# Non-Lyapunov annealed decay for 1d Anderson eigenfunctions

Davide Macera1 2

###### Abstract

In [10] Jitomirskaya, Krüger and Liu analysed the dynamical decay in
expectation for the supercritical almost-Mathieu operator in function of the
coupling parameter , showing that it is equal to the Lyapunov exponent of its
transfer matrix cocycle, and asked whether the same is true for the 1d
Anderson model. We show that this is essentially never true when the disorder
parameter is sufficiently large.

11footnotetext: Department of Mathematics, Durham University, Durham, UK.
Email<EMAIL_ADDRESS>
Supported by EPSRC EP\T004290\122footnotetext: This work was started when the
author was a PhD student at Roma Tre university.

##  1 Introduction

Consider the one-dimensional Anderson model, i.e. the operator $H$ acting on a
dense subset of $\ell^{2}(\mathbb{Z})$ via

$(H\psi)(x)=\psi(x+1)+V_{x}\psi(x)+\psi(x-1)\leavevmode\nobreak\ ,\quad
x\in\mathbb{Z}\leavevmode\nobreak\ ,$ (1)

where $V_{x}$ are i.i.d. random variables. We assume that the distribution of
$V_{0}$ is bounded and not concentrated at one point (in most of the
discussion below, the first assumption can be relaxed to the existence of a
finite fractional moment $\mathbb{E}|V_{0}|^{\eta}<\infty$).
Carmona–Klein–Martinelli [5] showed that under these assumptions $H$ exhibits
Anderson localisation, i.e. almost surely $H$ has pure point spectrum, and
moreover

$\mathbb{P}\left\\{\forall(\lambda,\psi)\in
E\,\,\,\limsup_{x\to\pm\infty}\frac{1}{|x|}\log|\psi(x)|=-\gamma(\lambda)\right\\}=1\leavevmode\nobreak\
,$ (2)

where $\mathcal{E}[H]$ is the collection of eigenpairs of $H$ (the spectrum is
almost surely simple, so $|\psi|$ is well-defined), and $\gamma(\lambda)$ is
the Lyapunov exponent of $H$ at energy $\lambda$. Under more restrictive
assumptions on $H$, the pure point nature of the spectrum was first proved by
Goldsheid–Molchanov–Pastur [8] and by Kunz-Souillard [12]; the exponential
decay of the eigenfunctions was first established by Molchanov [15].

While the proof of [5] employs multi-scale analysis, single-scale proofs have
recently been found by Bucaj et al. [4], Gorodetski–Kleptsyn [9], and
Jitomirskaya-Zhu [11]. Generalisations to models with off-diagonal disorder
and to matrix-valued potentials are studied in [16, 13].

A stronger notion of Anderson localisation involves the notion of
eigenfunction correlator, introduced by Aizenman [1]. Denote

$Q(x,y)=\sup\left\\{|f(H)(x,y)|\,\,:\,\,f:\mathbb{R}\to\mathbb{C}\leavevmode\nobreak\
,\,\,\sup|f|\leqslant 1\right\\}\leavevmode\nobreak\ ,$

where the supremum is taken over Borel functions. If $H$ has pure point
spectrum, the correlator takes the form

$Q(x,y)=\sum_{(\lambda,\,\psi)\kern 0.70004pt\in\kern
0.70004ptE}|\psi(x)||\psi(y)|\leavevmode\nobreak\ .$

Then there exists $\gamma>0$ such that for any $x$

$\mathbb{P}\left\\{\limsup_{y\to\pm\infty}\frac{1}{|y-x|}\log
Q(x,y)\leqslant-\gamma\right\\}=1\leavevmode\nobreak\ .$ (3)

In fact, in the current setting (3) holds with $\gamma=\gamma_{\inf}$, where

$\gamma_{\inf}=\inf_{\lambda\kern 0.70004pt\in\kern
0.70004pt\sigma(H)}\gamma(\lambda)$ (4)

and $\sigma(H)$ is the spectrum of $H$ (a deterministic set) – see [11]. This
strong form of (3) implies (2), as well as dynamical localisation, decay of
the Fermi projection as well as other properties of relevance in quantum
dynamics.

Ge and Zhao built on the work [11] and proved the following :

######  Theorem 1.1 (Ge–Zhao).

For the operator $H$ of (1) with $V_{0}$ bounded and not concentrated at one
point, one has, for any $x\in\mathbb{Z}$,

$\gamma^{\mathbb{E}}=-\limsup_{y\to\pm\infty}\frac{1}{|y-x|}\log\mathbb{E}Q(x,y)>0\leavevmode\nobreak\
.$ (5)

In Section 2 we give another, arguably, simpler, proof of this result,
adopting an argument from [6].

Jitomirskaya, Krüger and Liu [10] studied the validity of (5) in the almost-
periodic setting, namely, for the supercritical almost-Mathieu operator with
Diophantine frequency, and showed that in that setting $\gamma^{\mathbb{E}}$
can be taken to be equal to $\gamma_{\inf}$. They asked whether the same is
true for the Anderson model. We show that this is not the case. A first
counterexample comes from the Anderson-Bernoulli model:

######  Theorem 1.2.

For $a>0$, consider the operator $H_{a}=H_{0}+aV$ with $V_{x}$ being a random
variable having an atom at $0$.. Then $\gamma^{\mathbb{E}}$ is bounded from
above uniformly in $a$.

In particular, if $V_{x}$ is a Bernoulli random variable with parameter $p$,
by a result of Martinelli and Micheli [14], $\gamma_{\inf}\geqslant c\,\log a$
for sufficiently large $a$. Therefore, by the above theorem,
$\gamma^{\mathbb{E}}(H^{a})\neq\gamma_{I}(H^{a})$ for $a$ large enough.
Furthermore, the above theorem remains true for _any_ bounded random potential
with finite first moment, at sufficiently high disorder:

######  Theorem 1.3.

Let $V=\\{V_{i}\\}_{i\in\mathbb{Z}}$ be a nondeterministic, bounded, i.i.d.
random potential such that $\mathbb{E}[V_{0}]<\infty$ , and let
$H^{a}:=H_{0}+aV$.
Then, for any $a$ large enough

$\inf_{E\in\kern
0.70004pt\sigma(H^{a})}\gamma_{E}(H^{a})>\gamma^{\mathbb{E}}(H^{a})\leavevmode\nobreak\
.$

### Acknowledgements

I’m deeply indebted to my former PhD advisor, Sasha Sodin, for suggesting the
topic of this paper and for giving major contributions to its development. I
believe he should have been a co-author of this paper, but for reasons I don’t
understand he asked me to erase his name from it.
I’m also grateful to Alexander Elgart for many useful comments on a
preliminary version of this paper.

##  2 Proof of Theorem 1.1

For $a,b\in\mathbb{Z}$, denote by $H_{[a,b]}$ the restriction of $H$ to
$[a,b]$ (with Dirichlet boundary conditions), and let
$G_{E}[H_{[a,b]}]=(H_{[a,b]}-E)^{-1}$. Let $\tau>0$, $E\in\mathbb{R}$ and
$N\geq 1$. A site $x\in\mathbb{Z}$ is called $(\tau,E,N)$-nonresonant
($x\notin\mathrm{Res}(\tau,E,N)$) if

$|G_{E}[H_{[x-N,x+N]}](x,x\pm
N)|\leqslant\mathrm{e}^{-(\gamma(E)-\tau)N}\leavevmode\nobreak\ .$

Otherwise, $x$ is called $(\tau,E,N)$-resonant ($x\in\mathrm{Res}(\tau,E,N)$).
The proof of the theorem uses the following

######  Claim 1.

Assume that $V_{0}$ is bounded and not concentrated at one point. Then for any
$\tau>0$ there exist $C,c>0$ such that

$\mathbb{P}\left\\{\forall
E\in\mathbb{R}\,\,\mathrm{diam}\mathrm{Res}(\tau.E,N)\cap[-N^{2},N^{2}]>2N\right\\}\leqslant
C\mathrm{e}^{-cN}\leavevmode\nobreak\ .$

See [13, Proposition 2.1] for this formulation (in the more general case of
matrix potentials) and [11] for similar statements.

Next, we need a representation for the eigenfunction correlator as a singular
integral (see [2]):

$Q(x,y)=\lim_{L\to\infty}Q^{L}(x,y)\leavevmode\nobreak\ ,\quad
Q^{L}(x,y)=\lim_{\epsilon\to
0^{+}}\frac{\epsilon}{2}\int|G_{E}[H_{[-L,L]}](x,y)|^{1-\epsilon}\mathrm{d}E\leqslant
1\leavevmode\nobreak\ .$ (6)

Having these two ingredients, we argue as follows. Without loss of generality
we can assume that $x=0$. Set $\tau=\frac{1}{2}\min_{E\in\sigma(H)}\gamma(E)$
and $N=\lfloor\frac{y}{10}\rfloor$, and consider the event

$\mathcal{R}=\left\\{\exists
E\in\mathbb{R}\,\,:\,\,0,y\in\mathrm{Res}(\tau,E,N)\right\\}\leavevmode\nobreak\
.$

According to Claim 1 , $\mathbb{P}(\mathcal{R})\leqslant C\mathrm{e}^{-cN}$.
On the complement $\Omega\setminus\mathcal{R}$, we have $\mathbb{R}=A\cup B$,
where

$A=\left\\{E\in\mathbb{R}\,\,:\,\,0\notin\mathrm{Res}(\tau,E,N)\right\\}\leavevmode\nobreak\
,\quad
B=\left\\{E\in\mathbb{R}\,\,:\,\,y\notin\mathrm{Res}(\tau,E,N)\right\\}\leavevmode\nobreak\
.$

For $E\in A$,

$\begin{split}G_{E}[H_{[-L,L]}](0,y)=\,&G_{E}[H_{[-N,N]}](0,N)G_{E}[H_{[-L,L]}](N+1,y)\,+\\\
+\,&G_{E}[H_{[-N,N]}](0,-N)G_{E}[H_{[-L,L]}](-N-1,y)\leavevmode\nobreak\
,\end{split}$

whence

$|G_{E}[H_{[-L,L]}](0,y)|\leqslant\ \mathrm{e}^{-\tau
N}(|G_{E}[H_{[-L,L]}](N+1,y)|+|G_{E}[H_{[-L,L]}](-N-1,y)|\leavevmode\nobreak\
.$

and an analogous bound can be deduced for $B$.
Thus

$\lim_{\epsilon\to+0}\frac{\epsilon}{2}\bigg{[}\int_{A}+\int_{B}\bigg{]}|G_{E}[H_{[-L,L]}](0,y)|^{1-\epsilon}\mathrm{d}E\leqslant
4\mathrm{e}^{-\tau N}\leavevmode\nobreak\ .$

Finally,

$\begin{split}\mathbb{E}Q(0,y)&=\mathbb{E}(Q(0,y)\kern 1.00006pt|\kern
1.00006pt\mathcal{R})\kern
1.00006pt\mathbb{P}(\mathcal{R})+\mathbb{E}(Q(0,y)\kern 1.00006pt|\kern
1.00006pt\Omega\setminus\mathcal{R})(1-\mathbb{P}(\mathcal{R}))\\\
&\leqslant\mathbb{P}(\mathcal{R})+\mathbb{E}(Q(0,y)\kern 1.00006pt|\kern
1.00006pt\Omega\setminus\mathcal{R})\leqslant
C\mathrm{e}^{-cN}+2\mathrm{e}^{-\tau N}\leavevmode\nobreak\ .\end{split}$

Thus $\gamma^{\mathbb{E}}\geqslant\min(c,\tau)>0$.

######  Remark.

This proof can be extended to quasi-one-dimensional operator, such as the
Anderson model on the strip of width $W$ or the more general model studied in
[13]. A slightly weaker version of (6) is still true in this case (see [6]):

$Q^{L}(x,y)\leqslant\lim_{\epsilon\to
0^{+}}\frac{\epsilon}{2}\int\|G_{E}[H_{[-L,L]}](x,y)\|^{1-\epsilon}\mathrm{d}E\leqslant
W\leavevmode\nobreak\ .$

and the argument above follows with minor modifications.

##  3 Proof of Theorem 1.2

Let $K>0$ be a large numerical constant (independent of any parameters), to be
specified later. For $x>0$, consider the event

$\Omega_{K,x}=\left\\{\forall
x\in[-Kx,(K+1)x],\,\,V_{x}=0\right\\}\leavevmode\nobreak\ .$

We shall prove the following: for any $\epsilon>0$, one has on $\Omega_{K,x}$
for sufficiently large $x$:

$Q(0,x)\geqslant\mathrm{e}^{-\epsilon|x|}\leavevmode\nobreak\ .$ (7)

Since $\mathbb{P}(\Omega_{K,x})=(1-p)^{2Kx+1}$, this this would imply that

$\gamma^{\mathbb{E}}\leqslant-\lim_{x\to\infty}\dfrac{1}{x}\log(1-p)^{2Kx+1}\mathrm{e}^{-cx}\leqslant
c-2K\log(1-p)$ (8)

as claimed.

We now turn to the proof of (7). Observe that for any $\delta>0$

$Q(0,x)\geqslant\delta\,|G_{\delta}[H](0,x)|\geqslant\delta|G_{\delta}[H-2\cdot\mathbf{1}](0,x)|\leavevmode\nobreak\
.$ (9)

Let $T=H_{0}$ be the free Laplacian (obtained by setting $V_{x}\equiv 0$ in
(1)), and let $T_{K,x}$ be the restriction of $T$ to the finite volume
$[-Kx,(K+1)x]$. Then

$\begin{split}&|G_{\delta}[H-2\cdot\mathbf{1}](0,x)|\geqslant\\\
&\quad\geqslant|G_{\delta}[T_{K,x}-2\cdot\mathbf{1}](0,x)|-\\\
&\quad\qquad-|G_{\delta}[T_{K,x}-2\cdot\mathbf{1}](0,-Kx)||G_{\delta}[H-2\cdot\mathbf{1}](-Kx-1,x)|-\\\
&\quad\qquad-|G_{\delta}[T_{K,x}-2\cdot\mathbf{1}](0,(K+1)x)||G_{\delta}[H-2\cdot\mathbf{1}]((K+1)x+1,x)|\\\
&\quad\geqslant|G_{\delta}[T](0,x)|-\\\
&\quad\qquad-|G_{\delta}[T_{K,x}-2\cdot\mathbf{1}](0,-Kx)||G_{\delta}[H-2\cdot\mathbf{1}](-Kx-1,x)|-\\\
&\quad\qquad-|G_{\delta}[T_{K,x}-2\cdot\mathbf{1}](0,(K+1)x)||G_{\delta}[H-2\cdot\mathbf{1}]((K+1)x+1,x)|-\\\
&\quad\qquad-|G_{\delta}[T_{K,x}-2\cdot\mathbf{1}](0,-Kx)||G_{\delta}[T-2\cdot\mathbf{1}](-Kx-1,x)|-\\\
&\quad\qquad-|G_{\delta}[T_{K,x}-2\cdot\mathbf{1}](0,(K+1)x)||G_{\delta}[T-2\cdot\mathbf{1}]((K+1)x+1,x)|\leavevmode\nobreak\
.\end{split}$

By the Combes–Thomas estimate ([2], Theorem 10.5, statement 2), noticing that
$H-2\cdot\mathbf{1}$ and $T-2\cdot\mathbf{1}$ are negative operators, and thus
$\delta$ sits above their spectra, we deduce that for $\delta\in(0,1)$

$|G_{\delta}[H-2\cdot\mathbf{1}](0,x)|\geqslant|G_{\delta}[T-2\cdot\mathbf{1}](0,x)|-\tfrac{C_{1}}{\delta}\kern
1.00006pt\mathrm{e}^{-c_{1}(2K+1)\sqrt{\delta}\kern
0.70004pt|x|}\leavevmode\nobreak\ .$

Now,

$|G_{\delta}[T-2\cdot\mathbf{1}](0,x)|\geqslant
C_{2}\mathrm{e}^{-c_{2}\sqrt{\delta}\kern 0.70004pt|x|}\leavevmode\nobreak\ .$
(10)

Indeed, $g(x)=G_{\delta}[T+2\cdot\mathbf{1}](0,x)$ is by definition the
square-summable solution to the equation

$g(x+1)+g(x-1)+(2+\delta)g(x)=\delta_{x,0}\leavevmode\nobreak\ .$

Plugging in the ansatz $g(x)=\alpha\mathrm{e}^{-\xi|x|}$, we find that this is
indeed a solution provided that

$\xi=\operatorname{arccosh}\Big{(}1+\dfrac{\delta}{2}\Big{)};\qquad\alpha=(2+2\mathrm{e}^{-\xi}+\delta)^{-1}$

hence $|g(x)|\geqslant|\alpha|\mathrm{e}^{\kern 0.70004pt-\xi|x|}$, where
$\xi\geqslant c\sqrt{\delta}$. Having set $\delta=(\epsilon/c_{2})^{2}$ and
$2K+1=\lceil 100\kern 1.00006ptc_{2}/c_{1}\rceil$, we obtain (7). ∎

##  4 Proof of Theorem 1.3: logarithmic divergence for general potentials

A version of the Martinelli-Micheli bound the 1d Anderson model with
absolutely continuous, bounded potential has been proven in 1983 by Avron,
Craig and Simon in [3].

######  Theorem 4.1.

(Avron, Craig, Simon, [3])
Let $H^{a}=H_{0}+aV$ be a random Schrödinger operator where $V$ is a bounded
random potential with absolutely continuous density. Then the Lyapunov
exponent $\gamma_{a}$ of $H_{a}$ is such that

$\gamma_{a}\geqslant\log(a)-K$

where $K$ is a finite constant.

We will adapt the proof in [3] to any potential having finite first moment
(not necessarily absolutely continuous). Avron, Craig and Simon’s proof relies
on the Thouless’ formula for the Lyapunov exponent of a Schrödinger operator
$H$, stating that

$\gamma_{H}(E)=\int\log|E-E^{\prime}|\,\mathrm{d}\kappa_{H}(E^{\prime})$ (11)

where $\kappa_{H}(E^{\prime})$ denotes the _integrated density of states_ of
the operator $H$. They proceed then to bound the negative part of the
logarithm in 11 using the _Wegner estimate_ : If $H=H_{0}+V$ is a random
Schrödinger operator with i.i.d. potential, then

$\dfrac{\mathrm{d}\kappa_{H}(E)}{\mathrm{d}E}\leqslant\|\rho_{V}\|_{\infty}$

which is unfortunately proven true only when the distribution $\rho_{V}$ of
the potential is absolutely continuous.
Fortunately, Shubin, Vakilian and Wolff proved in [17] a slightly weaker bound
for the IDS of a random Schrödinger operator whose potential satisfies the
conditions of Theorem 1.3: if $\mathbb{E}[|V_{0}|]<\infty$, then

$|\kappa_{H}(E)-\kappa_{H}(E^{\prime})|\leqslant C|E-E^{\prime}|^{\alpha}$
(12)

for some $\alpha\in(0,1)$ and some constant $C>0$. This bound is sufficient to
let the Avron-Craig-Simon argument work in the present generality.
By applying the Thouless formula to $H^{a}$ and splitting the logarithm into
its positive and negative parts, we get that

$\begin{split}&\int\log|E-E^{\prime}|\,\mathrm{d}\kappa_{H^{a}}(E^{\prime})=\\\
=\
&\int\log_{+}|E-E^{\prime}|\,\mathrm{d}\kappa_{H^{a}}(E^{\prime})-\int\log_{-}|E-E^{\prime}|\,\mathrm{d}\kappa_{H^{a}}(E^{\prime})\\\
\geqslant\ &\int_{\kern
0.70004pta/2}^{\infty}\log_{+}|E-E^{\prime}|\,\mathrm{d}\kappa_{H^{a}}(E^{\prime})-\int_{\kern
0.70004pt0}^{\infty}\kern
1.00006pt\kappa_{H^{a}}\\{E^{\prime}:|E-E^{\prime}|\leqslant\mathrm{e}^{-t}\\}\kern
1.00006pt\mathrm{d}t\end{split}$

We claim that

$\int_{\kern 0.70004pt0}^{\infty}\kern
1.00006pt\kappa_{H^{a}}\\{E^{\prime}:|E-E^{\prime}|\leqslant\mathrm{e}^{-t}\\}\kern
1.00006pt\mathrm{d}t\leqslant C\leavevmode\nobreak\ ;$ (13)

for some $C>0$ uniform in $a$, and that

$\int_{\kern
0.70004pta/2}^{\infty}\log_{+}|E-E^{\prime}|\,\mathrm{d}\kappa_{H^{a}}(E^{\prime})\geqslant
c\log a\leavevmode\nobreak\ .$ (14)

for some positive constant $c$.
The first bound (13) is proven by using inequality 12 :

$\begin{split}\int_{\kern 0.70004pt0}^{\infty}\kern
1.00006pt\kappa_{H^{a}}\\{E^{\prime}:|E-E^{\prime}|\leqslant\mathrm{e}^{-t}\\}\kern
1.00006pt\mathrm{d}t\leqslant\ &\|V\|_{\infty}\int_{\kern
0.70004pt0}^{\infty}\ \mathrm{e}^{-t}\mathrm{d}t^{\alpha}\\\ \leqslant\
&C\int_{0}^{\infty}\alpha t^{\alpha-1}\mathrm{e}^{-t}\mathrm{d}t\\\ \leqslant\
&C\kern 1.00006pt\Gamma(\alpha)\leqslant C^{\prime}\leavevmode\nobreak\
.\end{split}$

######  Remark.

We strongly believe that the main result of this section (and thus Theorem 1.3
as a whole) can be extended to the most general setting for which 1-d
localisation has been proven (nondeterministic potential with any finite
fractional moment). However, generalising [12] to the case where one only has
a generic fractional moment appears to be nasty .
An alternative to this approach could be extending the proof of the
logarithmic divergence for the Anderson-Bernoulli model in [14] to the general
case; however, even if this seems to be doable and should not present major
technical difficulties, additional estimates would be needed to make
Martinelli’s and Micheli’s already six page long proof work for generic
potentials, and many formulas would get much longer and nastier.
In conclusion, the length of the present paper would likely get doubled by
such an attempt, therefore we avoid it to keep the paper short and more
readable while keeping the result reasonably general.

##  5 Proof of Theorem 1.3: Green function estimates

Since the above argument uses crucially the fact that a Bernoulli random
variable is $0$ with positive probability, one might suspect that the presence
of an atom at zero is required for the annealed dynamical decay to be non-
Lyapunov. However, in this section we will use a simple trick to generalise
the result to any bounded potential.The trick relies on the observation that
it is possible to decompose any dilated random variable $aX$ having an atom as
a sum of two (not necessarily independent ) random variables, one of which is
bounded in $a$, and the other has basically the same distribution as $aX$ with
the difference that the atom has been subtracted some mass. If we subtract in
this way mass from the atom at a sufficient rate, and control the error given
by the bounded addend, then we can show that the growth of the annealed decay
rate in $a$ is much slower than the logarithmic lower bound prescribed by the
the results of Avron-Craig-Simon and Shubin-Vakilian-Wolff. We first state a
relevant result by Avron, Craig and Simon on logarithmic divergence in the
coupling of the Lyapunov exponent for 1d Anderson models that have absolutely
continuous bounded potentials.

###### Proof.

In order to exploit the Bernoulli case, we will make use of the following
observation,

######  Observation 5.1.

Let $X$ be a bounded random variable, and let $\bar{x}\in\mathrm{supp}(X)$.
Suppose that $X$ is absolutely continuous in a neighborhood of $\bar{x}$ and
denote by $\widetilde{X}^{\epsilon}$ the random variable having the same
density as $X$, except for the fact that it has an atom at $\bar{x}$ of mass
$\epsilon$, suitably renormalised. Then there exists a bounded random variable
$\tilde{\eta}$ (not necessarily independent on $X$) such that

$aX\stackrel{{\scriptstyle\mathrm{d}}}{{=}}a^{\prime}\widetilde{X}^{\epsilon}+\tilde{\eta}$

for some $a^{\prime}>0$.

This observation basically asserts that we can remove (or, by extension,
subtract mass to) an atom from the distribution of a random variable at the
cost of adding another (dependent) random variable uniformly bounded in the
coupling.
It follows by simply observing that if $\tilde{\eta}$ has the same
distribution of $X$ and $\widetilde{X}^{\epsilon}$ is chosen to take the same
values as $\tilde{\eta}$ (so that $\widetilde{X}^{\epsilon}$ would retain its
usual law and its atom at $\bar{x}$, but becoming totally dependent on
$\tilde{\eta}$), then
$a\widetilde{X}^{\epsilon}+\tilde{\eta}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}(a+1)X$.
We use this observation on the potential $V$ with $\epsilon=\epsilon(a)\gg
a^{-\beta}$ for all $\beta>0$, and decompose
$aV=a\widetilde{V}^{\epsilon(a)}+\tilde{\eta}_{R}$, where $\tilde{\eta}_{R}$
is supported on the interval $(-R,R)$. Then
$H_{a}=T^{R}+a\widetilde{V}^{\epsilon(a)}$ where
$T^{R}=T_{0}+\tilde{\eta}_{R}$ , and $\widetilde{V}^{\epsilon(a)}$ is a random
variable having the same distribution as
$V\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\tilde{\eta}_{R}$ except for having
an atom at $0$ of mass $\epsilon(a)$.
Then again, if $Q(0,x)\geqslant\mathrm{e}^{-\epsilon|x|}$ , we get, as in (8),

$\gamma^{\mathbb{E}}\leqslant c+2K\log(\epsilon(a))\ll\log(a)$ (15)

Furthermore, we call $T^{R}_{K,x}$ the restriction of $T^{R}$ to the box
$[-Kx,(K+1)x]$, analogously as before.
Bounding the eigenfunction correlator of $H^{a}$ with the Green function of a
negative operator obtained by shifting $H^{a}$, a double resolvent expansion
analogue to the one performed in the proof of Theorem 1.2 and the Combes-
Thomas bound yield

$\begin{split}Q(0,x)\geqslant\,&\delta\kern
1.00006ptG_{\delta}[H^{a}-(2+R)\cdot\mathbf{1}](0,x)\\\
\geqslant\,&\delta\kern
1.00006ptG_{\delta}[T^{R}-(2+R)\cdot\mathbf{1}](0,x)-C_{3}\mathrm{e}^{-c_{3}(2K+1)\sqrt{\delta}|x|}\end{split}$

on the event

$\Omega_{x,\kern 0.70004ptK;\kern 0.70004pta}:=\\{V_{x}^{\epsilon(a)}\equiv 0\
\ \forall\,x\in[-Kx,(K+1)x]\\},\ \quad\mathbb{P}\\{\Omega_{x,\kern
0.70004ptK;\kern 0.70004pta}\\}=[\epsilon(a)]^{(2K+1)x}.$

Note that the fact that we are subtracting $(2+R)\cdot\mathbf{1}$ makes sure
that $\delta$ is always above the spectrum of $H^{a}$.
Eventually, the only thing left to us to show is that

$G_{\delta}[T^{R}-(2+R)\cdot\mathbf{1}](0,x)\geqslant\widetilde{C}\mathrm{e}^{-\tilde{c}\delta|x|}\leavevmode\nobreak\
.$

By writing down the Neumann series for
$G_{\delta}[T^{R}-(2+R)\cdot\mathbf{1}]$ we get the following inequalities:

$\begin{split}G_{\delta}[T^{R}-(2+R)\cdot\mathbf{1}]=\,&\dfrac{1}{\mathbf{1}-T_{0}(2+R+\delta-V^{R})^{-1}}\cdot\dfrac{1}{2+R+\delta-V^{R}}\\\
=\,&\Big{(}\sum_{n=0}^{\infty}T_{0}^{n}(2+R+\delta-V^{R})^{-n}\Big{)}\cdot\dfrac{1}{2+R+\delta-V^{R}}\\\
\geqslant\,&\dfrac{1}{2+\delta}\sum_{n=0}^{\infty}T_{0}^{n}(2+\delta)^{-n}\\\
=\,&G_{\delta}[T_{0}-2\cdot\mathbf{1}]\end{split}$

due to the positivity of $T_{0}-V^{R}+(2+R)\cdot\mathbf{1}$. In particular

$G_{\delta}[T^{R}-(2+R)\cdot\mathbf{1}](0,x)\geqslant
G_{\delta}(T_{0}-2\cdot\mathbf{1})](0,x)\geqslant\widetilde{C}\mathrm{e}^{-\tilde{c}\sqrt{\delta}\kern
0.70004pt|x|}\leavevmode\nobreak\ .$

Setting, again, $K$ large enough so that $2K+1=\lceil 100\kern
1.00006pt\tilde{c}/c_{3}\rceil$, and setting
$\delta=(\epsilon/\tilde{c})^{2}$, we finally conclude that

$Q(0,x)\geqslant\delta
G_{\delta}[T^{R}-(2+R)\cdot\mathbf{1}](0,x)\geqslant\mathrm{e}^{-\epsilon|x|}.$

This, combined with (15) and the Avron-Craig-Simon bound for general bounded
potentials proven in Paragraph 4 implies the thesis. ∎

## References

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# Task Adaptive Feature Transformation for One-Shot Learning

###### Abstract

We introduce a simple non-linear embedding adaptation layer, which is fine-
tuned on top of fixed pre-trained features for one-shot tasks, improving
significantly transductive entropy-based inference for low-shot regimes. Our
norm-induced transformation could be understood as a re-parametrization of the
feature space to disentangle the representations of different classes in a
task specific manner. It focuses on the relevant feature dimensions while
hindering the effects of non-relevant dimensions that may cause overfitting in
a one-shot setting. We also provide an interpretation of our proposed feature
transformation in the basic case of few-shot inference with K-means
clustering. Furthermore, we give an interesting bound-optimization link
between K-means and entropy minimization. This emphasizes why our feature
transformation is useful in the context of entropy minimization. We report
comprehensive experiments, which show consistent improvements over a variety
of one-shot benchmarks, outperforming recent state-of-the-art methods.

Index Terms—  Few-Shot Learning, Domain adaptation

##  1 Introduction

Deep learning models have achieved impressive success in a breadth of
applications. However, these successes mostly rely on learning from huge
amounts of annotated data, which requires a time-consuming and expensive
process. Deep learning models still have difficulty generalizing to novel
classes unseen during training, given only a few labeled instances for these
new classes. In this context, few-shot learning research has attracted wide
interest recently. For example in a one-shot learning setting, a model is
first trained on substantial labeled data over an initial set of classes,
commonly called the base classes. Then, supervision is confined to one labeled
example per novel class, which is not observed during base training. The model
is then fine-tuned on these labeled examples from the novel classes (the
support set) and evaluated on the unlabeled samples (the query set).
Traditional fine-tuning would result in over-fitting in such low-data regimes.
A large body of works investigated few-shot learning via meta-learning
strategies, such as the very popular prototypical networks [1]. Meta-learning
creates a set of few-shot tasks (or episodes), with support and query samples
that simulate generalization difficulties during testing and train the model
to generalize well on these tasks.

Fig. 1: TSNE plots depicting the feature space with or without the proposed
feature transformation in (1). The support images of the 1-shot task are
provided (leftmost). The bigger markers correspond to the support image in
each class.

Related works: Our method is in line with recent transductive methods in the
few-shot learning literature, e.g. [2, 3, 4, 5, 6], among others. Transductive
inference performs class predictions jointly for all the unlabeled query
samples of the task, rather than one sample at a time as in inductive
inference. For instance, TPN [3] uses label propagation along with episodic
training and a specific network architecture; the goal was to learn how to
propagate labels from the support to the query samples. CAN-T [7] is another
meta-learning based transductive method, which uses attention mechanisms to
propagate labels to unlabeled samples. The authors of [5] proposed a method
based on graph clustering, which regularizes the inductive predictions of
query samples with a Laplacian term.

Many transductive few-shot methods focused on strategies for fine-tuning a
pre-trained model during inference. For instance, the entropy fine-tuning in
[4] re-trains the whole network, performing costly gradient updates over all
the parameters during inference. Transductive Information Maximization (TIM)
[6] proposes an entropy-based fine-tuning loss, which maximizes the mutual
information between the query features and their label predictions for a few-
shot task at inference, while minimizing the cross-entropy loss on the support
set. However, instead of retraining the whole network, [6] only fine-tunes the
softmax layer on top of the fixed pre-trained features. This showed
substantial improvements over retraining the whole network. In addition to its
recent successful use in transductive few-shot classification [6, 4], it is
worth noting that entropy minimization is widely used in semi-supervised
learning [8, 9], and has been successfully used recently in unsupervised
domain adaptation [10] and unsupervised representation learning [11].

Our Contribution: Fine-tuning the classifier on top of fixed pre-trained
features from the base classes may not take full advantage of the expressive
power of the task-specific feature space. A standard linear transformation
causes overfitting when dealing with limited supervision. In this regard, we
propose a simple yet effective norm-induced feature transformation, which is
fine-tuned to emphasize class-specific feature dimensions while hindering the
effect of non-relevant dimensions that may cause overfitting in a few-shot
setting. Our non-linear transformation could be understood as a re-
parametrization of the feature space, which disentangles the representations
of the different classes in a task-specific manner. While our motivation is
conceptually similar to early kernel-based metric-learning methods [12], in
which non-linear transformations are implicit, our transformation is explicit.
We provide an interpretation of our transformation in the basic case of few-
shot inference with K-means clustering. Furthermore, we give an interesting
bound-optimization link between K-means and entropy minimization. This
emphasizes why our feature transformation is useful in the context of entropy
minimization, which is widely used in learning, even beyond few-shot
classification. We report comprehensive experiments, showing that the proposed
transformation could yield consistent improvements over various one-shot
benchmarks, outperforming recent state-of-the-art methods.

##  2 Task Adaptive Feature Transformation for One-Shot Learning

In the one-shot setting, we are given a labeled support set $\mathcal{S}$ with
$C$ novel test classes, where each novel class has one labeled example. The
objective is to accurately classify unlabeled unseen query sample set
$\mathcal{Q}$ from these $C$ classes. Let $f_{\phi}$ denote the embedding
function of a deep convolutional neural network, with parameters $\phi$ and
$\mathbf{x}_{i}=f_{\phi}(\mathbf{y}_{i})\in\mathbb{R}^{d}$ is the features of
a given sample $\mathbf{y}_{i}$. $f_{\phi}$ is pre-trained from a labeled set
$\mathbb{X}_{\text{base}}$, via a standard cross-entropy loss, with base
classes that are different from the test classes of $\mathcal{S}$ and
$\mathcal{Q}$.

Proposed Transformation: The proposed feature transformation is performed
during fine-tuning, which is derived from minimizing an entropy-based loss
function for the target one-shot task. We will detail the entropy-based loss
function below, and draw an interesting connection to the basic K-means
objective through bound optimization. For now, let us introduce our non-linear
transformation, which reads as follows for each $L_{2}$-normalized pre-trained
feature vector $\mathbf{x}_{i}$ of a given target few-shot task:

$\displaystyle\begin{split}\mathbf{g}(\mathbf{x}_{i},\mathbf{W})&=-\frac{1}{2}\left(\|\mathbf{x}_{i}-\mathbf{w}_{1}\|^{2},\ldots,\|\mathbf{x}_{i}-\mathbf{w}_{d}\|^{2}\right)^{T}\end{split}$
(1)

where $\mathbf{x}_{i}$ is the initial feature vector either from the support
set $\mathcal{S}$ or query set $\mathcal{Q}$, and superscript $T$ denotes the
transpose operator. We introduce a learnable transformation matrix
$\mathbf{W}=[\mathbf{w}_{1}^{T}\ldots\mathbf{w}_{d}^{T}]\in\mathbb{R}^{d\times
d}$, which is updated during the fine-tuning procedure. To understand the
effect of our transformation, let us first consider a transductive inference
with a basic K-means clustering of the transformed features of the query set.
This could be done by optimizing the following mixed objective:

$\mathcal{J}(\mathbf{W},\bm{\theta},\bm{Q})=\sum_{i\in\mathcal{Q}}\sum_{c=1}^{C}q_{ic}\|\bm{\theta}_{c}-\mathbf{g}(\mathbf{x}_{i},\mathbf{W})\|^{2}$
(2)

where $\bm{\theta}=(\bm{\theta}_{c})_{1\leq c\leq C}$ represent class
prototypes, and $\bm{Q}$ is the $|\mathcal{Q}|$-by-$C$ matrix whose rows are
given by binary assignment simplex vectors $\bm{q}_{i}=(q_{ic})_{1\leq c\leq
C}\in\\{0,1\\}^{C}$: $q_{ic}=1$ if sample $\mathbf{x}_{i}$ is assigned to
class $c$ and $q_{ic}=0$ otherwise. Alternating iterative minimization of the
mixed K-means objective in Eq. (2) with respect to $\mathbf{W}$, $\bm{\theta}$
and $\bm{Q}$ could be viewed as a joint task-adaptive metric learning and
clustering, and has a clear interpretation of the effect of the proposed
transformation. Optimization with respect to transformation parameters
$\mathbf{W}$ encourages new features $\mathbf{g}(\mathbf{x}_{i},\mathbf{W})$
to approach their current class prototypes (or means) $\bm{\theta}_{c}$,
thereby disentangling the class representations; see the TSNE-plot
illustrations in Figure 1. Clearly, given current assignments $q_{ic}^{j}$ at
iteration $j$, the optimal $\bm{\theta}_{c}$ minimizing (2) corresponds to the
mean of features within class $c$:
$\bm{\theta}_{c}^{j}=\frac{\sum_{i\in\mathcal{Q}}\mathbf{g}(\mathbf{x}_{i},\mathbf{W})}{|\mathcal{Q}|}$.
Also, given both $q_{ic}^{j}$ and $\bm{\theta}_{c}^{j}$, it is clear that the
objective in Eq. (2) contains, for each sample, an $L_{2}$ distance between
the transformed feature of the sample and its current-class prototype.
Therefore, optimization with respect to $\mathbf{W}$ encourages the
transformed feature to align with its current-class prototype.

It is important to note that the specific norm-induced form of $\mathbf{g}$
that we propose in (1) constrains implicitly the transformation, hindering the
effects of non-relevant dimensions that may cause over-fitting. An
unconstrained transformation $\mathbf{g}$, such as a neural net for instance,
trained jointly with K-means, might yield trivial solutions i.e. bringing all
of the transformed features in the same cluster. This difficulty is known in
the context of deep clustering [13]. In fact, the norm-based form of each
component in transformation (1) forces some dimensions to approach zero, when
aligning with the prototypes. Identity, for instance, is not recoverable under
this constrained form, unlike an unconstrained neural-net transformation.

Let us now give more details in terms of the TSNE plots in Figure 1. We
randomly sample a 1-shot task with 5 test classes from each of the
miniImageNet and Aircraft datasets. The task from miniImageNet contains three
fine-grained classes sampled from the generic dog category: ‘Golden
retriever’, ‘Dalmatian’, and ‘African hunting dog’. The other two classes are
from generic categories: ‘Nematode’ and ‘Crate’. The task from the Aircraft
dataset contains the fine-grained categories of 5 different airplane models,
which are visually quite similar to each other. The leftmost plots refer to
the entropy-based loss in (3) fine-tuned on top of the initial fixed pre-
trained feature vector $\mathbf{x}_{i}$ without the feature transformation. In
this case, only the classifier weights $\\{\bm{\theta}_{c}\\}_{c=1}^{C}$ are
updated, which results in accuracies of 77.3% for miniImagenet and 69.33% for
the Aircraft. Note that the support images (with bigger markers) are not well
separated along with their corresponding query samples in the pre-trained
feature space learned from the base classes. Linear transformation for feature
transformation during fine-tuning causes overfitting with limited supervision,
as can be seen in figures 1(b) and 1(e). In this case, features from different
classes are brought in the same cluster and the estimates of errors could be
found from the respective classification accuracy given at the top of each
plot. Finally, if we utilize the proposed transformation in Eq. (1) on top of
the initial pre-trained features, we achieve a better spread-out task adaptive
feature space as shown in the rightmost TSNE plot 1(c) and 1(f), with boosted
accuracies of 93.33% (16% improvement) and 84.00% (13% improvement),
respectively for miniImagenet and Aircraft dataset.

Entropy-based loss function: In our case, the transformation matrix
$\mathbf{W}$ is learned by fine-tuning a transductive information maximization
(TIM) loss [6]. TIM loss is a combination of cross-entropy defined over the
support set $\mathcal{S}$ and a mutual information term, which includes two
Shannon entropies: The entropy of posterior predictions (i.e., the softmax
outputs of the network) and the entropy of the marginal probabilities of the
classes, both defined over the query set $\mathcal{Q}$:

$\displaystyle\begin{split}\mathcal{L}(\mathbf{W},\bm{\theta})=\overbrace{-\frac{\lambda}{|\mathcal{S}|}\sum_{i\in\mathcal{S}}\sum_{c=1}^{C}y_{ic}\log(p_{ic})}^{\text{cross-
entropy}}\\\
\underbrace{-\frac{\alpha}{|\mathcal{Q}|}\sum_{i\in\mathcal{Q}}\sum_{c=1}^{C}p_{ic}\log(p_{ic})}_{\text{conditional
entropy}}&+\underbrace{\sum_{c=1}^{C}\widehat{p}_{c}\log\widehat{p}_{c}}_{\text{marginal
entropy}}\end{split}$ (3)

where

$p_{ic}=\text{s}(\bm{\theta}_{c},\mathbf{W},\mathbf{x}_{i})=\frac{\exp\left(-\frac{\tau}{2}\|\bm{\theta}_{c}-\mathbf{g}(\mathbf{x}_{i},\mathbf{W})\|^{2}\right)}{\sum_{k}\exp\left(-\frac{\tau}{2}\|\bm{\theta}_{k}-\mathbf{g}(\mathbf{x}_{i},\mathbf{W})\|^{2}\right)}$

denotes the softmax probability outputs,
$\widehat{p}_{c}=\frac{1}{|\mathcal{Q}|}\sum_{i\in\mathcal{Q}}p_{ic}$ is the
marginal probability of class $c$, and $y_{ic}\in\\{0,1\\}$ the ground-truth
labels for the support samples and $\tau$ is the temperature parameter.
Minimizing the conditional entropy pushes the network probability predictions
toward the vertices of the simplex, yielding confident predictions. The
marginal entropy term avoids the trivial single-class solutions that might
result from conditional entropy minimization.

On the link between entropy and K-means: We now change gear and show an
interesting bound-optimization link between the conditional entropy in Eq. (3)
and K-means in Eq. (2). This link further clarifies why our feature
transformation is useful in the context of entropy minimization. In addition
to its recent successful use in few-shot classification [6, 4], entropy is
widely used in semi-supervised learning [8, 9], and has been successfully used
recently in unsupervised domain adaptation [10] and unsupervised
representation learning [11]. Therefore, connecting entropy minimization to
K-means could provide interesting insights even beyond few-shot
classification. To show the link, let us first decompose the conditional
entropy in (3):

$\underbrace{\sum_{i,c}\text{s}(\bm{\theta}_{c},\mathbf{W},\mathbf{x}_{i})\|\bm{\theta}_{c}-\mathbf{g}(\mathbf{x}_{i},\mathbf{W})\|^{2}}_{\mathcal{H}(\mathbf{W},\bm{\theta}):\,\text{Clustering}}+\underbrace{\sum_{i}l(\bm{\theta},\mathbf{W},\mathbf{x}_{i})}_{\text{Prototype}~{}\text{dispersion}}$
(4)

where
$l(\bm{\theta},\mathbf{W},\mathbf{x}_{i})=\log\sum_{c}\exp\left(-\frac{\tau}{2}\|\bm{\theta}_{c}-\mathbf{g}(\mathbf{x}_{i},\mathbf{W})\|^{2}\right)$.
Minimizing the prototype dispersion encourages large distances between the
prototypes and the features of all data points. Term
$\mathcal{H}(\mathbf{W},\bm{\theta})$ in Eq. (4) is closely related to basic
K-means (2) from a bound-optimization perspective, although it seems more
complex. The following shows that optimizing a soft K-means could be viewed as
an approximate Majorize-Minimize (MM) algorithm for optimizing
$\mathcal{H}(\mathbf{W},\bm{\theta})$.

Given a function $\mathcal{H}(\mathbf{W},\bm{\theta})$, the general MM
paradigm minimizes iteratively a tight upper bound on $\mathcal{H}$:

$\displaystyle\mathcal{H}(\mathbf{W},\bm{\theta})$
$\displaystyle\leq\mathcal{A}^{j}(\mathbf{W},\bm{\theta})~{}\forall~{}\mathbf{W},\bm{\theta}$
$\displaystyle\mathcal{H}(\mathbf{W}^{j},\bm{\theta}^{j})$
$\displaystyle=\mathcal{A}^{j}(\mathbf{W}^{j},\bm{\theta}^{j})$ (5)

where $j$ is the current iteration index. An upper bound satisfying the
tightness condition in (2) is often referred to as auxiliary function of the
original objective $\mathcal{H}$. It is straightforward to verify that
minimizing $\mathcal{A}^{j}$ iteratively guarantees the original objective
does not increase:
$\mathcal{H}(\mathbf{W}^{j+1},\bm{\theta}^{j+1})\leq\mathcal{A}^{j}(\mathbf{W}^{j+1},\bm{\theta}^{j+1})\leq\mathcal{A}^{j}(\mathbf{W}^{j},\bm{\theta}^{j})=\mathcal{H}(\mathbf{W}^{j},\bm{\theta}^{j})$.

Algorithm 1 FT-TIM inference

Input: Pre-trained encoder $f_{\phi}$, One-shot task
$\\{\mathcal{S},\mathcal{Q}\\}$
TIM Parameters: Number of Fine-tuning iterations $I$, temperature $\tau$, loss
weights $\\{\lambda,\alpha\\}$,learning rate for
$\\{\bm{\theta}_{c}\\}_{c=1}^{C}$
FT Parameters: Learning rate for transformation matrix $\mathbf{W}$, iteration
number $I_{W}$ to start transformation

1: L2 normalization: $\mathbf{x}_{i}=f_{\phi}(x_{i})/\left\lVert
f_{\phi}(x_{i})\right\rVert_{2}$, $\forall
x_{i}\in{\mathcal{S}\cup\mathcal{Q}}$

2: Initialize $\mathbf{W}=\mathbf{X}_{s}^{T}\mathbf{X}_{s}$, where
$\mathbf{X}_{s}=\\{\mathbf{x}_{s}\\},\forall\mathbf{x}_{s}\in\mathcal{S}$

3: Initialize $iter=0$

4: while $iter\leq I$ do

5: $\mathbf{x}^{{}^{\prime}}_{i}=\mathbf{x}_{i}$

6: if $iter\geq I_{W}$ then

7: Compute transformed feature $\mathbf{g}(\mathbf{x}_{i})$ from (1)

8: L2 normalization:
$\mathbf{x}^{{}^{\prime}}_{i}=\frac{g(\mathbf{x}_{i})}{\|g(\mathbf{x}_{i})\|_{2}}$
.

9: end if

10: Compute
$p_{ic}=\text{softmax}(-\frac{\tau}{2}\|\bm{\theta}_{c}-\mathbf{x}^{{}^{\prime}}_{i}\|^{2}),\forall
i,\forall c$

11: Compute marginal
$\widehat{p}_{c}=\frac{1}{|\mathcal{Q}|}\sum_{i\in\mathcal{Q}}p_{ic},\forall
c$

12: Compute Loss in (3)

13: if $iter\geq I_{W}$ then

14: Update $\mathbf{W}$

15: end if

16: Update $\\{\bm{\theta}_{c}\\}_{c=1}^{C}$

17: end while

18: return Query predictions
$\widehat{y}_{i}=\operatorname*{arg\,max}_{c}p_{ic},\forall i\in\mathcal{Q}$

######  Proposition 1.

$\mathcal{H}(\mathbf{W},\bm{\theta})$ is upper bounded by the following soft
K-means objective for any set of soft simplex assignment variables
$\bm{q}_{i}=(q_{ic})_{1\leq c\leq C}\in[0,1]^{C}$, $i\in\mathcal{Q}$:

$\mathcal{H}(\mathbf{W},\bm{\theta})\leq\mathcal{J}(\mathbf{W},\bm{\theta},\bm{Q})+\frac{\tau}{2}\sum_{i}\bm{q}_{i}^{T}\log\bm{q}_{i}$
(6)

Furthermore, given parameters $\mathbf{W}^{j}$ and prototype
$\bm{\theta}^{j}=(\bm{\theta}_{c}^{j})_{1\leq c\leq C}$ at iteration $j$,
choosing specific expressions
$q_{ic}=\text{s}(\bm{\theta}_{c}^{j},\mathbf{W}^{j},\mathbf{x}_{i})$ in upper
bound (6) yields an approximate auxiliary function on
$\mathcal{H}(\mathbf{W},\bm{\theta})$ when $\tau$ is small ($\tau\rightarrow
0$).

###### Proof.

The upper bound in (6) is convex w.r.t $\bm{Q}$ as it is the sum of linear and
convex functions. Solving the KKT conditions for minimizing this bound, s.t.
simplex constraint on each $\bm{q}_{i}$, yields closed-form solutions:
$\tilde{q}_{ic}=\text{s}(\bm{\theta}_{c},\mathbf{W},\mathbf{x}_{i})$ The
inequality in (6) follows directly from plugging these optimal solutions in
the upper-bound in (6) and using the fact that $\tau$ is small
($\tau\rightarrow 0$). Finally, it is straightforward to verify that the
specific choice
$q_{ic}=\text{s}(\bm{\theta}_{c}^{j},\mathbf{W}^{j},\mathbf{x}_{i})$ makes the
upper bound in (6) tight at the current solution and, hence, an auxiliary
function, when temperature $\tau\rightarrow 0$. ∎

Methods Network miniImageNet tieredImageNet CUB Aircraft MAML [14] ResNet-18
49.61 - 68.42 - TPN [3] ResNet-12 59.46 - - - Entropy-min [4] ResNet-12 62.35
68.36 - - DPGN [15] ResNet-18 66.63 70.46 - - CAN+T [7] ResNet-18 67.19 73.21
- - DSN-MR [16] ResNet-18 64.60 67.39 - - MetaoptNet [17] ResNet-18 62.64
65.99 - - LaplacianShot [5] ResNet-18 70.89 77.60 79.93 - TIM [6] ResNet-18
72.77 80.80 82.24 83.06 RAP-LaplacianShot [2] ResNet-12 74.29 - 83.59 - FT-TIM
(ours) ResNet-18 75.00 83.45 85.54 84.47 AWGIM [18] WRN 63.12 67.69 - -
Entropy-min [4] WRN 65.73 73.34 - - SIB [19] WRN 70.0 70.90 - - BD-CSPN [20]
WRN 70.31 78.74 - - SIB+E3BM [21] WRN 71.4 75.6 - - LaplacianShot [5] WRN
73.44 78.80 - - IFSL [22] WRN 73.51 83.07 - - TIM [6] WRN 77.8 82.1 - - FT-TIM
(ours) WRN 79.22 85.06 - -

Table 1: Average one-shot accuracy (in %) for the standard benchmark.

##  3 Experiments

Datasets: We used four one-shot benchmarks, including both the fine-grained
classification settings (CUB and Aircraft) and standard one-shot
classification setting (miniImageNet and tieredImageNet). miniImageNet is a
subset of the larger ILSVRC-12 dataset [23]. We use the standard split of 64
classes for base training, 16 for validation, and 20 for testing.
tieredImageNet [24] is also a subset of the ILSVRC-12 dataset, but with 608
classes instead. We split the dataset into 351 classes for base training, 97
for validation and 160 for testing. CUB [25] is a fine-grained image
classification dataset with 200 categories. We split it into 100 classes for
base training, 50 for validation and 50 for testing. Aircraft or FGVCAircraft
[26] is a fine-grained image classification dataset with 100 airplane models.
Following the same ratio as CUB, we split classes into 50 base classes for
training, 25 validation, and 25 test classes. Images are resized to $84\times
84$ pixels.

Implementation Details: The results of the proposed FT-TIM is reproduced and
evaluated in the same settings as in [5, 6] for fair comparisons. The network
models are trained with cross-entropy loss on the base classes. We utilize the
same publicly available pre-trained models of [5, 6] for miniImageNet,
tieredImageNet, and CUB. For the Aircraft dataset, we train the model
according to the same protocol. The evaluation is done on two different setups
of 5-way one-shot benchmark: 1) Standard one-shot benchmark, 15 samples per
class in the query set for each task, and the average accuracy over query sets
are reported. 2) Semi-supervised one-shot benchmark, where we treat 15 samples
per class as the additional unlabeled samples along with the support set and
report the accuracy on a separate held out test set containing 5 test samples
from each class. In this setup, we compare the results with and without the
proposed task adaptive feature transformation while fine-tuning entropy loss
(TIM) [6] in (3). The average accuracy over 600 one-shot tasks are reported.
In case of FT-TIM, transformation matrix $\mathbf{W}$ is fine-tuned with
$0.01$ learning rate, starting from the 200th fine-tuning iteration, which we
decide from mini-Imagenet validation set accuracy. The feature transformation
weights $\mathbf{W}$ is initialized from the cosine similarity matrix formed
with the $L_{2}$ normalized initial pre trained support set features.

Methods miniImageNet tieredImageNet CUB TIM [6] 72.40 80.70 82.60 FT-TIM
(ours) 73.96 83.45 84.69

Table 2: Average accuracy (in %) for the semi-supervised one-shot learning
setup. The best results are highlighted in bold font.

###  3.1 Results

The results of the general one-shot classification are highlighted in Table 1.
It can be observed that in each of the datasets and network models, the
proposed FT-TIM which includes the proposed feature transformation
consistently improves the 1-shot accuracy by $~{}1-3$% in comparison to the
baseline TIM [6] without the proposed transformation. Note that, the proposed
FT-TIM also outperforms the other recent transductive methods such as ICIR
[27] and RAP-LaplacianShot [2] by simply fine-tuning the feature
transformation during evaluation. The similar consistent improvement is also
reflected in the case of fine-grained classification on both of the CUB and
Aircraft datasets in Table 1. These results clearly demonstrate that the
proposed feature transformation can bring out the expressive power of the task
adaptive feature space in one-shot learning.

We again evaluate the efficacy of the proposed feature transformation in semi-
supervised one-shot tasks, where additional unlabeled samples are provided
along with the one-shot labeled data per novel class. The transformation
weights and the classifier weights are updated during the fine-tuning with the
labeled data and the additional unlabeled data in the one-shot task. Finally,
the inference is performed on a separate held out test set. To observe the
benefit of plugging the proposed feature transformation during fine-tuning
entropy based loss, we compare the proposed FT-TIM with respect to baseline
TIM [6] without the proposed transformation. From the results in Table 2, we
can observe that consistent improvements are achieved by FT-TIM across
different datasets, number of shots, and network models. These results
indicate that the proposed transformation layer, while fine-tuned on top of
pre-trained features jointly with the classifier, helps to disentangle the
representations of different classes in a task-specific manner.

##  4 Conclusion

In this paper, we present a simple yet effective feature transformation layer,
which brings consistent improvements in transductive one-shot learning while
fine-tuned on top of pre-trained features. The proposed transformation takes
full advantage of the expressive power of the task-specific feature space. It
could be understood as a re-parametrization of the feature space, which
disentangles the representations of different classes in a task-specific
manner. We further provided an interpretation of our transformation in the
basic case of few-shot inference with K-means clustering, along with an
interesting bound-optimization link between K-means and entropy minimization.
This emphasizes why our feature transformation is useful in the context of
entropy minimization, which is widely used in learning.

## References

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# QualityBLE: A QoS Aware Implementation for BLE Mesh Networks

Jimmy Fernandez Landivar1 0000-0002-4904-5256 Pieter Crombez2 Sofie Pollin1
and Hazem Sallouha1
0000-0002-1470-2076 0000-0002-1288-1023 1Departement of Electrical Engineering
(ESAT) - WaveCoRE  KU Leuven  Belgium

2Televic Healthcare, Belgium<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Bluetooth Low Energy (BLE) Mesh is widely recognized as a driver technology
for IoT applications. However, the lack of quality of service (QoS) in BLE
Mesh, represented by packet prioritization, significantly limits its
potential. This work implements a quality-of-service (QoS) method for BLE Mesh
to prioritize the data packets and provide them with different network
transmission settings according to their assigned priority. Unlike existing
works on QoS for BLE Mesh, our proposed approach does not require any
modifications to the BLE Mesh protocol and can be smoothly adopted in existing
BLE Mesh networks. We conducted an extensive measurement campaign to evaluate
our solution over a 15-node BLE Mesh network deployed to emulate a smart
healthcare scenario where 45 sensors with an assigned priority transmit
information over the network. The experiments provide performance results for
single and multi channel network scenarios. The obtained results validate our
solution, showing the difference between the established priorities and
providing insights and guidelines to conduct further research on QoS over BLE
Mesh and broadcast-based networks.

##  1 Introduction

The Internet of Things (IoT) applications space is constantly increasing,
covering many areas, from home, transportation, and agriculture to healthcare
and industry [1, 2]. As a result, the number of IoT devices is rising, setting
a diverse range of requirements on quality-of-service (QoS), power
consumption, coverage, and spectral efficiency [2]. Bluetooth Low Energy (BLE)
is one of the most popular IoT technologies which suffices IoT applications
with a limited number of nodes. In order to scale up BLE networks, a multi-
hop-based mesh BLE standard was released in 2017[3].

The BLE Mesh standard, which is built on top of the BLE lower protocol layers,
uses BLE’s advertising/scanning states to broadcast messages to other devices,
allowing them to communicate in a mesh structure [3]. A BLE Mesh network can
theoretically include up to 32767 devices, and a BLE Mesh packet can perform
up to 127 hops to reach its destination [4]. These characteristics positioned
BLE Mesh as a favorable technology for several applications where network
power consumption and scalability, in terms of the number of devices and
range, are more critical than latency [5]. These applications include home
automation [3], healthcare [5], and smart infrastructures [6]. While BLE Mesh
meets the connectivity requirements of these applications, it sends all
packets without any prioritization, failing to provide any QoS needed in
modern IoT networks. For instance, in modern smart hospitals where patients
have wearable devices, transmitting important vital signs or nurse calls must
have a higher priority than an asset tag that sends a control signal or a
command to turn on the illumination system.

The use of IoT in the area of healthcare applications has recently attracted
considerable research focus. The work presented in [7] showed the benefits of
IoT in healthcare, highlighting the necessity of supporting multiple types of
packet priorities in the network in big health infrastructures such as
hospitals. In [5], the authors demonstrated IoT networks’ essential role in
monitoring applications in general and for healthcare in particular. However,
the work was focused on BLE implementations only and did not address the
scalability problem faced in massive deployments such as in the case of
hospitals. The urgent need for a QoS implementation that can be integrated in
the BLE Mesh standard motivates our work in this paper, which is, to the best
of our knowledge, an open research question.

This paper proposes a QoS implementation for BLE Mesh networks, which can be
integrated into the network while staying in line with the BLE Mesh standard.
In particular, our QoS implementation uses only 1-Byte of the packet sequence
number (SEQ) field on the network Protocol Data Unit (PDU) structure to
incorporate a priority class to every packet. Different network transmission
parameters are automatically set for each priority on the nodes. We evaluate
the performance of our solution in a 15-node BLE Mesh testbed deployed in a
full building floor, emulating a smart healthcare infrastructure use case. Our
main contributions are summarized as follows:

  * •

We introduce a QoS implementation method for the BLE Mesh protocol stack.
Specifically, this method adds a priority class (up to 256 values) to the
network layer PDU. It is associated with various network transmission
configurations that modify the Transmission Power, Time to Live (TTL), Number
of rebroadcasts, and packet Advertising Interval. This implementation enables
the possibility to differentiate the network transmission services for every
priority class.

  * •

An extensive measurements campaign is performed on a 15-node (nRF52 System-On-
Chip and Zephyr RTOS-based) BLE Mesh network. Each of our nodes virtualizes
three sensors, representing a 45-sensor network in total.

The rest of this paper is organized as follows: Section 2 defines the related
work and state-of-the-art. Section 3 presents the BLE Mesh networking process.
Section 4 describes the proposed method for QoS implementation in BLE Mesh.
Section 5 details the data collection and experiment campaign. Finally,
sections 6 and 7 present the performance evaluation results, conclusions, and
future work.

##  2 Related Work

Since the BLE Mesh standard’s official release, several works have studied its
performance regarding scalability, reliability, and latency. The author in [8]
implemented a low-cost BLE Mesh testbed with multiple configurable nodes,
including the possibility to test flooding and route-based mesh protocols. The
work in [9] presented the results of massive experimentation on BLE Mesh,
showing results related to the network performance, latency, and size in an
environment with coexistence of other wireless technologies. The performance
of a BLE Mesh network composed of heterogeneous devices deployed in an area of
1100 m2 was tested and analyzed in [10]. The work in [11] suggested methods to
improve BLE Mesh and its node features by applying self-organizing network
concepts to the BLE Mesh node roles. The authors in [12] suggested deployment
strategies and node configuration parameters in order to improve the
performance of BLE Mesh. However, none of these aforementioned works
introduced a QoS solution for BLE Mesh network as they mainly focused on a
signle traffic class. Only a few existing works addressed the QoS problem in
BLE Mesh. These works either require frequent manual configurations [13],
propose different strategies for different network configurations [14], or
make changes on the inner default functions on the protocol itself [15].
Moreover, a framework for introducing QoS in BLE Mesh by varying some features
of the radio standard, such as the backoff time mechanism and channel
selection, is proposed in [15]. However, solutions modifying critical elements
in the standard limit the possibility of their adoption on a large scale,
where following a standard protocol is a key requirement. The urgent need for
a QoS implementation in BLE Mesh networks, that stays in line with the
technology protocol motivates our work in this paper.

##  3 BLE Mesh Background

This section presents the principal characteristics of the BLE Mesh
communication process, network architecture, and protocol stack.

###  3.1 Elements, models, and node roles

Packets in BLE Mesh are sent using a controlled flooding protocol where the
nodes broadcast the packets using BLE’s advertising and scanning channels. BLE
Mesh nodes receive the packets and rebroadcast them until reaching the
destination or traverse all nodes in the mesh network. This mechanism is
called relaying and allows the packets to do multiple hops between devices by
being relayed. The number of hops a packet can do is controlled by a parameter
called Time to Live (TTL) [1, 3]. The TTL decreases with every hop made by the
packet, limiting the number of hops a BLE Mesh packet can perform. When TTL
reaches 0, the packet will not be relayed [1].

The BLE Mesh network integrates different types of nodes that can send, relay,
and receive messages [1]. The standard defines four types based on their role:
First, The Relay Nodes can receive and re-transmit messages. Second, The Proxy
Nodes enables the communication between BLE devices and the mesh network.
Third,  The Low Power (LP) Nodes are in energy-saving mode until they awake
and communicate with the mesh network through the Friend nodes. Fourth, The
Friend Nodes enables the communication between the LP nodes and the mesh
network, receiving and sending information collected from the LP nodes and
relaying it to the other nodes. Every BLE Mesh node supports the four roles,
and they can be set by software according to the network design. Figure 1
depicts the BLE Mesh network architecture.

The BLE Mesh nodes can have one or more roles, and every node has a unicast
address. In their architecture, the nodes can have multiple constituent parts
that establish communication with the mesh network jointly or independently.
These parts are called elements[16]. A node can have one or more elements, and
every element has a different unicast address. A group of nodes including
different elements can have a group address[1]. The functions of every element
to interact with the mesh network are defined by a structure called a
model[16, 4], as illustrated in Figure 1. The models communicate with
different nodes using mesh messages. The messages are part of the BLE Mesh
packet and are transmitted by nodes from and to an address. These addresses
could be unicast or group addresses[1]. A node can only send messages to an
address it has been registered to publish and receive them only if it has
subscribed. Each message is defined by a unique opcode that allows
communication between nodes in the mesh network, identifying the type of
information and assigning a model to process it[16].

Figure 1: BLE Mesh network architecture.

###  3.2 BLE Mesh protocol stack architecture

The BLE Mesh protocol comprises seven layers on top of the BLE protocol stack.
Here, the BLE’s physical and link layers establish radio communication through
advertising and scanning channels 37, 38, and 39 [4]. From the bottom to the
top, the seven different layers and their functions are:

  1. 1.

The bearer layer composed of the advertising bearer and the GATT bearer. The
advertising bearer is the one that uses the BLE functions of scanning and
advertising to send and receive packets exclusively for BLE Mesh.

  2. 2.

The Network Layer is in charge of sending the encrypted transport PDUs to the
bearer layer, adding the addresses and extra packet information to be sent.
The relay process is carried out here. If a packet is received and if it
belongs to the node, it goes to the upper layer; otherwise, it is discarded or
relayed.

  3. 3.

The Lower Transport Layer sends the PDU to the lower transport layer in the
peer devices. Here, the data is segmented if necessary.

  4. 4.

The Upper Transport Layer is where the application data is encapsulated and
encrypted to be sent to the lower transport layer.

  5. 5.

The Access Layer verifies the data to be encrypted and sent it to the
transport layer. It also verifies that the data coming from the transport
layer is correct to be sent to the upper foundation models’ and models’
layers.

  6. 6.

The Foundation Models’ Layer is where the models and their different states
are interpreted and defined.

  7. 7.

The Models Layer is the one that associates the different models and states
with the user applications.

##  4 QoS Implementation for BLE Mesh

In this section, we introduce our novel QoS implementation for BLE Mesh to
enable priority and differentiate services for the data packets flooding the
network. Our method is performed over the
nRFConnectSDK111https://www.nordicsemi.com/ BLE Mesh protocol stack of the
Zephyr RTOS 222https://www.zephyrproject.org/. In the following, we detail our
approach and the functions we implemented in the BLE Mesh stack to enable QoS
in BLE Mesh networks.

###  4.1 Adding the priority class to mesh packets

The proposed method enables QoS by adding the priority value to the mesh
packets every time a model sends a message. Here, the unique opcode of each
model is used to define the packet’s priority. This opcode is sent as an
argument through the access and transport layer functions to the network
layer, where the opcode is translated to a priority class of 1-byte size,
allowing a maximum of 256 priority classes.

Once the priority value is received on the network layer, it is added as
another field of the network PDU. The extra 8 bits of the priority class are
allocated into the sequence number (SEQ) field. Here the 24 bits of the SEQ
field are reduced to 16, and the remaining 8 bits are assigned to the priority
class. Figure 2 shows the BLE Mesh network PDU structures with the priority
field. It is important to note that the 16 bits assigned to the SEQ number
provide enough packet counts to not compromise the security of the BLE Mesh
standard against replay attacks [4]. If necessary, the SEQ field size could be
increased to 20 bits.

Figure 2:  QoS priority class allocated into the sequence field (SEQ) in the
BLE Mesh network-layer protocol data unit (PDU).

###  4.2 Configuring parameters for QoS

The configurable transmission parameters represent a group of parameters
located in the different structures of the BLE Mesh protocol. The parameters
can be set to optimize the network performance and give personalized
transmission characteristics to BLE Mesh packets. Table 1 summarizes each
parameter and its range of values.

Table 1: BLE Mesh transmission parameters Parameter |  Description |  Values
---|---|---
Number of Rebroadcast Repetitions (N.Rep) |  Number of times a packet is rebroadcast by a network node |  0 to 1000
Advertising Interval (Adv. Interval) |  Time between advertising intervals |  20 ms to 10.24 s
Time to Live (TTL) |  The max number of hops a packet can perform |  0 to 127
Transmission Power (Tx Power) |  The transmission power of the BLE radio connection∗ |  (4, 0, -8, -20, -40 ) dBm
∗Changing the transmission power is a feature in the Nordic nRF52832 System-
On-Chip.

When a new node is added to the BLE Mesh network, the network transmission
parameters, the addresses, and different publish/subscription rights are set
during a process called provisioning. After this process, only the number of
rebroadcasts, advertising interval, and TTL can be changed by executing manual
runtime configurations from an external order of the provisioner node. In
contrast, the Transmission Power cannot be changed during or after the
provisioning process. Thus, changing the network parameters from an external
order each time a new packet is sent would be effort- and time-inefficient for
the source nodes and impossible for the relay nodes. Accordingly, the first
step of our solution is to vary these parameters for each packet directly at
the nodes, enabling the possibility of having network transmission settings
for each packet according to its given priority.

###  4.3 Transmitting packets with priority over BLE Mesh networks

The Packet priority class determines the network transmission parameters for
the source and relay nodes in the mesh network. When a packet is transmitted
for the first time, the source node elements transmitting it uses a model for
every priority. This model sets the initial network transmission parameters
and the message opcode according to the priority. As explained in Subsection
4.1, the packet is broadcast with the priority field added at the network
layer. For each packet received at a relay node, the network layer determines
if this packet is addressed to the node unicast address or to the group
address and needs to be relayed to the next node. In the second case, if the
packet needs to be relayed, the implemented functions on the network layer
determine its priority, set the correct node transmission parameters, and
forward the packet to the next node. The BLE Mesh packets without a priority
(default) are set with second-priority transmission parameters. Algorithm 1
summarizes the relay process with our QoS implementation.

Algorithm 1 Network-Layer packet relay process with QoS.

1: while True do

2: newPacket $\leftarrow$ receivePacket( )

3: packetAddress = getAddress(newPacket)

4: if packetAddress == nodeAddress then

5: toTransportLayer( )

6: else if packetAddress == groupAddress then

7: toTransportLayer( )

8: if toRelay == True then

9: checkPriority( )

10: setNetworkPara ( )

11: relay( )

12: else

13: discard( )

14: end if

15: end if

16: end while

##  5 Experimental setup and data collection

To test the performance of BLE Mesh with the proposed QoS implementation, we
deployed an experimental BLE Mesh network. This section presents its
implementation and the phases of the experiment campaign.

###  5.1 BLE Mesh experimental network

The experimental BLE Mesh network is deployed on a full floor of an office
building with a total area of around 400 meters. Here, 15 BLE Mesh nodes are
placed in equal distance distribution to represent a smart healthcare sensor
network. Every node represents a sensor device generating unsegmented 11-byte
BLE Mesh data packets and pushing them in the network. This network setup
comprises 15 Nordic Semiconductor nRF52DK development kits with the nRF52832
System-On-Chip333https://www.nordicsemi.com/products/nrf52832. The nodes are
programmed using version 1.8.0 of the nRF Connect SDK with Zephyr RTOS real-
time operating system and version 5.2 of the Bluetooth core specification. For
the data collection process, 6 Raspberry Pi
4B444https://www.raspberrypi.com/products/raspberry-pi-4-model-b/ (RPi)
single-board computers are placed next to the nodes. The RPis are synchronized
using Network-Time-Protocol. They collect the nodes’ information through a
serial connection and upload it to a server. Figure 3 shows the experimental
network.

Figure 3: BLE Mesh experimental network deployment.

The nodes are provisioned and set with the relay feature, enabling them to
send, receive and relay messages. Each node has three elements, each one with
a unicast address. Each element represents a sensor in our experiments, and
each of the three sensors will have a different network configuration. This
way, the source nodes provide three priority classes, one for each sensor.
Table 2 summarizes the network transmission characteristics for each priority.

Table 2: BLE Mesh QoS priority list Priority |  N. Rep |  TTL |  Adv. Interval |  Tx Power
---|---|---|---|---
1 - Element 1 - Sensor 1 |  2∗ |  7 |  20 ms |  4 dBm
2 - Element 2 - Sensor 2 |  2∗ |  5 |  100 ms |  -8 dBm
3 - Element 3 - Sensor 3 |  2∗ |  3 |  200 ms |  -20 dBm
∗The BLE Mesh standard recommends setting the Number of Rebroadcasts (N.rep)
to 2 [17].

###  5.2 Experiment campaign

Our experiment campaign consists of two experiments. For every experiment, the
data is collected by a Raspberry Pi at the source and the destination nodes.
Table 3 illustrates the structure of the database table. The two experiments
are detailed as follows:

  1. 1.

Experiment 1: Single-channel scenario:
This experiment aims to evaluate the BLE Mesh network performance in a limited
congestion network scenario. Therefore, the designated source node is A, the
destination node is H, and the other 13 mesh nodes work as receivers and
relays (See Figure 3). A sends to H 6000 unsegmented data packets with random
priorities with a packet generation rate of 2 seconds.

  2. 2.

Experiment 2: Multi-channel scenario:
Experiment two aims to evaluate the BLE Mesh network performance in a
relatively high congestion network scenario. To this end, we defined 2 traffic
channels. The first channel’s source node is A, and the destination node is H.
The second channel’s source node is N, and the destination node is G (See
Figure 3). The other 11 nodes work as receivers and relays. Each source node
sends 6000 unsegmented data packets with random priorities and a generation
rate of 2 seconds.

Table 3: BLE Mesh experiment dataset structure Timestamp (ms) | Test Id | Packet Id |  | Sender
---
Address
| Receiver
---
Address
TTL |  | Tx Power
---
(dBm)
| Priority
---
Class
Delivered∗ |  | Number
---
of hops∗
PDT∗ (ms)
1670585825695 | 1 | 18 | 0x0091 | 0x00C4 | 7 | 4 | 1 | 1 | 1 | 17
1670585837784 | 1 | 24 | 0x0091 | 0x00C4 | 7 | 4 | 1 | 1 | 0 | 12
1670585847856 | 1 | 29 | 0x0092 | 0x00C4 | 5 | -8 | 2 | 1 | 1 | 20
1670585851885 | 1 | 31 | 0x0092 | 0x00C4 | 5 | -8 | 2 | 1 | 2 | 29
1670585880137 | 1 | 45 | 0x0093 | 0x00C4 | 3 | -20 | 3 | 1 | 2 | 233
1670585884168 | 1 | 47 | 0x0093 | 0x00C4 | 3 | -20 | 3 | 1 | 2 | 227
∗The variables Delivered, Number of hops, and PDT (ms) are variables generated
after data collection to evaluate the network performance.

##  6 Results

This section summarizes the most important insights obtained from our
experiment campaign. Three Key Performance Indicators (KPIs) are defined to
evaluate the performance of our QoS implementation.

  * •

The Packet Delivery Time (PDT) measures the elapsed time in milliseconds of a
message from the source to the destination.

  * •

The Packet Delivery Rate (PDR) indicates the delivered packets over the total
number of packets sent.

  * •

The Number of Hops (N.Hops) counts the number of hops that a packet performed
to reach its destination.

###  6.1 Experiment 1: Single-channel scenario.

In this experiment, the three defined priorities can be clearly distinguished.
Figure 4 shows the empirical cumulative distribution function (eCDF) for the
PDT corresponding to each priority. As shown in the figure, 80% of the first
priority data packets were delivered in less than 20 ms. For the second and
third priorities, the same 80% of the packets were delivered in less than 100
ms and 300 ms, respectively. These results confirm that our QoS implementation
shows a clear difference in terms of PDT between the three priorities
considered. Table 4 presents the detailed statistics of the experiment
results, clearly showing the difference in performance between the three
adopted priority classes. It is worth noting that this experiment analyzes the
ideal case, in which the traffic is only generated in one channel without
interference from other sources or channels. This ideal traffic scenario is
also validated with the PDR analysis. As shown in Table 4, the first and
second priorities have a PDR of 1, and the third priority have a PDR of 0.905.
The number of hops is also related to the assigned priority, with a minimum
average of 1.882 and 3 hops for the first and third priorities, respectively.
For the third priority, the PDR (0.905) according to the number of hops can be
interpreted as a border effect of the experiment and used to define the
correct TTL limit values for each application.

Table 4: Experiment 1: Key Performance Indicators | Key Performance
---
Indicator
Priority 1 | Priority 2 | Priority 3
PDR | 1 | 1 | 0.905
Number of hops Avg | 1.882 | 2.495 | 3
PDT Avg (ms) | 19.606 | 42.163 | 237.814
PDT Std. Dev (ms) | 14.742 | 43.693 | 185.764
PDT Min (ms) | 3 | 9 | 16
PDT Max (ms) | 383 | 494 | 1251
Figure 4: Experiment 1 results for single-channel scenario (A to H).

###  6.2 Experiment 2: Multi-channel scenario.

The three defined priorities can be clearly distinguished in this experiment
for channel 1 and channel 2. The first important point to note is that the
average PDT values are higher for the two channels of this experiment (See
Table 5). Second, there are differences in the performance between channels 1
and 2. Figures 5 and 6 show the PDT eCDFs of both channels, where for the
first priority, the difference between channels in average PDT is 22.536 ms,
and for the second and third priorities, the difference is 31,133 and 98,821
ms, respectively. Table 5 shows the detailed statistics of the second
experiment’s results. As shown in the table, the PDR remains 1 for both
channels’ first and second priority. However, for the third priority, channel
2 performs 2,3% better. For both channels, the number of hops gradually
increases depending on the priority, e.g., in channel 2, from 1.643 to 2.939
hops executed by the first and third priority, respectively. The results of
experiment 2 also validate the effectiveness of our QoS implementation. It
shows the differences between the channels and the differences between
priorities in a more realistic network scenario, where more than one channel
is exchanging data packets.

Table 5: Experiment 2: Key Performance Indicators | Key Performance
---
Indicator
Priority 1 | Priority 2 | Priority 3
Channel 1 (A to H)
PDR | 1 | 1 | 0.947
Number of hops Avg | 1.885 | 2.187 | 2.888
PDT Avg (ms) | 30.579 | 52.392 | 227.004
PDT Std. Dev (ms) | 54.170 | 77.935 | 221.675
PDT Min (ms) | 9 | 7 | 21
PDT Max (ms) | 489 | 591 | 1464
Channel 2 (N to G)
PDR | 1 | 1 | 0.923
Number of hops Avg | 1.643 | 2.478 | 2.939
PDT Avg (ms) | 53.115 | 83.525 | 325.825
PDT Std. Dev (ms) | 96.850 | 112.398 | 261.839
PDT Min (ms) | 8 | 13 | 18
PDT Max (ms) | 473 | 609 | 1635
Figure 5: Experiment 2 results for channel 1 (A to H) in a multi-channel
scenario. Figure 6: Experiment 2 results for channel 2 (N to G) in a multi-
channel scenario.

##  7 Conclusions

We presented a QoS method for BLE Mesh networks implemented into the network
layer of BLE Mesh without interfering with the standard itself. We conducted
two extensive measurements to evaluate the performance of the proposed QoS
implementation. Both experiments clearly confirmed the differences between the
adopted priority classes. We show that a PDR of 1 is always guaranteed for the
first and the second traffic classes. The PDT also confirmed the priority
differentiation, presenting values of approximately 20 ms and 325 ms for the
first and third priority classes, respectively, showing a gap of approximately
300 ms between them. Moreover, the number of hops is also associated with the
priorities, varying from 1.8 to 3 hops from the first to the third priority.
Exploring machine-learning-based parameters selection to improve the
parameters selection process for the different QoS classes in BLE Mesh
networks is a potential future work.

##  8 Acknowledgments

The present work has received funding from the European Union’s Horizon 2020
Marie Skłodowska Curie Innovative Training Network Greenedge (GA. No. 953775).
The work of Hazem Sallouha was funded by the Research Foundation Flanders
(FWO), Postdoctoral Fellowship No. 12ZE222N.

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