NEW SEMIREGULAR VARIABLE STAR NEAR THE WIZARD NEBULA

EVOLUTION OF THE RED GIANT “MACOMP_V1”

G. Conzo1; M. Moriconi1; P.G. Marotta1

${1})$ Gruppo Astrofili Palidoro, Fiumicino, Italy
<EMAIL_ADDRESS>

Abstract: The red giant MaCoMP_V1 in Cepheus at coordinates RA (J2000)
$22:49:05.49$ DEC (J2000) $+57:52:41.6$ is a semiregular variable star
classified as SRS, number 2225960 in the AAVSO VSX database. Using the Fourier
transform, the period $P=(24.751\pm 0.062)$ d was evaluated and, with the
support of the ASAS-SN and ZTF surveys, a well-defined light curve was made.
The analysis resulted in the fundamental physical parameters of MaCoMP_V1,
such as the mass $M=(4.97\pm 0.38)$ M⊙ and radius $R=(40.5\pm 6.7)$ R⊙, with
consistent values suggesting the characteristics of a semiregular red giant.
In addition, the effective temperature $T_{eff}=(4500\pm 135)$ K from the
$Gaia$ catalog and the stellar evolution based on the
$Sch\ddot{o}nberg$-$Chandraskehar$ limit was estimated.
---

##  1 Introduction

Semiregular variables are the most widespread stars in the galaxy and are very
interesting to study because of their evolutionary complexity. After leaving
the Main Sequence, they pass into the Cepheid instability region, transforming
into pulsating variables of the $\delta$ Cephei type.

These stars can be described as semiregular giants and supergiant variables
with spectral classes ranging from F to K, referred to as SRd. If these
variables show high luminosity in the evolutionary process, they fall into the
red supergiant region, becoming SRc type. In contrast, they turn into SRab
semiregular variables with late spectral classes if they show lower
luminosity. Finally, in the classification of semiregular variables, there are
the pulsating low-mass supergiants, which may be in a thermal evolutionary
phase such that they move from red giant to protoplanetary nebula (Kudashkina,
2019).

A month-long photo session was conducted in 2019 on the Wizard Nebula, an
object identified in the open cluster classified NGC 3780 in the New General
Catalog (Dreyer, 1888). It is an emission nebula and is known for its
ionization due to the presence of the DH Cephei binary star system. Therefore,
it has proven to be of great interest in both photographic and photometric
studies, in fact the star field is known to have a high presence of giant
stars (Chavarria-K. _et al._ , 1994).

Many stars were measured using a 60mm Refractor apochromatic telescope and an
ASI-385MM camera. Adopting the Aladin (Centre de Données Astronomiques de
Strasbourg , CDS) software, only red stars were chosen and studied because
these types have a late evolutionary state, and they are interesting to
observe. A star at about 20’ from the Wizard Nebula was found through
photometric measurements, showing possible brightness variations. To confirm
these results, it was necessary to continue observations until the end of 2021
with a better setup and to compare them with automatic surveys using ASAS-SN
(Kochanek _et al._ , 2017) and ZTF (Masci _et al._ , 2019).

At the end of January 2022, the American Association of Variable Star
Observers (AAVSO) approved the discovery of 2MASS J22490550 +5752417 at
coordinates RA (J2000) $22:49:05.49$ DEC (J2000) $+57:52:41.6$, and named it
MaCoMP_V1. This nomenclature derives from the authors: Ma (Mara Moriconi), Co
(Conzo Giuseppe) and MP (Marotta Paolo). Finally, V1 indicates the first
variable star discovered by the team.

##  2 Photometric observations

The collected images were calibrated appropriately, and photometry was carried
out with Maxim DL v6.20 software. The reference and check stars (Check) used
for our purposes are listed in Table 1 and are identified in Figure 1.

Table 1: Reference and Check stars used for the photometry. Type | Name | Coordinates (J2000)
---|---|---
Reference | 2MASS J22491561+5754412 | 22 49 15.62 +57 54 41.2
Check | 2MASS J22485978+5755182 | 22 48 59.78 +57 55 18.2
Figure 1: Star field with the target, reference, and check stars.

The period was obtained with the Fourier Transform, using the Period04
software (Lenz and Breger, 2014), and then the final period was calculated.

$P=24.751\mathrm{d}$ (1)

With this value, it was possible to merge the observed data with those from
ASAS-SN (Kochanek _et al._ , 2017) and ZTF (Masci _et al._ , 2019) surveys. A
luminosity variation in cycles of 24 days 18 hours and 14 minutes identified
in the light curve was obtained, the profile of which shows a typical
semiregular red giant variation of SRS type (Watson, Henden, and Price, 2006).

Figure 2: Light curve of MaCoMP_V1, ZTF Survey (Sloan $R$ and $G$ filters) and
ASAS-SN Survey (Johnson $V$ filter) compared with GAP data (clear).

The plot shows four sets of data centred on HJD epoch $2458726,641$ at values
0, 1, and 2 in the phase domain when the star is brightest. The data observed
by GAP (black dots in Fig. 2) were made in clear using two different
telescopes:

  * •

Apochromatic Refractor Telescope with 600mm focal length and f/7.5 ratio;

  * •

Schmidt Cassegrain Telescope 2000mm focal length and f/10 ratio.

The ZTF survey data (red and green dots in Fig. 2) are splitted into the Red
Sloan filter (Brauneck _et al._ , 2018) and the Green Sloan filter (Brauneck
_et al._ , 2018). The $R$ filter shows an uncertainty $\sigma=0.01$ mag, while
the $G$ filter, with a larger scatter, shows an uncertainty $\sigma=0.02$ mag.
The data from the ASAS-SN Survey (gray dots in Fig. 2) were made in the
Johnson $V$ filter (Bessell, 1990) and have an uncertainty $sigma=0.08$ mag.

Since no filter was used during the photometric sessions, it was necessary to
adjust the magnitudes to the standard Johnson $V$ magnitude, using the average
magnitude of the $Gaia$ catalogue (Riello _et al._ , 2021). The photometric
filters have different passbands than the Johnson-Cousins ones, so switching
to a different photometric system was necessary. This was done by adopting the
Equ. 2 and using the coefficients from the corresponding conversions webpage
(Jordi _et al._ , 2010).

$C_{1}=a+b\cdot C_{2}+c\cdot{C_{2}}^{2}+d\cdot{C_{1}}^{3}$ (2)

With this method, the reference value of stellar magnitude was obtained,
defining it as “zero point” and expressing all magnitudes in $V$, in fact, the
GAP data differ by $1.724$ mag, the red ZTF data differ by $1.297$ mag, and
the green ZTF data differ by $1.069$ mag.

##  3 Data analysis

###  3.1 Fundamental parameters studying

From photometric observations, fundamental physical parameters were estimated,
starting with the relative magnitude and stellar distance. The absolute
magnitude was obtained, using the parallax parameter $RUWE=0.971$ (Gaia
Collaboration, 2020) to adopt a reliable distance from $Gaia$. This resulted
in a parallax $plx=(0.1423\pm 0.0129)$ mas and a photometric magnitude
$V=15.29$ mag, which makes it possible to use the magnitude-distance relation
(Hughes, 2006). We considered the extinction $A_{v}=(3.29\ pm0.16)$ mag
(Teplitz _et al._ , 2018) relative to the direction of the star and the
distance $d=(7027\pm 637)$ pc from the $Gaia$ parallax, to calculate the
absolute magnitude

$M_{v}=m-5log(d)+5-A_{v}=(-2.24\pm 0.48)\mathrm{mag}$ (3)

The brightness of MaCoMP_V1 can be estimated, from absolute magnitude and
$Gaia$ distance, using the brightness-magnitude relationship, like a standard
candle at $10$ pc (Unsöld, 2013).

$L=10^{\frac{M_{v}-M_{\odot}}{-2.5}}=(609\pm 185)\mathrm{L_{\odot}}$ (4)

where $M_{v}$ is the absolute magnitude obtained in relation (3) and
$M_{\odot}$ is the absolute magnitude of Sun. Therefore, using
$T_{eff}=(4500\pm 135)$ K from $Gaia$ EDR3 (Gaia Collaboration, 2020) and with
Stefan-Boltzmann law (Narimanov and Smolyaninov, 2011), the star radius

$R=\sqrt{\frac{L}{4\pi\sigma T_{eff}^{4}}}=(40.57\pm 6.67)\mathrm{R_{\odot}}$
(5)

is obtained. Finally, using the luminosity and the temperature, the mass was
calculated (Eker _et al._ , 2015) as

$M=L^{1/4}=T_{eff}\sqrt{R}=(4.96\pm 0.38)\mathrm{M_{\odot}}.$ (6)

This confirms the correctness of the photometry in relation to the temperature
suggested by $Gaia$, but it should be noted that the estimation of the
parameters is done analytically. To achieve higher accuracy and veracity,
spectrometric observations leading to an estimate of actual temperature and
metallicity would be required.

###  3.2 MaCoMP_V1 evolution based on Schönberg-Chandraskehar limit

In the case of stars with masses $M>2$ M⊙ there is a limiting mass value of
the core at which accretion must stop before hydrogen is converted into
helium. Consequently, the main sequence corresponds to the melting of the
initial mass of hydrogen. This can be proved by the virial theorem (Collins,
1978) which defines the surface pressure term which must be less than the
maximum value for equilibrium conditions to exist

$P_{sup,max}\propto\frac{T_{c}^{4}}{\mu_{c}^{4}M_{c}^{2}}$ (7)

where c refers to the stellar core. Stability condition can be written as
average weight per particle between the surrounding envelope and the core.

$\frac{M_{c}}{M}\leq 0.37\left(\frac{\mu_{e}}{\mu_{c}}\right)^{2}$ (8)

where $M$ is the total mass, $\mu_{c}$ and $\mu_{e}$ are the average weights
for particles in nucleus and envelope which are $\mu_{c}=1.3$ and
$\mu_{e}=0.6$, respectively. This results in $\simeq 0.10M$ critical mass in
solar units.

This procedure is known as the $Sch\ddot{o}nberg$-$Chandraskehar$ limit (SC)
(Ziółkowski and Zdziarski, 2020) and it studies the collapse of the nucleus on
a Kelvin-Helmholtz ($\simeq$ 30 Myr) time scale (MacDonald and Mullan, 2003).
The layers outside the nucleus increase the star size by $\simeq$ 10,
transforming it into a red giant.

In the case of MaCoMP_V1, to interpret the relation (8) it is necessary to
find the value of the core mass, starting from the star brightness

$L\simeq 200L_{\odot}\left(\frac{M_{core}}{0.3M_{\odot}}\right)^{7.6}$ (9)

Reversing the relationship (9), we get $M_{c}=(0.35\pm 0.03)M_{\odot}$, so by
evaluating the mass ratio, it is possible to define whether the stars core is
close to instability.

$\frac{M_{c}}{M}=0.070\pm 0.008$ (10)
$0.37\left(\frac{\mu_{e}}{\mu_{c}}\right)^{2}=0.079\pm 0.007$ (11)

The result of the relation (10) is smaller than the result of the relation
(11), then MaCoMP_V1 meets the mass limit value defined in the law (8),
ensuring the stability of the stellar core. MaCoMP_V1, as a red giant star, is
very bright and, according to stellar evolution, has completed hydrogen-
burning but has not yet collapsed. It has not finished nuclear reactions
within the core, and helium burning has begun (Ziółkowski and Zdziarski,
2020).

##  4 Results

Observations conducted near the Wizard Nebula between 2019 and 2021 revealed a
suspected change in the star’s brightness. Photometry sessions led to define
that the star 2MASS J22490550 +5752417 at coordinates RA (J2000) $22:49:05.49$
DEC (J2000) $+57:52:41.6$ is an SR-type variable (Kudashkina, 2019). After
data collection, it was approved and registered under VSX code 2225960 in the
Variable Star Database (Watson, Henden, and Price, 2022).

MaCoMP_V1 was shown to be a semiregular red giant with brightness variation
over the period $P=24.751$ d, describing typical pulsating stars trend shown
in Figure 2. The variability, the star stability and, according to the SC
limit, core stability was analysed. Data analysis led to the physical
characteristics defined in Table 2.

Table 2: Fundamental parameters of MaCoMP_V1 Name | MaCoMP_V1
---|---
Variability type | SRS
Period |  24.751 $\pm$ 0.062 d
Distance |  7027 $\pm$ 637 pc
Absolute Magnitude |  -2.24 $\pm$ 0.48 mag
Luminosity |  609 $\pm$ 185 L⊙
Temperature |  4500 $\pm$ 1385 K
Radius |  40.57 $\pm$ 6.67 R⊙
Mass |  4.69 $\pm$ 0.38 M⊙

Evolutionary tracks are defined by the physical processes that take place
inside stars. Objects with different masses will have different evolutionary
paths (Kippenhahn, 1994). MaCoMP_V1 has more than two solar masses, so it
should have left the Main Sequence and should arrive in the Red Giant branch.
The found parameters allow to place the star in the diagram shown in Figure 3.

Figure 3: MaCoMP_V1 (red circle) in stellar evolutionary tracks diagram
(Lithopsian, 2012).

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On the Idempotent Conjecture
for Sidki Doubles
Indira Chatterji
LJAD, UMR 7351 CNRS, Université de Nice
Guido Mislin
Department of Mathematics ETHZ, Zürich

To Ruth Charney

Let $G$ be a group and $X(G)$ its Sidki Double. The idempotent conjecture says that there should be no non-trivial idempotent in the complex group ring of a torsion-free group. We investigate this conjecture for the Sidki double of a torsion-free group, and obtain a partial result.

§ INTRODUCTION

Let $G$ be a group and $\psi: G\to G^\psi$ an isomorphism
between (disjoint) groups.
Write $g^\psi$ for $\psi(g)$ and denote by $[x,y]$ for the commutator
$x^{-1}y^{-1}xy$ of two group elements $x$ and $y$.
The Sidki Double $X(G)$ of $G$
is the group defined by the following presentation:

$$ X(G)=\left<G, G^\psi\,|\, [g,g^\psi]\,, g\in G\right>\,.$$

From the definition we see that $G\mapsto X(G)$ is a functor
from groups to groups. Also, $X(G)$ maps onto $G\times G$ and is a
factor group of the free product
$G*G$ :

$$G*G \longrightarrow X(G)\longrightarrow G\times G\,.$$

It follows that we can view $G$ as well as $G^\psi$ as subgroups of $X(G)$.
We then write $g,g^\psi\in X(G)$ for $g\in G$, if no confusion is possible.
As elements of $X(G)$, $g$ and $g^\psi$ commute but are distinct, if $g\neq e$.
The doubled group $X(G)$ inherits many properties from the original group $G$.
For instance, if $G$ is solvable then
$X(G)$ is solvable too (cf. Sidki <cit.>, Corollary 4.8). More generally, it is straightforward to see (cf. Section <ref>) that for $G$ amenable, then $X(G)$ is amenable as well. In this note, we investigate the stability of various idempotent conjectures when taking Sidki doubles.

Recall that an idempotent in a unital ring is called trivial if it is equal to $0$ or $1$. In case where the group $G$ has an element $g\in G$ of $n$-torsion, one checks that $p=\frac{1}{n}(1+g+\cdots+g^{n-1})$ is an idempotent and
Kaplansky conjectured that for a torsion-free group $G$ and $\K$ any field,
the only idempotent of $\K G$ are the trivial ones (cf. <cit.>). Two other famous conjectures due to Kaplansky and others (concerning torsion-free groups) are
the Unit Conjecture, stating that all units of $\K G$
are of the form $\lambda g$ with $\lambda\in\K$ and $g\in G$,
and the Zero-Divisor Conjecture, stating that $\K G$ is a domain. Both of these conjectures imply the Idempotent Conjecture.  For
the Unit Conjecture over fields of finite characteristic, there are now counterexamples in characteristic 2 according to Gardam <cit.> and in general
prime characteristic according to Murray <cit.>.
In the sequel, we will mainly deal with the case of $\K=\C$, the field of complex numbers.

We are interested in proving that $\C X(G)$ has no non-trivial idempotent in case where $\C G$ has none. Of course, a necessary condition for that to hold is that $X(G)$ is torsion free, and according to <cit.> this is in general not the case as for instance $X({\bf Z}^n)$ has 2-torsion as soon as $n\geq 3$. We shall see more examples in Proposition  <ref>, but we don't know for instance whether $X({\bf F}_2)$ is torsion-free for ${\bf F}_2$ the free group on two generators. Our main theorem is the following.

Let $G$ be a group and assume that its Sidki double $X(G)$ is torsion-free.
If $\C (G\times G\times G)$ has no non-trivial idempotent, then
$\C X(G)$ does  not contain any idempotent other that $0$ and $1$.

It is not true in general that if $\C G$ satisfies the idempotent conjecture, then so does its Sidki double $X(G)$, since the groups in Proposition <ref> can be chosen to satisfy the idempotent conjecture.
There are
many classes of groups $G$ for which it is known that $\C(G\times G\times G)$
has no non-trivial idempotent, for instance if $G$ is a torsion-free a-T-menable
group. This follows from the fact that such groups satisfy the Bass Conjecture over ${\C}$,
cf. <cit.>.

The proof of the Main Theorem relies on Kaplansky's Theorem, which states that for any group $H$ and idempotent $x=x^2=\sum x_h\cdot h \in \C H$, then

$$x\in \{0,1\} \Longleftrightarrow x_e\in\{0,1\}\,. $$

We will refer to the coefficient $x_e$ of $x$ as the Kaplansky Trace of $x$
and denote it by $\kappa(x)$.
One can define $\kappa$ on arbitrary
elements of $\C H$ by putting

$$\kappa(a)=a_e\,,\quad \text{for any} \quad a=\sum a_h\cdot h\in \C H\,.$$

The function $\kappa: \C H\to\C$ is a trace, meaning that it
is $\C$-linear and satisfies

for all $x,y\in\C H$.
We will also make use of another trace function
on $\C H$, the augmentation trace  $\epsilon:\C H\to\C$
defined by

$$ \epsilon(\sum a_h\cdot h)=\sum a_h\,.$$

Note that $\epsilon (xy)=\epsilon(yx)$ and $\epsilon$ is $\C$-linear and moreover $\epsilon$ is multiplicative: $\epsilon(ab)=\epsilon(a)\epsilon(b)$, making it a ring homomorphism. Therefore, if $x=x^2\in\C H$, then $\epsilon(x)$ is an idempotent
in $\C$, thus $\epsilon(x)\in \{0,1\}$. One concludes that if an idempotent $x$
in $\C H$ satisfies $\kappa(x)=\epsilon(x)$ then $\kappa(x)\in\{0,1\}$
and therefore, by Kaplansky's Theorem, $x$ equals $0$ or $1$. That $\kappa(x)=\epsilon(x)$ when those traces are extended to matrices is the content of the Weak Bass Trace Conjecture. Indeed, the definition of these traces can be extended to matrices as follows. Let
$A=(a_{ij})\in M_n(\C H)$ with $a_{ij}=\sum_{x\in H}a_{ij}(h)\cdot h$, and $a_{ij}(h)\in\C$ for any $h\in H$. We then set

$$\kappa(A)=\sum_{1\le i\le n}a_{ii}(e)\,,\quad \epsilon(A)=\sum_{h\in H}
(\sum_{1\le i\le n}a_{ii}(h))\,.$$

According to Zaleskii <cit.>, for any idempotent matrix $A$, $\kappa(A)$ is a rational number
$\ge 0\,,$ and $\epsilon(A)$ is an integer satisfying
$0\leq \epsilon(A)\leq n$, because $\epsilon(A)$ is the
trace of an idempotent matrix in $M_n(\C)$.

Bass considered in <cit.> the Hattori-Stallings Trace,
defined for an idempotent matrix (or a finitely generated projective $\C H$-module). It is a
class function $H\to \C $ and gives rise to trace functions on idempotent
matrices as follows. If $A=A^2\in M_n(\C H)$ as above, one gets a finitely generated
projective (left) $\C H$-module $P:=(\C H)^n\cdot A$ and for $x \in H$ the
Hattori-Stallings Trace of $A$ is given by

$$r_A(x):= \sum_{y\in [x], 1\le i\le n}a_{ii}(y).$$

Here $[x]$ stands for the conjugacy class of $x$.
The Bass Trace Conjecture for $\C H$ is the statement that $r_A(x)=0$
for $x\in H$ of infinite order.

The Weak Bass Trace Conjecture asserts that for a torsion-free
group $H$ and idempotent matrix $A\in M_n(\C H)$
one has $\kappa(A)=\epsilon(A)$, or equivalently, $\sum_{[x], x \neq e } r_A(x)=0$. This can
thus be viewed as a generalization of the Idempotent Conjecture for $\C H$, which is
the case of $(1\times 1)$-matrices.

To prove the Main Theorem we establish
that under the conditions stated, idempotent matrices $A\in M_n(\C X(G))$
satisfy $\kappa(A)=\epsilon(A)$. So in fact we more generally show that if $\C (G\times G\times G)$ satisfies weak Bass trace conjecture, then so does $\C H$ for any torsion-free subgroup $H$ of  $X(G)$, see Proposition <ref>.

In Section 1  we will review some basic properties of Sidki Doubles.
In Section 2 we prove the Main Theorem and in Section 3 we construct
examples of Sidki Doubles to illustrate some of their features.

In <cit.> Rocco defined a group $\mathcal{V}(G)$, which has features similar
to $X(G)$, as follows (for group elements $x,y$ we write $x^y$ for the conjugate $y^{-1}xy$)

$$\mathcal{V}(G)=\left< G, G^\psi\,|\,[g, h^\psi]^{k^\epsilon}=[g^k,(h^k)^\psi],\,g, h, k\in G ,\epsilon\in\{1,\psi\}\right>.$$

From the definition, one can see that $\mathcal{V}(G)$ fits in between
the free product and the cartesian product of $G$ with itself:

$$ G\ast G\longrightarrow\!\!\!\!\to\mathcal{V}(G)
\longrightarrow\!\!\!\!\to G\times G\,,$$

both maps being surjective. According to Rocco <cit.>*Section 2 the group $\mathcal{V}(G)$ maps onto a quotient of $X(G)$, with central kernel $\Delta(G)<\mathcal{V}(G)$, the subgroup $\Delta(G)$
being generated by all $[g,g^\psi]$ with $g\in G$. More precisely, there is a stem-extension (those are discussed in Section <ref>)

$$\Delta(G)\to \mathcal{V}(G)\to X(G)/R(G)\,,$$

with $R(G)=[G,L(G),G^\psi]<X(G)$ the subgroup generated by triple commutators, where $L(G)=\langle g^{-1}g^\psi| g\in G\rangle<X(G)$.

Our methods could also be applied to the study of idempotent elements in
the group ring $\mathbb{C}\, \mathcal{V}(G)$.

We thank the referee for valuable suggestions and for pointing out the reference
We also thank Martin Bridson for interesting conversations, this paper has been written inspired by his lecture.

§ HATTORI-STALLINGS TRACE ON SIDKI DOUBLES

We will use the notation introduced by Kochloukova and Sidki
in <cit.>. For the Sidki Double $X(G)$, as defined in the Introduction,
there is a natural homomorphism

$$\rho: X(G)\longrightarrow G\times G\times G$$

defined on generators by $\rho(g)=(g,g,1)$ and $\rho(g^\psi)=(1,g,g)$ and whose kernel is denoted by $\ker(\rho)=W(G)$, which is a normal subgroup in $X(G)$, abelian according to Sidki in <cit.>. In this section we prove that the Hattori-Stallings trace for an integral or complex group ring of a Sidki double $X(G)$ is always trivial on the elements of infinite order from the subgroup $W(G)<X(G)$.

Let $G$ be a finitely generated group, $R=\Z$ or $\C$ and $A\in M_n(R X(G))$ an
idempotent matrix. If $y\in W(G) < X(G)$ is any element of infinite order, then

To prove this proposition, we first study the conjugation action of $X(G)$ on $W(G)$.

The conjugation action of $X(G)$ on $W(G)$ factors through an
abelian group $A(G)$, which is finitely generated in case $G$ is finitely generated.

Let $\mu: G\times G\times G \to G, (x,y,z)\mapsto y$
denote the middle projection. Then we have
a short exact sequence

$$\operatorname{ker}{(\mu\rho)} =:L(G)\longrightarrow X(G)\stackrel{\mu\rho}\longrightarrow G\,.

The subgroup $L(G)$ is generated by elements of the form $g^{-1}g^\psi\in X(G)$, where $g\in G$.
Similarly, with

$$\omega: G\times  G\times G\to G\times G, (x,y,z)\mapsto (x,z)$$

the left-rightprojection and writing $D(G)=\ker(\omega\rho)$ for the
kernel of $\omega\rho$, one can check that

$$W(G)=D(G)\cap L(G)\lhd X(G)\,.$$

A crucial property we will use is the fact that $L(G)$ centralizes $D(G)$
(cf. Lemma 4.1.6 (ii) of <cit.>). Therefore, $W(G)$ is central in $L(G)D(G)$:

$$W(G)=L(G)\cap D(G)\xrightarrow{\text{central}}L(G)D(G)
\xrightarrow{\text{normal}} X(G)\,.$$

We first claim that the commutator subgroup $G'$ of $G$ satisfies

$$G'\times G'\times G'<\rho(L(G)D(G))\,.$$

This can be seen from the following equations in $G\times G\times G$:

\begin{eqnarray*}([u,v],e,e)& =&(u^{-1},e,u)(v^{-1},e,v))(uv,e,(uv)^{-1})\\
&=&\rho(u^{-1}u^\psi\cdot v^{-1}v^\psi\cdot uv((uv)^{-1})^\psi)\in \rho(L(G))\,,\end{eqnarray*}

using the fact that $L(G)$ is generated by elements of the form
$g^{-1}g^\psi$, $g\in G$. Similarly $(e,e,[u,v])\in \rho(L(G))$. Because $D(G)$  is generated
by elements of the form $[x,y^\psi]$ with $x,y\in G$,

\begin{eqnarray*}\rho([x,y^\psi])&=&\rho(x^{-1}(y^\psi)^{-1}xy^\psi)\\

we see that  $(e,[x,y],e)\in \rho(D(G))$. One concludes that $\rho(L(G)D(G))$
contains the commutator subgroup of $G\times G\times G$, finishing our first claim.
Now, since $W(G)$ is central in $L(G)D(G)$, the conjugation action of $X(G)$ on $W(G)$ factors through

$$X(G)/L(G)D(G)\cong \rho(X(G))/\rho(L(G)D(G))=:A(G)\,,$$

which is an abelian group, as
$G'\times G'\times G' <\rho(L(G)D(G))$.
If $G$ is finitely
generated, then so is $X(G)$ and its factor group $A(G)$.

We can turn to the proof of Proposition <ref>, which is now a direct consequence of Bass' theorem <cit.>*Theorem 8.1, (c).

Take $y\in W(G)$ of
infinite order.  We want to show that $r_A(y)=0$. Because of Bass' Theorem
<cit.>*Theorem 8.1, (c) applied to the
finitely generated projective $R X(G)$-module $P=(R X(G))^n\cdot A$,
it suffices to show that
there can be only finitely many primes $p$ such that $y$ is conjugate in
$X(G)$  to $y^{p^n}$ for some positive integer $n$.
Assume that $y$ is conjugate to $y^{p^n}$ and denote the set of such primes by $\Pi$. According to Lemma <ref>, the conjugation  action of $X(G)$ on $W(G)$ factors through an abelian group $A(G)$. Therefore, we can choose for any such prime an automorphism $\phi_p\in A(G)$ for which $\phi_p(y)=y^{p^n}$ and since $y$ has infinite order, the set $\{\phi_p\}$ generates a free abelian subgroup
in $A(G)$ of rank equal to the cardinality of $\Pi$ in $A(G)$. But $A(G)$ is a finitely generated abelian group according to the second part of Lemma <ref>. Hence $y$ can be conjugate to $y^{p^n}$ for finitely many primes $p$ only, which concludes the proof.

§ REDUCTION TO FINITELY GENERATED GROUPS

We first prove some lemmas which imply that in the proof of the Main Theorem
for $X(G)$, it suffices to consider the case of a finitely generated group $G$. First, notice that the map $X(H)\to X(G)$ induced by the inclusion of a subgroup $H<G$, is in general not injective (cf. Example <ref>). However, the following lemma allows us to restrict our study to finitely generated subgroups.

Let $G$ be a group and $A\in M_n(R X(G))$ an idempotent
matrix, where $R$ is a commutative unital ring. Then
there exist a finitely generated subgroup $H<G$ such that $A$
can be lifted to an idempotent $B\in M_n(R X(H))$.

In order to prove this lemma, we first need to see that Sidki doubles behave well under direct limits.

Let $G$ be a group and denote by $\{G_\alpha\}_{\alpha\in I}$ the family
of its finitely generated subgroups. Let $R$ be a commutative unital ring.
Then there is a natural isomorphism

$$\varinjlim_{\alpha\in I} (X(G_\alpha)) \stackrel{\cong}\to X(G),$$

which induces an isomorphism $\varinjlim_{\alpha\in I} (M_n(R X(G_\alpha))) \stackrel{\cong}\to M_n(R X(G))$.

For $G_\alpha <G_\beta$ we write $\iota_{\alpha\beta}$
for the inclusion map $G_\alpha\to G_\beta$. By functoriality, there
are natural (not
necessarily injective) maps

$$X_{\alpha\beta}:=X(\iota_{\alpha\beta}): X(G_{\alpha})\to X(G_\beta)\,,$$

which on generators $x_\alpha,x_\alpha^\psi\in X(G_\alpha)$, $x_\alpha\in G_\alpha$ and $x_\alpha^\psi\in G_\alpha^\psi$, are given by $X_{\alpha\beta}(x_\alpha)= \iota_{\alpha\beta}(x_\alpha)\in X(G_\beta)$
and $X_{\alpha\beta}(x^\psi_\alpha)=\iota_{\alpha\beta}(x_\alpha^\psi)$. The resulting natural map

$$X_* : \varinjlim _{\alpha\in I} X(G_{\alpha}) \longrightarrow X(G)$$

is  surjective, because an element $z$ in $X(G)$ involves only
finitely many generators $x, x^\psi\in X(G)$, and these can be viewed as elements in
some $X(G_\alpha)$, giving rise to an element $z_\alpha\in X(G_\alpha)$ which maps to $z\in X(G)$.
To prove the injectivity, we take $z\in \varinjlim X(G_\alpha)$ which maps to $e$ in $X(G)$.
Write $K_\alpha$ for the kernel of $G_\alpha * G_\alpha \to X(G_\alpha)$. The element $z$ lifts to an element $z_\alpha\in G_\alpha * G_\alpha$. Writing $K$ for the kernel of the natural map $G * G\to X(G)$, we see that $z_\alpha$ maps to $K$ under the inclusion $G_\alpha *G_\alpha \to G * G$. Because $K$ is generated by conjugates of elements of the form $[g,g^\psi]$, we can find $\beta>\alpha$ such that $z_\alpha$ maps already to the kernel $K_\beta$ of
the natural map $G_\beta * G_\beta \to  X(G_\beta)$. We conclude that $z_\alpha$ maps to $e\in X(G_\beta)$. Thus its image  $z\in \varinjlim X(G_\alpha)$ is equal to $e$.

Because taking group rings and matrix rings commutes with direct limits over directed indexing
sets, we also have an isomorphism

$$M_n(R X_*): \varinjlim_{\alpha\in I} M_n(R X(G_\alpha))\longrightarrow
M_n(R X(G))\,.$$

Just as the map $X(H)\to X(G)$ induced by a subgroup $H<G$ is in general not injective, the map $W(H)\to W(G)$ is also in general not injective (also Example <ref>). However for $\{G_\alpha\}_{\alpha\in I}$
the family of finitely generated subgroups of $G$ the natural map

$$\varinjlim _{\alpha\in I}W(G_\alpha)\to W(G)$$

is an isomorphism. Indeed, the functor $\underset{\alpha\in I}{\varinjlim}$ is left exact and the exact sequences

$$\{e\}\to W(G_\alpha)\to X(G_\alpha)\stackrel{\rho_\alpha}\rightarrow
G_\alpha\times G_\alpha \times G_\alpha\,$$

together with Lemma <ref> allows us to conclude.

Let $A^2=A=\{a_{ij}\}\in M_n(R X(G))$ with $a_{ij}=
\sum_{x\in X(G)} a_{ij}(x)\cdot x$. Let
$\{G_\alpha\}_{\alpha\in I}$
be the family of finitely generated subgroups of $G$. Because (cf. Lemma <ref>)

$$\varinjlim_{\alpha\in I} M_n(R  X(G_\alpha))\stackrel{\cong}{\rightarrow}
M_n(R X(G))\,,$$

we can find an index $\beta\in I$ and a matrix $B_\beta \in M_n(R  X(G_\beta))$,
such that $B_\beta$ maps to $A$. If $B_\beta$ is an idempotent, then $B=B_\beta$ proves the lemma. Otherwise $B_\beta$ is not an idempotent, so define
$D=B_\beta^2- B_\beta$. Because $D$ maps to $0\in M_n(R  X(G))$
we can find
an index $\gamma\in I$, with $\gamma\geq\beta$, and such that the natural map

$$\theta_{\beta\gamma}: M_n(R X(G_\beta)) \rightarrow M_n(R X(G_\gamma))$$

yields $\theta_{\beta\gamma}(D)=0$. It follows that $B:=
\theta_{\beta\gamma}(B_\beta)$ is  an idempotent in $M_n(R X(G_\gamma))$
which maps to $A$.

§ PROOF OF THE MAIN THEOREM

We have now all the ingredients to prove our main result.
Note that if $X(G)$ is torsion-free then so is $G$, since $G$ is a retract of $X(G)$.
Because for a torsion-free group, the idempotent conjecture is equivalent to the weak Bass Trace Conjecture for the case of $(1\times 1)$-matrices, the proof of our
Main Theorem is a special case of the proof of the following result
on idempotent $(n\times n)$-matrices.

Let $G$ be a group and $X(G)$ its Sidki double. If $X(G)$ is torsion-free
and $\C(G\times G\times G)$ satisfies the weak Bass Trace Conjecure,
then $\C X(G)$ satisfies the weak Bass Trace Conjecture too.

$A=(a_{ij})\in M_n(\C X(G))$ be an idempotent with $a_{ij}=\sum_{x\in X(G)} a_{ij}(x)\cdot x$. To prove the weak Bass Trace Conjecture for $A$ amounts
to show
that $\epsilon(A)=\kappa(A)$. This is equivalent to show that the sum
$\Sigma:=\sum_{[x]\in [X(G)\setminus \{e\}]}r_A(x)=0$. For
this we write
$\Sigma=U + V$ where

$$U=\sum_{[x]\in\ [X(G)],\, \rho(x)\neq e}r_A(x), \,\quad \text{and}\quad V=\Sigma -U\,.$$

We first show that $U=0$.
Write $B$ for the image of $A$ by the map
$M_n(\C X(G))\to M_n( \C (G\times G\times G))$ induced by $\rho$. The sum $U$ defined above is also given by

$$W = \sum_{[q]\in [G\times G\times G], q\neq e}r_B(q)$$

and this sum equals zero, because $G\times G\times G$ is torsion-free and by assumption $B$
satisfies that $\epsilon(B)=\kappa(B)$. We conclude that $W=U=0$.

Next, we show that $V=0$: The idempotent $A$ can be lifted to an
idempotent $D\in M_n(\C X(H))$ for some finitely generated subgroup $H<G$
(cf. Lemma <ref>). Write $D=(d_{ij})$ with
$d_{ij}=\sum_h d_{ij}(h)\cdot h$, with $h\in X(H)$.
If $v\in X(H)$ satisfies $\rho (v)=e$
and is of infinite order, then
$r_D(v)=0$ by Proposition <ref>.
Because $X(G)$ is assumed to be torsion-free, none of the terms $r_D(v)$ with
$v\in X(H)$ of finite order, can contribute to $V$, because such a $v$ maps to $e$ in the torsion-free group $X(G)$. Thus, $V$ is a sum of terms of the form
$r_D(v)$ with $v$ of infinite order and these are all individually equal to 0.
This shows that $V$ must be equal to $0$.

The following variation of the Main Theorem and the Proposition holds, if we replace the
ground ring $\C$ by $\Z$. The assumption that $X(G)$ is torsion-free
can then be dropped, because for an arbitrary group $H$,
$\Z H$ does not have
any idempotent besides of $0$ and $1$, since an idempotent $x\in \Z H$
obviously satisfies $\kappa(x)=x_e\in \Z$.

Let $G$ be a group and assume that $\Z(G\times G\times G)$ satisfies the weak Bass Trace Conjecture and let $X(G)$ be the Sidki double of $G$. Then
$\Z X(G)$ satisfies the weak Bass Trace Conjecture as well.

We follow the line of proof of the Proposition <ref>. First, we
observe that we can lift an idempotent matrix $A\in M_n(\Z X(G))$
to an idempotent $B\in M_n(\Z X(K))$
for some finitely generated subgroup $K<G$ (cf. <ref>).
As we can see $B$ as a matrix in $M_n(\Z X(K))$ we just need to
observe that elements $x\neq e$ in $X(K)$ of finite order have
$r_B(x)=0$ by Linnell's result <cit.>*Lemma 4.1.
Also, by assumption $G\times G\times G$ satisfy the weak Bass Conjecture
for $\Z (G\times G\times G)$. It follows that the image
of $\rho: X(G) \to G\times G\times G$ satisfies the weak Bass Trace
Conjecture as well. The rest of the proof is then as in the case of the Proposition <ref>.

§ PERFECT GROUPS AND STEM-EXTENSIONS

A concise reference for this section is Kervaire's note
<cit.>, and a more comprehensive treatment can be found in Gruenberg's book
<cit.>*Chapter 9,  9.9. The article by Eckmann-Hilton-Stammbach
<cit.> contains everything on stem-extensions we need here.

A central extension
$C\to H\to Q$ is called a stem-extension, if $C<[H,H]$.

Central extensions ${\mathcal E}:  C\to H\to Q$ are classified by elements in $H^2(Q,C)$.
The 5-term sequence of $\mathcal{E}$ has the form

$$ H_2(H,\Z)\to H_2(Q,\Z)\stackrel{\partial{(\mathcal E)}}\longrightarrow H_1(C,\Z)\to H_1(H,\Z)\to
H_1(Q,\Z)\to \{e\}\,.$$

A group $G$ is called perfect if $G=[G,G]$, or equivalently, $H_1(G,\Z)=\{e\}$. A group $G$ is called super-perfect if it is perfect and moreover $H_2(G,\Z)=\{e\}$.

For $Q$ perfect, the Universal Coefficient Theorem implies that

$$H^2(Q, C)\cong \operatorname{Hom}(H_2(Q,\Z), C)$$

so that the central extension $\mathcal E$ corresponds to a homomorphism $\mathcal E _*: H_2(Q,\Z)\to C\,.$
The homomorphism $\mathcal E_*$ can be identified with $\partial(\mathcal E)$ in the 5-term sequence,
if one identifies $H_1(C,\Z)$ with $C$ .

[Eckmann-Hilton-Stammbach <cit.>*Prop. 4.1] Let $\mathcal E: C\to G\to Q$ be a central extension with $Q$ a perfect group. Then the
following are equivalent:

(1) the classifying homomorphism $\mathcal E_*: H_2(Q, \Z)\to C$ is surjective ;

(2) $\mathcal E$ is a stem-extension;

(3) $G$ is perfect.

Because $Q$ is perfect, the 5-term sequence for $\mathcal E$
ends with

$$H_2(Q,\Z)\stackrel{\partial{(\mathcal E)}}\longrightarrow C \to G/[G,G]\to \{e\}\,.$$

Identifying $\mathcal E_*$ with $\partial(\mathcal E)$ we see that $(1)\Longleftrightarrow (3)$.
If $G$ is perfect, $C<[G,G]=G$, so $(3) \Rightarrow (2)$. Since $Q$ is
assumed to be perfect,
$G=C[G,G]$ and assuming (2), i.e.,  $C<[G,G]$, we have $G=[G,G]$.
Thus $(2)\Rightarrow (3)$.

The universal stem-extension $\mathcal U$ of a perfect group
$Q$ is obtained by choosing
$C=H_2(Q,\Z)$ and for $\mathcal U_*: H_2(Q,\Z)\to C$ the identity homomorphism, yielding

$$\mathcal U:  H_2(Q,\Z)\longrightarrow \tilde{H}\longrightarrow Q\,.$$

It follows from Lemma <ref>,
$(1)\Rightarrow (2)$, that
$\mathcal U$ is indeed a stem-extension, i.e. that $H_2(Q,\Z)$ is contained in ${[\tilde{H},\tilde{H}]}$. Moreover, the universal stem-extension has the following property.

[Kervaire <cit.>*Proposition 1]
Let $Q$ be a perfect group and

$$\mathcal E: C\longrightarrow H \longrightarrow Q$$

a stem-extension. If
$\mathcal U: H_2(Q,\Z)\to \tilde{H}\to Q$ denotes the universal stem-extension,
then $\tilde{H}$ is super-perfect and $\mathcal U$ maps onto $\mathcal E$ to yield a commutative diagram

{ H_2(Q,\Z) \ar@{->>}[d]^{\partial(\mathcal E)}\ar[r]& \tilde{H} \ar@{->>}[d]
\ar[r] &Q\ar@{=}[d]\\
C\ar[r]&H\ar[r] & Q.\\

For $G$ a perfect group, the Sidki Double $X(G)$ is particularly simple to

Let $G$ be a perfect group. Then

$$W(G)\to X(G)\stackrel{\rho}\longrightarrow G\times G\times G$$

is a stem-extension, i.e., $\rho$ is surjective  and
$W(G)$ is a central subgroup of $X(G)$ contained in $[X(G),X(G)]$.

First we check that $\rho: X(G)\to G\times G\times G$ is surjective.
It is known that $\rho$ maps the subgroup $D(G)L(G)<X(G)$ to a subgroup
of $G\times G\times G$ which contains the commutator subgroup of
$G\times G\times G$ (see Proof of Lemma <ref> above).  Because $G$
and therefore also $G\times G\times G$ is perfect, this implies that

$$\rho(L(G)D(G)) = \rho(X(G))=G\times G\times G$$

and in particular $\rho$
is surjective. Next, we check that $W(G)<X(G)$ is central. We know already that
$W(G)$ is central as a subgroup of $L(G)D(G)$, so it suffices to see that
$X(G)=L(G)D(G)$. Because
the kernel of $\rho$ lies in $L(G)D(G)$
it follows that

$$X(G)/L(G)D(G)\simeq \rho(X(G))/\rho(L(G)D(G))=\{e\}$$

and thus $L(G)D(G)=X(G)$. To see that $W(G)<[X(G),X(G)]$, we observe that
as $W(G)<D(G)$ and $D(G)=[G,G^\psi]<[X(G),X(G)]$.

Let $G$ be a super-perfect group. Then

$$X(G)\cong G\times G\times G\,.$$

The assumption on $G$ being perfect implies that $G\times G\times G$ is a perfect group.
According to Proposition <ref>, the extension

$$W(G)\to X(G)\to G\times G\times G$$

is a stem-extension. The associated
universal stem-extension has the form

$$H_2(G\times G\times G,\Z)\longrightarrow \widetilde{X(G)}\longrightarrow G\times G\times G\,.$$

From Lemma <ref> we see that  there is a surjective homomorphism

$$H_2(G\times G\times G,\Z)\longrightarrow W(G)\,.$$

But as $G$ is super-perfect, the $K\ddot{u}nneth$-Formula shows that
$G\times G$ as well as $G\times G\times G$ are super-perfect too. Thus $H_2(G\times G\times G,\Z)=\{e\}$ and therefore $W(G)=\{e\}$.

§ EXAMPLES

General theorems on the structure of Sidki Doubles are difficult to obtain.
However, it is easy to see that the Sidki Double of an amenable group is amenable using the map $\rho: X(G)\to G\times G\times G$ and observing that both the kernel $W(G)$ of $\rho$ and the image $\rho(X(G))$ are
amenable. But, for instance,  we don't know whether for
an a-T-menable group $G$ its Sidki Double is a-T-menable as well.
We give an example of groups $H<G$ for which the induced map of the Sidki doubles $X(H)\to X(G)$
fails to be injective (Example <ref>) and we show that for $G$ of finite
cohomological dimension, $X(G)$ can have infinite rational
cohomological dimension (Example <ref>). We also give an example
of a torsion-free finitely presented group $G$ with Sidki Double $X(G)$ admitting a finitely generated non-free, projective $\Q X(G)$-module for
$\Q$ the rational numbers (Example <ref>).

In Section 2 we proved results on $X(G)$ by passing to $X(H)$ for a finitely generated subgroup $H<G$; the induced map $X(H)\to X(G)$ will not be
injective in general  as the following example shows.

There is a finitely presented group $G$ and a finitely generated subgroup $H<G$
such that the natural maps $X(H)\to X(G)$ and $W(H)\to W(G)$
are not injective.

It was proved in <cit.>*Lemma 9.1 that for $\Gamma=\Z\times \Z$,
there is a natural isomorphism $W(\Gamma)\cong H_2(\Gamma,\Z)=\Z$. Therefore,
by writing $\Q\times\Q$ as a direct limit of groups $\Z\times\Z$
and using Lemma <ref>, we see that $W(\Q\times\Q)\cong\Q$.
One can embed $\Q$
into a finitely presented group $T$ which is simple and has type $FP_\infty$
(cf. Belk, Hyde and Matucci <cit.>*Theorem 3 and 4). Thus $\Q\times\Q<T\times T$ with $T\times T$
perfect and of type $FP_\infty$. By Theorem B of <cit.> it follows that
$W(T\times T)$ is a finitely generated abelian group. We conclude that $W(\Q\times \Q)\cong \Q$
cannot map injectively to $W(T\times T)$.
Now $\Q\times\Q$ is the union
of subgroups $\{S_\alpha, \alpha\in I\}$ with every $S_\alpha$
isomorphic to $\Z\times\Z$. Therefore we can find an  index $\beta\in I$ and
$S_\beta<\Q\times\Q$ such that the map $W(S_\beta)\to W(T\times T)$ induced by $S_\beta\to \Q\times\Q\to T\times T$ is not injective either.  Taking

$$H:=S_\beta<\Q\times\Q<T\times T=:G$$
yields the desired example.

Bridson and Kochloukova asked in <cit.>*Question 5.3, whether
for a finitely generated group $F$ of cohomological dimension one, $X(F)$ has finite cohomological dimension. We give
an example of a group $G$ with cohomological dimension 2
but with $X(G)$ having infinite rational cohomological dimension.

There is a group $G$ of cohomological dimension 2 whose
Sidki Double has infinite rational cohomological dimension.

Take a group $\Sigma^2$ with classifying space $K(\Sigma^2,1)$ a finite 2-dimensional
$CW$-complex with the same homology as the 2-sphere $S^2$ (for the construction of such a group see for instance  Maunder <cit.>).
Put $G=\ast_\N\Sigma^2$ a countably infinite free product of groups $\Sigma^2$.
$G$ has cohomological dimension 2 and $H_2(G,\Z)=\oplus_\N \Z $. According to <cit.> (see also <cit.>), the abelian group $W(G)$ maps onto $H_2(G,\Z)$ and we see that $W(G)$
and hence
$X(G)$ contains a free abelian subgroup of infinite rank. We conclude that
the rational cohomological dimension of $G$ must be infinite.

The final example concerns a group $G$ with torsion-free Sidki Double $X(G)$,
admitting a finitely generated projective $\Q X(G)$-module which is not free. For a group homomorphism $f:K\to L$, we write
$f_*$ for the functor from ${\Q}K$-modules to ${\Q}L$-modules defined by $f_*(M)= {\Q}L\otimes_{{\Q}K}M$, noting that the functor $f_*$ maps projectives to projectives.

Let $G$ be the Baumslag-Solitar group $BS(1,n)$ with $n>1$.
Then $X(G)$ is torsion-free and admits a projective, non-free $\Q X(G)$-module $\overline{P}$
satisfying $\epsilon(\overline{P})=\kappa(\overline{P})=1$.

Levin proved that for $n>1$,  $G=BS(1,n)$ admits a rank 1 projective, non-free
$\Q G$-module $P$, see <cit.>*Example, p. 63.
Also $W(BS(1,n))=0$ for $n>1$, see <cit.>*Lemma 9.1,(3) and
it follows that $X(BS(1,n))$ is torsion-free.
There is an injection $\sigma: G\to X(G)$, with retraction $\tau: X(G)\to G$.
Because $\tau_* \sigma_*(P)=P$ we conclude that $\overline{P}=\sigma_* P=\Q X(G)\otimes_{\Q G} P$ is a projective, non-free
module over $\Q X(G)$. Because $P\oplus \Q G\cong \Q G\oplus \Q G$
(cf. <cit.>), we also have $\sigma_*(P)\oplus \Q X(G)\cong
\Q X(G)\oplus \Q X(G)$ and thus

We now explain
how to construct torsion-free geometrically finite groups $G$ (groups admitting a $K(G,1)$ which is homotopy equivalent to a finite CW-complex), such that $X(G)$
contains non-trivial elements of finite order.

Let $G$ be a perfect group.
If $H_2(G,\Z)$ is a torsion group and contains an element of order $n>1$,
then $X(G)$ contains an element of order $n$ too.

We observe that for a perfect group $G$, the $K\ddot{u}nneth$-Formula
implies that

$$H_2(G\times G\times G,\Z)\cong H_2(G,\Z)\oplus H_2(G,\Z)\oplus H_2(G,\Z)\,.$$

Because $W(G)\to X(G)\to G\times G\times G$ is a stem-extension, we see from Lemma <ref>, that the universal stem-extension of
$G\times G\times G$ yields a surjective homomorphism

$$H_2(G,\Z)\oplus H_2(G,\Z)\oplus H_2(G,\Z)\to W(G)\,.$$

As $H_2(G,\Z)$ is a torsion group, this implies that $W(G)$ is a torsion group too.
On the other hand, it is known
that $W(G)$ maps onto $H_2(G,\Z)$, see for instance <cit.> Lemma 2.2. It follows that if $H_2(G,\Z)$ contains
an element $x$
of order $n$ and $y\in W(G)$ is a counter-image of $x$, then $y$ is a torsion
element and some multiple of $y$ will have order $n$.

We now can construct concrete examples of torsion-free groups with
torsion in their Sidki doubles.

Let $n>1$ be an integer.
There exists a finitely presented torsion-free group $G(n)$ with
classifying space $K(G(n),1)$ a finite $CW$-complex such that
$X(G(n))$ contains an element of order $n$. Moreover, the group $G(n)$ can be chosen so that $\C G(n)$ does not contain any non-trivial idempotent.

According to Leary <cit.>*Theorem A, for any $n\in\N$ there is $G(n)$ a cubical CAT(0)-group with $K(G(n),1)$ a finite CW-complex with the same
homology as the mapping cone $C(\phi)$ of a degree $n$ map $\phi: S^2\to S^2$;
($C(\phi)$ is a Moore space $M(\Z/n\Z,2)$).
Then $G(n)$ is perfect and $H_2(G(n),\Z)\cong \Z/n\Z$.
It follows from Proposition <ref> that $X(G(n))$ contains an element of order $n$.
Because cubical CAT(0)-groups satisfy the Bass Conjecture and $G(n)$ is torsion-free,
$\C G(n)$ does not have any non-trivial idempotent.

We can also follow Maunder <cit.> and take for $G(n)$ a group with $K(G(n),1)$ a finite
CW-complex of dimension three, with the same homology as the mapping cone $C(\phi)$ of a degree $n$ map $\phi: S^2\to S^2$. This would also give a torsion-free group $G(n)$ with torsion in $X(G(n))$, but then we don't know if $\C G(n)$ is known to not have any non-trivial idempotent.

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SparseStep | 4.360 | 41.900 | 0.017 | 0.030 | 0.015 | 84.450 | 0.847 | 0.844 | 0.148 | 0.108
Best Subset | 5.820 | 47.700 | 0.106 | 0.040 | 0.020 | 84.975 | 0.852 | 0.852 | 0.098 | 0.107
SCAD | 24.680 | 66.300 | 1.823 | 0.030 | 0.020 | 86.700 | 0.867 | 0.870 | 0.035 | 0.093
MCP | 11.880 | 61.400 | 0.580 | 0.030 | 0.020 | 86.850 | 0.870 | 0.871 | 0.027 | 0.092
SIS | 6.080 | 39.600 | 0.214 | 0.040 | 0.020 | 81.025 | 0.816 | 0.806 | 0.001 | 0.132
ISIS | 6.970 | 50.000 | 0.199 | 0.020 | 0.010 | 84.150 | 0.843 | 0.844 | 0.020 | 0.117
SIVS | 4.340 | 39.600 | 0.038 | 0.070 | 0.020 | 83.275 | 0.839 | 0.827 | 6.325 | 0.114
varSelRF | 37.540 | 48.400 | 3.303 |  |  | 73.575 | 0.737 | 0.750 | 0.515 | 0.190
PIMP | 83.430 | 48.100 | 7.942 |  |  | 71.775 | 0.728 | 0.723 | 3.479 | 0.207
Boruta | 9.770 | 37.200 | 0.611 |  |  | 75.325 | 0.755 | 0.763 | 0.736 | 0.168
VSURF | 9.730 | 38.300 | 0.596 |  |  | 76.650 | 0.767 | 0.772 | 3.623 | 0.167
RRF | 3.900 | 26.200 | 0.129 |  |  | 75.950 | 0.764 | 0.761 | 0.075 | 0.169
NTA | 81.060 | 51.400 | 7.669 |  |  | 71.425 | 0.720 | 0.735 | 0.544 | 0.210
| Correlated case |
Method | Selected | Imp% | Unimp% | MSE | MAE | Accuracy | Precision | Recall | Time | Brier
LASSO | 54.320 | 68.700 | 4.793 | 0.060 | 0.025 | 82.500 | 0.829 | 0.815 | 0.005 | 0.127
ElasticNet | 57.290 | 61.200 | 5.169 | 0.050 | 0.030 | 78.825 | 0.797 | 0.783 | 0.374 | 0.180
ALASSO | 41.280 | 64.400 | 3.519 | 0.040 | 0.020 | 82.325 | 0.828 | 0.817 | 0.023 | 0.128
SparseStep | 4.380 | 41.900 | 0.019 | 0.030 | 0.010 | 83.925 | 0.847 | 0.832 | 0.022 | 0.113
Best Subset | 5.740 | 48.000 | 0.095 | 0.040 | 0.020 | 84.375 | 0.853 | 0.835 | 0.019 | 0.109
SCAD | 24.580 | 66.200 | 1.814 | 0.030 | 0.020 | 85.650 | 0.862 | 0.852 | 0.020 | 0.103
MCP | 11.760 | 60.000 | 0.582 | 0.040 | 0.020 | 85.400 | 0.861 | 0.848 | 0.014 | 0.102
SIS | 4.990 | 36.300 | 0.137 | 0.050 | 0.020 | 80.925 | 0.818 | 0.800 | 0.003 | 0.138
ISIS | 6.940 | 49.100 | 0.205 | 0.020 | 0.020 | 83.800 | 0.842 | 0.833 | 0.020 | 0.122
SIVS | 4.240 | 39.100 | 0.033 | 0.060 | 0.020 | 83.200 | 0.838 | 0.826 | 6.063 | 0.120
varSelRF | 37.500 | 48.700 | 3.296 |  |  | 74.400 | 0.756 | 0.731 | 0.446 | 0.189
PIMP | 86.970 | 47.800 | 8.302 |  |  | 71.375 | 0.737 | 0.693 | 3.014 | 0.206
Boruta | 11.060 | 36.800 | 0.745 |  |  | 75.025 | 0.755 | 0.743 | 0.908 | 0.173
VSURF | 9.690 | 35.800 | 0.617 |  |  | 74.450 | 0.750 | 0.737 | 5.053 | 0.175
RRF | 4.240 | 26.400 | 0.162 |  |  | 74.475 | 0.754 | 0.737 | 0.226 | 0.176
NTA | 84.070 | 51.800 | 7.969 |  |  | 72.075 | 0.743 | 0.696 | 0.480 | 0.205

Table A9: performance metrics of the sixteen methods in the high dimensional
simulation setting ($n=500,m=1000,p=10$).

| Independent case |
---|---|---
Method | Selected | Imp% | Unimp% | MSE | MAE | Accuracy | Precision | Recall | Time | Brier
LASSO | 98.360 | 91.100 | 9.015 | 0.030 | 0.020 | 88.400 | 0.880 | 0.884 | 0.016 | 0.085
ElasticNet | 79.460 | 77.800 | 7.240 | 0.040 | 0.030 | 86.710 | 0.868 | 0.863 | 0.898 | 0.146
ALASSO | 48.740 | 82.600 | 4.089 | 0.020 | 0.020 | 89.060 | 0.886 | 0.890 | 0.052 | 0.077
SparseStep | 7.310 | 70.400 | 0.027 | 0.010 | 0.010 | 90.840 | 0.904 | 0.908 | 0.228 | 0.067
Best Subset | 8.160 | 74.600 | 0.071 | 0.010 | 0.010 | 90.930 | 0.904 | 0.911 | 0.267 | 0.066
SCAD | 29.380 | 87.100 | 2.088 | 0.010 | 0.010 | 91.000 | 0.904 | 0.911 | 0.036 | 0.066
MCP | 14.950 | 82.800 | 0.674 | 0.010 | 0.010 | 91.230 | 0.909 | 0.911 | 0.053 | 0.064
SIS | 8.080 | 63.900 | 0.171 | 0.020 | 0.010 | 88.850 | 0.884 | 0.887 | 0.002 | 0.078
ISIS | 14.990 | 80.400 | 0.702 | 0.040 | 0.020 | 88.890 | 0.885 | 0.890 | 0.084 | 0.091
SIVS | 4.350 | 42.900 | 0.006 | 0.050 | 0.020 | 84.730 | 0.841 | 0.850 | 14.583 | 0.104
varSelRF | 28.590 | 62.500 | 2.257 |  |  | 81.680 | 0.814 | 0.815 | 1.541 | 0.154
PIMP | 89.880 | 61.300 | 8.460 |  |  | 78.680 | 0.788 | 0.784 | 10.999 | 0.186
Boruta | 8.430 | 51.900 | 0.327 |  |  | 82.440 | 0.821 | 0.824 | 0.859 | 0.132
VSURF | 9.580 | 56.100 | 0.401 |  |  | 82.560 | 0.827 | 0.818 | 8.952 | 0.132
RRF | 4.121 | 32.828 | 0.085 |  |  | 78.737 | 0.780 | 0.792 | 0.226 | 0.149
NTA | 73.660 | 64.800 | 6.786 |  |  | 79.180 | 0.789 | 0.793 | 1.450 | 0.182
| Correlated case |
Method | Selected | Imp% | Unimp% | MSE | MAE | Accuracy | Precision | Recall | Time | Brier
LASSO | 92.090 | 90.700 | 8.386 | 0.040 | 0.020 | 88.280 | 0.877 | 0.886 | 0.015 | 0.084
ElasticNet | 92.870 | 80.200 | 8.571 | 0.040 | 0.030 | 85.910 | 0.859 | 0.856 | 0.899 | 0.144
ALASSO | 47.060 | 83.400 | 3.911 | 0.020 | 0.020 | 89.130 | 0.884 | 0.896 | 0.053 | 0.077
SparseStep | 7.360 | 71.300 | 0.023 | 0.010 | 0.010 | 90.380 | 0.901 | 0.903 | 0.014 | 0.068
Best Subset | 7.790 | 73.500 | 0.044 | 0.010 | 0.010 | 90.730 | 0.903 | 0.909 | 0.029 | 0.066
SCAD | 26.260 | 86.700 | 1.777 | 0.010 | 0.010 | 90.810 | 0.903 | 0.912 | 0.030 | 0.066
MCP | 13.800 | 81.700 | 0.569 | 0.010 | 0.010 | 90.800 | 0.902 | 0.912 | 0.039 | 0.065
SIS | 6.940 | 61.400 | 0.081 | 0.020 | 0.010 | 88.910 | 0.887 | 0.888 | 0.006 | 0.079
ISIS | 14.640 | 80.600 | 0.665 | 0.070 | 0.030 | 88.850 | 0.886 | 0.887 | 0.139 | 0.095
SIVS | 4.360 | 43.400 | 0.002 | 0.050 | 0.020 | 85.210 | 0.846 | 0.856 | 13.534 | 0.104
varSelRF | 30.810 | 61.100 | 2.495 |  |  | 81.460 | 0.813 | 0.814 | 1.441 | 0.151
PIMP | 91.290 | 61.900 | 8.596 |  |  | 79.200 | 0.793 | 0.794 | 10.411 | 0.180
Boruta | 12.050 | 54.800 | 0.664 |  |  | 82.120 | 0.819 | 0.821 | 1.946 | 0.135
VSURF | 9.540 | 55.800 | 0.400 |  |  | 82.460 | 0.819 | 0.829 | 8.680 | 0.131
RRF | 4.260 | 32.000 | 0.107 |  |  | 78.760 | 0.782 | 0.790 | 0.210 | 0.147
NTA | 80.210 | 66.900 | 7.427 |  |  | 79.550 | 0.794 | 0.799 | 1.536 | 0.178

Figure A1: Boxplots of the prediction Brier scores across various iterations
in case of $n=100$ for different dimensions. Figure A2: Boxplots of the
prediction Brier scores across various iterations in case of $n=200$ for
different dimensions. Figure A3: Boxplots of the prediction Brier scores
across various iterations in case of $n=500$ for different dimensions.


# Convergence Rate Maximization for Split Learning-based Control of EMG
Prosthetic Devices

1st Matea Marinova  Faculty of Electrical Engineering and
Information Technologies
Ss. Cyril and Methodius University
Skopje, R. North Macedonia
<EMAIL_ADDRESS>2nd Daniel Denkovski  Faculty of Electrical
Engineering and
Information Technologies
Ss. Cyril and Methodius University
Skopje, R. North Macedonia
<EMAIL_ADDRESS>3rd Hristijan Gjoreski  Faculty of Electrical
Engineering and
Information Technologies
Ss. Cyril and Methodius University
Skopje, R. North Macedonia
<EMAIL_ADDRESS>4th Zoran Hadzi-Velkov  Faculty of Electrical
Engineering and
Information Technologies
Ss. Cyril and Methodius University
Skopje, R. North Macedonia
<EMAIL_ADDRESS>5th Valentin Rakovic  Faculty of Electrical
Engineering and
Information Technologies
Ss. Cyril and Methodius University
Skopje, R. North Macedonia
<EMAIL_ADDRESS>
###### Abstract

Split Learning (SL) is a promising Distributed Learning approach in
electromyography (EMG) based prosthetic control, due to its applicability
within resource-constrained environments. Other learning approaches, such as
Deep Learning and Federated Learning (FL), provide suboptimal solutions, since
prosthetic devices are extremely limited in terms of processing power and
battery life. The viability of implementing SL in such scenarios is caused by
its inherent model partitioning, with clients executing the smaller model
segment. However, selecting an inadequate cut layer hinders the training
process in SL systems. This paper presents an algorithm for optimal cut layer
selection in terms of maximizing the convergence rate of the model. The
performance evaluation demonstrates that the proposed algorithm substantially
accelerates the convergence in an EMG pattern recognition task for improving
prosthetic device control.

###### Index Terms:

Convergence rate maximization, Optimal cut layer, Split learning, Prosthetic
control, Electromyography, Deep learning

##  I Introduction

Centralized machine learning involves transmitting a vast amount of raw data,
which may cause both increased latency and potential network congestion.
Distributed learning techniques, such as Federated Learning and Split
Learning, actively address these issues by conducting distributed training
without transmission of raw data between the clients and the server
[References]. Therefore, these distributed learning approaches are adequate
for scenarios involving resource-constrained and time-varying systems.

FL is a distributed learning paradigm in which clients perform collaborative
training of a global model. Each of the participating clients maintains a copy
of the complete model and trains it using the local dataset. Subsequently, the
central entity receives these local models in order to generate a global one.
This implies that in FL, the server performs low-demanding tasks in terms of
computation, which leads to suboptimal resource utilization, assuming that it
has higher computational capabilities compared to the clients [References].

SL is another distributed learning strategy that has emerged as an
alternative, which successfully resolves the considered challenges faced by
FL. In SL, the model is partitioned into separate client-side and server-side
segments at a layer known as cut layer or split layer. By having the server
process a larger fraction of the model, its resources are employed more
efficiently [References]. Furthermore, the training involves transmission of
the cut layer’ activations and gradients only rather than the entire model
parameters. This fundamental property provides further privacy preservation
compared to FL by restricting the entities’ access only to their designated
sub-networks. As such, SL is a promising approach within domains where the
involved clients require a higher degree of privacy, for example, in
healthcare [References].

Employing SL is also suitable in scenarios featuring clients with limited
computational resources. The wearable devices utilized in both EMG-based
prosthetic control and Human Activity Recognition (HAR) belong to this
category. Specifically, a significant issue caused by the low processing power
of EMG controlled prostheses is their inability to incorporate large models.
This property makes the application of deep learning and FL impractical in
such systems. Moreover, implementing small models does not satisfy the strict
requirements regarding the recognition accuracy. Deployment of SL can
potentially solve the specified problem by assigning only a small model
segment to the prosthetic device.

One of the primary challenges in SL is choosing the cut layer, as this
decision can have a profound impact on the effectiveness of the deployed ML
model. In real-world deployments, system resources are limited and exhibit
frequent variations. Hence, SL must employ dynamic cut layer selection, that
optimizes the learning capabilities of the underlying model, under specific
constraints. This paper does not only demonstrate the viability of
incorporating SL in EMG controlled prosthetic technology, but also develops a
cut layer selection algorithm for maximizing the convergence rate, considering
the limited and time-varying system resources.

The paper is structured as follows. Section II provides related work regarding
deep learning based HAR solutions, mostly focusing on EMG pattern recognition
for improving prosthetic control, and algorithms for cut layer selection in SL
systems. In Section III, a detailed overview of the considered SL system model
is presented. Section IV gives a thorough explanation of the proposed optimal
cut layer selection algorithm. Section V features a performance comparison
between the proposed and the benchmark algorithm in an EMG pattern recognition
task. Section VI concludes the paper.

##  II Related Work

According to [References], EMG pattern recognition based control of prosthetic
devices relies on the hypothesis that EMG patterns contain significant
information about the intended movements. Deep learning techniques can
effectively address some weaknesses of conventional machine learning
approaches in EMG pattern recognition [References] and HAR, such as their
dependence on statistical features, and domain knowledge of the system.
Furthermore, traditional techniques incorporate shallow features unable to
recognize complex activities. Deep learning mitigates these setbacks through
feature learning directly from raw sensors’ data without feature engineering,
and domain knowledge as well [References]. Hence, deep neural networks are
becoming increasingly favored in these domains, especially convolutional
neural networks (CNNs), which are known for their ability to capture complex
features by combining previously learnt simpler ones.

These studies [References], [References], [References] demonstrate that CNN
models exhibit superior performance in both EMG pattern recognition and HAR-
specific tasks compared to statistical machine learning techniques, such as
KNN, SVM, RF, etc. In addition, [References] provides a proof of concept for
implementation of deep learning models on edge devices (e.g. smartphones). In
[References], a comparison of deep neural network architectures is provided,
which shows that CNNs outperform LSTM, biLSTM and DBN models, and their
results are competitive with the scores of GRU networks. Furthermore, the
results reveal that CNN architectures are more adequate for tasks in resource-
constrained environments, regarding both speed and memory requirements. The
prospect of employing deep learning solutions in the field of EMG-based
prosthetic device control implies that SL can be incorporated within the
specified domain as well. Moreover, this research direction has yet to be
explored, since only [References] combines SL and HAR, using large Bi-RNN for
indoor activity recognition.

The effectiveness of SL solutions is intricately connected to the cut layer
decision. Attempts in the existing literature have been made in terms of
creating algorithms for cut layer selection. The prevalent strategy relies on
implementing the brute-force method [References],[References]. The authors in
[References] propose an efficient algorithm for multi-split scenarios, however
its relevance diminishes when considering single-split SL systems, since the
graph transformations and pruning are not applicable. In [References], the
Kalai–Smorodinsky bargaining solution for finding the optimal cut layer is
employed, which is an advanced game theory algorithm, based on assumption
approximations. As such, it is not only complex, but may also result in
suboptimal solutions. In [References], to solve the cut layer selection
problem, the authors implement the branch-and-bound algorithm. However,
bounding techniques possess a critical role in its performance. Additionally,
this algorithm demonstrates high computational complexity when used for
solving large problems.

##  III System Model

The system model assumes a SL setup composed of one server and multiple
heterogeneous clients with limited computation and communication resources,
such as the SL configuration illustrated in fig. 1. EMG controlled prostheses
fall into this category of resource-constrained devices. The learning is
performed in collaborative settings and sequential manner.

Figure 1: Generic split learning architecture composed of N clients, which
contain the first three layers of the neural network; the remaining four
layers are assigned to the server

In SL, prior to the training procedure, the model parameters are initialized
randomly, and the neural network is divided into two segments, with one
segment assigned to the client, while the other segment resides on the server
side. For each batch of samples, the procedure involves Forward Propagation
(FP) and Backpropagation (BP). In that regard, during FP the samples are
processed by the client up to the cut layer, which subsequently sends the
resulting cut layer’s outputs (i.e. smashed data) to the server. Through this
approach, the server can proceed further with the FP process using solely the
smashed data, without requiring access to the client’s raw data. Then, in BP
the gradients are computed with respect to the considered loss function. The
smashed data’s gradients are transmitted to the client, which continues and
terminates the BP process accordingly.

Additionally, weight synchronization is conducted between the clients
participating in the training process. This implies that before the start of
the client’s epoch, it receives the client-side model parameters from the
server, which were updated during the training epoch of the previous client.
It is possible to implement SL without weight synchronization, however this
variant does not guarantee that the model will converge [References].

The entire SL procedure is illustrated in algorithm 1. It is assumed that in
one SL round each client trains on its local dataset for a single epoch,
before the server proceeds to train the next client’s model. In algorithm 1,
an epoch is described with lines 8-14, where $D_{k}$ and $B_{k}$ denote the
client’s dataset size and batch size, respectively.

1 for  _$training$ $round$ $t=1,2,...,T$_ do

2 for  _$client$ $c=1,2,...,N$_ do

3 Divide the neural network into client-side and server-side segments;

4 if  _t=1 and c=1_ then

5 Randomly initialize the parameters;

6 else

7 The server sends an appropriate model segment to client $c$ for weight
synchronization;

8

9 for  _$each$ $batch$ $b=1,2,...,\frac{D_{k}}{B_{k}}$_ do

10 The client performs FP on the client-side model;

11 The client sends the smashed data to the server;

12 The server performs FP on the server-side model;

13 The server performs BP on the server-side model;

14 The server transmits the smashed data’s gradients to the client;

15 The client performs BP on the client-side model;

16

17 if  _$t\neq T$ or $c\neq N$_ then

18 Client $c$ sends its model parameters to the server, enabling weight
synchronization for the subsequent client;

19

20

21

Algorithm 1 Split learning procedure

The system model is built on the premise of wireless transmission over
reciprocal uplink and downlink fading channels. As such, the specified model
considers the inevitable channel state variations inherent to wireless
channels. Furthermore, the fluctuations in the computational resources of the
participating entities are considered as well, since they may occur due to
other applications running in parallel. However, during a single epoch, system
stability is assumed, i.e. the specified system resources do not exhibit any
fluctuations.

This study focuses on selecting the optimal cut layer in order to accelerate
the convergence rate of the model, i.e. to minimize the overall training delay
composed of four main components: i) the computation time of the client
($\tau_{k}$), ii) the computation time of the server ($\tau_{s}$), iii) the
time required for transmitting activations/gradients during FP/BP ($t_{0}$),
and iv) the weight synchronization transmission delay ($t_{p}$). To simplify
the analysis while still preserving its generality, it is assumed that the
forward propagation and backpropagation delays ($\tau_{k}$, $\tau_{s}$,
$t_{0}$) are equal [References],[References],[References]. Thus, for client’s
dataset size $D_{k}$ and batch size $B_{k}$, the training delay per epoch is
given by:

$T=\frac{2D_{k}}{B_{k}}(\tau_{k}+t_{0}+\tau_{s})+2t_{p},\vspace{-1mm}$ (1)

where each of the components depends on the position of the split layer. It
should be noted that at the beginning of the learning process, the first
client does not receive any parameters from the server. However, this
exception applies only to the first training round. Another exception occurs
in the final round, i.e. the last client does not send its model parameters to
the server for weight synchronization.

The device-side and server-side computational loads for one sample can be
defined as:

$\displaystyle L_{k}(i)=\sum_{j=1}^{i}l(j)$ (2a) $\displaystyle
L_{s}(i)=L_{total}-L_{k}(i),$ (2b)

where $i$ is the position of the selected cut layer, and $l(j)$ denotes the
computational load per sample for layer $j$ calculated by multiplying the
number of outputs by the number of FLOPs needed for processing each output.
Therefore, the computation delays of the client and the server are given by:

$\displaystyle\tau_{k}(i)=\frac{L_{k}(i)B_{k}}{f_{k}}$ (3a)
$\displaystyle\tau_{s}(i)=\frac{L_{s}(i)B_{k}}{f_{s}}$ (3b)

where $f_{k}$ and $f_{s}$ are the computing speeds of the client and the
server, respectively, measured in FLOPS (floating-point operations per
second).

Transmitting $N_{k}(i)$ activations/gradients of the split layer $i$ for
$B_{k}$ samples per batch through a wireless channel with transmission rate
$R$, introduces the following transmission delay:

$t_{0}(i)=\frac{N_{k}(i)B_{k}}{R}.$ (4)

The delay induced by the transmission of client-side model parameters between
the server and the client for weight synchronization is:

$t_{p}(i)=\frac{N_{c}(i)}{R},$ (5)

where $N_{c}(i)=\sum_{j=1}^{i}N_{p}(j)$ represents the cumulative function of
the client-side model parameters, and $N_{p}(j)$ denotes the number of
parameters in a given layer $j$.

It is evident that choosing a split layer from the initial layers in the
neural network is computationally more efficient, under the assumption that
the server has superior computing capability. Moreover, these split layer
candidates result in lesser communication overhead caused by the weight
synchronization. Conversely, the gradual activation size reduction in a
convolutional neural network, indicates that opting for a split layer from the
latter part of the network is more beneficial in terms of reducing the
communication overhead stemming from the transmission of activations and
gradients. This signifies that identifying the optimal cut layer in a specific
time frame, is dependent on the interrelation between the available
fluctuating system resources and the architectural aspects of the considered
neural network. Given these factors, the following section outlines the
proposed algorithm for optimal cut layer selection.

##  IV Optimal Cut Layer Selection Algorithm

This section elaborates on the proposed Optimal Cut Layer Selection Algorithm
(OCLA), which consists of two separate phases. The first phase is conducted
offline, while the second one operates in online mode.

A prerequisite for the offline phase of the algorithm is having a thorough
understanding of the neural network architecture, as well as the size of the
dataset that the model will be trained on. In essence, for a given neural
network, the pool of potential cut layers can be determined entirely based on
the profiling functions111The illustrated profiling functions are derived from
the neural network specified in Section V. depicted in fig. 2, fig. 3 and fig.
4. This indicates that the elimination of inadequate candidates from the
selection process is feasible without prior information about the SL system
conditions, i.e. the available system resources. The specified pruning method
is composed of two distinct steps.

Figure 2: Cumulative client-side computational load per layer Figure 3: Per-
layer activation size Figure 4: Cumulative client-side model parameters per
layer

It can be observed from fig. 2 that the client-side computational load is a
cumulative function. As such, it dictates that only layers which satisfy
progressively decreasing communication overhead function of the neural network
are considered as potential candidates in the selection process. This analysis
leads to the formulation of the first pruning step called profile function
pruning, which eliminates the layer on position $i+1$ if it complies with the
following condition:

$N_{k}(i+1)+\frac{N_{c}(i+1)-N_{c}(i)}{D_{k}}\geq N_{k}(i),\ \forall
i\in\\{1,2,...,M-1\\}.$

(6)

###### Proof:

Please refer to Appendix A. ∎

In (6), M denotes the final layer in the considered neural network. This layer
is always excluded from the selection process, since choosing it would imply
that the entire model is assigned to the client, which contradicts the core
definition of SL. This condition indicates that the activation size, depicted
in fig. 3, affects to a greater extent the cut layer decision compared to the
weight synchronization parameters. Moreover, it indicates that as the dataset
size grows, the impact of these parameters (fig. 4) on the cut layer selection
process further diminishes.

The second pruning step is performed using the communication-computation
trade-off function for consecutive layers, which is defined as:

$\displaystyle\Delta(j,j+1)=\frac{N_{k}(j)-N_{k}(j+1)-\frac{N_{c}(j+1)-N_{c}(j)}{D_{k}}}{L_{k}(j+1)-L_{k}(j)}\vspace{+2mm}$
(7a)
$\displaystyle\Delta(j-1,j)=\frac{N_{k}(j-1)-N_{k}(j)-\frac{N_{c}(j)-N_{c}(j-1)}{D_{k}}}{L_{k}(j)-L_{k}(j-1)},$
(7b)

$\forall j\in\\{1,2,...,P\\}$, where P is the last layer in the pool of
potential cut layers after completing the first pruning step.

For proper execution of OCLA, two modifications to the $\Delta$ function are
applied. First, it is assumed that $\Delta(0,1)\rightarrow\infty$. This aligns
with the observation that the first layer is the most efficient candidate
regarding the processing aspect. It is important to consider it as a potential
cut layer, since one of the main advantages of using the SL approach is
exactly its capability to decrease the necessary computation performed by the
resource-constrained clients. The second alteration involves adding a virtual
final layer $P+1$ characterized by activation size $N_{k,v}(P+1)=0$,
computational load $L_{v}(P+1)=0$ and cumulative function of the parameters
$N_{c,v}(P+1)=0$, which leads to $\Delta(P,P+1)<0$.

The second step in the offline phase is performed by pruning the layers that
do not satisfy the following condition:

$\Delta(j-1,j)>\Delta(j,j+1),\ \forall j\in\\{1,2,...,P\\}$ (8)

In fact, condition (8) ensures that the communication-computation trade-off
function is a strictly decreasing one. It should be noted that this step might
require to be executed more than once, until the desired behavior of $\Delta$
is achieved.

An important advantage of the discussed pruning steps is that they can be
applied before using other algorithms for cut layer selection as well.
Therefore, they are capable of decreasing substantially the computational
complexity of the given algorithm, even when employed in conjunction with
computationally inefficient methods such as exhaustive search.

After the pruning is completed, OCLA inspects whether a given layer fulfills
the concluding condition for optimality, which is based on the available
system resources. This condition stems from the fact that the optimal cut
layer $n$ results in lower training delay compared to both of his adjacent
layers $n-1$ and $n+1$:

$\displaystyle T(n)<T(n+1)$ (9a) $\displaystyle T(n)<T(n-1),\forall
n\in\\{1,2,...,K\\}$ (9b)

where $T(n)$ represents the overall delay if $n$ is selected as the cut layer,
and $K$ is the number of elements in the set of potential cut layers after the
discussed pruning steps are performed.

######  Lemma 1.1.

Assume that the communication-computation trade-off between layer $n$ and
layer $n+1$ is known and specified as in eq.(7a), and that eq.(9a) always
holds. If $n$ is the optimal cut layer, then the respective communication-
computation trade-off is bounded by:

$\Delta(n,n+1)<\frac{\beta R}{f_{k}}.\vspace{-1mm}$ (10)

######  Lemma 1.2.

Assume that the communication-computation trade-off between layer $n$ and
layer $n-1$, is known and specified as in eq.(7b) and that eq.(9b) always
holds. If $n$ is the optimal cut layer, then the respective communication
computation trade-off is bounded by:

$\Delta(n-1,n)>\frac{\beta R}{f_{k}}.\vspace{-1mm}$ (11)

###### Proof:

Please refer to Appendix B. ∎

The ultimate criterion, which if satisfied ensures the optimality of the
considered cut layer candidate depending on the available system parameters
$f_{k}$, $f_{s}$ and $R$, can be stated as:

$\Delta(n,n+1)<\frac{\beta R}{f_{k}}<\Delta(n-1,n),\ \forall
n\in\\{1,2,...,K\\}$ (12)

where $a=\frac{f_{s}}{f_{k}}$, and $\beta=\frac{a-1}{a}$ is the relative ratio
between the computing speeds of the server and the client.

Based on the specified optimality condition, the final offline step includes
creating a database which contains the split regions, defined by the
communication-computation trade-off functions for the K remaining potential
cut layers.

A key advantage of OCLA lies in its ability to determine all of the potential
cut layers in the absence of information about the available system resources.
Specifically, OCLA identifies the set of K remaining layers solely based on
the neural network architecture and other input parameters, such as the
dataset size. However, only one layer satisfies the optimality condition (12)
for a given neural network and system parameters, and the subsequent online
phase is dedicated to identifying it.

In essence, the profiling is an offline optimization step performed only once
for a given neural network architecture and dataset size, and the obtained
results are used during the online phase. The online part of OCLA consists of
merely reading the position of the cut layer from the previously generated
database.

##  V Results

The performance of OCLA is analyzed in a SL scenario where the training is
performed sequentially between the server and N=10 heterogeneous clients.
Specifically, OCLA is applied per each client’s epoch, in order to accelerate
the overall convergence rate. A detailed list of the simulation parameters is
provided in Table I. The analysis considers the time-varying inherent property
of the computation and communication resources by modeling the transmission
rate and the server-client computing speed ratios as random variables that
follow folded normal distributions. The performance evaluation of OCLA
features a comparison with a naive algorithm as the baseline approach, which
persistently selects the same cut layer (e.g. the first layer), disregarding
the impact of the neural network architecture in the cut layer selection
process, and the variability of the system resources as well.

In order to achieve statistically accurate results, since the system
parameters are considered to be random variables, a Monte Carlo simulation is
performed as part of the evaluation process. Specifically, the simulation
involves a particular number of iterations (I), within each iteration J
samples are drawn from the folded normal distributions, and subsequently used
for comparing both algorithms in terms of the given performance metrics. In
this study, the comparative analysis is performed by using the performance
gain, and the model’s training accuracy and loss, as key performance metrics.

Our experiments are conducted using the dataset from [References], which
consists of EMG signals collected from 10 subjects with two EMG channels. The
data collection process involved 6 trials, and within each trial the
participants repeated movements classified in 10 different classes, i.e.
flexion of each of the individual fingers, pinching between each finger and
the thumb, and hand close. The collected data is partitioned into training and
test datasets, by allocating five trials of each subject into the training
dataset, and allocating the remaining trial to the test set. After performing
the segmentation, each client’s training dataset contains 9992 samples, and
their respective test sets contain 1992 samples. During model training, the
cross entropy loss and the Adamax optimizer are used. The specific 1D CNN
model architecture employed for obtaining the results is described in
[References], and shown in Table II. It should be noted that that the input
layer cannot be a cut layer in SL, as such its index is considered to be 0.

TABLE I: Simulation Parameters |  Parameter |  Value
---|---|---
|  Number of layers in the neural network |  $M=8$
|  Number of iterations |  $I=1000$
|  Number of samples |  $J=300$
|  Mean value of the computing speeds’ ratio |  $\mathbb{E}[1-\beta]=0.03$
|  Standard deviation of the computing speeds’ ratio |  $\sigma_{1-\beta}=0.0003-0.015$
|  Coefficient of variation of the computing speeds’ ratio |  $(1-\beta)_{cv}=0.01-0.5$
|  Mean value of the transmission rate |  $\mathbb{E}[R]=20$ Mbps
|  Standard deviation of the tranmission rate |  $\sigma_{R}=0.2-10$ Mbps
|  Coefficient of variation of the transmission rate |  $R_{cv}=0.01-0.5$
|  Number of training rounds |  $T=35$
|  Number of epochs per round |  $E=1$
|  Number of clients |  $N=10$
|  Client’s dataset size |  $D_{k}=9992$
|  Batch size |  $B_{k}=100$
TABLE II: Neural Network Architecture Index | Layer Name | Output Size | Activation Function
---|---|---|---
0 | Input layer | 800x2 | None
1 | CONV 1 | 793x200 | ReLU
2 | CONV 2 | 786x200 | ReLU
3 | POOL 1 | 198x200 | None
4 | CONV 3 | 91x200 | ReLU
5 | CONV 4 | 84x200 | ReLU
6 | Global Avg POOL | 1x200 | None
7 | DROPOUT | 1x200 | None
8 | FC | 10 | Softmax

Fig. 5 illustrates the performance gain of OCLA in comparison to the naive
algorithm considering a scenario in which it consistently select layer 3 as a
split layer. The performance gain of a given algorithm depends on the
coefficient of variations $(1-\beta)_{cv}$ and $R_{cv}$ defined as:

$\displaystyle(1-\beta)_{cv}$
$\displaystyle=\frac{\sigma_{1-\beta}}{\mathbb{E}[1-\beta]}$ (13a)
$\displaystyle\vspace{+2mm}R_{cv}$
$\displaystyle=\frac{\sigma_{R}}{\mathbb{E}[R]}.$ (13b)

Therefore, the performance gain metric is determined by:

$gain(R_{cv},(1-\beta)_{cv})=\frac{A_{OCLA}(R_{cv},(1-\beta)_{cv})}{A_{naive}(R_{cv},(1-\beta)_{cv})},$
(14)

where $A_{OCLA}$ and $A_{naive}$ denote the optimal cut layer selection rates
of OCLA and the naive algorithm, respectively. The optimal cut layer selection
rate is specified as:

$A=\frac{\sum_{i=1}^{P}I_{A}(y_{i}=\hat{y_{i}})}{P},$ (15)

where $y_{i}$ and $\hat{y_{i}}$ represent the optimal cut layer and the cut
layer selected by the considered algorithm; $I_{A}(\cdot)$ is an indicator
function; $P$ denotes the total number of predictions.

It is evident that for lower values of the coefficients of variations defined
by eq.(13), the performance gain of OCLA is lower. However, the gain
substantially increases if at least one of the coefficients increase, or both.
Specifically, for lower $(1-\beta)_{cv}$ and $R_{cv}$, the considered range of
values for the system parameters is too small, and focused on the split region
of layer 3. As such, they most frequently lead to optimality of cut layer 3,
causing a higher accuracy of the baseline algorithm. Conversely, by widening
the range of values of the observed system parameters, more layers emerge as
optimal cut layers, i.e. more split regions are included, which contributes to
a decrease in accuracy of the baseline algorithm. This property holds
significant importance, since non-deterministic time-varying parameters
exhibit higher coefficients of variation within practical real-world
environments.

Figure 5: Performance gain of OCLA regarding the coefficients of variations of
the transmission rate $R_{cv}$ and the computing speeds’ ratio
$(1-\beta)_{cv}$; the baseline algorithm consistently selects layer 3

The results illustrated in fig. 6, clearly show the importance of identifying
a fitting cut layer in terms of the optimization problem at hand. As
demonstrated, OCLA converges much faster than the baseline algorithm.
Specifically, the considerably different convergence rates when applying OCLA
and the given benchmarks, are due to the different cut layer decisions that
each of the algorithms establishes. As such, they result in dissimilar
training delays, despite having the same hyperparameter configuration. It is
evident that the proposed algorithm exhibits the highest convergence rate,
since it consistently and successfully determines the optimal split layer.
Conversely, fig. 6 shows that each benchmark severely hinders the learning
process by introducing high training delay caused by the frequently made
suboptimal cut layer decisions.

Figure 6: Comparison between OCLA and the baseline algorithm selecting various
cut layers in terms of the training loss vs. training time

In addition, the results depicted in fig. 7 demonstrate that using OCLA also
provides higher training accuracy and drastically accelerates the convergence
of the model.

Figure 7: Comparison between OCLA and the baseline algorithm selecting various
cut layers in terms of training accuracy vs. training time

As indicated by these findings, merely configuring adequate hyperparameters
does not lead to the desired model performance in practical scenarios that
involve time-varying and resource-limited systems. As such, optimal
performance can only be achieved through considering the interrelation between
the given optimization problem and the critical system properties, i.e. time-
variability and resource limitations.

##  VI Conclusion

This study suggests that split learning-based control of EMG prosthetics is a
viable and promising option. However, the cut layer decision in such resource-
constrained and time-varying SL systems profoundly impacts many system-related
aspects, including the model’s convergence rate. Hence, the paper develops an
algorithm for identifying the position of the optimal cut layer with regard to
convergence rate maximization, depending on the available system resources and
the neural network architecture at hand. While the optimal cut layer
drastically accelerates the learning, choosing a suboptimal one may lead to
severe consequences regarding the convergence rate of the model. This
conclusion implies that for ensuring the effectiveness of integrating SL in
EMG-driven prosthetic control, the system resources and the deep learning
model architecture must be considered.

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## Appendix A

Within this appendix we clarify the reasoning behind the profile function
pruning step.

The total communication overhead incorporates the transmission of
activations/gradients and the parameters for weight synchronization. As such,
when selecting layer $n$ as cut layer, the total communication overhead is
given by $\frac{2D_{k}}{B_{k}}N_{k}(n)B_{k}+2N_{c}(n)$, where
$\frac{D_{k}}{B_{k}}$ denotes the number of batches in the epoch, and
$N_{c}(n)$ is the cumulative function of the parameters.

Considering that setting layer $n+1$ leads to higher computation delay, in
order for it to remain in the pool of potential cut layers, layer $n+1$ needs
to have lower transmission delay compared to layer $n$. Accordingly, it needs
to have lower communication overhead as well. This leads to the conclusion
that layer $n+1$ will be eliminated from the selection process if the
following condition is satisfied:

$\displaystyle\begin{array}[]{ll}\frac{2D_{k}}{B_{k}}N_{k}(n+1)B_{k}+2N_{c}(n+1)\geq\frac{2D_{k}}{B_{k}}N_{k}(n)B_{k}+2N_{c}(n)\vspace{+2mm}\\\
N_{k}(n+1)D_{k}+N_{c}(n+1)\geq N_{k}(n)D_{k}+N_{c}(n),\end{array}$ (17)

which results in the concluding criterion for profile function pruning as
specified in (6):

$N_{k}(n+1)+\frac{N_{c}(n+1)-N_{c}(n)}{D_{k}}\geq N_{k}(n)$

## Appendix B

This appendix presents the derivations of Lemma 10 and Lemma 11.

## Appendix B.1

## Proof of Lemma 10

The optimal cut layer $n$ must comply with condition (9a). Following that,
some elementary transformations are applied according to eq.(1)-eq.(5):

$\displaystyle T(n+1)>T(n)$
$\displaystyle\frac{2D_{k}}{B_{k}}[\tau_{k}(n+1)+t_{0}(n+1)+\tau_{s}(n+1)]+2t_{p}(n+1)$
$\displaystyle>\frac{2D_{k}}{B_{k}}[\tau_{k}(n)+t_{0}(n)+\tau_{s}(n)]+2t_{p}(n)$
$\displaystyle\frac{2D_{k}}{B_{k}}[\frac{L_{k}(n+1)B_{k}}{f_{k}}+\frac{N_{k}(n+1)B_{k}}{R}+\frac{L_{s}(n+1)B_{k}}{f_{s}}]+\frac{2Nc(n+1)}{R}$
$\displaystyle>\frac{2D_{k}}{B_{k}}[\frac{L_{k}(n)B_{k}}{f_{k}}+\frac{N_{k}(n)B_{k}}{R}+\frac{L_{s}(n)B_{k}}{f_{s}}]+\frac{2N_{c}(n)}{R}$
$\displaystyle
D_{k}[\frac{L_{k}(n+1)}{f_{k}}+\frac{N_{k}(n+1)}{R}+\frac{L_{s}(n+1)}{f_{s}}]+\frac{N_{c}(n+1)}{R}$
$\displaystyle>D_{k}[\frac{L_{k}(n)}{f_{k}}+\frac{N_{k}(n)}{R}+\frac{L_{s}(n)}{f_{s}}]+\frac{N_{c}(n)}{R}$
$\displaystyle
D_{k}[\frac{L_{k}(n+1)-L_{k}(n)}{f_{k}}+\frac{L_{s}(n+1)-L_{s}(n)}{f_{s}}]$
$\displaystyle>D_{k}\frac{N_{k}(n)-N_{k}(n+1)}{R}-\frac{N_{c}(n+1)-N_{c}(n)}{R}$
$\displaystyle
D_{k}[\frac{L_{k}(n+1)-L_{k}(n)}{f_{k}}+\frac{L_{total}-L_{k}(n+1)-[L_{total}-L_{k}(n)]}{f_{s}}]$
$\displaystyle>D_{k}\frac{N_{k}(n)-N_{k}(n+1)}{R}-\frac{N_{c}(n+1)-N_{c}(n)}{R}$
$\displaystyle\frac{L_{k}(n+1)-L_{k}(n)}{f_{k}}-\frac{L_{k}(n+1)-L_{k}(n)}{af_{k}}$
$\displaystyle>\frac{N_{k}(n)-N_{k}(n+1)}{R}-\frac{N_{c}(n+1)-N_{c}(n)}{D_{k}R}$
$\displaystyle(a-1)\frac{L_{k}(n+1)-L_{k}(n)}{af_{k}}>\frac{N_{k}(n)-N_{k}(n+1)-\frac{N_{c}(n+1)-N_{c}(n)}{D_{k}}}{R}$

In the previous steps, the substitution $a=\frac{f_{s}}{f_{k}}$ is used. By
introducing another substitution regarding the computing speeds of the
entities, $\beta=\frac{a-1}{a}$, the inequation undergoes further
simplification:

$\displaystyle\frac{\beta[L_{k}(n+1)-L_{k}(n)]}{f_{k}}>\frac{N_{k}(n)-N_{k}(n+1)-\frac{N_{c}(n+1)-N_{c}(n)}{D_{k}}}{R}$
(18) $\displaystyle\frac{\beta
R}{f_{k}}>\frac{N_{k}(n)-N_{k}(n+1)-\frac{N_{c}(n+1)-N_{c}(n)}{D_{k}}}{L_{k}(n+1)-L_{k}(n)}$

The right side of the previous expression is defined by (7a), hence Lemma 10
is derived.

## Appendix B.2

## Proof of Lemma 11

The optimal cut layer $n$ must comply with condition (9b). Subsequently,
similar elementary transformations are performed based on eq.(1)-eq.(5):

$\displaystyle T(n-1)>T(n)\vspace{+1mm}$
$\displaystyle\frac{2D_{k}}{B_{k}}[\tau_{k}(n-1)+t_{0}(n-1)+\tau_{s}(n-1)]+2t_{p}(n-1)$
$\displaystyle>\frac{2D_{k}}{B_{k}}[\tau_{k}(n)+t_{0}(n)+\tau_{s}(n)]+2t_{p}(n)$
$\displaystyle\frac{2D_{k}}{B_{k}}[\frac{L_{k}(n-1)B_{k}}{f_{k}}+\frac{N_{k}(n-1)B_{k}}{R}+\frac{L_{s}(n-1)B_{k}}{f_{s}}]+\frac{2N_{c}(n-1)}{R}$
$\displaystyle>\frac{2D_{k}}{B_{k}}[\frac{L_{k}(n)B_{k}}{f_{k}}+\frac{N_{k}(n)B_{k}}{R}+\frac{L_{s}(n)B_{k}}{f_{s}}]+\frac{2N_{c}(n)}{R}$
$\displaystyle
D_{k}[\frac{L_{k}(n-1)}{f_{k}}+\frac{N_{k}(n-1)}{R}+\frac{L_{s}(n-1)}{f_{s}}]+\frac{N_{c}(n-1)}{R}$
$\displaystyle>D_{k}[\frac{L_{k}(n)}{f_{k}}+\frac{N_{k}(n)}{R}+\frac{L_{s}(n)}{f_{s}}]+\frac{N_{c}(n)}{R}$
$\displaystyle
D_{k}\frac{N_{k}(n-1)-N_{k}(n)}{R}-\frac{N_{c}(n)-N_{c}(n-1)}{R}$
$\displaystyle>D_{k}[\frac{L_{k}(n)-L_{k}(n-1)}{f_{k}}+\frac{L_{s}(n)-L_{s}(n-1)}{f_{s}}]$
$\displaystyle
D_{k}\frac{N_{k}(n-1)-N_{k}(n)}{R}-\frac{N_{c}(n)-N_{c}(n-1)}{R}$
$\displaystyle>D_{k}[\frac{L_{k}(n)-L_{k}(n-1)}{f_{k}}+\frac{L_{total}-L_{k}(n)-[L_{total}-L_{k}(n-1)]}{f_{s}}]$
$\displaystyle\frac{N_{k}(n-1)-N_{k}(n)}{R}-\frac{N_{c}(n)-N_{c}(n-1)}{D_{k}R}$
$\displaystyle>\frac{L_{k}(n)-L_{k}(n-1)}{f_{k}}-\frac{L_{k}(n)-L_{k}(n-1)}{af_{k}}$
$\displaystyle\frac{N_{k}(n-1)-N_{k}(n)-\frac{N_{c}(n)-N_{c}(n-1)}{D_{k}}}{R}>(a-1)\frac{L_{k}(n)-L_{k}(n-1)}{af_{k}}$

The derivation proceeds with substituting $\beta=\frac{a-1}{a}$, where
$a=\frac{f_{s}}{f_{k}}$. The substitution results in the following simplified
inequation:

$\displaystyle\frac{N_{k}(n-1)-N_{k}(n)-\frac{N_{c}(n)-N_{c}(n-1)}{D_{k}}}{R}>\frac{\beta[L_{k}(n)-L_{k}(n-1)]}{f_{k}}$
(19)
$\displaystyle\frac{N_{k}(n-1)-N_{k}(n)-\frac{N_{c}(n)-N_{c}(n-1)}{D_{k}}}{L_{k}(n)-L_{k}(n-1)}>\frac{\beta
R}{f_{k}}$

The left side of the previous expression is defined by (7b), thus the
derivation of Lemma 11 is obtained.