\psi(x)<\log x-\frac{1}{2x},\quad x>1,$ (3.8)
$\psi(x)>\log x-\frac{1}{x},\quad x>1,$ (3.9)
$\psi^{\prime}(x)<\frac{1}{x}+\frac{1}{x^{2}},\quad x>0,$ (3.10)
$\log(x+1)-\frac{1}{2(x+1)}\leq\sqrt{x},\quad x\geq 1,$ (3.11)
$\Big{(}\log(x+1)-\frac{1}{x+1}\Big{)}^{2}\geq\frac{19}{10},\quad x\geq 4,$
(3.12)
$2(r^{2}+c)(r^{2}-(2\mu+1)c)\sqrt{x}+x\big{(}r^{4}-2r^{2}(4\mu+3)c+(2\mu+1)^{2}c^{2}\big{)}\frac{19}{10}\\\
+x(r^{2}+c)(r^{2}-(2\mu+1)c)\Big{(}\frac{1}{x+1}+\frac{1}{(x+1)^{2}}\Big{)}\geq
0,\\\ \quad x\geq 4,c\geq\Gamma(5)^{2},\mu>0,0\leq r\leq\sqrt{\mu}.$

Note that (3.6) and (3.12) are polynomial inequalities with polynomial
constraints, which can be proven by cylindrical algebraic decomposition (CAD),
using a computer algebra system. The inequalities (3.10) and (3.11) are very
easy to show. The estimates (3.7) and (3.8) are found on p. 288 of [18], and
(3.9) is an easy consequence of (26) in [17].

For the rest of the proof, we may assume $x\geq 4.$ The factors in front of
$\psi(x+1)$ and $\psi^{\prime}(x+1)$ in (3.5) are clearly negative, whereas
the factor in front of $\psi(x+1)^{2}$ is non-negative, by (3.6). Thus,
(3.7)–(3.9) imply

$A(x)\geq
2(r^{2}+\Gamma(x+1)^{2})(r^{2}-(2\mu+1)\Gamma(x+1)^{2})\Big{(}\log(x+1)-\frac{1}{2(x+1)}\Big{)}\\\
+x(r^{4}-2r^{2}(4\mu+3)\Gamma(x+1)^{2}+(2\mu+1)^{2}\Gamma(x+1)^{4})\Big{(}\log(x+1)-\frac{1}{x+1}\Big{)}^{2}\\\
+x(r^{2}+\Gamma(x+1)^{2})(r^{2}-(2\mu+1)\Gamma(x+1)^{2})\Big{(}\frac{1}{x+1}+\frac{1}{(x+1)^{2}}\Big{)}.$

Using (3.10) and (3.11), we thus obtain

$A(x)\geq 2(r^{2}+\Gamma(x+1)^{2})(r^{2}-(2\mu+1)\Gamma(x+1)^{2})\sqrt{x}\\\
+x(r^{4}-2r^{2}(4\mu+3)\Gamma(x+1)^{2}+(2\mu+1)^{2}\Gamma(x+1)^{4})\frac{19}{10}\\\
+x(r^{2}+\Gamma(x+1)^{2})(r^{2}-(2\mu+1)\Gamma(x+1)^{2})\Big{(}\frac{1}{x+1}+\frac{1}{(x+1)^{2}}\Big{)}.$

This is non-negative by (3.12). The convexity of $g(x)$ for $x\geq 4$ is
established, which completes the proof. ∎

######  Theorem 3.3.

For $\mu>0$ and $0<r\leq\sqrt{\mu}$,

$\Re\bigg{(}{\frac{\mathbb{Q}_{\mu}(r;z)}{z}}\bigg{)}>\frac{1}{2},\quad
z\in\mathbb{U}.$

###### Proof.

We use Lemma 1.3. To show that

$\left\\{{{C_{n}}}\right\\}_{n=1}^{\infty}=\left\\{\frac{n!(r^{2}+1)^{\mu+1}}{((n!)^{2}+r^{2})^{\mu+1}}\right\\}_{n=1}^{\infty}$

is a decreasing sequence, we define

$\tilde{g}(x)=\frac{\Gamma(x+1)}{((\Gamma(x+1))^{2}+r^{2})^{\mu+1}},\quad
x\geq 1,\mu>0.$

We have

$\tilde{g}^{\prime}(x)=\frac{\Gamma^{\prime}(x+1)\left[r^{2}-(2\mu+1)(\Gamma(x+1))^{2}\right]}{\left[(\Gamma(x+1))^{2}+r^{2}\right]^{\mu+2}}\leq
0$

when $x\geq 1$, $\mu>0$ and $0<r\leq\sqrt{2\mu+1}$. Next we prove that
$\\{C_{n}\\}$ is a convex sequence. Define

$\tilde{h}(x)=\tilde{g}(x)-\tilde{g}(x+1).$

We first show that $\tilde{h}(1)\geq\tilde{h}(2)\geq\tilde{h}(3).$ We have

$\displaystyle\tilde{h}(1)-\tilde{h}(2)>\frac{1}{(r^{2}+1)^{\mu+1}}-\frac{4}{(r^{2}+4)^{\mu+1}}\geq
0,$

because the map $x\mapsto x/(r^{2}+x)^{\mu+1}$ decreases for $x\geq
r^{2}/\mu.$ For the same reason,

$\displaystyle\tilde{h}(2)-\tilde{h}(3)$
$\displaystyle>\frac{1}{2}\Big{(}\frac{4}{(r^{2}+4)^{\mu+1}}-\frac{24}{(r^{2}+36)^{\mu+1}}\Big{)}$
$\displaystyle>\frac{1}{2}\Big{(}\frac{4}{(r^{2}+4)^{\mu+1}}-\frac{36}{(r^{2}+36)^{\mu+1}}\Big{)}>0.$

We complete the proof by showing that the function $\tilde{g}(x)$ is convex
for $x\geq 3,$ similarly as in the proof of Theorem 3.2. We have

$\tilde{g}^{\prime\prime}(x)=\Gamma(x+1)(r^{2}+\Gamma(x+1)^{2})^{-\mu-3}\tilde{A}(x),$

where

$\tilde{A}(x):=\Big{(}r^{4}-2r^{2}(4\mu+3)\Gamma(x+1)^{2}+(2\mu+1)^{2}\Gamma(x+1)^{4}\Big{)}\psi(x+1)^{2}\\\
+\big{(}r^{2}+\Gamma(x+1)^{2}\big{)}\big{(}r^{2}-(2\mu+1)\Gamma(x+1)^{2}\big{)}\psi^{\prime}(x+1).$

We state the following inequalities:

(3.13) $-2r^{2}(4\mu+3)+(2\mu+1)^{2}c\geq 0,\quad\mu>0,0\leq
r\leq\sqrt{\mu},c\geq 2.$ (3.14)
$\Big{(}\log(x+1)-\frac{1}{x+1}\Big{)}^{2}\geq 1,\quad x\geq 3.$ (3.15)
$\frac{1}{x+1}+\frac{1}{(x+1)^{2}}\leq\frac{1}{2},\quad x\geq 3.$ (3.16)
$(2\mu+1)^{2}-\frac{2r^{2}(4\mu+3)}{c}-\frac{2\mu+1}{2}\Big{(}1+\frac{r^{2}}{c}\Big{)}>0,\quad\mu>0,0\leq
r\leq\sqrt{\mu},c\geq 5.$

It is very easy to show (3.14). The estimate (3.15) is obvious, and (3.13) and
(3.16) can be proven by computer algebra (see above). From now on we assume
$x\geq 3.$ Since $\psi^{2}(x+1)$ and $\psi^{\prime}(x+1)$ are non-negative, we
have

$\tilde{A}(x)\geq\big{[}{-2r^{2}(4\mu+3)\Gamma(x+1)^{2}}+(2\mu+1)^{2}\Gamma(x+1)^{4}\big{]}\psi(x+1)^{2}\\\
-\big{(}r^{2}+\Gamma(x+1)^{2}\big{)}(2\mu+1)\Gamma(x+1)^{2}\psi^{\prime}(x+1).$

By (3.13), the term $[\dots]$ is $\geq 0$. We then use (3.8) and (3.9) and
obtain

$\tilde{A}(x)\geq\big{[}{-2r^{2}(4\mu+3)\Gamma(x+1)^{2}}+(2\mu+1)^{2}\Gamma(x+1)^{4}\big{]}\Big{(}\log(x+1)-\frac{1}{x+1}\Big{)}^{2}\\\
-\big{(}r^{2}+\Gamma(x+1)^{2}\big{)}(2\mu+1)\Gamma(x+1)^{2}\Big{(}\frac{1}{x+1}+\frac{1}{(x+1)^{2}}\Big{)}.$

By (3.14) and (3.15), we further get

$\displaystyle\tilde{A}(x)$
$\displaystyle\geq\big{[}-2r^{2}(4\mu+3)\Gamma(x+1)^{2}+(2\mu+1)^{2}\Gamma(x+1)^{4}\big{]}$
$\displaystyle\qquad\qquad-\frac{1}{2}\big{(}r^{2}+\Gamma(x+1)^{2}\big{)}(2\mu+1)\Gamma(x+1)^{2}$
$\displaystyle=\Gamma(x+1)^{4}\Big{(}\Big{[}(2\mu+1)^{2}-\frac{2r^{2}(4\mu+3)}{\Gamma(x+1)^{2}}\Big{]}-\frac{2\mu+1}{2}\Big{[}1+\frac{r^{2}}{\Gamma(x+1)^{2}}\Big{]}\Big{)}.$

This is positive by (3.16), for $x\geq 3.$ ∎

######  Theorem 3.4.

For $\mu\geq 2$ and $0<r\leq\sqrt{\mu}$,

$\Re\big{(}{\mathbb{Q}^{\prime}_{\mu}(r;z)}\big{)}>\frac{1}{2},\quad
z\in\mathbb{U}.$

###### Proof.

From (1.3),

$\mathbb{Q}^{\prime}_{\mu}(r;z)=1+\sum\limits_{n=2}^{\infty}{\frac{n\,n!(r^{2}+1)^{\mu+1}}{((n!)^{2}+r^{2})^{\mu+1}}z^{n-1}}.$

We have shown in Theorems 3.1 and 3.2 that the sequence

$\frac{n\,n!(r^{2}+1)^{\mu+1}}{((n!)^{2}+r^{2})^{\mu+1}}$

is convex and non-increasing, and so the result follows from Lemma 1.3. ∎

##  4\. Two further examples

We conclude the paper with two more examples of starlike Mathieu-type power
series. Here, we apply the criterion that $z+\sum_{n=2}^{\infty}a_{n}z^{n}$ is
starlike if $\sum_{n=2}^{\infty}n|a_{n}|<1.$ See Goodman [9] and the
references given there.

######  Example 4.1.

Define

$\hat{S}(z)=z+\sum_{n=2}^{\infty}\frac{8n}{(n^{2}+1)^{3}}z^{n},\quad|z|<1.$

Using an inequality due to Diananda [6] and the above criterion, it can be
shown that

$\int_{0}^{z}\frac{\hat{S}(t)}{t}dt=z+4\sum_{n=2}^{\infty}\frac{2z^{n}}{(n^{2}+1)^{3}}$

is starlike. Indeed, by Theorem 1 in [6],

$4\sum_{n=2}^{\infty}\frac{2n}{(n^{2}+1)^{3}}=4\Big{(}\sum_{n=1}^{\infty}\frac{2n}{(n^{2}+1)^{3}}-\frac{1}{4}\Big{)}<4\Big{(}\frac{1}{2}-\frac{1}{4}\Big{)}=1.$

######  Example 4.2.

The power series

$z+4\sum_{n=2}^{\infty}\frac{(2n-1)!!}{[(2n+1)!!+1]^{2}}z^{n},\quad|z|<1,$

is starlike. Recall that the double factorial is defined by $(2n+1)!!=1\times
3\times\dots\times(2n+1).$ Starlikeness follows from the estimate

$\displaystyle\sum_{n=1}^{\infty}\frac{4n(2n-1)!!}{[(2n+1)!!+1]^{2}}$
$\displaystyle<2\sum_{n=1}^{\infty}\frac{(2n+1)(2n-1)!!-(2n-1)!!}{[(2n+1)!!+1][(2n-1)!!+1]}$
$\displaystyle=2\sum_{n=1}^{\infty}\Big{[}\frac{1}{(2n-1)!!+1}-\frac{1}{(2n+1)!!+1}\Big{]}$
$\displaystyle=2\cdot\frac{1}{(2\cdot 1-1)!!+1}=1,$

i.e.

$\sum_{n=2}^{\infty}\frac{4n(2n-1)!!}{[(2n+1)!!+1]^{2}}<1-\frac{1}{4}<1.$

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# FedPartWhole: Federated domain generalization via consistent part-whole
hierarchies

###### Abstract

Federated Domain Generalization (FedDG), aims to tackle the challenge of
generalizing to unseen domains at test time while catering to the data privacy
constraints that prevent centralized data storage from different domains
originating at various clients. Existing approaches can be broadly categorized
into four groups: domain alignment, data manipulation, learning strategies,
and optimization of model aggregation weights. This paper proposes a novel
approach to Federated Domain Generalization that tackles the problem from the
perspective of the backbone model architecture. The core principle is that
objects, even under substantial domain shifts and appearance variations,
maintain a consistent hierarchical structure of parts and wholes. For
instance, a photograph and a sketch of a dog share the same hierarchical
organization, consisting of a head, body, limbs, and so on. The introduced
architecture explicitly incorporates a feature representation for the image
parse tree. To the best of our knowledge, this is the first work to tackle
Federated Domain Generalization from a model architecture standpoint. Our
approach outperforms a convolutional architecture of comparable size by over
12%, despite utilizing fewer parameters. Additionally, it is inherently
interpretable, contrary to the black-box nature of CNNs, which fosters trust
in its predictions, a crucial asset in federated learning.

Index Terms—  Domain Generalization, Federated Learning, Foundational Models

Fig. 1: The Figure illustrates the motivation behind our approach. Objects
will maintain their compositional hierarchical structure despite any change in
appearance, which aids domain generalization. On the left, the parse tree of a
picture of a dog, while the right-hand side depicts the parse tree for a
sketch drawing of a dog. The figure illustrates that both images have the same
parse tree, despite the huge appearance discrepancy.

##  1 Introduction

Fig. 2: The Figure illustrates the CCNet architecture. a) the input image. b)
the input tokenizer. c) the initialization pipeline. d) the tokenized input e)
the hidden representation at t = 0. f) The network architecture. g) the hidden
representation at t = 1. h) the softmax classification output. i) the
structure of the classification head.

Generalization, the ability to apply learned concepts to unseen situations, is
a fundamental challenge in achieving human-level artificial intelligence.
Machine learning algorithms have achieved remarkable success under the
assumption of identically and independently distributed (i.i.d.) data, even
surpassing human level performance [1] on popular benchmarks like ImageNet
[2]. However, their performance often degrades significantly when faced with
domain shifts in the test data [3]. This challenge has motivated extensive
research on models that can generalize to unseen domains during testing. The
work in [4] was the earliest to propose a method to tackle this problem, which
is known as the Domain Generalization (DG) problem. Popular DG approaches rely
on exposing models to data from multiple source domains during the training
time. This allows the model to learn features that are invariant across these
domains, ideally leading to robustness against domain shifts at test time.
However, this relies on a centralized training setup where all source domains
reside on a single server, enabling the model to learn from data from
different domains within a single mini-batch [5] [6] [7]. This approach
presents practical limitations. In real-world scenarios, data from different
domains often originates from distributed devices or organizations, making
centralized storage infeasible. Additionally, data privacy regulations and
laws, such as GDPR [8], may restrict the sharing of raw data across entities.

Federated Domain Generalization (FedDG) tackles this challenge by enabling
models to learn from distributed data while preserving privacy. Existing FedDG
approaches can be broadly categorized into four main areas: 1) Federated
Domain Alignment, which focuses on aligning features from different domains to
reduce domain discrepancies. 2) Data Manipulation, which aims to improve model
performance by augmenting existing source domains or generating new synthetic
domains. 3) Learning Strategies, which explore novel training algorithms
specifically designed for the FedDG setting. 4) Aggregation Optimization,
which focuses on developing techniques for effectively aggregating model
parameters and gradients received from participating devices.

Pioneering work done in [9] introduced a theoretical framework called GLOM,
which is a proposal of how neural networks could learn to represent part-whole
hierarchies. [10] implemented the framework, dubbed as Agglomerator, and
showed promising results on toy datasets. While [11], showed how to pre-train
the network effectively to work on more complex natural scenes, benefitting
from the pre-trained knowledge of other commonly used architectures.

In this paper, we build upon the theoretical framework of GLOM and propose a
novel architecture, a Cortical Columns Network (CCNet), specifically designed
to enhance generalization capabilities and tackle the FedDG problem. The main
motivation behind this approach is the observation that objects across
different domains belonging to the same class share a consistent hierarchical
structure of parts and wholes, Figure 1 illustrates the concept. The proposed
architecture explicitly models the part-whole hierarchies in its feature maps.
Facilitating the classification decision based on the recognized parse tree.
Our main contributions can be summarized as follows:

  * •

We present a novel lightweight backbone architecture, CCNet. This architecture
explicitly models the compositional structure of visual objects within its
feature map. To the best of our knowledge, this is the first paper to address
the FedDG problem from a backbone architectural point of view.

  * •

The proposed backbone can be used with existing FedDG algorithms, making our
development orthogonal to the existing approaches. When used as a backbone in
different FedDG algorithms, the results show strong performance improvements
over the widely established CNN architecture, with fewer parameters.

  * •

Contrary to many existing backbones, the proposed backbone architecture does
not require a computationally expensive pre-training step thanks to its
explicit representation of the compositional structure of the objects.

##  2 Related Work

###  2.1 Federated Domain Generelization

Federated domain generalization (FDG) aims to tackle both the domain shift and
data privacy problem simultaneously. The research efforts on FDG can be mainly
categorized into 4 main groups. 1) Domain alignment, aims to align the
representations of multiple source domains, which helps the model mitigate
domain shifts and generalize to unseen domains at test time. Adversarial
domain alignment, proposed by [12], involves training a discriminator to
identify the feature distribution of the source domain, while a generator
learns to produce robust features to confuse the discriminator. While [13]
proposed an approach to minimize the maximum mean discrepancy between source
and target domains. 2) Data Manipulation, based techniques, aim to diversify
and enlarge the source domains as much as possible by manipulating the
available data. [14] applied style transfer augmentation to mimic the target
domain styles within the source images. While [15] uses a powerful
augmentation based on the frequency spectrum information. 3) Learning
strategies, encompasses a variety of techniques and algorithms that aim to
learn a robust global model. Techniques include channel decoupling [16], and
regularization strategies [17]. 4) Aggregation optimization, techniques aim to
find optimal ways to aggregate model parameters or gradient updates. [18]
proposed using distance metrics to decrease the influence of diverging
clients. While [19] optimizes gradient updates by introducing a loss on the
gradient alignment. Finally, our work represents both a novel research
direction and orthogonal to the existing directions in the realm of FedDG. We
tackle the problem by proposing an architecture that explicitly models the
scene’s parse tree, which should be invariant under domain shifts.

###  2.2 Related Foundational Models

Few endeavors have been made to develop architectures capable of representing
scene parse trees. Our work falls in line with such models.

####  2.2.1 Capsule Networks

One such architecture is the Capsule Network [20]. Capsule Networks create
layers of capsules, which are fundamental elements responsible for detecting
objects or their constituent parts. These capsules, employing dynamic routing
by agreement, establish connections with their corresponding parent capsules,
effectively forming a parse tree of the scene. While Capsule Networks have
demonstrated competitive performance on easier datasets, although they face
challenges in matching the performance of convolutional or transformer-based
approaches on more intricate datasets.

####  2.2.2 Agglomerator

The theoretical framework of GLOM was introduced [9] to remedy some of the
issues with capsule networks. Agglomerator [10] is an implementation of GLOM.
Agglomerator’s authors proposed training to undergo two distinct training
phases. Firstly, it is pre-trained using the supervised contrastive loss.
Subsequently, the network is trained with the cross entropy loss. Agglomerator
alleviated some of the issues with capsule networks. However, [11] showed that
it fails to deal with more complex natural images beyond toy datasets, such as
those in ImageNet [2]. To improve the performance, and pre-train the network
efficiently, they proposed a knowledge-distillation approach to pre-train
Agglomerator using the available knowledge in other pre-trained models.

####  2.2.3 Segment Anything

The Segment Anything Model, SAM, was recently introduced as a new foundation
model for image segmentation on images from any domain. Segmentation is one of
the most fundamental computer vision tasks and finds its way to many
applications. A segmentation foundation model, that offers zero-shot
generalization capabilities without requiring domain-specific training, is
very likely to have beneficial information. SAM was trained on more than a
billion segmentation masks, which explains its performance. SAM can be
prompted to generate segmentation masks using either a bounding box, text
description, a mask, a point prompt, or a combination of the prompts. Since
some prompts could be ambiguous, SAM offers a multi-output segmentation mask
to resolve the ambiguity. In our work, we leverage the domain generalized
knowledge in SAM and use it to generate an initial scene parse tree. We use
the embeddings of the generated parse tree to initialize the CCNet feature
map.

Fig. 3: The Figure depicts a sample of the PACS dataset from the four
different domains, illustrating the challenge of domain generalization under
significant domain shift.

##  3 Proposed Method

In this section, we describe our proposed system, inspired by the column-like
arrangement of neurons found in the human’s neocortex and by the theoretical
framework, GLOM. We will start by introducing the main building blocks of the
proposed architecture, dubbed CCNet. Firstly, the input image is passed
through a convolutional network to obtain $N=n\times n$ feature map patches
which will serve as the input to our system. The system represents the part-
whole hierarchy of the image using columns. Each Column, $C_{i}$, specializes
in processing a single spatial location and is assigned to process the
corresponding $i^{t}h$ input patch, with $i\in\\{1,\dots,N\\}$. A Column
comprises $L$ levels of embeddings. An embedding vector at a time step $t$,
and level $l$ of the $i^{t}h$ C is denoted by $Z_{t}^{l,i}$, where
$l\in\\{1,\dots,L\\}$.

The latent representation of the network, $Z$, is an explicit modeling of the
part-whole hierarchy of objects and their parts in a visual scene. Each
embedding level in the column is a latent representation of the image patch at
some level of abstraction. The lowest level of the column represents the low-
level information present in the patch and as we go towards the highest level
we expect to see a vector representing the object of which the patch is a part
of till we arrive at a scene-level representation.

###  3.1 Input Tokenization

The first step in our architecture is transforming the input image into
patches of input tokens. This is achieved by firstly passing the image through
a convolutional network that transforms an input image into $n\times n$
feature map. Each of the feature map patches will be embedded into the lowest
level of its corresponding column, where the $Z_{0}^{0,i}$ vector will come
from the $i^{t}h$ feature map patch.

Table 1: Leave-one-domain-out results on both Domain Generalization datasets
(PACS and VLCS) comparing CCNet and CNN as backbones for different algorithms

| PACS | VLCS
---|---|---
Method | P | A | C | S | Avg | V | L | C | S | AVG
FedAvg + CNN | 76.05% | 43.80% | 52.39% | 47.67% | 54.98% | 58.24% | 56.46% | 90.85% | 65.18% | 67.68%
FedAvg + CCNet (ours) | 91.92% | 75.10% | 65.10% | 70.78% | 75.73% | 66.24% | 56.71% | 93.63% | 70.36% | 71.74%
AM + CNN | 69.52% | 57.18% | 64.89% | 74.73% | 66.58% | 53.11% | 60.98% | 78.07% | 60.81% | 63.24%
AM + CCNet (ours) | 90.84% | 74.02% | 66.13% | 66.35% | 74.34% | 64.96% | 54.58% | 93.16% | 64.48% | 69.30%
FedProx + CNN | 73.35% | 35.55% | 53.11% | 28.76% | 47.69% | 56.27% | 59.85% | 85.14% | 61.12% | 65.60%
FedProx + CCNet (ours) | 92.16% | 77.29% | 67.15% | 67.63% | 76.06% | 68.11% | 56.84% | 94.58% | 68.02% | 71.89%
RSC + CNN | 74.79% | 62.60% | 70.05% | 73.96% | 70.35% | 60.51% | 61.10% | 91.27% | 68.02% | 70.23%
RSC + CCNet (ours) | 90.48% | 75.44% | 66.08% | 65.77% | 74.44% | 68.31% | 56.71% | 94.10% | 66.80% | 71.48%
Scaffold + CNN | 74.43% | 38.09% | 56.83% | 29.02% | 49.59% | 57.16% | 60.23% | 88.92% | 61.83% | 67.04%
Scaffold + CCNet (ours) | 91.02% | 75.20% | 66.60% | 66.40% | 74.81% | 65.15% | 57.09% | 93.40% | 64.97% | 70.15%

###  3.2 CCNet

CCNet consists of 4 main modules that are responsible for processing the
hidden representation at a time step t to obtain the representation at the
next time step, $Z_{t+1}$. Firstly, the bottom-up (BU) module predicts the
next level in a particular column from the previous level in the same column
in the previous time step. The BU module is a stack of $L$ MLP networks, where
each two consecutive levels are connected by an MLP network. Formally, the
$l^{t}h$ MLP in the BU module predicts $Z^{l,i}_{t+1}$ from $Z^{l-1,i}_{t}$.
Similarly, we have the Top Down (TD) module, which predicts a lower level from
a high level in a particular column in the previous time step. The TD module
consists of $l-1$ MLP networks, where the $l^{t}h$ network predicts
$Z^{l,i}_{t+1}$ from $Z^{l+1,i}_{t}$. The Identity ($I$) module simply copies
the value of a level in a column to the same level in the next time step.
Lastly, since each column represents a single patch from the image, a simple
Attention module is needed to aggregate spatial information from neighboring
columns. A vector obtains its attention weight from a weighted sum across the
neighboring columns at the same level, $l$. Formally, the attention weight
given at a location $i$ to a vector $Z^{l,j}$ is computed as

$\frac{e^{\beta Z^{l,i}.Z^{l,j}}}{\sum_{k}^{K}e^{\beta Z^{l,i}.Z^{l,k}}}$ (1)

Where $K$ is the set of all neighboring locations that $i$ attends to, $\beta$
is a parameter that acts like an inverse temperature to adjust the attention
sharpness. Finally, contributions from the 4 modules are averaged with
learnable weights to obtain the representation for a particular level vector.

###  3.3 Training

We train CCNet in a supervised manner on the source domains using the cross
entropy classification loss. We perform a single time step update from $t=0$
to $t=1$ as illustrated in Figure LABEL:fig:init to obtain the hierarchical
representation onto which we build the 3 classification heads. Instead of a
random initialization of the feature map at $t=0$, Segment Anything Model
(SAM) is utilized to obtain a hierarchical segmentation of the scene which is
encoded and used as an initialization for the hidden representation. While SAM
doesn’t directly offer a hierarchical segmentation of the scene, the next
subsection will explain the details of the process.

###  3.4 Generating SAM Masks

We capitalize on the fact that SAM can offer multi-level segmentation masks in
its output to disambiguate an ambiguous point prompt. To obtain the initial
hidden representation, $Z_{0}$, We start by dividing the image into several
patches consistent with the discussed Input Convolutional Tokenizer module.
For each patch, we use the middle point as an input prompt for SAM. SAM will
return 3 level segmentation masks where a mask typically represents the whole
object, a mask representing a part of the object, and finally a mask
representing a sub-part of the object. More concretely, for each image in a
training batch of size $B$ we divide the image into $N$ patches, and generate
$N$ prompts points corresponding to the center points of each image patch as
an input to SAM which in response will generate $L$ outputs per point prompt,
each output is a binary mask of dimensions $H\times W$, where $H$, and $W$ are
the input image dimensions. The output from SAM is finally given by the 5D
tensor $M\in\mathbb{R}^{B\times N\times L\times H\times W}$.

###  3.5 Initializing the Feature Maps

We harness a downsized variant of the pre-trained MaxVIT [21] image encoder,
referred to as pico MaxVIT, to generate embeddings for the obtained SAM masks.
The masks $M$ are input to the MaxVIT encoder, producing embedding
representations for $M$, denoted as $M_{e}\in\mathbb{R}^{B\times N\times
L\times D_{e}}$. Here, $D_{e}$ corresponds to the image embedding size that
aligns with the previously mentioned hidden representation size in $Z_{e}$. We
set $Z_{0}=M_{e}$, providing a strong prior on the representations CCNet
should obtain for the image. It is worth noting that this initialization can
be error-prone, and the network is supposed to learn normally from clients
data

###  3.6 Visualization

Following the approach of the original work on Agglomerator, we have carried
out a visual analysis of the islands of agreement across multiple images. The
results indicate that the model has effectively captured meaningful
hierarchical representations of the objects within the scenes. This is evident
in the embedding vectors, which display strong similarities across locations
that correspond to the same object or its components. Figure 4 illustrates
these islands of agreement at various levels. Patches with resembling
embedding vectors are depicted using similar colours. Notably, as we progress
up the hierarchy, towards the rightmost columns, we observe the convergence of
embedding vectors into two distinct ’islands.’ One represents the foreground
elements, and the other corresponds to the background.

##  4 Experiments

Fig. 4: The Figure illustrates the islands of agreements formed at $t=0$ in
CCNet. The right most columns show higher levels in the hirer achy, with the
last column being the input image. The Figure illustrates a similar
hierarchical structure of images belonging to different domains.  Table 2:
Leave-one-domain-out related backbones results on both PACS and VLCS datasets
using FedAvg method

|  | PACS | VLCS
---|---|---|---
Params (M) | Method | P | A | C | S | Avg | V | L | C | S | Avg
3 | Distill Part Whole [11] | 31.5 | 24.46 | 30.84 | 32.45 | 29.81 | 46.2 | 50.82 | 63.68 | 43.55 | 51.06
3 | Agglomerator [10] | 39.34 | 26.17 | 35.03 | 38.71 | 34.81 | 46.3 | 52.45 | 64.15 | 45.69 | 52.15
4 | CCNet (ours) | 91.92 | 75.1 | 65.1 | 70.78 | 75.725 | 66.24 | 56.71 | 93.63 | 70.36 | 71.74

###  4.1 Datasets

We evaluate our approach using two widely recognized benchmarks for domain
generalization.

1) The PACS dataset [22] is specifically curated for domain generalization
tasks. It comprises four distinct domains: Photo, Art Painting, Cartoon, and
Sketch. This dataset consists of 9,991 images distributed across these
domains, with each domain containing seven categories. The dataset’s diversity
stems from the variety of artistic styles represented, rendering it a
challenging benchmark for domain generalization. Figure 3 illustrates the
challenge.

2) The VLCS dataset [23] is commonly employed for assessing domain
generalization methods. It comprises large images categorized into five
classes (bird, car, chair, dog, and person), distributed equally across four
domains (Caltech101 , LabelMe , SUN09 and VOC2007). For each benchmark, we
conduct leave-one-out testing across all four domains. This entails reserving
one domain as the unseen domain in each experiment, training on the remaining
three domains, and evaluating the performance on the unseen domain without any
fine-tuning.

###  4.2 Implementation Details

We locally train the clients for 5 epochs to ensure local convergence,
employing a batch size of 256. Optimization is carried out using the Lion
optimizer [24], with a learning rate of 0.0003 and weight decay of 0.05. The
values of $\beta_{1}$ and $\beta_{2}$ are set to 0.95 and 0.98, respectively,
as recommended by the authors of Lion to mitigate training instability. We
observe that 10 communication rounds between clients are adequate for the
convergence of the global model, and this number is fixed for all experiments
conducted on all datasets. Our optimal variant of CCNet employs 3
classification heads with identical structures, utilizing the same
classification head as in the original Agglomerator.

###  4.3 Experiments and Results

To show the effectiveness of our proposed backbone, we compare it against
MobileNet [25], a widely adopted lightweight CNN architecture with a parameter
count similar to ours. Although our architecture was not pre-trained on
datasets such as ImageNet, it incorporates features from a pre-trained encoder
during the initialization step, as detailed in the methodology. Hence, to
ensure a fair comparison, we selected a MobileNet architecture pre-trained on
ImageNet1k. The backbone comparison is performed on 5 different FedDG
algorithms belonging to different categories. 1) FedAvg [26] is one of the
earliest and well established algorithms in Federated Learning. 2) AM [15] is
a technique that mixes the frequency information from different clients to
increase robustness. 3) FedProx [27] which offers parameterizes the weights
for different models to better handle data heterogeneity. 4) RSC [17] which is
a regularization based technique 5) Scaffold [28] makes use of data similarity
to tackle the client drift. Results are shown in Table 1. Our backbone using
fewer parameters outperforms the CNN backbone on both benchmarks on all the 5
different algorithms. On PACS, the average performance across the four domains
improved by more than 20% compared to the second-best approach, MobileNet.
However, on VLCS, it only led to an average performance improvement of around
4% compared to MobileNet. It is noteworthy that PACS presents more challenging
domain shifts than VLCS, and this is where our proposed approach demonstrates
its merits.

###  4.4 Ablation Study

We conduct an analysis that compares our proposed CCNet, and the
aforementioned related foundational models that attempt to model the scene
parse tree as well as MobileNet. The models are used as backbones for the
FedAvg algorithm that acts as a strong baseline for comparison. Results on
PACS, and VLCS are shown in Table 2.

Interestingly, it is observed that the pre-training approach proposed in [11]
performed worse than using the vanilla Agglomerator without any pre-training.
This contrasts with the superior performance reported for the same pre-
training methodology on a typical supervised classification task. This may
suggest that the pre-training solely teaches the model how to build the parse
tree with embeddings very specific to the source domain, but fails when
transferred to another domain.

We also investigate the impact of varying the number of classification heads
in our CCNet model. Specifically, we tested three configurations: 1) Using a
single classification head on the last level only, following the approach of
Agglomerator and distill part-whole. 2) Building an extra classification head,
one on top of the second level (2 heads). 3) Incorporating one more head on
top of the first level (3 heads). The final classification result is obtained
by averaging the predictions from all the heads. The results are summarized in
Table 3. Interestingly, while the last level in the feature representation
should encapsulate the most semantic representation of the image, we observed
that incorporating classification heads on the parts and sub-parts levels
contributed to better generalization and improved accuracy. This highlights
the significance of recognizing the structural components of the scene in
achieving accurate classification.

##  5 Conclusion

In this paper, we introduce CCNet, a novel approach in the domain of federated
domain generalization. This architecture is specifically designed to model the
part-whole hierarchies inherent in scenes, which are expected to remain
consistent across various domains despite changes in appearance or significant
shifts in domain. Our proposed approach showcases robust generalization
capabilities, surpassing the performance of both the original Agglomerator
model and the distill part-whole pre-training approach without relying on a
pre-training step. Additionally, the model surpasses MobileNet, a widely used
lightweight CNN architecture, while utilizing fewer parameters and offering
the advantage of explainability through the visualization of islands of
agreements. We hope that our work will inspire further research in developing
models that not only exhibit improved generalizability but also provide
enhanced interpretability.

Table 3: Ablation on the number of classification heads on top of the
different levels for our proposed method.

|  | PACS |
---|---|---|---
Params (M) | Method | P | A | C | S | Avg
3 | CCNet (1 head) | 89.64 | 72.31 | 64.33 | 63.37 | 72.4125
3.5 | CCNet (2 heads) | 87.01 | 72.51 | 64.08 | 68.9 | 73.125
4 | CCNet (3 heads) | 91.92 | 75.1 | 65.1 | 70.78 | 75.725

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