# Asymptotic behavior for anisotropic fractional energies

Julian Fernández Bonder  Instituto de Cálculo (UBA - CONICET) and Departamento
de Matemática, FCEyN, Universidad de Buenos Aires, Pabellón I, Ciudad
Universitaria (1428), Buenos Aires, Argentina<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Ariel Salort

###### Abstract.

In this paper we investigate the asymptotic behavior of anisotropic fractional
energies as the fractional parameter $s\in(0,1)$ approaches both $s\uparrow 1$
and $s\downarrow 0$ in the spirit of the celebrated papers of Bourgain-Brezis-
Mironescu [6] and Maz’ya-Shaposhnikova [20].

Then, focusing con the case $s\uparrow 1$ we analyze the behavior of solutions
to the corresponding minimization problems and finally, we also study the
problem where a homogenization effect is combined with the localization
phenomena that occurs when $s\uparrow 1$.

###### Key words and phrases:

Fractional energies, fractional order Sobolev spaces, homogenization

###### 2020 Mathematics Subject Classification:

35J92, 35R11, 35B27

This workd was partially supported by UBACYT Prog. 2018 20020170100445BA and
by ANPCyT PICT 2019-00985. J. Fernández Bonder and A. Salort are members of
CONICET

##  1\. Introduction

The celebrated result by Bourgain, Brezis and Mironescu establishes the
behavior of the so-called Gagliardo seminorm in fractional order Sobolev
spaces of order $s$ as $s\uparrow 1$, providing new characterizations for
functions in the Sobolev space $W^{1,p}(\Omega)$. More precisely, given a
smooth bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq 1$ and
$p\in[1,\infty)$, for any $u\in W^{1,p}(\Omega)$ in [6] it is proved that

$\lim_{s\uparrow
1}(1-s)\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy=\mathcal{K}_{p,n}\|\nabla
u\|_{p}^{p}$

where the constant $\mathcal{K}_{p,n}$ is given by

$\mathcal{K}_{p,n}=\frac{1}{p}\int_{\mathbb{S}^{n-1}}|\omega_{1}|^{p}\,d\mathcal{H}^{n-1}.$

Here $\mathbb{S}^{n-1}$ is the unit sphere in $\mathbb{R}^{n}$ and
$\mathcal{H}^{n-1}$ the Hausdorff $(n-1)-$dimensional measure.

The formula above was proved to hold with less assumptions on the domain. In
fact, in [18] it is established the validity of the BBM-formula for any open
set $\Omega\subset\mathbb{R}^{n}$, and recently, in [12], for any bounded
domain. This analysis was completed in [9, 23], where it was proven that a
similar formula holds for functions of bounded variation when
$\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz set.

Motivated with the results in [6], Maz’ya and Shaposhnikova complemented the
study by analyzing the behavior of the seminorm as $s\downarrow 0$. In fact,
the authors proved in [20] that for any $n\geq 1$ and $p\in[1,\infty)$

$\lim_{s\downarrow
0}s\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy=\mathcal{C}_{p,n}\|u\|_{p}^{p}$

whenever $u\in D^{s,p}(\mathbb{R}^{n})$ for some $s\in(0,1)$ where
$D^{s,p}(\mathbb{R}^{n})$ is the completion of
$C^{\infty}_{c}(\mathbb{R}^{n})$ with respect to the Gagliardo seminorm. The
constant $\mathcal{C}_{p,n}$ is given by

$\mathcal{C}_{p,n}=\frac{4\pi^{\frac{n}{2}}}{p\Gamma(\tfrac{n}{2})},$

where $\Gamma$ denotes the Gamma function.

The singular limits mentioned above are natural and have a physical relevance
in the framework of the theory of Lévy processes. This has led to the fact
that in the last years, a huge effort in trying to extend the asymptotic
results as $s\uparrow 1$ and $s\downarrow 0$ proved in [6, 20] to different
contexts has been carried out. We mention just some examples: for the theory
of fractional $s-$perimeters, the analysis of the asymptotic limits was
addressed in [5, 11]; the extension to functions allowing a behavior more
general than a power was done in [2, 3, 7, 15] in the context of fractional
Orlicz-Sobolev spaces; in the magnetic setting, the behavior of the
corresponding seminorms was studied [21, 22]; the extension of magnetic
fractional Orlicz-Sobolev spaces was dealt in [17, 19].

The purpose of this paper is to study the asymptotic behavior as $s\uparrow 1$
and $s\downarrow 0$ of anisotropic Gagliardo seminorms, that is, the quantity

$J_{m,s}(u):=\frac{1-s}{p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,x-y)\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy,$

where $m$ is a function bounded away from 0 and infinity satisfying some
suitable conditions (see hypotheses ($H_{1}$)–($H_{3}$) below).

Let us conclude this section by describing our main results. In Theorem 3.4 we
prove that given $u\in L^{p}(\mathbb{R}^{n})$ fixed,

$\lim_{s\uparrow 1}J_{m,s}(u)=\int_{\mathbb{R}^{n}}\mathcal{A}(x,\nabla
u)\,dx$

where

$\mathcal{A}(x,\xi)=\frac{1}{p}\int_{\mathbb{S}^{n-1}}a(x,w)|\xi\cdot
w|^{p}\,d\mathcal{H}^{n-1}$

and $a(x,\omega)$ is a radial limit of the weight function $m$ (see
($H_{3}$)).

In Theorem 3.5 we also treat the case of a sequence, i.e., the behavior of
$J_{m,s}(u_{s})$ as $s\uparrow 1$, where $\\{u_{s}\\}_{s}$ is a sequence of
functions in $L^{p}(\mathbb{R}^{n})$ such that
$(1-s)[u_{s}]_{s,p}^{p}+\|u_{s}\|_{p}^{p}$ is uniformly bounded.

When $u\in W^{s_{0},p}(\mathbb{R}^{n})$ for some $s_{0}\in(0,1)$, then in
Theorem 4.1 we prove that

$\lim_{s\downarrow 0}sJ_{m,s}(u)=\int_{\mathbb{R}^{n}}|u|^{p}b(x)\,dx,$

where $b(x)=\lim_{s\downarrow 0}b_{s}(x)$ a.e. $x\in\mathbb{R}^{n}$, and

$b_{s}(x):=2s\int_{\mathbb{S}^{n-1}}\int_{2|x|}^{\infty}\frac{m(x,r\omega)}{r^{sp+1}}\,dr\,d\mathcal{H}^{n-1}.$

In the last part of the paper we analyze whether homogenization and
localization processes can be interchanged. To be more precise, observe that
the functional $J_{m,s}$ of a function $u_{s}\in W^{s,p}_{0}(\Omega)$,
$\Omega\subset\mathbb{R}^{n}$, is related with weak solutions to

$\mathcal{L}_{m,s}u_{s}=f\quad\text{ in }\Omega$

where $f\in L^{p^{\prime}}(\Omega)$ and $\mathcal{L}_{m,s}$ is the Fréchet
derivative of $J_{m.s}$, i.e.,

$\mathcal{L}_{m,s}(u)=p.v.(1-s)\int_{\mathbb{R}^{n}}m(x,x-y)\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy.$

In Section 6 we consider solutions of a family of kernels
$m_{\varepsilon}(x,x-y)$, $\varepsilon>0$ having the form
$m_{\varepsilon}(x,x-y)=m(\tfrac{x}{\varepsilon},x-y)$, where $m(x,x-y)$
fullfills the previous assumptions and it is further a $Q-$periodic function
in the first variable, being $Q$ the unit cube in $\mathbb{R}^{n}$. Given a
solution of $u_{s,\varepsilon}\in W^{1,p}_{0}(\Omega)$ to
$\mathcal{L}_{m_{\varepsilon},s}u_{\varepsilon,s}=f$ in $\Omega$, in
Proposition 6.1 we prove that

$\lim_{\varepsilon\downarrow 0}\left(\lim_{s\uparrow
1}u_{s,\varepsilon}\right)=u^{*},$

(in the $L^{p}(\Omega)$ sense) where $u^{*}\in W^{1,p}_{0}(\Omega)$ is the
weak solution of

$-\mathop{\text{div}}(\nabla_{\xi}\mathcal{A}^{*}(\nabla
u^{*}))=f\quad\text{in }\Omega\qquad\text{ with
}\mathcal{A}^{*}(\xi)=\inf_{v\in
W^{1,1}_{per}(Q)}\int_{Q}\mathcal{A}(y,\xi+\nabla v(y))\,dy.$

On the other hand, in Propisition 6.2 we get that

$\lim_{s\uparrow 1}\left(\lim_{\varepsilon\downarrow
0}u_{s,\varepsilon}\right)=\bar{u}$

(in the $L^{p}(\Omega)$ sense), where $\bar{u}\in W^{1,p}_{0}(\Omega)$ is the
solution to

$-\mathop{\text{div}}(\nabla_{\xi}\bar{\mathcal{A}}(\nabla\bar{u}))=f\quad\text{in
}\Omega,\qquad\text{ with
}\bar{\mathcal{A}}(\xi)=\int_{Q}\mathcal{A}(y,\xi)\,dy.$

This shows that in general, homogenization and localization do not commute.

### Organization of the paper

After this introduction, in Section 2, we collect some preliminaries, and
establish some notation that will be used in the sequel.

In Section 3, we analyze the problems for $s\uparrow 1$, the so-called BBM-
type results in the spirit of Bourgains-Brezis-Mironescu’s paper [6].

In Section 4, we analyze the problem $s\downarrow 0$, the MS-type results, in
the spirit of Maz’ya-Shaposhnikova’s paper [20].

In Section 5 we connect the BBM-type results of Section 3 with the asymptotic
behavior of solutions to nonlocal problems and the transition to solutions to
local ones.

Finally, in Section 6, we investigate the interplay between localization (i.e.
$s\uparrow 1$) and homogenization.

##  2\. Preliminaries

###  2.1. Fractional order Sobolev spaces

Throughout this article we will use the standard Gagliardo definition of
fractional order Sobolev spaces. That is: Given a fractional parameter
$s\in(0,1)$ and an integrable parameter $p\in[1,\infty)$, the fractional order
Sobolev space, $W^{s,p}(\mathbb{R}^{n})$ is defined as

$W^{s,p}(\mathbb{R}^{n}):=\left\\{u\in
L^{p}(\mathbb{R}^{n})\colon[u]_{s,p}^{p}<\infty\right\\},$

where $[\,\cdot\,]_{s,p}$ is the so-called Gagliardo seminorm that is defined
as

$[u]_{s,p}^{p}:=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dxdh.$

This space is endowed with the norm

$\|u\|_{s,p}=\left(\|u\|_{p}^{p}+[u]_{s,p}^{p}\right)^{\frac{1}{p}}$

and $(W^{s,p}(\mathbb{R}^{n}),\|\cdot\|_{s,p})$ is a separable Banach space,
that is reflexive if $p>1$.

When considering domains $\Omega\subset\mathbb{R}^{n}$ we will use the
notation $W^{s,p}_{0}(\Omega)$ to denote the set of functions in
$W^{s,p}(\mathbb{R}^{n})$ that vanishes outside $\Omega$, namely

$W^{s,p}_{0}(\Omega):=\\{u\in W^{s,p}(\mathbb{R}^{n})\colon u=0\text{ a.e. in
}\mathbb{R}^{n}\setminus\Omega\\}.$

Observe that this space agrees with the closure of test functions in $\Omega$
if, for instance, $\Omega$ has Lipschitz boundary or if $s<\tfrac{1}{p}$.

For these spaces, the Rellich-Kondrashov compactness result holds true, i.e.

######  Theorem 2.1.

Assume that $s\in(0,1)$ and $p\in[1,\infty)$ and let
$\\{u_{k}\\}_{k\in\mathbb{N}}\subset W^{s,p}_{0}(\Omega)$ be a bounded
sequence. Then, there exists $u\in W^{s,p}_{0}(\Omega)$ and a subsequence
$\\{u_{k_{j}}\\}_{j\in\mathbb{N}}\subset\\{u_{k}\\}_{k\in\mathbb{N}}$ such
that

$u_{k_{j}}\to u\quad\text{ in }L^{p}_{\text{loc}}(\Omega).$

If $\Omega$ is bounded, the convergence is in $L^{p}(\Omega)$.

All of the above mentioned results are well known and can be found, for
instance, in [10].

###  2.2. Some notation

In several places of the paper, the following notation will be used:

  * •

The unit sphere in $\mathbb{R}^{n}$ will be denoted by $\mathbb{S}^{n-1}$.

  * •

The $(n-1)-$dimensional Hausdorff measure will be denoted by
$\mathcal{H}^{n-1}$.

  * •

The volume of the unit ball in $\mathbb{R}^{n}$ will be denoted by
$\omega_{n}$.

  * •

The volume of the $(n-1)-$dimensional unit sphere in $\mathbb{R}^{n}$ is then
$n\omega_{n}$.

###  2.3. Anisotropic fractional energies

We consider a kernel function $m=m(x,h)$, $m\in
L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ and for each function $m$ and
each fractional parameter $s\in(0,1)$ we define the functional $J_{m,s}\colon
W^{s,p}_{0}(\Omega)\to\mathbb{R}$,

$J_{m,s}(u):=\frac{(1-s)}{p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dxdh,$

where $\Omega$ is a domain in $\mathbb{R}^{n}$, not necessarily bounded.

If the kernel function $m$ is bounded below away from 0, the functional
$J_{m,s}$ is coercive, so we impose the following condition on $m$:

($H_{1}$) $m_{-}\leq m(x,h)\leq m_{+}$

for some $0<m_{-}\leq m_{+}<\infty$.

It is easy to see that $J_{m,s}$ is Fréchet differentiable. If we try to
obtain an integral representation of the derivative $J_{m,s}^{\prime}(u)\in
W^{-s,p^{\prime}}(\Omega)$, we need to impose some symmetry assumptions on the
kernel $m$, namely,

($H_{2}$) $m(x,h)=m(x-h,-h).$

Under this condition, it is easy to see (see for instance [14]), that the
derivative $J_{m,s}^{\prime}(u)$ has the following integral representation,

(2.1) $\begin{split}J^{\prime}_{m,s}(u):=&\mathcal{L}_{m,s}(u)\\\
=&p.v.(1-s)\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p-2}(u(x)-u(x-h))}{|h|^{n+sp}}\,dh,\end{split}$

where $p.v.$ stands for in principal value.

Observe that hypotheses ($H_{2}$) is by no means restrictive, since denoting

$m_{\text{sym}}(x,h)=\frac{m(x,h)+m(x-h,-h)}{2}$

we have that $m_{\text{sym}}$ satisfies ($H_{2}$) and

$J_{m,s}=J_{m_{\text{sym}},s}.$

In order to analyze the case where $s\uparrow 1$ in our functionals $J_{m,s}$
we need to assume some asymptotic behavior on the kernel $m$. This condition,
though it seems quite technical right now it will become apparent later on:

There exists a function $a:\mathbb{R}^{n}\times\mathbb{S}^{n-1}\to\mathbb{R}$
such that

($H_{3}$) $m(x,r\omega)=a(x,\omega)+O(r)$

uniformly in $\omega\in\mathbb{S}^{n-1}$.

This condition is saying that $m$ has some singular behavior on the diagonal
that is determined by the angle in which one approaches the origin.

Observe that $m\in C^{1}$ implies that the limit function $a$ in ($H_{3}$) is
independent of $\omega$. In fact, $a(x,\omega)=m(x,0)$ in this case and a
typical nontrivial example to keep in mind is the following

$m(x,h)=\left|M(x,h)\frac{h}{|h|}\right|^{\alpha},\ \alpha\neq 0$

where $M(x,h)\in\mathbb{R}^{n\times n}$ is a symmetric uniformly elliptic
matrix with the structural hypothesis

$M(x,h)=M(x-h,-h).$

In this case, the function $a(x,\omega)$ is given by

$a(x,\omega)=\left|M(x,0)\omega\right|^{\alpha}.$

##  3\. Limit as $s\uparrow 1$ of $J_{m,s}$

The purpose of this section is to analyze the behavior as $s\uparrow 1$ of the
functional $J_{m,s}$. This is the extension of the celebrated result of
Bourgain-Brezis-Mironescu to the anisotropic case.

First we begin by studying the pointwise limit of the funcionales that is much
simpler. Later on, we will deal with the Gamma-convergence of the funcional
that is more subtle.

###  3.1. Pointwise limit

The results in this subsection are inspired by [1] where the authors consider
some particular case of weight function for $p=2$.

To begin with, we cite a Lemma that can be found in [6].

######  Lemma 3.1.

Given $u\in W^{s,p}_{0}(\Omega)$ it holds that

$\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dh\leq
m_{+}[u]_{s,p}^{p}\leq\frac{n\omega_{n}m_{+}}{p}\left(\frac{1}{1-s}\|\nabla
u\|_{p}^{p}+\frac{2^{p}}{s}\|u\|_{p}^{p}\right).$

###### Proof.

Just combine ($H_{1}$) with [6, Theorem 1]. ∎

The following proposition is key in the proof of our main result.

######  Proposition 3.2.

Given $u\in C^{2}_{c}(\mathbb{R}^{n})$ and a fixed $x\in\mathbb{R}^{n}$ we
have that

$\lim_{s\uparrow
1}(1-s)\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dh=\mathcal{A}(x,\nabla
u)$

where $\mathcal{A}(x,\xi)$ is given by

(3.1) $\mathcal{A}(x,\xi)=\frac{1}{p}\int_{\mathbb{S}^{n-1}}a(x,w)|\xi\cdot
w|^{p}\,d\mathcal{H}^{n-1}.$

######  Remark 3.3.

In the linear case, that is when $p=2$, the operator $\mathcal{A}(x,\xi)$ has
a more explicit form,

$\mathcal{A}(x,\xi)=A(x)\xi\cdot\xi,$

where the matrix $A\in\mathbb{R}^{n\times n}$ is given by

$a_{ij}(x)=\frac{1}{2}\int_{\mathbb{S}^{n-1}}w_{i}w_{j}a(x,w)\,d\mathcal{H}^{n-1}.$

###### Proof of Proposition 3.2.

For each fixed $x\in\mathbb{R}^{n}$ we split the integral

$\displaystyle\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dh$
$\displaystyle=\left(\int_{|h|\geq
1}+\int_{|h|<1}\right)m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dh$
$\displaystyle=I_{1}+I_{2}.$

Since ($H_{1}$) holds,

$|I_{1}|\leq 2^{p-1}m_{+}\|u\|_{\infty}^{p}\int_{|h|\geq
1}\frac{1}{|h|^{n+sp}}\,dh<\infty,$

and we focus only on $I_{2}$. Since $x\mapsto|x|^{p}$ is locally Lipschitz and
$u\in C^{2}$, we have that

$\left|\frac{|u(x)-u(x-h)|^{p}}{|h|^{sp}}-\frac{|\nabla u(x)\cdot
h|^{p}}{|h|^{sp}}\right|\leq L\frac{|u(x)-u(x-h)-\nabla u(x)\cdot
h|}{|h|^{s}}\leq C|h|^{2-s}$

where $C$ depends of the $C^{2}-$norm of $u$.

Since the following integral vanishes

$\lim_{s\uparrow 1}(1-s)\int_{|h|\leq 1}|h|^{2-s-n}\,dh=\lim_{s\uparrow
1}(1-s)\frac{n\omega_{n}}{2-s}=0,$

it follows that

$\displaystyle\lim_{s\uparrow 1}(1-s)I_{2}$ $\displaystyle=\lim_{s\uparrow
1}(1-s)\int_{|h|\leq 1}m(x,h)\frac{|\nabla u(x)\cdot h|^{p}}{|h|^{n+sp}}\,dh$

Hence, by using polar coordinates we get

$\displaystyle\int_{|h|\leq 1}m(x,h)\frac{|\nabla u(x)\cdot
h|^{p}}{|h|^{n+sp}}\,dh$ $\displaystyle=\int_{|h|\leq 1}m(x,h)\frac{|\nabla
u(x)\cdot\tfrac{h}{|h|}|^{p}}{|h|^{n+sp-p}}\,dh$
$\displaystyle=\int_{0}^{1}\int_{\mathbb{S}^{n-1}}m(x,r\omega)|\nabla
u(x)\cdot\omega|^{p}r^{p(1-s)-1}\,d\mathcal{H}^{n-1}\,dr$
$\displaystyle=\int_{\mathbb{S}^{n-1}}|\nabla
u(x)\cdot\omega|^{p}\left(\int_{0}^{1}m(x,r\omega)r^{p(1-s)-1}\,dr\right)\,d\mathcal{H}^{n-1}.$

Observe that from ($H_{3}$) we have that $m(x,r\omega)=a(x,\omega)+O(r)$,
which implies that

$\displaystyle\int_{0}^{1}m(x,r\omega)r^{p(1-s)-1}\,dr$
$\displaystyle=\int_{0}^{1}a(x,\omega)r^{p(1-s)-1}\,dr+\int_{0}^{1}O(r^{p(1-s)})\,dr$
$\displaystyle=a(x,\omega)\frac{1}{p(1-s)}+O(1)$

and consequently

$\lim_{s\uparrow
1}(1-s)\int_{0}^{1}m(x,r\omega)r^{p(1-s)-1}\,dr=\frac{1}{p}a(x,\omega).$

Finally,

$\displaystyle\lim_{s\uparrow 1}(1-s)$ $\displaystyle\int_{|h|\leq
1}m(x,h)\frac{|\nabla u(x)\cdot h|^{p}}{|h|^{n+sp}}\,dh$
$\displaystyle=\lim_{s\uparrow 1}(1-s)\int_{\mathbb{S}^{n-1}}|\nabla
u(x)\cdot\omega|^{p}\left(\int_{0}^{1}m(x,r\omega)r^{p(1-s)-1}\,dr\right)\,d\mathcal{H}^{n-1}$
$\displaystyle=\frac{1}{p}\int_{\mathbb{S}^{n-1}}a(x,\omega)|\nabla
u(x)\cdot\omega|^{p}\,d\mathcal{H}^{n-1},$

which concludes the proof. ∎

We are ready to state and proof our main result in this subsection.

######  Theorem 3.4.

Given $u\in L^{p}(\mathbb{R}^{n})$ and a fixed $x\in\mathbb{R}^{n}$ we have
that

$\lim_{s\uparrow
1}(1-s)\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dxdh=\int_{\mathbb{R}^{n}}\mathcal{A}(x,\nabla
u)\,dx$

where $\mathcal{A}(x,\xi)$ is given in (3.1).

###### Proof.

Given $u\in C^{2}_{c}(\mathbb{R}^{n})$ with $\text{supp}(u)\subset B_{R}(0)$,
in view of Proposition 3.2, it only remains to show the existence of an
integrable majorant for $(1-s)F_{s}$, where $F_{s}$ is given by

$F_{s}(x):=\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dh.$

But, thanks to ($H_{1}$), this task is the same done in [6], more precisely,

$(1-s)|F_{s}(x)|\leq
Cm_{+}\left(\chi_{B_{R}(0)}(x)+|x|^{-(n+\tfrac{1}{2})}\chi_{B_{r}(0)^{c}}(x)\right)\in
L^{1}(\mathbb{R}^{n}).$

Then, from Proposition 3.2 and the Dominated Convergence Theorem the result
follows for any $u\in C^{2}_{c}(\mathbb{R}^{n})$.

Using Lemma 3.1 and [6, Theorem 2], the result is extended to an arbitrary
function $u\in W^{1,p}(\mathbb{R}^{n})$.

Finally, arguing as in [6, Theorem 2] (see also [15]) it holds that if

$\liminf_{s\uparrow
1}(1-s)\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dxdh<\infty,$

then $u\in W^{1,p}(\mathbb{R}^{n})$ and the result follows. ∎

###  3.2. The case of a sequence

In this subsection we deal with the case of a sequence that will imply, among
other things, the Gamma convergence of the functionals $J_{m,s}$.

######  Theorem 3.5.

Let $0\leq s_{k}\ \uparrow 1$ and $\\{u_{k}\\}_{k\in\mathbb{N}}\subset
L^{p}(\mathbb{R}^{n})$ be such that

$\sup_{k\in\mathbb{N}}(1-s_{k})[u_{k}]_{s_{k},p}^{p}<\infty\quad\text{and}\quad\sup_{k\in\mathbb{N}}\|u_{k}\|_{L^{p}(\mathbb{R}^{n})}<\infty.$

Then there exists $u\in L^{p}(\mathbb{R}^{n})$ and a subsequence
$\\{u_{k_{j}}\\}_{j\in\mathbb{N}}\subset\\{u_{k}\\}_{k\in\mathbb{N}}$ such
that $u_{k_{j}}\to u$ in $L^{p}_{loc}(\mathbb{R}^{n})$. Moreover $u\in
W^{1,p}(\mathbb{R}^{n})$ and the following estimate holds

$\int_{\mathbb{R}^{n}}\mathcal{A}(x,\nabla
u)\,dx\leq\liminf_{k\to\infty}J_{m,s_{k}}(u_{k}).$

The proof of the above result will be a direct consequence of the following
useful estimate:

######  Theorem 3.6.

Let $0<s_{1}<s_{2}<1$ and $u\in L^{p}(\mathbb{R}^{n})$. Then

$\displaystyle J_{m,s_{1}}(u)\leq
2^{p(1-s_{1})}J_{m,s_{2}}(u)+\frac{2^{p-1}m_{+}n\omega_{n}(1-s_{1})}{s_{1}}\|u\|_{p}^{p}.$

The key point in proving Theorem 3.6 is the following lemma that is proved in
[6].

######  Lemma 3.7 (Lemma 2, [6]).

Let $g,h:(0,1)\to\mathbb{R}^{+}$ measurable functions. Suppose that for some
constant $c>0$ it holds that $g(t)\leq cg(\tfrac{t}{2})$ for $t\in(0,1)$ and
that $h$ is decreasing. Then, given $r>-1$,

$\int_{0}^{1}t^{r}g(t)h(t)\,dt\geq\frac{r+1}{2^{r+1}}\int_{0}^{1}t^{r}g(t)\,dt\int_{0}^{1}t^{r}h(t)\,dt.$

Actually, the proof in [6] is done with $c=1$. The extension for general $c>0$
is immediate.

Now we proceed with the proof of the estimate.

###### Proof of Theorem 3.6.

The proof is very similar to that of [6, Theorem 4] (see also [15, Theorem
5.1]). We include some details in order to make the paper self contained.

Given $u\in L^{p}(\mathbb{R}^{n})$, we define for $t>0$ and $0<s<1$,

$\displaystyle F(t)$
$\displaystyle=\int_{\mathbb{S}^{n-1}}\int_{\mathbb{R}^{n}}m(x,tw)|u(x)-u(x-tw)|^{p}\,dx\,d\mathcal{H}^{n-1}$
$\displaystyle=\frac{1}{t^{n-1}}\int_{|h|=t}\int_{\mathbb{R}^{n}}m(x,h)|u(x)-u(x-h)|^{p}\,dx\,d\mathcal{H}^{n-1}$

and $g(t)=\frac{F(t)}{t^{p}}$.

From [6, p. 13] and assumption ($H_{1}$) if follows that

$g(2t)\leq\frac{m_{+}}{m_{-}}g(t).$

Then, observe that

(3.2)
$\displaystyle\begin{split}\int_{|h|<1}\int_{\mathbb{R}^{n}}&m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}dx\,dh\\\
&=\int_{0}^{1}\int_{|h|=t}\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{t^{n+sp}}dx\,d\mathcal{H}^{n-1}\,dt\\\
&=\int_{0}^{1}\frac{F(t)}{t^{1+sp}}\,dt=\int_{0}^{1}\frac{g(t)}{t^{1-p(1-s)}}\,dt.\end{split}$

Consider now $0<s_{1}<s_{2}<1$. Therefore,

$\displaystyle\int_{0}^{1}\frac{1}{t^{1-p(1-s_{2})}}g(t)\,dt=\int_{0}^{1}\frac{1}{t^{1-p(1-s_{1})}}g(t)\frac{1}{t^{p(s_{2}-s_{1})}}\,dt.$

Now, from Lemma 3.7 with $r=p(1-s_{1})-1$ and $h(t)=t^{-p(s_{2}-s_{1})}$ we
get

(3.3)
$\begin{split}\int_{0}^{1}\frac{1}{t^{1-p(1-s_{2})}}g(t)\,dt\geq&\frac{p(1-s_{1})}{2^{p(1-s_{1})}}\int_{0}^{1}\frac{1}{t^{1-p(1-s_{1})}}g(t)\,dt\int_{0}^{1}\frac{1}{t^{1-p(1-s_{2})}}\,dt\\\
=&\frac{1}{2^{p(1-s_{1})}}\frac{1-s_{1}}{1-s_{2}}\int_{0}^{1}\frac{1}{t^{1-p(1-s_{1})}}g(t)\,dt.\end{split}$

From (3.2) and (3.3) we deduce that

(3.4)
$\displaystyle\begin{split}\frac{(1-s_{1})}{2^{p(1-s_{1})}}&\int_{|h|<1}\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+s_{1}p}}\,dxdh\\\
&\leq(1-s_{2})\int_{\\{|h|<1\\}}\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+s_{2}p}}\,dxdh.\end{split}$

Finally we observe that

$\displaystyle\int_{\\{|h|\geq
1\\}}\int_{\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dxdh$
$\displaystyle\leq
2^{p}m_{+}n\omega_{n}\|u\|_{p}^{p}\int_{1}^{\infty}\frac{1}{t^{1+sp}}\,dt$
$\displaystyle=\frac{2^{p}m_{+}n\omega_{n}}{sp}\|u\|_{p}^{p}.$

The proof concludes combining this last inequality with (3.4). ∎

Now we can proceed with the proof of Theorem 3.5.

###### Proof of Theorem 3.5.

With the help of Theorem 3.6 the proof is the same as [6, Theorem 4] and [15,
Theorem 5.1]

We include some details for the reader’s convenience.

Let $0<s_{k}\uparrow 1$ and $\\{u_{k}\\}_{k\in\mathbb{N}}\subset
L^{p}(\mathbb{R}^{n})$ such that

$\sup_{k\in\mathbb{N}}(1-s_{k})[u_{k}]_{s_{k},p}^{p}<\infty\quad\text{and}\quad\sup_{k\in\mathbb{N}}\|u_{k}\|_{p}<\infty.$

For a fixed $0<t<1$ using Theorem 3.6 we have that
$\\{u_{k}\\}_{k\in\mathbb{N}}\subset W^{t,p}(\mathbb{R}^{n})$ is bounded and
so by the Rellich-Kondrashov compactness Theorem (Theorem 2.1), there exists a
subsequence (still denoted by $\\{u_{k}\\}_{k\in\mathbb{N}}$) and a limit
function $u\in L^{p}(\mathbb{R}^{n})$ such that $u_{k}\to u$ in
$L^{p}_{\text{loc}}(\mathbb{R}^{n})$. We can also assume that $u_{k}\to u$
a.e. in $\mathbb{R}^{n}$.

Now, by Fatou’s Lemma, we have

$\displaystyle\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)$
$\displaystyle\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+tp}}\,dxdy$
$\displaystyle\leq\liminf_{k\to\infty}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)\frac{|u_{k}(x)-u_{k}(x-h)|^{p}}{|h|^{n+tp}}\,dxdy$

and by Theorem 3.6 we obtain

$\displaystyle\frac{1-t}{2^{(1-t)p}}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}$
$\displaystyle m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+tp}}\,dxdy$
$\displaystyle\leq$
$\displaystyle\liminf_{k\to\infty}(1-s_{k})\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)\frac{|u_{k}(x)-u_{k}(x-h)|^{p}}{|h|^{n+s_{k}p}}\,dxdy$
$\displaystyle+\frac{n\omega_{n}2^{p}(1-t)m_{+}}{tp}M$

wher $M=\sup_{k\in\mathbb{N}}\|u_{k}\|_{p}^{p}$.

Finally the result follows taking the limit $t\uparrow 1$ and using Theorem
3.4. ∎

##  4\. Limit as $s\downarrow 0$ of $J_{m,s}$

In this section we analyze the limit case where $s\downarrow 0$ of the
functionals $J_{m,s}$. This is what is called a Maz’ya-Shaposhnikova type
result after the results obtained in [20]. That is we are interested in
studying the limit

$\lim_{s\downarrow 0}sJ_{m,s}(u).$

First we define the following weights depending on $m$ and $s$,

$b_{s}(x):=2s\int_{\mathbb{S}^{n-1}}\int_{2|x|}^{\infty}\frac{m(x,r\omega)}{r^{sp+1}}\,dr\,d\mathcal{H}^{n-1}.$

Observe that this weight has the following bounds

$\frac{2^{1-sp}n\omega_{n}}{p}\frac{m_{-}}{|x|^{sp}}\leq
b_{s}(x)\leq\frac{2^{1-sp}n\omega_{n}}{p}\frac{m_{+}}{|x|^{sp}},$

where $m_{\pm}$ are given in ($H_{1}$).

We will assume that there exists the limit function

$b(x)=\lim_{s\downarrow 0}b_{s}(x)\quad\text{a.e. }x\in\mathbb{R}^{n}.$

Our main result in the section is

######  Theorem 4.1.

Under the above assumptions and notations, if $u\in
W^{s_{0},p}(\mathbb{R}^{n})$ for some $s_{0}\in(0,1)$, then

$\lim_{s\downarrow 0}sJ_{m,s}(u)=\int_{\mathbb{R}^{n}}|u|^{p}b(x)\,dx.$

The proof of this result follows the general strategy developed in [20] but
also applies some ideas from [4].

The proof will be a direct consequence of the next two lemmas. The first one
is a Hardy-type inequality with weights

######  Lemma 4.2.

Let $u\in W^{s_{0},p}(\mathbb{R}^{n})$ for some $s_{0}\in(0,1)$, then

$\liminf_{s\downarrow 0}sJ_{m,s}(u)\geq\int_{\mathbb{R}^{n}}|u|^{p}b(x)\,dx.$

###### Proof.

Let us call

$I^{p}=\int_{\mathbb{R}^{n}}\int_{|h|>2|x|}m(x,h)\frac{|u(x)|^{p}}{|h|^{n+sp}}\,dhdx.$

Then

$\displaystyle I^{p}$ $\displaystyle=\int_{\mathbb{R}^{n}}\left(\int_{|h|\geq
2|x|}\frac{m(x,h)}{|h|^{n+sp}}dh\right)|u(x)|^{p}\,dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{S}^{n-1}}\int_{2|x|}^{\infty}\frac{m(x,r\omega)}{r^{1+sp}}\,drd\mathcal{H}^{n-1}\right)|u(x)|^{p}\,dx.$

Now, for any $\varepsilon>0$,

$\displaystyle I^{p}\leq$
$\displaystyle(1+\varepsilon)^{p-1}\int_{\mathbb{R}^{n}}\int_{|h|\geq
2|x|}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle+\left(\frac{1+\varepsilon}{\varepsilon}\right)^{p-1}\int_{\mathbb{R}^{n}}\int_{|h|\geq
2|x|}m(x,h)\frac{|u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$ $\displaystyle:=$
$\displaystyle(1+\varepsilon)^{p-1}(a)^{p}+\left(\frac{1+\varepsilon}{\varepsilon}\right)^{p-1}(b)^{p}.$

Let us first bound $(b)$. As $|h|>2|x|$ we have that
$\frac{2}{3}|x-h|<|h|<2|x-h|$. Hence

$\displaystyle(b)^{p}$ $\displaystyle=\int_{\mathbb{R}^{n}}\int_{|h|\geq
2|x|}\frac{|u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle\leq\left(\frac{3}{2}\right)^{n+sp}\int_{\mathbb{R}^{n}}\int_{\frac{2}{3}|y|<|x-y|<2|y|}\frac{|u(y)|^{p}}{|y|^{n+sp}}\,dydx$
$\displaystyle=\left(\frac{3}{2}\right)^{n+sp}\int_{\mathbb{R}^{n}}\frac{|u(y)|^{p}}{|y|^{n+sp}}\left(\int_{\frac{2}{3}|y|<|x-y|<2|y|}\,dx\right)dy$
$\displaystyle\leq\left(\frac{3}{2}\right)^{n+sp}\int_{\mathbb{R}^{n}}\frac{|u(y)|^{p}}{|y|^{n+sp}}n\omega_{n}2^{n}|y|^{n}dy$
$\displaystyle=n\omega_{n}\frac{3^{n+sp}}{2^{sp}}\int_{\mathbb{R}^{n}}\frac{|u(y)|^{p}}{|y|^{sp}}dy.$

Observe that this last quantity is finite by Hardy’s inequality.

For $(a)$, we observe that, changing variables,

$\displaystyle(a)^{p}$ $\displaystyle=\int_{\mathbb{R}^{n}}\int_{|h|\geq
2|x|}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{|h|\geq
2|x-h|}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx:=(\tilde{a})^{p},$

where we have used the symmetry assumption ($H_{2}$).

Observe that the sets $\\{|h|\geq 2|x-h|\\}$ and $\\{|h|\geq 2|x|\\}$ are
disjoints, so

$2(a)^{p}=(a)^{p}+(\tilde{a})^{p}\leq\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx.$

Next, observe that

$\displaystyle sJ_{m,s}(u)$ $\displaystyle\geq 2s(a)^{p}\geq
2s\left[\frac{1}{(1+\varepsilon)^{p-1}}I^{p}-\frac{1}{\varepsilon^{p-1}}(b)^{p}\right]$
$\displaystyle\geq\frac{1}{(1+\varepsilon)^{p-1}}\int_{\mathbb{R}^{n}}|u(x)|^{p}b_{s}(x)\,dx-\frac{2s}{\varepsilon^{p-1}}n\omega_{n}\frac{3^{n+sp}}{2^{sp}}\int_{\mathbb{R}^{n}}\frac{|u(y)|^{p}}{|y|^{sp}}dy.$

Finally, using Fatou’s Lemma,

$\liminf_{s\downarrow
0}sJ_{m,s}(u)\geq\frac{1}{(1+\varepsilon)^{p-1}}\int_{\mathbb{R}^{n}}|u(x)|^{p}b(x)\,dx,$

for any $\varepsilon>0$ and the result follows. ∎

The next lemma gives us the upper estimate.

######  Lemma 4.3 (Limsup estimate).

For any $u\in W_{0}^{s_{0},p}(\mathbb{R}^{n})$ for some $s_{0}\in(0,1)$, it
holds that

(4.1) $\limsup_{s\downarrow
0}sJ_{m,s}(u)\leq\int_{\mathbb{R}^{n}}|u(x)|^{p}b(x)\,dx.$

###### Proof.

Observe that our symmetry assumption ($H_{2}$) gives that

$\displaystyle\int_{\mathbb{R}^{n}}\int_{|x-h|<|x|}m(x,h)$
$\displaystyle\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{|x-h|>|x|}m(x-h,-h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{|x-h|>|x|}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx.$

Therefore

$\displaystyle\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)$
$\displaystyle\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle=2\int_{\mathbb{R}^{n}}\int_{|x-h|\geq|x|}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx.$

Arguing as in the previous lemma, given $\varepsilon>0$ we get

$\displaystyle sJ_{m,s}(u)=$ $\displaystyle
2s\left(\int_{\mathbb{R}^{n}}\int_{|x-h|\geq
2|x|}+\int_{\mathbb{R}^{n}}\int_{|x|<|x-h|<2|x|}\right)m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle\leq$
$\displaystyle(1+\varepsilon)^{p-1}\int_{\mathbb{R}^{n}}|u(x)|^{p}b_{s}(x)\,dx$
$\displaystyle+2sm^{+}\left[\left(\frac{1+\varepsilon}{\varepsilon}\right)^{p-1}\int_{\mathbb{R}^{n}}\int_{|x-h|\geq
2|x|}\frac{|u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx\right.$
$\displaystyle\qquad\qquad\left.+\int_{\mathbb{R}^{n}}\int_{|x|<|x-h|<2|x|}\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx\right]$
$\displaystyle:=(a)^{p}+2sm^{+}[(b)^{p}+(c)^{p}].$

We need to get uniform, in $s$, bounds on $(b)$ and $(c)$. For $(b)$, we use
that $|x-h|\geq 2|x|$ implies that $|h|>\frac{1}{2}|x-h|$ together with
Fubini’s Theorem to obtain

$\displaystyle(b)^{p}$ $\displaystyle\leq
2^{n+sp}\int_{\mathbb{R}^{n}}\left(\int_{|x-h|\geq|x|}\frac{|u(x-h)|^{p}}{|x-h|^{n+sp}}\,dh\right)dx$
$\displaystyle=2^{sp}n\omega_{n}\int_{\mathbb{R}^{n}}\frac{|u(x)|^{p}}{|x|^{sp}}\,dx<\infty.$

Hardy’s inequality gives us the desired uniform bound on $(b)$ and therefore

$\limsup_{s\to 0^{+}}\;s(b)^{p}=0.$

It remains to bound $(c)$. To begin with, we split the integral into two
parts, one where $|h|$ is large and another one where $|h|$ is bounded.

$\displaystyle(c)^{p}=$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{|x|<|x-h|<2|x|\atop|h|\leq
N}\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle+\int_{\mathbb{R}^{n}}\int_{|x|<|x-h|<2|x|\atop|h|>N}\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle:=$ $\displaystyle(c_{1})^{p}+(c_{2})^{p}.$

To bound $(c_{1})$ we proceed as follows

$\displaystyle(c_{1})^{p}$ $\displaystyle\leq
N^{p(\tau-s)}\int_{\mathbb{R}^{n}}\int_{|x|<|x-h|<2|x|\atop|h|\leq
N}\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+\tau p}}\,dhdx$ $\displaystyle\leq
N^{p(\tau-s)}[u]_{\tau,p}^{p},$

where $\tau>s$ is fixed. From this expression,

$\limsup_{s\to 0^{+}}\;s(c_{1})^{p}=0.$

It remains to get a bound for $(c_{2})$. First, we observe that, as
$|x|<|x-h|<2|x|$ and $|h|>N$, it follows that $|x|>\frac{N}{3}$ and
$|x-h|>\frac{N}{3}$. Hence

$\displaystyle(c_{2})^{p}$ $\displaystyle\leq
2^{p-1}\int_{\mathbb{R}^{n}}\int_{|x|<|x-h|<2|x|\atop|h|>N}\frac{|u(x)|^{p}}{|h|^{n+sp}}\,dhdx+2^{p-1}\int_{\mathbb{R}^{n}}\int_{|x|<|x-h|<2|x|\atop|h|>N}\frac{|u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle\leq
2^{p-1}\int_{\mathbb{R}^{n}}\int_{|x|>N/3\atop|h|>N}\frac{|u(x)|^{p}}{|h|^{n+sp}}\,dydx+2^{p-1}\int_{\mathbb{R}^{n}}\int_{|x-h|>N/3\atop|h|>N}\frac{|u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx$
$\displaystyle=2^{p}\int_{\mathbb{R}^{n}}\int_{|x|>N/3\atop|h|>N}\frac{|u(x)|^{p}}{|h|^{n+sp}}\,dydx$
$\displaystyle=2^{p}\int_{|x|>\frac{N}{3}}|u(x)|^{p}\left(n\omega_{n}\int_{N}^{\infty}\frac{r^{n-1}}{r^{n+sp}}\,dr\right)dx$
$\displaystyle=\frac{n\omega_{n}2^{p}}{sp}\frac{1}{N^{sp}}\int_{|x|>\frac{N}{3}}|u(x)|^{p}dx$

from where it follows that

$\limsup_{s\to
0+}s(c_{s})^{p}\leq\frac{n\omega_{n}2^{p}}{p}\int_{|x|>\frac{N}{3}}|u(x)|^{p}dx,$

and this quantity is arbitrary small if $N$ is large. ∎

With the help of Lemmas 4.2 and 4.3 we can deduce the main result of the
section.

###### Proof of Theorem 4.1.

Immediate from Lemmas 4.2 and 4.3. ∎

##  5\. Anisotropic nonlocal and local problems

One application of the results in Section 3 is to analyze the asymptotic
behavior of the solutions to anisotropic nonlocal problems. That is, given a
domain $\Omega\subset\mathbb{R}^{n}$ and a source term $f\in
L^{p^{\prime}}(\Omega)$ one wants to analyze the limit as $s\uparrow 1$ of the
solutions to

(5.1) $\begin{cases}\mathcal{L}_{m,s}u_{s}=f&\text{in }\Omega\\\
u_{s}=0&\text{in }\mathbb{R}^{n}\setminus\Omega,\end{cases}$

where $\mathcal{L}_{m,s}$ is the Fréchet derivative of $J_{m,n}$ given by
(2.1).

In the case where $m=1$ this problem is well understood since the seminal
works of [6] and for some recent results regarding this problem, even in the
semilinear-type case (that is when $f=f(u)$) we refer to [15].

In this general case, the results in Section 3, suggest that the limit problem
for (5.1) when $s\uparrow 1$ is

(5.2) $\begin{cases}\mathcal{L}_{\mathcal{A}}u=f&\text{in }\Omega\\\
u=0&\text{on }\partial\Omega,\end{cases}$

where
$\mathcal{L}_{\mathcal{A}}u=-\mathop{\text{div}}(\nabla_{\xi}\mathcal{A}(x,\nabla
u))$.

Recall that $\mathcal{L}_{\mathcal{A}}$ is the Fréchet derivative of the
functional

$J(u)=\int_{\mathbb{R}^{n}}\mathcal{A}(x,\nabla u)\,dx.$

The results of Section 3, immediately gives:

######  Theorem 5.1.

Assume that $m$ verifies ($H_{1}$)–($H_{3}$) and let $\mathcal{A}$ be defined
by (3.1). Define the functionals $J_{m,s},J\colon
L^{p}(\Omega)\to\bar{\mathbb{R}}$ as

$\displaystyle
J_{m,s}(u)=\begin{cases}\displaystyle\frac{1-s}{p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(x,h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{n+sp}}\,dhdx&\text{if
}u\in W^{s,p}_{0}(\Omega)\\\ \infty&\text{else}\end{cases}$ $\displaystyle
J(u)=\begin{cases}\displaystyle\int_{\mathbb{R}^{n}}\mathcal{A}(x,\nabla
u)\,dx&\text{if }u\in W^{1,p}_{0}(\Omega)\\\ \infty&\text{else.}\end{cases}$

Then $J_{m,s}$ Gamma-converges to $J$ as $s\uparrow 1$.

The definition and properties of Gamma-convergence can be seen in [8], The
proof of Theorem 5.1 is straightforward from Section 3 and the details are
completely analogous as in [16].

The main feature of Gamma-convergence is that it implies the following result

######  Theorem 5.2.

Let $J_{m,s}$ and $J$ be defined as in Theorem 5.1. Then, if $f\in
L^{p^{\prime}}(\Omega)$, there exists a unique minimum $u_{s}\in
W^{s,p}_{0}(\Omega)$ of

$J_{m,s}(v)-\int_{\Omega}fv\,dx$

a unique minimum $u\in W^{1,p}_{0}(\Omega)$ of

$J(v)-\int_{\Omega}fv\,dx$

and $u_{s}\to u$ in $L^{p}(\Omega)$.

Again, the details of Theorem 5.2 with the obvious modifications, can be found
in [16].

As a corollary of Theorem 5.2 we get the connection between the solution to
(5.1) with the solution to (5.2).

######  Corollary 5.3.

For each $s\in(0,1)$, there exists a unique solution $u_{s}\in
W^{s,p}_{0}(\Omega)$ to (5.1). This sequence of solutions
$\\{u_{s}\\}_{s\in(0,1)}$ converges, as $s\uparrow 1$, in $L^{p}(\Omega)$ to
some function $u\in W^{1,p}_{0}(\Omega)$ and this function $u$ is the unique
solution to (5.2).

##  6\. Remarks on homogenization

The purpose of this section is to investigate the simultaneous effect that
localization (i.e. $s\uparrow 1$) and homogenization can have in some
problems.

To be precise, assume that now we have a family of kernels
$m_{\varepsilon}(x,h)$ satisfying ($H_{1}$)–($H_{3}$). Then, for each
$\varepsilon$ when we take $s\uparrow 1$, we obtain a limit function
$\mathcal{A}_{\varepsilon}(x,\xi)$ as defined in (3.1).

The results of the previous section tell us that the solutions
$u_{s,\varepsilon}$ of (5.1) with $m=m_{\varepsilon}$ are converging as
$s\uparrow 1$ to $u_{\varepsilon}$, the solution to (5.2) with
$\mathcal{A}=\mathcal{A}_{\varepsilon}$.

The problem that we want to address is what happens when
$\varepsilon\downarrow 0$.

In order to understand this question, we focus on the model problem where
$m_{\varepsilon}(x,h)$ is obtain from a single kernel $m$ in the form

$m_{\varepsilon}(x,h)=m(\tfrac{x}{\varepsilon},h),$

and $m(x,h)$ is a periodic function in $x$ of period 1 in each $x_{i}$,
$i=1,\dots,n$.

To keep things even simpler, we start with the one dimensional problem.

###  6.1. The one-dimensional case

In the $1-$dimensional case explicit formulas describing the behavior of the
limit problems are available. Indeed, consider a $1-$periodic function in $x$,
$m(x,h)$ satisfying ($H_{1}$)–($H_{3}$). According to Theorem 3.4 we have that

$\lim_{s\uparrow
1}(1-s)\iint_{\mathbb{R}\times\mathbb{R}}m(\tfrac{x}{\varepsilon},h)\frac{|u(x)-u(x-h)|^{p}}{|h|^{1+sp}}\,dxdh=\int_{\mathbb{R}^{n}}\mathcal{A}(\tfrac{x}{\varepsilon},|u^{\prime}(x)|)\,dx=:J^{\varepsilon}(u).$

In this one-dimensional case, the limit function $\mathcal{A}(y,\xi)$ can be
easily computed as

$\mathcal{A}(y,\xi)=\frac{1}{p}\left(a(y,-1)+a(y,1)\right)|\xi|^{p}=:A(y)|\xi|^{p}.$

When $\varepsilon$ vanishes, it is well known that (see [13, Proposition 3.7])
$J^{\varepsilon}$ Gamma-converges to $J^{*}$, where

$J^{*}(u)=\int_{\mathbb{R}}A^{*}|u^{\prime}(x)|^{p}\,dx,$

and $A^{*}$ is a constant coefficient given by

$A^{*}:=\left(\int_{0}^{1}A(t)^{-1/(p-1)}\,dt\right)^{-1/(p-1)}.$

This fact gives as a result that minimizers $u_{s,\varepsilon}$ of (5.1) with
coefficients $m_{\varepsilon}$ verifiy that

$\displaystyle\lim_{\varepsilon\downarrow 0}\left(\lim_{s\uparrow
1}u_{s,\varepsilon}\right)=u^{*},$

where $u^{*}$ is the solution to

(6.1) $\begin{cases}-(A^{*}|u^{\prime}|^{p-2}u^{\prime})^{\prime}=f&\text{in
}\Omega\\\ u=0&\text{on }\partial\Omega.\end{cases}$

On the other hand, it is well-known (see, for instance [14]) that as
$\varepsilon\downarrow 0$ it holds that solutions $u_{s,\varepsilon}$ of (5.1)
with coefficients $m_{\varepsilon}$ converge to the solution $\bar{u}_{s}$ of
(5.1) with coefficient $\bar{m}(h)$ given by

$\bar{m}(h)=\int_{0}^{1}m(t,h)\,dt.$

Finally, applying Theorem 3.4 we arrive at

$\displaystyle\lim_{s\uparrow 1}\left(\lim_{\varepsilon\downarrow
0}u_{s,\varepsilon}\right)=\bar{u},$

where $\bar{u}$ is the solution to the problem

(6.2) $\begin{cases}-(\bar{A}|u^{\prime}|^{p-2}u^{\prime})^{\prime}=f&\text{in
}\Omega\\\ u=0&\text{on }\partial\Omega.\end{cases}$

In this case, $\bar{A}$ is given by

$\bar{A}=\frac{1}{p}(\bar{m}(-1)+\bar{m}(1))=\frac{1}{p}\int_{0}^{1}m(t,-1)+m(t,1)\,dt.$

From these simple formulas one can immediately see that the localization
process and the homogenization process are not interchangeables.

###  6.2. The general case

The computations of the previous subsection can be extended with some care to
the $n-$dimensional case.

Let now $m(x,h)$ be a $Q-$periodic function in its first variable, $Q$ being
the unit cube in $\mathbb{R}^{n}$, and satisfying hypothesis
($H_{1}$)–($H_{3}$).

Let us now state the problem in a precise way.

A function $u_{s,\varepsilon}\in W^{s,p}_{0}(\Omega)$ is a weak solution of

$\begin{cases}\mathcal{L}_{m_{\varepsilon},s}u=f&\mbox{in }\Omega\\\
u=0&\mbox{in }\mathbb{R}^{n}\setminus\Omega,\end{cases}$

if

$\langle\mathcal{L}_{m_{\varepsilon},s}u_{s,\varepsilon},v\rangle=\int_{\Omega}fv\
dx,$

for every $v\in W^{s,p}_{0}(\Omega)$ where $f\in W^{-s,p^{\prime}}(\Omega)$
and $\mathcal{L}_{m_{\varepsilon},s}$ is the Fréchet derivative of
$J_{m_{\varepsilon},s}$ given in (2.1). Observe that it follows that
$\langle\mathcal{L}_{m_{\varepsilon},s}u,v\rangle$ is given by

$(1-s)\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}m(\tfrac{x}{\varepsilon},h)\frac{|u(x)-u(x-h)|^{p-2}(u(x)-u(x-h))(v(x)-v(x-h))}{|h|^{n+sp}}\,dhdx.$

Our first result concerns with the problem of first localizing and then
homogenizing, that is, we first take the limit as $s\uparrow 1$ and then the
limit as $\varepsilon\downarrow 0$.

######  Proposition 6.1.

It holds that

$\lim_{\varepsilon\downarrow 0}\left(\lim_{s\uparrow
1}u_{s,\varepsilon}\right)=u^{*},$

(in the $L^{p}(\Omega)$ sense) where $u^{*}\in W^{1,p}_{0}(\Omega)$ is the
weak solution of

$\begin{cases}-\mathop{\text{div}}(\nabla_{\xi}\mathcal{A}^{*}(\nabla
u^{*}))=f&\text{in }\Omega\\\ u^{*}=0&\text{on }\partial\Omega,\end{cases}$

and

$\mathcal{A}^{*}(\xi)=\inf_{v\in
W^{1,1}_{\text{per}}(Q)}\int_{Q}\mathcal{A}(y,\xi+\nabla v(y))\,dy$

###### Proof.

First, we have to take the limit as $s\uparrow 1$ for fixed $\varepsilon>0$,
but this was carried out in Section 5, and it holds that

$\lim_{s\uparrow 1}u_{s,\varepsilon}=u_{\varepsilon},$

where $u_{\varepsilon}\in W^{1,p}_{0}(\Omega)$ is the solution to

$\begin{cases}-\mathop{\text{div}}(\nabla_{\xi}\mathcal{A}_{\varepsilon}(x,\nabla
u_{\varepsilon}))=f&\text{in }\Omega\\\ u_{\varepsilon}=0&\text{on
}\partial\Omega.\end{cases}$

Now, we can apply the results of [8, Chapter 24] to conclude the desired
result as $\varepsilon\downarrow 0$. ∎

To finish the section, we now deal with the case where first we homogenize and
then localize, i.e. first take the limit $\varepsilon\downarrow 0$ and then
the limit $s\uparrow 1$.

######  Proposition 6.2.

It holds that

$\lim_{s\uparrow 1}\left(\lim_{\varepsilon\downarrow
0}u_{s,\varepsilon}\right)=\bar{u}$

(in the $L^{p}(\Omega)$ sense), where $\bar{u}\in W^{1,p}_{0}(\Omega)$ is the
solution to

$\begin{cases}-\mathop{\text{div}}(\nabla_{\xi}\bar{\mathcal{A}}(\nabla\bar{u}))=f&\text{in
}\Omega\\\ \bar{u}=0&\text{on }\partial\Omega,\end{cases}$

where $\bar{\mathcal{A}}$ is given by

$\bar{\mathcal{A}}(\xi)=\int_{Q}\mathcal{A}(y,\xi)\,dy,$

and $\mathcal{A}$ is the one given by (3.1).

###### Proof.

We first have to take the limit as $\varepsilon\downarrow 0$ for fixed
$s\in(0,1)$. But this problem was already solved in [14] and what is known is
that

$\lim_{\varepsilon\to 0}u_{s,\varepsilon}=\bar{u}_{s},$

where $\bar{u}_{s}$ is the solution to

$\begin{cases}\mathcal{L}_{\bar{m},s}\bar{u}_{s}=f&\text{in }\Omega\\\
\bar{u}_{s}=0&\text{in }\mathbb{R}^{n}\setminus\Omega,\end{cases}$

and

$\bar{m}(h)=\int_{Q}m(y,h)\,dy.$

Now, we can take the limit as $s\uparrow 1$ using the results of Section 5 to
conclude the desired result. ∎

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# Assessing the Quality of the Datasets by Identifying Mislabeled Samples

Vaibhav Pulastya, Gaurav Nuti, Yash Kumar Atri, Tanmoy Chakraborty
IIIT-Delhi, New Delhi, India
{vaibhav17271, gaurav17050, yashk<EMAIL_ADDRESS>
###### Abstract

Due to the over-emphasize of the quantity of data, the data quality has often
been overlooked. However, not all training data points contribute equally to
learning. In particular, if mislabeled, it might actively damage the
performance of the model and the ability to generalize out of distribution, as
the model might end up learning spurious artifacts present in the dataset.
This problem gets compounded by the prevalence of heavily parameterized and
complex deep neural networks, which can, with their high capacity, end up
memorizing the noise present in the dataset. This paper proposes a novel
statistic – noise score, as a measure for the quality of each data point to
identify such mislabeled samples based on the variations in the latent space
representation. In our work, we use the representations derived by the
inference network of data quality supervised variational autoencoder (AQUAVS).
Our method leverages the fact that samples belonging to the same class will
have similar latent representations. Therefore, by identifying the outliers in
the latent space, we can find the mislabeled samples. We validate our proposed
statistic through experimentation by corrupting MNIST, FashionMNIST, and
CIFAR10/100 datasets in different noise settings for the task of identifying
mislabelled samples. We further show significant improvements in accuracy for
the classification task for each dataset.

###### Index Terms:

Noise reduction, variational autoencoders, label error, dataset quality

##  I Introduction

Deep learning models have seen great practical success in achieving state-of-
the-art performance for various tasks. However, such models generally are
heavy, with a large number of trainable (or learnable) parameters, and are
intrinsically constrained by the data they are trained on. This often leads to
one of the major concerns for such deep learning models as due to their high
capacity, they often overfit the training data provided to them irrespective
of any artefacts that the data might contain [1]. Therefore, if the training
data is noisy (i.e., consisting of mislabeled data points), this results in
hampering the model’s generalizability. Additionally, due to the demand for
large-scale datasets followed by the common dictum that more labeled data
results in better performance, dataset creation is generally automated,
emphasising the quantity of the data rather than the quality of data. This
increases the scope for noise in datasets, as in many settings, they might
contain data that is "weakly labeled" through web-scraping or other similar
techniques [2]. Although the data might be inexpensive to collect, these
methods often lead to errors in labeling; thereby, training on such data
instead ends up hampering the model performance. Furthermore, crowdsourcing
platforms, which employ human annotators or rely on community feedback or
survey for data labeling are also susceptible to making mistakes [3]. Hence,
in a real-world setting, it becomes inevitable to have a source of noise for
large datasets.

This paper aims to automatically identify such mislabelled or “hard” data
points without any availability of all “clean” a priori subset of data or any
knowledge about the type of noise present in the dataset. Upon identification
and filtration of such samples, we expect the training on the derived clean
subset to improve the generalization of the model. We validate this by
conducting a series of experiments on different datasets and against two
different noise settings - random and systematic. A random sample of a class
is mislabeled uniformly to any other class in case of random noise. Whereas in
systematic noise, samples belonging to a specific class are mislabeled to
another fixed class, which intends to imitate a scenario wherein samples
belonging to a class may be confused for some other class during the
annotation process.

To combat the mentioned problem, we quantify the quality of each data point
relative to the whole dataset. We introduce AQUAVS, a variational autoencoder
based architecture with an auxiliary network. AQUAVS incorporates data labels
to effectively learn the latent representations of input data. We propose a
novel method that utilizes the variations in latent representations. Our
primary intuition is that latent space representations of samples belonging to
the same class will be similar. Therefore, our metric measures how far off
(outlying) is the latent representation of a sample to that of samples
belonging to the same class. We define noise score given by the number of
outlying latent features of a sample. We discuss our proposed approach in
detail in Section III. As illustrated in Figure 1, we expect points with a
high noise score to be mislabeled and apply threshold to filter out the
resultant clean subset.

Lastly, we validate our method by conducting extensive experiments on four
different datasets, namely MNIST [4], FashionMNIST [5], CIFAR-10 [6], and
CIFAR-100 [6]. We assume the original datasets to contain all clean samples,
corrupt them for our experiments using different noise models, and then report
the results for each dataset. For our baseline, we discuss an approach that
uses training dynamics of entropy of data sample. Recent empirical studies [7,
8] suggested how training dynamics contain information that can be used to
differentiate between samples vis-a-vis their contribution to learning.

Figure 1: Examples of images and their corresponding assigned labels from the
three datasets – MNIST, FashionMNIST, and CIFAR-10, having high noise score
values. Such points are classified and expected to be mislabeled.

The main contributions of this paper are summarized below:

  * •

We propose a novel algorithm that makes use of latent space representation of
an input observation and returns a quality measure (or noise score) for it. We
use this to classify high noise score samples as mislabelled or hard-to-learn
data points.

  * •

Our method identifies noisy samples without any access to a clean subsample
for deduction, or any prior knowledge on the type of noise present in the
dataset.

  * •

We conduct extensive experimentation for the task of identifying mislabelled
points under different noise settings – uniform and systematic noise, on
multiple well-known datasets, wherein our method shows significant improvement
over baselines.

Reproducibility: We have made the code and datasets public at
https://github.com/LCS2-IIITD/AQUAVS.

##  II Related Works

The challenges with learning a noisy dataset have been well-explored [9, 10,
11]. A detailed survey of existing literature can be found in [12, 13]. To
counteract noisy labels, various approaches such as forward or backward
correction [14] and S-model [15], suggest estimation of the noise distribution
in terms of noise transition matrices for classes and further rectify the loss
function based on it. The primary challenge is to accurately estimate the
aforementioned matrices. Sabzevari et al. [16] identified the noise samples by
training a classifier on subsampled data and utilizing the threshold to filter
the mislabeled data points. Another technique applied in [17] and [18]
involves rectifying the said noise by using the prediction of deep neural
networks (DNN). However, this method poses a challenge of overfitting. To
account for the issue of overfitting, in joint optimization, Tanaka et al.
[19] introduced a regularization term that takes prior knowledge about noise
into account, which is not always readily available in real-world settings.
Another approach that has gained traction identifies samples that are
important to training. This involves training models selectively on such
samples or to weight samples according to the importance measure (e.g. [20,
21, 22]). For this, the primary challenge is to design robust and compelling
criteria to score the training samples.

One of the recent approaches, Labelfix [23] proposes such criterion given by
the probability of the assigned original label for an input. Similarly, our
proposed method does segregation by describing a novel measure – noise score
as the standard for selectively filtering out the clean data points. The
proposed pipeline follows a structure as the one by Brodley and Friedl [24].
The two-step pipeline includes (i) identifying mislabeled samples and (ii)
training models on the derived subset after discarding mislabeled samples.
Additionally, many deep learning methods have been proposed to identify
mislabeled samples (at times implicitly). These include Iterative Noisy Cross-
Validation (INCV) [25], which uses cross-validation; MentorNet [20] implements
auxiliary networks; AUM [26] achieves this by observing the training dynamics
to similarly come up with a score, and then apply threshold to discard “noisy”
samples. Other suggested methods involve using a guaranteed clean smaller
subsample of the entire data to draw deductions about the dataset and using
them to filter out anomalous entities from the dataset (e.g. [27, 28, 29,
30]). We tackle the task constrained on the strict setting by identifying
required data points without any access to clean subsample and prior knowledge
of the type of noise present in the data.

Furthermore, recent works in anomaly detection use posterior inferences drawn
by training variational autoencoders. These methods propose altercations to
the architecture and loss functions of standard variational autoencoder to
learn latent space representations of the data. Pol et al. [31] described a
metric based on autoencoder loss function for the task of anomaly detection.
Beggel et al. [32] identified the said points during training based on the
likelihood estimate of the latent variable such that anomalous points are
expected to get a lower estimate comparative to regular data points. However,
our work primarily focuses on deriving a cleaner subset by identifying
mislabeled samples in a considerably high noise setting.

Figure 2: A standard VAE architecture consists of an encoder model (inference)
and a decoder model (generative). The $\mu$ and $\sigma$ layers encode the
mean and standard deviation of the Gaussian prior of the latent variables. The
latent vector $z$ is then sampled using the reparameterization trick [33, 34].
Figure 3: An architecture of a supervised VAE: AQUAVS extends standard VAE by
adding an auxiliary discriminative network.

##  III Methodology

We denote the dataset by $D=\\{x_{i},y_{i}\\}^{N}_{i=1}$ for our task. It
consists of two types of samples – correctly labeled and mislabeled samples. A
mislabeled sample is one whose actual label (i.e., ground-truth) does not
match the assigned label. We follow a general assumption that such samples
hamper the model generalizability. For large-scale datasets, correctly labeled
sample could also possibly contribute negatively and might be confusing to the
model; but a simplifying assumption is made in this case that it does not hurt
model’s generalization. We first evaluate our method for the task of noise
reduction through identification of mislabeled points. Based on our
assumption, we remove such points and evaluate them for the classification
task of each dataset using clean test data.

Variational Autoencoder: It is often employed for efficient approximation of
probabilistic inference of the continuous latent variable $z$ given an
observed value $x$. The framework of variational autoencoders (VAEs) [35]
allows us to explore variations in the data, by learning latent space
representation through probabilistic inferences.

For a dataset $D$, standard VAE learns an unsupervised task of stochastic
mapping between an observed input ($x$-space) and a continuous latent
($z$-space). Figure 2 shows an overview of a standard VAE architecture. For
our approach, we employ a modification to the standard VAE, extending an
auxiliary discriminative network that learns y-labels of data.

Figure 4: Detailed architecture of the supervised VAE used for learning latent
representations for MNIST and FashionMNIST datasets. Figure 5: Detailed
architecture of the supervised VAE used for learning latent representations
for CIFAR-10/100 datasets.

###  III-A Proposed Architecture

Recent studies [36, 9] suggested supervised and semi-supervised architecture
for VAE to learn latent space representations, utilizing y-labels
corresponding to each observation. The authors showed that the availability of
labels improves the VAE encoder inference. Therefore, for our experiment, we
implement a supervised VAE model, called AQUAVS. Figure 3 shows an overview of
the AQUAVS model. The architecture is largely inspired by the latent-feature
discriminative model (M1) proposed by [37]. However, unlike the M1 model,
where the classifier is trained after learning feature representation, we
jointly train for both classification and feature representation.

Specifically, for an input observation $x$, the goal of AQUAVS is to train a
network given by $p(x,\,z,\,y)=p(z)\,p(x\,|\,z)\,p(y\,|\,z)$, where $p(z)$ is
the prior distribution over latent variables $z$, $p(x\,|\,z)$ is the
likelihood function for decoder that generates $x$ given latent variable $z$,
and $p(y\,|\,z)$ is the likelihood function for classifier which predicts the
label $y$ corresponding to input $x$ given latent variable $z$. This consists
of posterior inference network $q_{\phi}(z\,|\,x)$ to infer latent variable
$z$, a generative network $p_{\theta}(x|z)$ to reconstruct input $x$, and a
discriminative network $q_{\phi}(y\,|\,z)$ to predict label $y$. Therefore,
for an input $x_{i}$ belonging to $D$, AQUAVS returns a pair of reconstructed
input image and output label i.e. $(x_{i_{recon}},\,y_{i}*)$.

###  III-B Model Objective

We denote $y$ to be the label of the input datapoint $x$. We describe the
following loss functions for optimizing model and variational parameters
denoted by $\theta$ and $\phi$, respectively as follows:

  * •

Autoencoder Loss is given by evidence lower bound (ELBO) loss function. Here,
$z$ denotes the latent variables.

$\begin{split}L_{ELBO}=&E_{q_{\phi}(z\,|\,x)}[\,logp_{\theta}(x\,|\,z)\,]\,-\,KL[\,q_{\phi}(z\,|\,x)\,||\,p_{\theta}(z)\,]\end{split}$
(1)

The ELBO loss has two terms - first term corresponds to the reconstruction
loss, and the second is a regularization term given by Kullback-Leibler
divergence.

  * •

Classifier Loss is given by categorical cross-entropy loss function.

$L_{cl}=\sum\limits_{i}{y_{i}\cdot logp(y_{i})}$ (2)

Our AQUAVS model is trained to minimize the training objective given by,

$L=L_{ELBO}+L_{cl}$ (3)

Figure 6: Illustration for comparing noise scores of correctly labeled and
mislabelled samples for MNIST, FashionMNIST, CIFAR-10, and CIFAR-100. The
AQUAVS latent space dimension for each dataset has a size of 100 (hence, noise
score $<=100$). The box plots distinguish between the values computed for
correctly and mislabeled data points as we observe mislabelled samples getting
higher noise scores comparative to correctly labeled samples.

###  III-C Computing Noise Scores

We train AQUAVS to learn data distributions in a latent space. Latent space
representation or embedding for an input observation is obtained via an
inference network (encoder) of AQUAVS. The proposed metric for the quality of
datapoint noise score exploits the distribution of latent variables of the
data points belonging to the same class as that of the data point. The
intuition behind the proposed method is that data points belonging to the same
class should have similar latent representations. Inspired by a univariate
outlier detection approach [38] to work on a multivariate setting of latent
vectors, we define noise score as the number of outlying latent variables
differing from the observations belonging to the same class. Algorithm 1 gives
the pseudo-code for computing noise score for each data point of a dataset $P$
using median and median absolute deviation statistic on a multivariate
distribution (i.e., over latent vectors). A constant $\alpha$ is a
hyperparameter which adjusts the stringency of outlier detection on each
latent variable. We set a default value of $\alpha$ as $1.5$. Upon obtaining
these noise scores, following our prior assumption that mislabelled points
hurt the model’s performance, we set a hyperparameter threshold based on noise
scores distribution to dump points with high scores.

Algorithm 1 Pseudo-code for computing Noise Score

0: Encoder function $f$, Dataset $P$, Constant $\alpha=1.5$

Group the dataset $P=\\{x_{i},y_{i}\\}^{N}_{i=1}$ by the $y-$labels. Obtaining
a list of disjoint sets $S=[S_{1},S_{2},...,S_{c}]$, such that, $P=S_{1}\cup
S_{2}...\cup S_{c}$

$NScore\leftarrow\phi$

for all $S_{i}$ in $S$ do

$X_{i}\leftarrow\\{x_{j}\;|\;(x_{j},\,y_{j})\in S_{i}\\}$

$XEnc\leftarrow f(X_{i})$

$m\leftarrow Median(XEnc,\;axis=0)$

$M\leftarrow MedianAbsoluteDeviation(XEnc,\;axis=0)$

for all $latent\\_vector_{x}$ in $XEnc$ do

$score\leftarrow 0$

for all $x_{j}$ in $latent\\_vector_{x}$ do

if $x_{j}>m_{j}+\alpha*M_{j}$ then

$score\leftarrow score+1$

end if

end for

$NScore[x]\leftarrow score$

end for

end for

return $NScore$

TABLE I: Dataset Statistics. Dataset | Size (Train/Test, (Height $\times$ Width) | #Classes
---|---|---
MNIST | 60000/10000, (28 $\times$ 28) | 10
FashionMNIST | 60000/10000, (28 $\times$ 28) | 10
CIFAR10 | 50000/10000, (32 $\times$ 32) | 10
CIFAR100 | 50000/10000, (32 $\times$ 32) | 100

##  IV Competing Methods

Standard: This is a baseline for robust training of classification model. For
this, we do naive training on the complete training dataset.

LabelFix: Muller and Merkert [23] scored data points by the probability of an
input observation to be assigned the original label. A hyperparameter
threshold is then used to identify mislabeled data points by discarding those
with low probability. We took the publicly-available
implementation111https://github.com/mueller91/labelfix. to identify mislabeled
data points. We then perform robust training on the filtered subset with the
same model setting as ours for the purpose of comparison.

Entropy Training Dynamics (Entropy TD): For this baseline, we differentiate
samples based on training dynamics of entropy metric. The Shannon entropy
measure of a data point $x$ at epoch $t$ is given by:

$E^{(t)}(x,y)=\sum_{i}p_{i}^{(t)}\cdot log(p_{i}^{(t)})$

where $p_{i}^{(t)}$ is an output of the softmax layer at epoch $t$, which
corresponds to the probability of $x$ belonging to the class $i$.

We take account of training dynamics by averaging entropy over all training
epochs. This is given by:

$E(x,y)=\frac{1}{T}\cdot\sum\limits_{t=1}^{T}E^{(t)}(x,y)$

Thresholding is applied to the above metric to classify points with a high
value of averaged entropy as mislabeled to obtain the filtered dataset used
for training the classification model.

Oracle:  This serves as an upper bound for the model performance by training
on all guaranteed clean samples of the training dataset.

TABLE II: Results of mislabel identification on MNIST and FashionMNIST
dataset. We present the benchmarks for the following baselines – Entropy TD,
LabelFix and AQUAVS. We skip standard baseline and oracle as they are
undefined for this task. We highlight the best performing method in bold.

Dataset | MNIST | MNIST | FashionMNIST | FashionMNIST
---|---|---|---|---
Noise Type | Uniform Noise | Systematic Noise | Uniform Noise | Systematic Noise
Noise % | Method | Precision | Recall | Accuracy | Precision | Recall | Accuracy | Precision | Recall | Accuracy | Precision | Recall | Accuracy
0.1 | Entropy TD | 0.9 | 0.9 | 82.3 | 0.89 | 0.95 | 85.2 | 0.92 | 0.92 | 83.2 | 0.91 | 0.91 | 82.7
LabelFix | 1 | 1 | 99.3 | 0.99 | 0.99 | 98.2 | 0.99 | 0.99 | 98.1 | 0.96 | 0.96 | 94.5
AQUAVS | 0.99 | 0.99 | 99.1 | 0.99 | 0.99 | 98.8 | 0.98 | 0.98 | 96.7 | 0.99 | 0.98 | 97.8
0.2 | Entropy TD | 0.81 | 0.82 | 75.1 | 0.81 | 0.91 | 80.2 | 0.8 | 0.9 | 75.2 | 0.73 | 0.81 | 0.77
LabelFix | 1 | 0.99 | 98.9 | 0.99 | 0.99 | 97.9 | 0.99 | 0.98 | 97.3 | 0.93 | 0.93 | 89.3
AQUAVS | 0.99 | 0.99 | 98.3 | 0.99 | 0.99 | 98.2 | 0.97 | 0.98 | 95.9 | 0.97 | 0.96 | 94.1
0.3 | Entropy TD | 0.73 | 0.93 | 70.1 | 0.72 | 0.8 | 74.3 | 0.7 | 0.91 | 67.3 | 0.7 | 0.81 | 62.7
LabelFix | 0.99 | 0.99 | 97.8 | 0.97 | 0.95 | 94.5 | 0.96 | 0.96 | 94.9 | 0.88 | 0.88 | 84.2
AQUAVS | 0.99 | 0.99 | 98.2 | 0.99 | 0.98 | 97.5 | 0.98 | 0.95 | 94.7 | 0.93 | 0.92 | 89.5
0.4 | Entropy TD | 0.71 | 0.94 | 65.2 | 0.7 | 0.8 | 62.8 | 0.61 | 0.91 | 59.4 | 0.6 | 0.8 | 57.3
LabelFix | 0.97 | 0.98 | 96.1 | 0.93 | 0.93 | 91.7 | 0.93 | 0.93 | 94.1 | 0.79 | 0.81 | 76.7
AQUAVS | 0.97 | 0.95 | 95.5 | 0.97 | 0.93 | 94.1 | 0.93 | 0.94 | 91.5 | 0.87 | 0.84 | 83.1

TABLE III: Results of mislabel identification on CIFAR10 and CIFAR100 dataset.
We present the benchmarks for the following baselines – Entropy TD, LabelFix
and AQUAVS. We skip standard baseline and oracle as they are undefined for
this task. We highlight the best performing method in bold.

Dataset | CIFAR10 | CIFAR10 | CIFAR100 | CIFAR100
---|---|---|---|---
Noise Type | Uniform Noise | Systematic Noise | Uniform Noise | Systematic Noise
Noise % | Method | Precision | Recall | Accuracy | Precision | Recall | Accuracy | Precision | Recall | Accuracy | Precision | Recall | Accuracy
20% | Entropy TD | 0.92 | 0.69 | 71.2 | 0.81 | 0.91 | 76.2 | 0.86 | 0.86 | 77.4 | 0.8 | 0.9 | 75.4
LabelFix | 0.94 | 0.96 | 92.7 | 0.9 | 0.92 | 86.2 | 0.95 | 0.95 | 92.4 | 0.85 | 0.85 | 75.16
AQUAVS | 0.93 | 0.96 | 90.7 | 0.93 | 0.8 | 82.4 | 0.9 | 0.95 | 85.1 | 0.9 | 0.75 | 72.2
40% | Entropy TD | 0.8 | 0.8 | 75.1 | 0.79 | 0.79 | 74.2 | 0.68 | 0.9 | 68.3 | 0.6 | 0.9 | 58.2
LabelFix | 0.91 | 0.91 | 87.9 | 0.67 | 0.67 | 58.9 | 0.9 | 0.9 | 87.4 | 0.63 | 0.63 | 55.73
AQUAVS | 0.85 | 0.91 | 84.8 | 0.82 | 0.6 | 69.2 | 0.84 | 0.9 | 83.5 | 0.66 | 0.75 | 61.2

##  V Experiments

We conduct our experiments on the following datasets: MNIST, FashionMNIST,
CIFAR10, and CIFAR100 as shown in Table I. Since these datasets are well
known, we assume them to be correctly labeled i.e., original labels as the
true labels. We add noise to the training set using two noise models – (i)
uniform noise model adds mislabeling by assigning a random label such that
each class has an equal probability of getting mislabeled to any other class,
and (ii) systematic noise model wherein data points belonging to the same true
class get assigned the same incorrect label. The latter simulates a scenario
where one class has a predisposition to be mistaken for another class, e.g.,
handwritten digit 8 often gets confused with the digit 3. We describe the
mislabeling of the systematic noise model for a dataset with $L$ classes given
by $\\{0,...,(L\,-\,1)\\}$ by assigning random data points with the initial
true label as $l$ to class $(l\,+\,1)\%L$.

###  V-A Implementation Details

We implement two different AQUAVS networks for our experiments. Detailed
architecture is illustrated in Figure 4 used for MNIST and FashionMNIST, and
Figure 5 used for CIFAR-10/100. The dimension of 100 is used for latent space.
The encoder design implemented for CIFAR (in Figure 5) has been inspired by
VGG-like networks [39], and the decoder design uses transposed convolutional
layer to reconstruct the input observation. AQUAVS is trained on the noisy
training dataset $D$. To prevent overtraining on the noisy training dataset,
we apply early termination on the validation loss given by Equation 3. Upon
training, the encoder (or inference) network is then used to compute the
required noise scores for each data point through Algorithm 1 on dataset $D$.
Figure 6 presents the box plots to discern between the obtained noise scores
for correctly labeled and mislabeled points. A hyperparameter threshold is
deduced based on the distribution of noise scores of $D$ to filter out points
with high noise scores as mislabelled/hard data points.

TABLE IV: Robust training performance on MNIST and FashionMNIST dataset. We
present the baselines over all the methods. Oracle method serves as the upper
bound of models performance. We highlight the best performing method in bold.
Uniform Noise
---
Dataset | MNIST (5% subsampled) | FashionMNIST
Noise % | Oracle | Standard | AQUAVS | LabelFix | Entropy TD | Oracle | Standard | AQUAVS | LabelFix | Entropy TD
10% | 95.4 | 94.3 | 96.2 | 95.6 | 94.4 | 90.1 | 89.5 | 89.7 | 89.7 | 88.5
20% | 94.5 | 92.3 | 95.12 | 94.9 | 93.5 | 89.8 | 88.9 | 89.4 | 89.6 | 87.4
30% | 94.9 | 91.9 | 94.3 | 93.9 | 92.5 | 89.8 | 88.1 | 88.5 | 88.3 | 87.2
40% | 94.5 | 90.3 | 93.3 | 93.6 | 91.3 | 89.2 | 86.5 | 87.4 | 87.9 | 86.3
Systematic Noise
Dataset | MNIST (5% subsampled) | FashionMNIST
Noise % | Oracle | Standard | AQUAVS | LabelFix | Entropy TD | Oracle | Standard | AQUAVS | LabelFix | Entropy TD
10% | 95.7 | 92.6 | 95.6 | 95.3 | 93.7 | 90.1 | 89.3 | 89.8 | 89.4 | 89.1
20% | 94.8 | 91.2 | 94.8 | 94.6 | 90.6 | 90.2 | 88.9 | 89.1 | 88.7 | 87.5
30% | 94.3 | 91.7 | 94.6 | 93.7 | 89.2 | 89.1 | 85.3 | 87.5 | 86.2 | 84.6
40% | 94.2 | 88.8 | 93.1 | 91.2 | 88.2 | 88.8 | 74.5 | 85.1 | 82.4 | 78.6
TABLE V: Robust training performance on CIFAR10 and CIFAR100 dataset. We
present the baselines over all the methods. Oracle method serves as the upper
bound of models performance. We highlight the best performing method in bold.
Uniform Noise
---
Dataset | CIFAR10 | CIFAR100
Noise % | Oracle | Standard | AQUAVS | LabelFix | Entropy TD | Oracle | Standard | AQUAVS | LabelFix | Entropy TD
20% | 89.4 | 72.6 | 82.5 | 84.2 | 77.5 | 61.1 | 47.7 | 50.7 | 52.3 | 51.1
40% | 85.5 | 57.5 | 71.2 | 76.5 | 64.8 | 57.5 | 39.7 | 40.1 | 43.1 | 39.6
Systematic Noise
Dataset | CIFAR10 | CIFAR100
Noise % | Oracle | Standard | AQUAVS | LabelFix | Entropy TD | Oracle | Standard | AQUAVS | LabelFix | Entropy TD
20% | 89.5 | 69.1 | 81.5 | 81.2 | 72 | 61.7 | 52.2 | 53.1 | 51.4 | 52.4
40% | 85.8 | 51.9 | 68.7 | 56.2 | 53.74 | 58.1 | 38.9 | 39.1 | 33.9 | 38.2

###  V-B Identification of Mislabeled Data

For the first task of our pipeline i.e., filtering out correctly labeled
points from the dataset, we use precision, recall, and accuracy as the
evaluation metrics. For a resultant filtered subset, precision indicates the
percentage of clean data points among all the filtered data points, measuring
the quality of the filtered dataset. Recall measures the percentage of clean
data points in the filtered subset among all clean data points of the original
dataset. Lastly, accuracy measures the correctness of the method’s
segregation.

Tables II and III present results for the task of mislabel identification. Our
method achieves high precision, recall, and accuracy (consistently $>=0.94$)
for MNIST and against both noise models for varying amounts of corruption up
to $40\%$. In general, for all methods, systematic noise is observed to be
harder to filter compared to uniform noise. This task becomes more challenging
for CIFAR datasets. We note CIFAR100 with high noise as the most challenging
setting, due to it being a more complex dataset compared to others. In
particular, our method shows improvement to the results of baselines against
systematic noise for MNIST, FashionMNIST, and CIFAR10.

###  V-C Robust Training

We discard the identified mislabeled samples and train for the classification
task of these datasets. We evaluate models performance on a clean test set. We
compare our results against all the methods described in Section IV. We
implement RESNET32 [40] for CIFAR10/100 dataset and 2-layer vanilla CNN
network for MNIST and FashionMNIST datasets. Since MNIST is a relatively
simpler dataset to model, in order to distinguish all results, we train the
model on a 5% sample of the training dataset. Tables IV and V present results
for robust training of classification model. In all cases, We note significant
improvements against the standard baseline. Moreover, in case of MNIST, the
accuracy of our method exceeds that of oracle for 10% and 20% noise.
Additionally, for a more challenging setting of CIFAR-100 as shown in Table V,
we only see slight improvements to the standard baselines. These results are
indicative that the quality of datasets could bring more improvements to
having more data.

##  VI Conclusion

In our paper, we introduced AQUAVS, a VAE based architecture with an auxiliary
discriminative network. AQUAVS combines the data labels to better learn the
latent space representations improving the models encoder inference. We also
introduced a novel noise score metric, which exploits the variations in the
latent representation of data points belonging to the same class. This measure
is further used to order data points on the quality and segregate noisy or
mislabeled ones. We tested our technique for varying amounts of label error
using two noise models – uniform and systematic. We showed that our method is
successfully able to distinguish mislabeled and correctly labeled points. Note
that our method does not use any prior information on the type of noise
present in data. Due to the prevalence of large parameterized deep neural
networks, they end up learning this noise, which hurts the generalization
ability of the model. Hence, we remove the identified mislabeled data points
and observe significant improvements in the performance of the classification
model. Furthermore, since our algorithm relies on an inference network, we
believe that exploring other autoencoder architectures could provide further
improvements to existing results. In particular, for large datasets, where
noise invariably creeps up in real-world settings, we believe our proposed
pipeline will be useful in identifying and removing low-quality data points.

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