# Controlled finite continuous frames

Hafida Massit1, Mohamed Rossafi2 and Choonkil Park3∗ 1Department of
Mathematics, Faculty Of Sciences, University of Ibn Tofail, Kenitra, Morocco
<EMAIL_ADDRESS>2LaSMA Laboratory, Department of Mathematics, Faculty of
Sciences, Dhar El Mahraz University Sidi Mohamed Ben Abdellah, Fes, Morocco
<EMAIL_ADDRESS><EMAIL_ADDRESS>3Research Institute for
Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea
<EMAIL_ADDRESS>
(Date:
∗Corresponding author: Choonkil Park (email<EMAIL_ADDRESS>fax:
+82-2-2281-0019, orcid: 0000-0001-6329-8228).)

###### Abstract.

In this paper, we present controlled finite continuous frames in a finite
dimensional Hilbert space and we study some properties of them. Parseval
controlled integral frames are presented and we characterize operators that
construct controlled integral finite frames.

###### Key words and phrases:

finite frame; controlled continuous finite frame; Parseval controlled frame.

###### 2010 Mathematics Subject Classification:

41A58, 42C15, 46L05.

##  1\. Introduction and preliminaries

The concept of frames in Hilbert spaces has been introduced by Duffin and
Schaffer [7] in 1952 to study some deep problems in nonharmonic Fourier
series, after the fundamental paper [6] by Daubechies, Grossman and Meyer,
frame theory began to be widely used, particularly in the more specialized
context of wavelet frames and Gabor frames. The majority of these applications
requires frames in finite-dimensional spaces. For example, Jamali et al [12]
and Javanshiri et al [13], were obtained results that are interesting in
applications of frames.

Recently, controlled frames were introduced by Balzas [4], Antoine and Grybos
to improve the numerical efficiency of iterative algorithms for inverting the
frame operator on abstract Hilbert spaces [3], however they are used earlier
in [5] for spherical wavelets. For more details, the reader can refer to [4,
8, 10, 14].

The concept of a generalization of frames to a family indexed by some locally
compact space endowed with a Radon measure was proposed by Kaisar [9] and
independently by Ali, Antoine and Gazeau [1]. In this paper we try to give a
generalization of the results given in [2] moving from the discrete case to
the continuous case.

For more information on frame theory and its applications, we refer the
readers to [15, 16, 17, 18, 19].

Throughout this paper, assume that $(\mathfrak{A},\mu)$ is a measure space
with positive measure $\mu$, $\mathcal{H}$ and $\mathcal{H}^{N}$ are used for
showing a Hilbert space and a finite-dimensional Hilbert space, respectively,
$GL(H)$ denotes the set of all bounded linear operators with a bounded
inverse, and $GL^{+}(H)$ is the set of positive operators in $GL(H)$.

######  Definition 1.1.

[16] Let $\mathcal{H}^{N}$ be an $N$-dimentional Hilbert space, and
$(\mathfrak{A},\mu)$ be a measure space. Then a map
$F:\mathfrak{A}\rightarrow\mathcal{H}^{N}$ is called an integral frame in
$\mathcal{H}^{N}$ if there exist $0<A\leq B<\infty$ such that

$A\|f\|^{2}\leq\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle\langle
F_{\varsigma},f\rangle d\mu(\varsigma)\leq B\|f\|^{2}\quad\forall
f\in\mathcal{H}^{N}.$ (1.1)

The elements $A$ and $B$ are called the integral frame bounds. If $A=B$, we
call this an integral tight frame. If $A=B=1$, it is called an integral
Parseval frame. If only the right hand inequality of (2.3) is satisfied, we
call $F$ a controlled integral Bessel map with bound $B$.

If $F$ is a Bessel map, then
$T_{F}:L^{2}(\mathfrak{A},\mu)\rightarrow\mathcal{H}^{N}$, defined by
$T_{F}(f)=\int_{\mathfrak{A}}\langle f,F(\varsigma)\rangle
F(\varsigma)d\mu(\varsigma)$, is a bounded linear operator. $T_{F}$ is
surjective and bounded if and only if $F$ is an integral frame. This operator
is called the synthesis operator.

The adjoint of $T_{F}$, which is called the analysis operator, is defined by

$T^{\ast}_{F}:\mathcal{H}^{N}\rightarrow
L^{2}(\mathfrak{A},\mu),\;\;\;\;T^{\ast}_{F}(f)(\varsigma)=\langle
f,F(\varsigma)\rangle,\;\varsigma\in\mathfrak{A}.$

The continuous frame operator is defined to be $S_{F}=T_{F}T^{\ast}_{F}$, it
is invertible and positive.

Recall that a Bessel map $F$ is a frame if and only if there exists a
continuous Bessel mapping $G$ is a dual of F if for any
$f,g\in\mathcal{H}^{N}$

$\langle f,g\rangle=\int_{\mathfrak{A}}\langle f,G(\varsigma)\rangle\langle
g,F(\varsigma)\rangle d\mu(\varsigma),\;f,g\in\mathcal{H}^{N},$

$G$ is called a dual frame for $F$ and $S^{-1}_{F}F$ is a dual of $F$.

##  2\. Main results

We consider some properties of controlled continuous frames in finite Hilbert
spaces.

######  Definition 2.1.

Let $\mathcal{H}^{N}$ be an $N$-dimensional Hilbert space and
$(\mathfrak{A},\mu)$ be a measure space. Then a family
$\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is called a $V$-controlled
integral frame for an invertible operator $V$ on $\mathcal{H}^{N}$ if there
exist $0<A\leq B<\infty$ such that

$A\|f\|^{2}\leq\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle\langle
VF_{\varsigma},f\rangle d\mu(\varsigma)\leq B\|f\|^{2}\quad\forall
f\in\mathcal{H}^{N}.$ (2.1)

The elements $A$ and $B$ are called the $V$-controlled integral frame bounds.
If $A=B$, we call this a $V$-controlled integral tight frame. If $A=B=1$, it
is called a $V$-controlled integral Parseval frame. If only the right hand
inequality of (2.1) is satisfied, we call $F$ a $V$-controlled integral Bessel
map with bound $B$.

Similar to ordinary frames, the controlled integral frame operator is defined
for a controlled frame on $\mathcal{H}^{N}$ by
$S_{VF}f=\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
VF_{\varsigma}d\mu(\varsigma)$.Wihch assumed in weak sense.

The controlled synthesis operator
$T^{\ast}_{VF}:L^{2}(\mathfrak{A},\mu)\rightarrow\mathcal{H}^{N}$ is defined
by $TT^{\ast}_{VF}(f)=\int_{\mathfrak{A}}\langle f,F(\varsigma)\rangle
VF(\varsigma)d\mu(\varsigma)$ and $S_{UF}=T^{\ast}_{VF}T_{F}$, where $T_{F}$
is the analysis operator of $\\{F_{\varsigma}\\}_{\varsigma}$.

######  Proposition 2.2.

Let $F:\mathfrak{A}\rightarrow\mathcal{H}^{N}$ such that
$\int_{\mathfrak{A}}\|VF((\varsigma))\|^{2}d\mu(\varsigma)<\infty$. Then $VF$
is a Bessel map.

###### Proof.

Using Cauchy-Schwarz inequality, we have

$\int_{\mathfrak{A}}|\langle
f,VF(\varsigma)\rangle|^{2}d\mu(\varsigma)\leq\int_{\mathfrak{A}}\|f\|^{2}\|VF(\varsigma)\|^{2}d\mu(\varsigma)\leq
B\|f\|^{2}\;with\;B=\int_{\mathfrak{A}}\|VF((\varsigma))\|^{2}d\mu(\varsigma).$

This completes the proof. ∎

We prove that the converse of Proposition 2.2 holds if $\mathcal{H}^{N}$ is
finite dimensional.

######  Proposition 2.3.

Let $\mathcal{H}^{N}$ be an $N$-dimensional Hilbert space and
$F:\mathfrak{A}\rightarrow\mathcal{H}^{N}$ be a Bessel map. Then
$\int_{\mathfrak{A}}\|VF((\varsigma))\|^{2}d\mu(\varsigma)<\infty$.

###### Proof.

Let $\\{e_{k}\\}_{k\in\\{1,2,...,n\\}}$ be an orthonormal basis for
$\mathcal{H}^{N}$. Then we have $\|VF(\varsigma)\|^{2}=\sum_{k=1}^{n}|\langle
VF(\varsigma),e_{k}\rangle|^{2}$. So

$\displaystyle\int_{\mathfrak{A}}\|VF(\varsigma)\|^{2}d\mu(\varsigma)$
$\displaystyle=\sum_{k=1}^{n}\int_{\mathfrak{A}}|\langle
VF(\varsigma),e_{k}\rangle|^{2}d\mu(\varsigma)$
$\displaystyle\leq\sum_{k=1}^{n}B\|e_{k}\|^{2}=Bn<\infty.$

This completes the proof. ∎

We give a new identity for controlled integral frames in finite dimensional
Hilbert spaces.

######  Proposition 2.4.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V$-controlled
integral frame where $V$ is an invertible operator on $\mathcal{H}^{N}$. Then
the following statements are equivalent.

  * (1)

$\\{F_{\varsigma}\\}_{\varsigma\in\mathcal{A}}$ is a $V-$controlled integral
frame with bounds $A$ and $B$.

  * (2)

$S_{VF}(f)=\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
VF_{\varsigma}d\mu(\varsigma)$ is an invertible and positive operator on
$\mathcal{H}^{N}$.

###### Proof.

$(1)\Rightarrow(2)$ is immediately from the definition of $V-$ controlled
integral frame operator.

$(2)\Rightarrow(1)$ for any $f\in\mathcal{H}^{N}$, suppose that $S_{VF}$ is
positive and invertible.

Then

$\langle S_{VF}f,f\rangle=\langle\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle
VF_{\varsigma}f,fd\mu(\varsigma)\rangle=\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle\langle VF_{\varsigma}f,f\rangle d\mu(\varsigma).$

This implies that

$\|\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle\langle
VF_{\varsigma}f,f\rangle d\mu(\varsigma)\|=\|\langle
S_{VF}f,f\rangle\|=\|S_{VF}^{\frac{1}{2}}f\|^{2},$

there exists $0<m$ such that

$m\langle f,f\rangle\leq\langle S_{VF}f,f\rangle.$ (2.2)

On other hand, for all $f\in\mathcal{H}$, there exists $0<m^{\prime}$ such
that

$\langle S_{VF}f,f\rangle\leq m^{\prime}\langle f,f\rangle$ (2.3)

From 2.2 and 2.3, we conclude that
$\\{F_{\varsigma}\\}_{\varsigma\in\mathcal{A}}$ is a $V-$controlled integral
frame ∎

######  Theorem 2.5.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a continuous frame
with the frame operator $S_{F}$. If $V\in GL^{+}(\mathcal{H}^{N})$ is self-
adjoint operator with $VS_{F}=S_{F}V$, then
$\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a $V-$controlled integral
frame.

###### Proof.

For $f\in\mathcal{H}$, we have

$\langle S_{V}f,f\rangle=\langle\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle
VF_{\varsigma}f,fd\mu(\varsigma)\rangle=\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle\langle VF_{\varsigma}f,f\rangle d\mu(\varsigma).$

So, we have

$A\|f\|^{2}\leq\langle S_{V}f,f\rangle.$

Then, The operator $S_{VF}$ is positive, also it’s selfadjoint. Let
$S_{VF}=VS_{F}$. The operator $S_{VF}$ is invertible. By Proposition 2.4,
$\\{F_{\varsigma}\\}_{\varsigma}$ is a $V-$controlled integral frame. ∎

######  Proposition 2.6.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V-$controlled
integral frame for $\mathcal{H}$ and $V\in GL(\mathcal{H})$. Then
$\\{F_{\varsigma}\\}_{\varsigma}$ is a continuous frame and $VS_{F}=S_{F}V$,
with

$\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
VF_{\varsigma}d\mu(\varsigma)=\int_{\mathfrak{A}}\langle
f,VF_{\varsigma}\rangle F_{\varsigma}d\mu(\varsigma).$

###### Proof.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathcal{A}}$ is a $V-$controlled
integral frame with bounds $A$ and $B$.

We have

$A\langle f,f\rangle\leq\langle S_{V}f,f\rangle=\langle VSf,f\rangle=\langle
V^{\frac{1}{2}}Sf,V^{\frac{1}{2}}f\rangle\leq\|V^{\frac{1}{2}}\|^{2}\langle
Sf,f\rangle.$

So,

$A\|V^{\frac{1}{2}}\|^{-2}\langle f,f\rangle\leq\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle\langle F_{\varsigma},f\rangle d\mu(\varsigma).$ (2.4)

On other hand, for all $f\in\mathcal{H}$ we have

$\displaystyle\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle\langle
F_{\varsigma},f\rangle d\mu(\varsigma)$ $\displaystyle=\langle Sf,f\rangle$
$\displaystyle=\langle V^{-1}VSf,f\rangle$
$\displaystyle=\langle(V^{-1}VS)^{\frac{1}{2}}f,(V^{-1}VS)^{\frac{1}{2}}f\rangle$
$\displaystyle\leq\|V^{\frac{-1}{2}}\|^{2}\langle(VS)^{\frac{1}{2}}f,(VS)^{\frac{1}{2}}f\rangle$
$\displaystyle=\|V^{\frac{-1}{2}}\|^{2}\langle(S_{V})^{\frac{1}{2}}f,(S_{V})^{\frac{1}{2}}f\rangle$
$\displaystyle=\|V^{\frac{-1}{2}}\|^{2}\langle S_{V}f,f\rangle$
$\displaystyle\leq\|V^{\frac{-1}{2}}\|^{2}B\langle f,f\rangle.$

Then,

$\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle\langle
F_{\varsigma},f\rangle d\mu(\varsigma)\leq\|V^{\frac{-1}{2}}\|^{2}B\langle
f,f\rangle.$ (2.5)

From 2.4 and 2.5 we conclude that $\\{F_{\varsigma}\\}$ is a continuous frame.
∎

We show that the condition $VS_{F}=S_{F}V$ is not given in a finite-
dimensional real Hilbert space in the following example.

######  Example 2.7.

Consider the frame
$\\{F_{\varsigma}\\}_{\varsigma}=\\{\binom{1}{\varsigma}\\}$ for
$\mathbb{R}^{2}$ and $\mathfrak{A}=[0,1]$ endewed with the Lebesgue measure.
With the operator $V=\begin{pmatrix}\begin{array}[]{cc}1&1\\\
-1&1\end{array}\end{pmatrix},$ it is clear that $V$ is positive and
invertible. By definition of frame operator, we have
$S_{F}=\begin{pmatrix}\begin{array}[]{cc}1&1/2\\\
-1/2&1/3\end{array}\end{pmatrix}.$

For all $f=\binom{x}{y}\in\mathbb{R}^{2}$, we have

$\displaystyle\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle\langle
VF_{\varsigma},f\rangle d\mu(\varsigma)$ $\displaystyle=\langle
S_{VF}f,f\rangle$ $\displaystyle=\langle
VS_{F}\binom{x}{y},\binom{x}{y}\rangle$
$\displaystyle=\dfrac{1}{2}x^{2}-\dfrac{y^{2}}{2}-\dfrac{2}{3}xy.$

We obtain that the frame $\\{F_{\varsigma}\\}$ is a $V-$controlled integral
frame such that $VS_{F}\not=S_{F}V$.

######  Proposition 2.8.

If $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a $V-$controlled
integral frame for $\mathcal{H}^{N}$ with the frame operator $S_{VF}$, then
$\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a continuous frame for
$\mathcal{H}$ with the frame operator $V^{-1}S_{VF}$.

###### Proof.

Let $S_{F}f=\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
F_{\varsigma}d\mu(\varsigma),\forall f\in\mathcal{H}^{N}$ and
$S_{VF}f=\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
VF_{\varsigma}d\mu(\varsigma)=VS_{F}f,\forall f\in\mathcal{H}^{N}$. Then
$V^{-1}S_{VF}f=S_{F}f$. The operator $S_{F}$ is an injective operator on
finite dimensional Hilbert space and $S_{F}$ is invertible. Therefore, for
$f\in\mathcal{H}^{N}$ we have

$f=\int_{\mathfrak{A}}\langle S_{F}^{-1}f,F_{\varsigma}\rangle
F_{\varsigma}d\mu(\varsigma)=\int_{\mathfrak{A}}\langle
f,(S_{F}^{-1})^{\ast}F_{\varsigma}\rangle F_{\varsigma}d\mu(\varsigma).$

This shows that $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a
continuous frame generator for $\mathcal{H}^{N}$ with the frame operator
$V^{-1}S_{VF}$ and $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a
generator for $\mathcal{H}^{N}$. ∎

######  Theorem 2.9.

Let $F=\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V-$controlled
integral frame for $\mathcal{H}^{N}$ where $V$ is an invertible operator and
the controlled frame operator $S_{VF}$ be a normal operator with
$VS_{VF}=S_{VF}V$. Then $V$ is a positive operator.

###### Proof.

By Proposition 2.8, $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a
continuous frame with the frame operator $S_{F}=V^{-1}S_{VF}$. We have
$S_{VF}V=VS_{VF}$ and so $S_{VF}S_{F}=VS_{F}S_{F}=S_{F}VS_{F}=S_{F}S_{VF}$.
There exists a set of common orthonormal eigenvectors of $S_{VF}$ and $S_{F}$
as $\\{e_{k}\\}_{k\in\\{1,2,...,N\\}}$.

Let $\\{\alpha_{k}\\}_{k\in\\{1,2,...,N\\}}$ and
$\\{\beta_{k}\\}_{k\in\\{1,2,...,N\\}}$ be eigenvalues of operators $S_{VF}$
and $S_{F}$, respectively.

For $\varsigma\in\mathfrak{A}$, we have

$Ve_{k}=(S_{VF}S_{F}^{-1})(e_{k})=S_{VF}(\beta_{k}^{-1}e_{k})=\beta_{k}^{-1}\alpha_{k}e_{k}.$

Then

$Vf=\sum_{k=1}^{N}\beta_{k}^{-1}\alpha_{k}\langle f,e_{k}\rangle$

Which follows $V$ is a positive operator. ∎

######  Proposition 2.10.

Let $F=\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a continuous frame
for $\mathcal{H}^{N}$ with the frame operator $S_{F}$. If
$\\{e_{k}\\}_{k\in\\{1,2,...,N\\}}$ and
$\\{\beta_{k}\\}_{k\in\\{1,2,...,N\\}}$ are the set of orthonormal
eigenvectors and the set of eigenvalues of $S_{F}$, respectively, then for
every set $\\{\alpha_{k}\\}_{k\in\\{1,2,...,N\\}}\subseteq(0,+\infty)$,
$F=\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a $V-$controlled frame,
where $V$ is defined by $Ve_{k}=\alpha_{k}e_{k}$, for $k=1,\cdots,N$.

###### Proof.

Let $f\in\mathcal{H}^{N}$. Then we have

$\displaystyle VS_{F}f=VS_{F}(\sum_{k=1}^{N}\langle f,e_{k}\rangle e_{k}))$
$\displaystyle=V(\sum_{k=1}^{N}\langle f,e_{k}\rangle S_{F}e_{k})$
$\displaystyle=\sum_{k=1}^{N}\alpha_{k}\langle f,e_{k}\rangle Ve_{k}$
$\displaystyle=\sum_{k=1}^{N}\beta_{k}\langle f,e_{k}\rangle\alpha_{k}e_{k}$
$\displaystyle=\sum_{k=1}^{N}\alpha_{k}\langle f,e_{k}\rangle S_{F}e_{k}$
$\displaystyle=S_{F}\sum_{k=1}^{N}\langle f,e_{k}\rangle\alpha_{k}e_{k}$
$\displaystyle=S_{F}\sum_{k=1}^{N}\langle f,e_{k}\rangle Ve_{k}$
$\displaystyle=S_{F}Vf.$

Since $\\{\alpha_{k}\\}\subset(0,\infty)$, so $V$ is positive and invertible
and also $V$ and $VS_{F}$ commute with each other. So $VS_{F}$ is an
invertible and positive operator with

$VS_{F}f=V(\sum_{k=1}^{N}\langle f,F_{k}\rangle F_{k})=\sum_{k=1}^{N}\langle
f,F_{k}\rangle VF_{k}.$

Therefore, $\\{F_{\varsigma}\\}_{\varsigma}$ is a $V-$controlled integral
frame with the frame operator $VS_{F}$. ∎

######  Theorem 2.11.

Let $F=\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V-$ controlled
integral frame with frame operator $S_{VF}$ and $L\in GL(\mathcal{H}^{N})$
($L$ is positive and so it is self -adjoint) such that $LV=VL$. Then
$\\{LF_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a $V-$controlled frame
with frame operator $LS_{VF}L^{\ast}$. Moreover,
$\\{L^{k}F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a $V-$ controlled
integral frame for $k\in\mathbb{R}$ with frame operator
$L^{k}S_{VF}(L^{k})^{\ast}$.

###### Proof.

We have

$S_{VLF}(f)=\int_{\mathfrak{A}}\langle f,LF_{\varsigma}\rangle
VLF_{\varsigma}d\mu(\varsigma)=\int_{\mathfrak{A}}\langle
f,LF_{\varsigma}\rangle LVF_{\varsigma}d\mu(\varsigma)=LS_{VF}L^{\ast}f.$

Thus $S_{VLF}=LS_{VF}L^{\ast}$ is invertible and

$\langle LS_{VF}L^{\ast}f,f\rangle=\langle S_{VF}L^{\ast},L^{\ast}f\geq
0,\;\forall f\in\mathcal{H}^{N},\;i.e.,\;S_{VF}\geq 0.$

This gives that $S_{VLF}$ is positive. Hence
$\\{LF_{\varsigma}\\}_{\varsigma}$ is a $V-$ controlled integral frame. Also
we have $L^{k}V=Vl^{k}$. Thus $\\{L^{k}F_{\varsigma}\\}_{\varsigma}$ is a
$V-$controlled integral frame with the frame operator
$L^{k}S_{VF}(L^{k})^{\ast}$. ∎

######  Corollary 2.12.

If $\\{F_{\varsigma}\\}_{\varsigma}$ is a $V-$ controlled integral frame such
that $VS_{F}=S_{F}V$, then
$\\{S_{F}^{\frac{\beta-1}{2}}F_{\varsigma}\\}_{\varsigma}$ is a $V-$controlled
integral frame for any $\beta\in\mathbb{R}$, with frame operator $VS_{F}$.

Gramian operator or Gramian matrix for a $V-$controlled frame has been defined
in [2] and we introduce Gramian operator for $V-$controlled frame in finite
Hilbert spaces, and we consider its properties.

######  Definition 2.13.

Let $\\{F_{\varsigma}\\}_{\varsigma}$ be a $V-$controlled integral frame with
analysis operator $T_{F}$ and synthesis operator $T^{\ast}_{VF}$. Then the
operator $G_{VF}:=T_{F}T^{\ast}_{VF}$ is called a $V-$Gramian operator. The
canonical matrix representation of Gramian operator of a $V-$controlled
integral frame $\\{F_{\varsigma}\\}$ is obtained by

$G=(\langle VF_{i},F_{j}\rangle)_{i,j\in\mathfrak{A}}.$

The following theorem investigates the Gramian matrix of the transferred
$V-$controlled integral frames.

######  Theorem 2.14.

Let $\\{F_{\varsigma}\\}_{\varsigma}$ be a $V-$controlled integral frame for
$\mathcal{H}^{N}$ and $T$ be a linear operator that commutes with $V$. Then
$T$ is unitary if and only if the $V-$Gramian matrix of $\\{TF_{\varsigma}\\}$
is equal to $G_{VF}$.

###### Proof.

Suppose that $T$ is unitary. Then we have

$G_{V(TF)}=\\{\langle
TVF_{\beta},TF_{\alpha}\rangle\\}_{\beta,\alpha}=\\{\langle
VF_{\beta},F_{\alpha}\rangle\\}=G_{VF}.$

Conversely, let $G_{VF}=G_{V(TF)}$. Then

$\langle VTF_{\beta},TF_{\alpha}\rangle=\langle VF_{\beta},F_{\alpha}\rangle$

and

$\langle T^{\ast}VTF_{\beta}-VF_{\beta},F_{\alpha}\rangle=0.$

For $f\in\mathcal{H}^{N}$, we have

$f=\int_{\mathfrak{A}}\langle f,S_{F}^{-1}F_{\varsigma}\rangle
d\mu(\varsigma).$

Then

$\displaystyle(T^{\ast}VT-V)f$
$\displaystyle=(T^{\ast}VT-V)\int_{\mathfrak{A}}\langle
f,S_{F}^{-1}F_{\varsigma}\rangle F_{\varsigma}d\mu(\varsigma)$
$\displaystyle=\int_{\mathfrak{A}}\langle
f,S_{F}^{-1}F_{\varsigma}\rangle(T^{\ast}VT-V)F_{\varsigma}d\mu(\varsigma)$
$\displaystyle=0.$

Thus we have $T^{\ast}VT=V$ and $T^{\ast}T=I$. ∎

Parseval frames are the closest family to orthonormal bases. We present and
study some properties of Parseval controlled integral frames in a finite-
dimensional Hilbert space.

######  Lemma 2.15.

[11] Let $\mathcal{H}^{N}$ be an $N-$dimensional Hilbert space, and
$G,L:\mathfrak{A}\rightarrow\mathcal{H}^{N}$ be continuous Parseval frames and
$K\in L(\mathcal{H}^{N})$ be self-adjoint. Then

$\int_{\mathfrak{A}}\|KG(\varsigma)\|^{2}d\mu(\varsigma)=\int_{\mathfrak{A}}\|KL(\varsigma)\|^{2}d\mu(\varsigma).$

######  Theorem 2.16.

Let $\mathcal{H}^{N}$ be an $N-$dimensional Hilbert space, and $F$ be a frame
for $\mathcal{H}^{N}$ and $G$ be a Parseval $V-$ controlled integral frame for
$\mathcal{H}^{N}$. Then

$\displaystyle\int_{\mathfrak{A}}\|V(G(\varsigma)-F(\varsigma))\|^{2}d\mu(\varsigma)$
$\displaystyle=$
$\displaystyle\int_{\mathfrak{A}}\|VS^{-1/2}F(\varsigma)-VF(\varsigma)\|^{2}d\mu(\varsigma)$
$\displaystyle+$
$\displaystyle\int_{\mathfrak{A}}\|V(S^{1/4}G(\varsigma)-S^{-1/4}F(\varsigma))\|^{2}d\mu(\varsigma).$

###### Proof.

By Lemma 2.15 with $L(\varsigma)=VS^{1/2}F(\varsigma)$, we have

$\int_{\mathfrak{A}}\|G(\varsigma)\|^{2}d\mu(\varsigma)=\int_{\mathfrak{A}}\|VS^{-1/2}F(\varsigma)\|^{2}d\mu(\varsigma)$

and

$\int_{\mathfrak{A}}\|S^{1/4}G(\varsigma)\|^{2}d\mu(\varsigma)=\int_{\mathfrak{A}}\|S^{1/4}F(\varsigma)\|^{2}d\mu(\varsigma)=\int_{\mathfrak{A}}\|S^{-1/4}F(\varsigma)\|^{2}d\mu(\varsigma).$

Thus

$\displaystyle\int_{\mathfrak{A}}\|G(\varsigma)-F(\varsigma)\|^{2}d\mu(\varsigma)-\int_{\mathfrak{A}}\|VS^{-1/2}F(\varsigma)-VF(\varsigma)\|^{2}d\mu(\varsigma)$
$\displaystyle=-2Re\int_{\mathfrak{A}}\langle G(\varsigma),F(\varsigma)\rangle
d\mu(\varsigma)+2\int_{\mathfrak{A}}\langle
VS^{-1/2}F(\varsigma),F(\varsigma)\rangle d\mu(\varsigma)$
$\displaystyle=-2Re\int_{\mathfrak{A}}\langle
VS^{1/4}G(\varsigma),S^{-1/4}F(\varsigma)\rangle
d\mu(\varsigma)+\int_{\mathfrak{A}}\|VS^{-1/4}F(\varsigma)\|^{2}d\mu(\varsigma)+\int_{\mathfrak{A}}\|VS^{-1/4}G(\varsigma)\|^{2}d\mu(\varsigma)$
$\displaystyle=\int_{\mathfrak{A}}\|V(S^{1/4}G(\varsigma)-S^{-1/4}F(\varsigma))\|^{2}d\mu(\varsigma).$

This completes the proof. ∎

######  Corollary 2.17.

Let $\mathcal{H}^{N}$ be an $N-$dimensional Hilbert space and $F$ be a frame
for $\mathcal{H}^{N}$ with frame operator $S$. For every Parseval
$V-$controlled integral frame $G$ of $\mathcal{H}^{N}$, we have

$\int_{\mathfrak{A}}\|V(G(\varsigma)-F(\varsigma))\|^{2}d\mu(\varsigma)\geq\int_{\mathfrak{A}}\|VS^{-1/2}F(\varsigma)-VF(\varsigma)\|^{2}d\mu(\varsigma)$

and we have equality if and only if

$G(\varsigma)=VS^{-1/2}F(\varsigma)\;,\varsigma\in\mathfrak{A}.$

###### Proof.

The first part follows immediately from Theorem 2.16.

We have equality if and only if

$\displaystyle
S^{1/4}G(\varsigma)=VS^{-1/4}F(\varsigma)\;,\varsigma\in\mathfrak{A}$
$\displaystyle\Leftrightarrow G(\varsigma)=VS^{-1/2}F(\varsigma).$

This completes the proof. ∎

######  Proposition 2.18.

Let $F=\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V-$ controlled
integral frame for $\mathcal{H}^{N}$. Then

$\int_{\mathfrak{A}}\langle VF_{\varsigma},F_{\varsigma}\rangle
d\mu(\varsigma)=N.$

###### Proof.

Let $\\{e_{k}\\}_{k=1}^{N}$be an orthonormal basis for $\mathcal{H}^{N}$. We
have

$e_{k}=S_{VF}e_{k}=\int_{\mathfrak{A}}\langle e_{k},F_{\varsigma}\rangle
VF_{\varsigma}d\mu(\varsigma).$

Thus

$\displaystyle N=\sum_{k=1}^{N}\|e_{k}\|^{2}$
$\displaystyle=\sum_{k=1}^{N}\int_{\mathfrak{A}}\langle
e_{k},F_{\varsigma}\rangle\langle VF_{\varsigma},e_{k}\rangle d\mu(\varsigma)$
$\displaystyle=\int_{\mathfrak{A}}\sum_{k=1}^{N}\langle
e_{k},F_{\varsigma}\rangle\langle VF_{\varsigma},e_{k}\rangle d\mu(\varsigma)$
$\displaystyle=\int_{\mathfrak{A}}\langle VF_{\varsigma},F_{\varsigma}\rangle
d\mu(\varsigma).$

This completes the proof. ∎

The following proposition illustrates that the orthogonal projections can be
preserved controlled frames in a finite-dimensional Hilbert space.

######  Proposition 2.19.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V-$ controlled
integral frame for $\mathcal{H}^{N}$, $E$ be a subspace of $\mathcal{H}^{N}$
and $U$ be an orthonogonal projection of $\mathcal{H}^{N}$ onto $E$ such that
$VU=UV$. Then $\\{UF_{\varsigma}\\}_{\varsigma}$ is a $V-$controlled frame for
$E$. If $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a Parseval
$V-$controlled integral frame for $\mathcal{H}^{N}$, then
$\\{UF_{\varsigma}\\}_{\varsigma}$ is a Parseval $V-$controlled integral frame
for $E$.

###### Proof.

For all $f\in E$, we have

$A\|f\|^{2}=A\|Uf\|^{2}\leq\int_{\mathfrak{A}}\langle
Uf,F_{\varsigma}\rangle\langle VF_{\varsigma},Uf\rangle d\mu(\varsigma)\leq
B\|Uf\|^{2}=B\|f\|^{2}$

and

$A\|f\|^{2}\leq\int_{\mathfrak{A}}\langle f,UF_{\varsigma}\rangle\langle
UVF_{\varsigma},f\rangle d\mu(\varsigma)\leq B\|f\|^{2},$

which implies that

$A\|f\|^{2}\leq\int_{\mathfrak{A}}\langle f,UF_{\varsigma}\rangle\langle
VUF_{\varsigma},f\rangle d\mu(\varsigma)\leq B\|f\|^{2}.$

Therefore, $\\{UF_{\varsigma}\\}_{\varsigma}$ is a $V-$ controlled integral
frame for $E$.

Suppose $\\{F_{\varsigma}\\}_{\varsigma}$ is a Parseval $V-$controlled
integral frame. Then for every $f\in E$,

$\displaystyle S_{VUF}(f)$ $\displaystyle=\int_{\mathfrak{A}}\langle
f,UF_{\varsigma}\rangle VUF_{\varsigma}d\mu(\varsigma)$
$\displaystyle=\int_{\mathfrak{A}}\langle Uf,F_{\varsigma}\rangle
UVF_{\varsigma}d\mu(\varsigma)$ $\displaystyle=U\int_{\mathfrak{A}}\langle
Uf,F_{\varsigma}\rangle VF_{\varsigma}d\mu(\varsigma)$ $\displaystyle=U^{2}f$
$\displaystyle=f.$

Therefore, $\\{UF_{\varsigma}\\}_{\varsigma}$ is a Parseval $V-$controlled
integral frame for $E$. ∎

If $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a $V-$controlled
integral frame with the controlled frame operator $S_{VF}$, then
$S_{VF}=VS_{F}$ and $f=\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle(S_{VF}^{-1}V)fd\mu(\varsigma)$ for every
$f\in\mathcal{H}^{N}$. This gives that
$\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a Parseval
$S_{VF}^{-1}V-$controlled integral frame, and
$\\{S_{VF}^{-1}VF_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$.

######  Theorem 2.20.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a continuous frame
with the frame operator $S_{F}$. Then every tight controlled integral frame
$\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is exactly an $\alpha-$tight
$\alpha S_{F}^{-1}$ controlled integral frame for $\alpha\in\mathbb{C}$.

###### Proof.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be an $\alpha-$tight
$V-$controlled integral frame, for $\alpha\in\mathbb{C}$. Then for
$f\in\mathcal{H}^{N}$, $\alpha f=\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle VF_{\varsigma}d\mu(\varsigma)$ and so $\alpha
I=S_{VF}=VS_{F}$ and $V=\alpha S^{-1}_{F}$, i.e.,
$\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ is a $\alpha-$tight $\alpha
S_{F}^{-1}$ controlled integral frame. ∎

We need to recall properties of the trace of linear operators on
$\mathcal{H}^{N}$ and then consider trace of an operator by controlled
integral frames.

The trace of a linear operator $L\in L(\mathcal{H}^{N})$ is defined by

$Tr(L)=\sum_{k=1}^{N}\langle Le_{k},e_{k}\rangle,$

where $\\{e_{k}\\}_{k=1}^{N}$ is an orthonormal basis for $\mathcal{H}^{N}$.
If $L_{1}$ and $L_{2}$ are self-adjoint positive operators, then $0\leq
Tr(L_{1}L_{2})\leq Tr(L_{1})\cdot Tr(L_{2})$.

######  Proposition 2.21.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V-$controlled
integral frame such that $V\in GL(\mathcal{H}^{N})$ is a self-adjoint
operator. Then

$Tr(S_{VF})\leq Tr(V)\int_{\mathfrak{A}}\|F_{\varsigma}\|^{2}d\mu(\varsigma).$

###### Proof.

Let $\\{\alpha_{k}\\}_{k=1}^{N}$ be the set of eigenvalues of the operator
frame $S_{F}$. Then
$Tr(S_{F})=\sum_{k=1}^{N}\alpha_{k}=\int_{\mathfrak{A}}\|F_{\varsigma}\|^{2}d\mu(\varsigma)$.

Therefore,

$Tr(S_{VF})=Tr(VS_{F})\leq
Tr(V)Tr(S_{F})=Tr(V)\sum_{k=1}^{N}\alpha_{k}=Tr(V)\int_{\mathfrak{A}}\|F_{\varsigma}\|^{2}d\mu(\varsigma).$

This completes the proof. ∎

######  Proposition 2.22.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a Parseval
$V-$controlled integral frame for $\mathcal{H}^{N}$ and $G$ be a linear
operator on $\mathcal{H}^{N}$. Then $Tr(G)=\int_{\mathfrak{A}}\langle
GVF_{\varsigma},F_{\varsigma}\rangle d\mu(\varsigma)$.

###### Proof.

Let $\\{e_{k}\\}_{k=1}^{N}$ be an orthonormal basis for $\mathcal{H}^{N}$.
Then $Tr(G)=\sum_{k=1}^{N}\langle Ge_{k},e_{k}\rangle$. Since
$\\{F_{\varsigma}\\}_{\varsigma}$ is a Parseval $V-$controlled integral frame,
for $k\in\\{1,2,3,...\\}$, we have

$Ge_{k}=\int_{\mathfrak{A}}\langle Ge_{k},F_{\varsigma}\rangle
VF_{\varsigma}d\mu(\varsigma)$

and

$\displaystyle Tr(G)$
$\displaystyle=\sum_{k=1}^{N}\langle\int_{\mathfrak{A}}\langle
Ge_{k},F_{\varsigma}\rangle VF_{\varsigma},e_{k}\rangle d\mu(\varsigma)$
$\displaystyle=\sum_{k=1}^{N}\int_{\mathfrak{A}}\langle
e_{k},G^{\ast}F_{\varsigma}\rangle\langle VF_{\varsigma},e_{k}\rangle
d\mu(\varsigma)$
$\displaystyle=\int_{\mathfrak{A}}\langle\sum_{k=1}^{N}\langle
VF_{\varsigma},e_{k}\rangle e_{k},G^{\ast}F_{\varsigma}\rangle
d\mu(\varsigma)$ $\displaystyle=\int_{\mathfrak{A}}\langle
VF_{\varsigma},G^{\ast}F_{\varsigma}\rangle d\mu(\varsigma)$
$\displaystyle=\int_{\mathfrak{A}}\langle GVF_{\varsigma},F_{\varsigma}\rangle
d\mu(\varsigma).$

This completes the proof. ∎

######  Remark 2.23.

Every $\alpha-$ tight $V-$controlled integral frame
$\\{F_{\varsigma}\\}_{\varsigma}$ induces a Parseval controlled integral
frame. We have for every $f\in\mathcal{H}^{N}$

$\alpha f=\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
VF_{\varsigma}d\mu(\varsigma),$

and then

$f=\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle(\alpha^{-1}V)F_{\varsigma}d\mu(\varsigma).$

This means that $\\{F_{\varsigma}\\}_{\varsigma}$ is a Parseval
$\alpha^{-1}V-$ controlled frame.

Also, $\\{F_{\varsigma}\\}$ is equivalent to
$\\{(\alpha^{-1}V)F_{\varsigma}\\}_{\varsigma}$ and
$\\{(\alpha^{-1}V)F_{\varsigma}\\}_{\varsigma}$ is a dual for
$\\{F_{\varsigma}\\}$. We recall that a frame
$\\{F_{\varsigma}\\}_{\varsigma}$ is equivalent to a frame
$\\{G_{\varsigma}\\}_{\varsigma}$ if there exists an invertible operator $P\in
B(\mathcal{H}^{N})$ such that $F_{\varsigma}=PG_{\varsigma}$.

######  Example 2.24.

Consider the Hilbert space $\L^{2}(\mathbb{R})$ and $\mathfrak{A}=[0,1]$
endewed with the Lebesgue measure. Let $F=\\{F_{\varsigma}\\}_{\varsigma}$ and
$G=\\{G_{\varsigma}\\}_{\varsigma}$ with
$F_{\varsigma}(x_{1},x_{2},x_{3},...)=\dfrac{x_{\varsigma}}{3}$ and
$G_{\varsigma}(x_{1},x_{2},x_{3},...)=\dfrac{x_{\varsigma}}{2}$. Then we
obtain

$\int_{\mathfrak{A}}\|F_{\varsigma}(x)\|^{2}d\mu(\varsigma)=\dfrac{1}{9}\|x\|^{2}$

and

$\int_{\mathfrak{A}}\|G_{\varsigma}(x)\|^{2}d\mu(\varsigma)=\dfrac{1}{4}\|x\|^{2}.$

Define $P:\mathcal{H}\rightarrow\mathcal{H}$ by
$P(x_{1},x_{2},x_{3},...)=(\dfrac{2}{3}x_{1},\dfrac{2}{3}x_{2},\dfrac{2}{3}x_{3},...)$.
Then $F_{\varsigma}P=G_{\varsigma}$ for all $\varsigma\in\mathfrak{A}$.

######  Theorem 2.25.

Let $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ be a $V-$controlled
integral frame. Then $\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$ has a
dual frame that is equivalent to
$\\{F_{\varsigma}\\}_{\varsigma\in\mathfrak{A}}$.

be

###### Proof.

Let $S_{VG}$ be the frame operator of $\\{F_{\varsigma}\\}_{\varsigma}$. Then
for $f\in\mathcal{H}^{N}$,

$S_{VG}f=\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
VF_{\varsigma}.d\mu(\varsigma)$

Since $S_{VG}$ is invertible, we have

$f=\int_{\mathfrak{A}}\langle
f,F_{\varsigma}\rangle(S^{-1}_{VG}V)F_{\varsigma}d\mu(\varsigma).$

Thus $\\{F_{\varsigma}\\}_{\varsigma}$ is a Parseval controlled integral frame
and the frame $\\{S_{VF}^{-1}VF_{\varsigma}\\}_{\varsigma}$ is a dual frame
for $\\{F_{\varsigma}\\}_{\varsigma}$ such that it is equivalent to
$\\{F_{\varsigma}\\}_{\varsigma}$. ∎

######  Proposition 2.26.

If $\\{G_{\varsigma}\\}_{\varsigma}$ is a dual of
$\\{F_{\varsigma}\\}_{\varsigma}$ such that it is equivalent to
$\\{F_{\varsigma}\\}_{\varsigma}$, then $\\{G_{\varsigma}\\}_{\varsigma}$
induces a Parseval controlled integral frame of
$\\{F_{\varsigma}\\}_{\varsigma}$.

###### Proof.

Let $\\{G_{\varsigma}\\}_{\varsigma}$ be a dual of
$\\{F_{\varsigma}\\}_{\varsigma}$ such that it is equivalent to
$\\{F_{\varsigma}\\}_{\varsigma}$. Then there exists an invertible operator
$V$ such that $G_{\varsigma}=VF^{\varsigma}$ for every
$\varsigma\in\mathfrak{A}$. We have

$f=\int_{\mathfrak{A}}\langle f,F_{\varsigma}\rangle
G_{\varsigma}d\mu(\varsigma),\;\forall f\in\mathcal{H}^{N}.$

Then $\\{F_{\varsigma}\\}_{\varsigma}$ is a Parseval $V-$controlled integral
frame. ∎

##  3\. Conclusion

In this manuscript we introduced and characterized controlled finite
continuous frames particularly Parseval controlled finite continuous frames as
a subset of dual frames and we reviewed some notions and properties of
operators and frames in Hilbert spaces. Also, we defined controlled finite
continuous frames and we gave their properties. Gramian matrix and its
properties for controlled finite continuous frames are examined. In the end we
studied controlled finite continuous frames as a proper subset of dual frames
is presented by the equivalent frames.

We will apply these results in a future work in Hardy and Sobolev spaces.

## Declarations

Availablity of data and materials
Not applicable.

Competing interest
The authors declare that they have no competing interests.

Fundings
Authors declare that there is no funding available for this article.

Authors’ contributions
The authors equally conceived of the study, participated in its design and
coordination, drafted the manuscript, participated in the sequence alignment,
and read and approved the final manuscript.

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# SimCGNN: Simple Contrastive Graph Neural Network for Session-based
Recommendation

Yuan Cao1,2 Xudong Zhang1,2 Fan Zhang1,2 Feifei Kou1 Josiah Poon3 Xiongnan
Jin4 Yongheng Wang4 &Jinpeng Chen1,2,∗ 1School of Computer Science (National
Pilot Software Engineering School), Beijing University of Posts and
Telecommunications, Beijing, China
2Key Laboratory of Trustworthy Distributed Computing and Service (BUPT),
Ministry of Education, Beijing, China
3 School of Computer Science, The University of Sydney, Sydney, Australia
4 Zhejiang Lab, Zhejiang, China {caoyuanboy, zhangxudong, zhang_fan,
koufeifei000<EMAIL_ADDRESS><EMAIL_ADDRESS>{xiongnan.jin,
<EMAIL_ADDRESS>
###### Abstract

Session-based recommendation (SBR) problem, which focuses on next-item
prediction for anonymous users, has received increasingly more attention from
researchers. Existing graph-based SBR methods all lack the ability to
differentiate between sessions with the same last item, and suffer from severe
popularity bias. Inspired by nowadays emerging contrastive learning methods,
this paper presents a Simple Contrastive Graph Neural Network for Session-
based Recommendation (SimCGNN). In SimCGNN, we first obtain normalized session
embeddings on constructed session graphs. We next construct positive and
negative samples of the sessions by two forward propagation and a novel
negative sample selection strategy, and then calculate the constructive loss.
Finally, session embeddings are used to give prediction. Extensive experiments
conducted on two real-word datasets show our SimCGNN achieves a significant
improvement over state-of-the-art methods.

##  1 Introduction

With the progressive development of the contemporary internet and the
explosion of information on the Internet, recommender systems have become an
essential component. Sequential recommender systems consider the dynamic
preference development and take both user-level information and item-level
information into consideration. In certain scenarios, however, the user can be
anonymous, which means that we only have access to item-level features and
user-level features are not visible. Session-based recommender, which observes
only the current session rather than sufficient historical user-item
interaction records, has drawn attention in recent years.

Figure 1: We trained SR-GNN on the diginetica dataset and selected a set of
sessions (127 in total) with the same last-interacted item (item 33889, the
most commonly occurred last-item) from the diginetica dataset for prediction.
This line graph counts the number of occurrences of different items on the
ground truth and model predicted values.

To capture the sequential relationship, Markov Chain (MC)-based sequential
recommendersHe and McAuley (2016)Rendle et al. (2010) are the first to be
proposed. However, due to limited representation ability and the strong
reliance upon the last interacted item in each session, their performances are
limited. Recurrent neural networks (RNNs) are then introduced into session-
based recommendation thanks to their natural ability for modeling sequential
information. With the introduction of GRU4RecHidasi et al. (2016), RNN turned
out to be the structure of choice for solving the session recommendation
problem, for example, NARMLi et al. (2017) designs two RNNs to capture user’s
global and local sequential preference correspondingly. It was not until the
existence of SR-GNNWu et al. (2019), which utilized a gated graph neural
network (GGNN) to extract sequential information on a session graph. Since
then, works following the basic schema of SR-GNN have been proposed. TAGNNYu
et al. (2020) added a target-attentive mechanism onto SR-GNN and gain
promising performance. Also, as SR-GNN focuses only on a local session,
methods like FGNNQiu et al. (2019) and GCE-GNNWang et al. (2020) both utilize
global information in session-based recommendation but in different ways.

Although all methods mentioned above have favorable results, they maintain a
drawback inherited from the MC-based approach to the present day. That is the
strong dependency upon the last interacted item. Take the classical SR-GNN as
an example, when assembling the final session, it simply takes linear
transformation over the concatenation of the embedding vector of the last-item
and the global embedding vector. And as a result, these methods may have
trouble in distinguishing between sessions that have the same last interacted
item. As shown in Fig.1, we first trained an SR-GNN on the Diginetica dataset
and then visualized the distribution of predicted labels and ground-truth
labels of sessions that shares the same last-interacted item. This indicates a
phenomenon that methods with this assembling techniques tends to predict the
same items for sessions with the same last-item, despite of the underlying
diversity. As we have mentioned earlier, a large branch of existing graph-
based methods (Yu et al. (2020); Wang et al. (2020) for example) refers to the
same approach as SR-GNN in the final assembly of the session embedding,
resulting in this phenomenon will also be widespread in the current advanced
graph-based SBR method. In this paper, we refer to this phenomenon as the
”same last-item confusion” problem. Apart from this, Fig.1 also proved the
hypothesis proposed by Gupta et al. (2019) that the prediction of SR-GNN is
likely to be affected by popularity bias.

To cope with the last-item, intuitively we tend to increase the discrepancy
between sessions with the same last-item, which naturally fits the schema of
constractive learning methods by considering sessions with the same last-item
as negative samples. However, as positive samples are hard to define, we
simply follows the idea of Gao et al. (2021) by treating the session itself as
a positive sample through dropoutSrivastava et al. (2014) techniques. To this
end, we proposed a novel session-based recommender namely Simple Contrastive
Graph Neural Network for Session-based Recommendation (SimCGNN). Firstly, in
order deal with the ”same last-item confusion” problem, we designed a novel
contrastive module to increase the discrepancy between sessions with the same
last interacted item. Secondly, to eliminate the advantage of popular items in
the final prediction and to predict the underlying interest of the users
themselves as much as possible, we normalised both the embedding of items and
sessions.

Our main contributions in this paper are listed as follows.

1) We introduce a contrast learning approach to solve the last-item
interaction confusion problem and propose a novel negative sample selection
strategy.

2) To alleviate the popularity bias, we proposed normalized item embeddings
and session embeddings.

3) Extensive experiments conducted on real-world datasets show that SimCGNN
outperforms the sota methods and additional experiments also demonstrate the
effectiveness of our approach for both of these two problems.

##  2 Related Work

###  2.1 Traditional Methods

A series of approaches Sarwar et al. (2001a); Salakhutdinov and Mnih (2007);
Koren et al. (2009) based on matrix factorization (MF) are representatives of
the traditional methods. MF-based methods consider that the user-item
interaction matrix is too sparse and then decompose the matrix into two low-
rank dense matrices, which correspond to users and items respectively.
However, MF-based approaches do not model users’ sequential interaction
behavior, and therefore not suitable for making sequential recommendations.

Given that static matrix decomposition methods do not model sequence behavior,
FPMCRendle et al. (2010) models user behavior as a Markov Chain and combines
Markov Chains with MF. FOSSILHe and McAuley (2016) improves on FPMC by
introducing factorized sequential prediction with an item similarity model and
higher-order Markov Chains, thus both long-term and short-term user behaviour
are taken into account. However, due to expressive capability limitation of
their shallow network, they do not perform well in nowadays more complex
recommendation scenarios.

###  2.2 Deep Learning-Based Methods

In recent years, as deep-learning methods emerge, deep learning-based
recommendation methodsSun et al. (2019); de Souza Pereira Moreira et al.
(2021) utilizing recurrent neural networks (RNNs) are on the rise.
GRU4RecHidasi et al. (2016) is a typical pioneer in utilizing RNN structure
for making sequential recommendations. After that, more attempts have been
made to perform sequential recommendations on RNNs. Tan et al.Tan et al.
(2016) improved the performance of RNN recommendation models by proposing
several data enhancements and training tricks. Li et al.Li et al. (2017)
proposed a neural attentive recommendation machine with an encoder-decoder
structure based on RNN to capture the user’s sequential preference. STAMPLiu
et al. (2018), which deposits the RNN structure by using simple MLPs and
attention mechanisms, proved to be efficient in capturing both users’ static
and dynamic interests. Although deep learning-based models have more powerful
representation capabilities than traditional methods, these approaches caanot
still model complex item relationships, for example, non-adjacent item
transitions.

###  2.3 Neural Network on Graphs

Recently, thanks to the rise of graph neural networksVelickovic et al. (2017);
Chang et al. (2015); Dong et al. (2020), many approachesXu et al. (2019); Chen
et al. (2022); He et al. (2020) use GNN-based network structures to solve SBR
problems. SR-GNNWu et al. (2019) is one of the earliest and most
representative ones, which models each session as a session graph, and applies
a gated-GNNLi et al. (2016) to finally get a representative embedding of the
session. After the great success of SR-GNN, many variants of SR-GNN have been
proposed. For example, TAGNNYu et al. (2020) proposed a target-aware attention
mechanism upon SR-GNN, which adaptively activates different user interests
concerning varied target items. FGNNQiu et al. (2019) takes both sequence
order and the latent order in the session graph into consideration. Disen-
GNNLi et al. (2022) constructs a disentangled session graph to discover
underlying session purpose. $S^{2}$-DHCNXia et al. (2021) leverages hypergraph
techniques to represent each session and utilizes constractive learning
techniques to perform self-supervised learning. Instead of utilizing local
session graphs only, methods such as GCE-GNNWang et al. (2020) construct
global graphs to obtain global information on the dataset. However, as
mentioned in the previous section, all of these methods have serious ”same
last-item confusion” problem and thus suffer from performance degradation.

##  3 Methodology

###  3.1 Problem Formulation

A session-based recommender is supposed to give recommendations to users based
on the inputs of their anonymous historical interaction sequences, e.g. clicks
or purchases. Given $\mathcal{V}=\left\\{v_{1},v_{2},...,v_{m}\right\\}$ as
the item set, an anonymous interaction session $s$ is an item sequence
$s_{i}=[v_{1}^{i},v_{2}^{i},...,v_{|s_{i}|}^{i}],s_{i}\in\mathcal{S}$, where
$v_{j}^{i}$ is the $j$-th interacted item of the $i$-th session. Given session
$s_{i}$, our goal is to predict $v_{|s_{i}|+1}^{i}$.

###  3.2 Overview

Figure 2: The overall network architecture of _SimCGNN_.

The overall workflow of our proposed SimCGNN is illustrated in Fig.2.

###  3.3 Session Feature Extraction Module

####  3.3.1 Normalized Item Embedding

Since the one-hot vectors of items are sparse and high-dimensional and do not
carry pairwise distance information, we first embed each item
$v_{i}\in\mathcal{V}$ into a $d$-dimensional representation
$\bm{v}_{i}\in\mathbb{R}^{d}$.

As some existing methods directly utilize $\bm{v}_{i}$ for downstream tasks,
we argue that the direct use of embedded vectors would lead to popularity bias
Gupta et al. (2019). To this end, we intuitively add $L_{2}$ normalization to
the raw feature vector. Apart from this, we also add Dropout directly on
embedding layer for the downstream contrastive module. The normalized vectors
$\hat{\bm{v}}_{i}$ are given as follows:

$\hat{\bm{v}}_{i}=Dropout(Norm(\bm{v}_{i}),p)$ (1)

where $p$ is the dropout probability, $Norm(\cdot)$ is the L2-normalization
function.

####  3.3.2 Session Graph Construction

To explore rich transitions among items and generate accurate latent vectors
of items, we model each session $s$ as a directed graph
$\mathcal{G}_{s}=(\mathcal{V}_{s},\mathcal{E}_{s})$, where each node
$v_{s,i}\in\mathcal{V}_{s}$ represents an item and each edge
$(v_{s,i-1},v_{s,i})\in\mathcal{E}_{s}$ represents a user interact $v_{s,i}$
after the interaction of $v_{s,i-1}$. To deal with items that occurs multiple
times in sessions, we assign a normalized weight to each edge. The weight is
calculated by the occurrence of the corresponding interaction
($(v_{s,i-1},v_{s,i})$) divides the out-degree of the starting node
($v_{s,i-1}$). We use the previously obtained embedding vector
$\hat{\bm{v}}_{i}$ as the initial state of the nodes in the constructed
session graph. After the construction, we obtain the outgoing adjacency matrix
$\mathbf{A}_{i}^{out}\in\mathbb{R}^{d\times d}$ and incoming adjacency matrix
$\mathbf{A}_{i}^{in}\in\mathbb{R}^{d\times d}$, where $n=|V|$. By
concatenating $\mathbf{A}_{i}^{out}$ and $\mathbf{A}_{i}^{in}$, we obtain the
final connection matrix $\mathbf{A}_{s}\in\mathbb{R}^{d\times 2d}$ of the
session graph.

####  3.3.3 Learning Graph-based Item Embeddings

Once the corresponding session graph has been constructed, we will then need
to use a graph neural network to extract the structured information from the
graph. In SimCGNN, we leverage a gated graph neural network (GGNN) to learn
node vectors in a session graph. Formally, for node $v_{s,i}$ in graph
$\mathcal{G}_{s}$, the update function can be formulated as,

$\displaystyle\bm{a}_{s,i}^{t}$
$\displaystyle=\mathbf{A}_{s,i:}\left[\bm{v}_{1}^{t-1},\bm{v}_{2}^{t-1},...,\bm{v}_{|s_{i}|}^{t-1}\right]^{\top}\mathbf{H}+\bm{b},$
(2) $\displaystyle\bm{z}_{s,i}^{t}$
$\displaystyle=\sigma\left(\mathbf{W}_{z}\bm{a}_{s,i}^{t}+\mathbf{U}_{z}\bm{v}_{i}^{t-1}\right),$
(3) $\displaystyle\bm{r}_{s,i}^{t}$
$\displaystyle=\sigma\left(\mathbf{W}_{r}\bm{a}_{s,i}^{t}+\mathbf{U}_{r}\bm{v}_{i}^{t-1}\right),$
(4) $\displaystyle\overline{\bm{v}}_{i}^{t}$
$\displaystyle=\mathrm{tanh}\left(\mathbf{W}_{o}\bm{a}_{s,i}^{t}+\mathbf{U}_{o}\left(\bm{r}_{s,i}^{t}\odot\bm{v}_{i}^{t-1}\right)\right),$
(5) $\displaystyle\bm{v}_{i}^{t}$
$\displaystyle=\left(1-\bm{z}_{s,i}^{t}\right)\odot\bm{v}_{i}^{t-1}+\bm{z}_{s,i}^{t}\odot\overline{\bm{v}}_{i}^{t},$
(6)

where $t$ is the training step, $\mathbf{A}_{s,i:}$ is the $i$-th row of
matrix $\mathbf{A}_{s}$ corresponding to $v_{s,i}$,
$\mathbf{H}\in\mathbb{R}^{d\times 2d}$ is trainable parameter,
$\bm{z}_{s,i}^{t}$ and $\bm{r}_{s,i}^{t}$ are the reset and update gates
respectively,
$\left[\bm{v}_{1}^{t-1},\bm{v}_{2}^{t-1},...,\bm{v}_{|s_{i}|}^{t-1}\right]$ is
the list of item vectors in $s$, $\sigma(\cdot)$ is the sigmoid function and
$\odot$ is the Hadamard product operator.

In order to add the necessary noise to the resulting graph item representation
vector, we also applied Dropout to the final output vector $\bm{v}_{i}^{l}$
after $l$-layers GNN. Also, since there is no location information embedded in
the graph neural network, we do the usual positional embedding on the
generated item vectors. The final graph-based item embeddings
$\widetilde{\bm{v}_{i}}$ can be calculated as,

$\widetilde{\bm{v}_{i}}=\bm{w}_{pos(v_{i})}+Dropout(\bm{v}_{i},p),$ (7)

where $\bm{w}_{t}\in\mathbb{R}^{d}$ is the trainable positional embedding,
$pos(v_{i})$ is the absolute position of the item in the session., $p$ is the
Dropout probability.

####  3.3.4 Generating Session Embeddings

In this section, we consider constructing the session representation vector by
combining long-term preference and short-term preference.

First, intuitively, we can represent the session’s short-term preference by
its last-interacted item $\bm{v}_{s,|s_{i}|}$. Thus, for session
$s_{i}=[v_{s,1},v_{s,2},...,v_{s,n}]$, the short-term session embedding
$\bm{s}_{i,s}$ of session $s$ can be defined as $\widetilde{\bm{v}}_{n}$.

As for the long-term preference, we consider the long-term session embedding
of session graph $\mathcal{G}_{s}$ by aggregating all node vectors. Meanwhile,
as different interaction records may own different levels of priority, we
utilize the attention mechanism to gain the long-term session preference
$\bm{s}_{i,l}$ as follows,

$\displaystyle\alpha_{i}$
$\displaystyle=\mathrm{softmax}(\bm{q}^{\top}\sigma(\mathbf{W}_{1}\widetilde{\bm{v}}_{n}+\mathbf{W}_{2}\widetilde{\bm{v}}_{i}+\bm{b})),$
(8) $\displaystyle\bm{s}_{i,l}$
$\displaystyle=\sum_{i=1}^{n}\alpha_{i}\widetilde{\bm{v}}_{i},$ (9)

where $\bm{q}\in\mathbb{R}^{d}$ and
$\mathbf{W}_{1},\mathbf{W}_{2}\in\mathbb{R}^{d\times d}$ are trainable
weights.

Finally, we get the final hybrid embedding $\bm{s}_{i,h}$ by simply linear
transform over the concatenation of the short-term and the long-term session
embeddings.

$\bm{s}_{i,h}=\mathbf{W}_{3}[\bm{s}_{i,l};\bm{s}_{i,s}],$ (10)

where $\mathbf{W}_{3}\in\mathbb{R}^{2d\times d}$ is trainable parameter and
$[\cdot;\cdot]$ represents concatenation operation.

###  3.4 Prediction Module

After extracting session hybrid embeddings, we adopt an MF layer to predict
the relevance between the given session $s_{i}$ and each candidate item
$v_{n}\in V$ by multiplying the normalized session representation
$Norm(\bm{s}_{i,h})$ and normalized item embedding $Norm(\bm{v}_{n})$ for the
avoidance of popularity bias, which can be defined as,

$\hat{h_{n}}=r*Norm(\bm{s}_{i,h})*Norm(\bm{v}_{n}),$ (11)

As $Norm(\bm{s}_{i,h})*Norm(\bm{v}_{n})$ is equal to the cosine similarity
between $\bm{s}_{i,h}$ and $\bm{v}_{n}$, the predicted logits are restricted
to $[-1,1]$, and the softmax score is likely to get saturated at high values
for the training set. To this end, we add a scaling factor $r>1$, which is
useful in practice to allow for better convergence.

Then we apply a softmax function on the output logit to get the final scaled
output probability vector $\hat{\bm{y}}$,

$\hat{\bm{y}}=\mathrm{softmax}(\hat{\bm{h}}),$ (12)

where $\hat{\bm{h}}$ denotes the recommendation scores of all candidate items
$v_{n}\in V$.

The prediction loss $\mathcal{L}_{pred}$ is defined by calculating the cross-
entropy of the prediction and ground truth,

$\mathcal{L}_{pred}=-\sum_{i=1}^{|D|}\bm{y}_{i}\log(\hat{\bm{y}})+(1-\bm{y}_{i})\log(1-\hat{\bm{y}}),$
(13)

where $\bm{y}$ denotes the one-hot encoding vector for the ground truth item.

###  3.5 Contrastive Module

As we mentioned before, simply using the hybrid session embedding
$\bm{s}_{i,h}$ for item prediction inevitably leads to the same last-item
confusion problem.

To address this, the most intuitive idea is to separate session
representations with the same last-item as much as possible. Naturally, we
came up with the idea of using comparative learning to solve this problem. We
refer to the schema of Gao et al. (2021), and since we have added the Dropout
layer above, we can simply perform twice forward propagation for each session
$s_{i}$ to produce two hybrid embeddings. For the embedding vector obtained by
the first forward propagation, we denote it by $\bm{s}_{i,h}^{1}$ and the
second by $\bm{s}_{i,h}^{2}$, which constructs a ”positive pair”. We use all
embeddings of sessions with the same last item as negative samples, which can
be represented by $\bm{s}^{-}_{j,h},j\in 1,2,...,N$, $N$ indicates the number
of negative samples. In this way, the contrastive loss $\mathcal{L}_{con}$ can
be calculated as,

$\mathcal{L}_{con}=-\sum_{i=1}^{|D|}\log\frac{e^{\mathrm{sim}(\bm{s}_{i,h}^{1},\bm{s}^{2}_{i,h})/\tau}}{\sum_{j=1}^{N}\left(e^{\mathrm{sim}(\bm{s}_{i,h}^{1},\bm{s}^{2}_{i,h})/\tau}+e^{\mathrm{sim}(\bm{s}_{i,h}^{1},\bm{s}^{-}_{j,h})/\tau}\right)},$
(14)

where $\tau$ is the temperature hyperparameter, $D$ is the training dataset,
and we utilize cosine similarity for $\mathrm{sim}(\cdot,\cdot)$ as follows,

$\mathrm{sim}(\bm{s}_{1},\bm{s}_{2})=\frac{\bm{s}_{1}\bm{s}_{2}}{||\bm{s}_{1}||||\bm{s}_{2}||}$
(15)

However, unlike Gao et al. (2021), since there are not always adequate
negative sessions with the same last-item in the same training batch, we
cannot directly use sessions in the same training batch as negative samples.
And if we forward the corresponding negative sessions each time we calculate
the contrastive loss, it would result in a huge waste of computational
resources. As a result, instead of exhaustively computing these
representations, similar to Wu et al. (2018), we maintain a memory bank
$M=\\{(\bm{f}^{1}_{i},\bm{f}^{2}_{i})\\}$. During each learning iteration, the
two hybrid embeddings $\bm{s}_{i,h}^{1},\bm{s}_{i,h}^{2}$ are updated to $M$
at the corresponding session entry
$\bm{s}_{i,h}^{1}\longrightarrow\bm{f}^{1}_{i},\bm{s}_{i,h}^{2}\longrightarrow\bm{f}^{2}_{i}$,
and negative session embeddings can be sampled from $M$, i.e.,
$\bm{s}_{j,1,h}^{-}\longleftarrow\bm{f}_{j}^{1},\bm{s}_{j,2,h}^{-}\longleftarrow\bm{f}_{j}^{2}$.
This allows us to compute the comparison loss without additional forward
propagation, but only with the corresponding vector in the memory bank $M$.

###  3.6 Model Optimization

In the previous section, we defined the prediction loss $\mathcal{L}_{pred}$
and the contrastive loss $\mathcal{L}_{con}$, we define our final loss
function with the integration of both of them as follows:

$\mathcal{L}=\mathcal{L}_{pred}+\beta\mathcal{L}_{con}+\lambda||\Theta||_{2}^{2},$
(16)

where $\Theta$ is all trainable parameters, $\beta$ is the hyper-parameter for
balancing the contrastive module, $\lambda$ is the regularization hyper-
parameter.

##  4 Experiments

###  4.1 Datasets

Following previous work, we choose two commonly used session recommendation
datasets, that is, Yoochoose
1/64111http://2015.recsyschallenge.com/challege.html and
Diginetica222http://cikm2016.cs.iupui.edu/cikm-cup. The Yoochoose dataset is
from the RecSys Challenge 2015 and consists of six months of interaction
sessions from an E-commercial website. We only make use of the most recent
fractions 1/64 of the training sequences of Yoochoose denoted. The Diginetica
dataset comes from CIKM Cup 2016, and only the transaction data is used.

For fairness consideration, we use the same preprocessing techniques as Wu et
al. (2019), which filter out all sessions of length 1 and items that appears
less than 5 times in both datasets. Moreover, we do data augmentation on a
session to obtain sequences and corresponding labels. For example, for session
$s=[v_{1},v_{2},...,v_{n}]$, we split it in to
$([v_{1}],v_{2}),([v_{1},v_{2}],v_{3}),...,([v_{1},v_{2},...,v_{n-1}],v_{n})$.
The statistics of the two datasets are shown in Table 1.

Table 1: Statistic of the Datasets Dataset | Yoochoose 1/64 | Diginetica
---|---|---
#click | 557,248 | 982,961
#training sessions | 369,859 | 719,470
#test sessions | 55,898 | 60,858
#items | 16,766 | 43,097
Average Length | 6.16 | 5.12

###  4.2 Evaluation Metrics

According to previous works, Recall@20 and MRR@20 are selected to evaluate the
performance of our method and baselines.

###  4.3 Baselines

To evident the effectiveness of our proposed SimCGNN, we compare it with the
following representative baselines.

  * •

POP and SPOP recommend the top-K popular item in training dataset and in the
current predicting session respectively.

  * •

Item-KNNSarwar et al. (2001b) leverages item-to-item collaborative filtering
techniques, which recommends items similar to the previously interacted items
by consine similarity.

  * •

BPRRendle et al. (2009) is a classical matrix factorization (MF) methods, and
is optimized by a pairwise ranking loss function.

  * •

FPMCRendle et al. (2010) is a sequential prediction method combining the
Markov chain and MF.

  * •

GRU4RECHidasi et al. (2016) utilizes the RNN structure to model the sequential
interaction of users and leverages multiple tricks to help the RNN converge to
the session recommendation problem.

  * •

NARMLi et al. (2017) improves GRU4REC by incorporating an attention mechanism
into RNN.

  * •

STAMPLiu et al. (2018) replaces the RNN structures by employing attention
mechanism.

  * •

SR-GNNWu et al. (2019) first introduces session graph structure into a
session-based recommendation. By utilizing a Gated GNN to extract on-graph
item embeddings, the last item together with a weighted sum of all the session
embeddings are concatenated for prediction.

  * •

FGNNQiu et al. (2019) formulates the next item recommendation within the
session as a graph classification problem.

  * •

GCE-GNNWang et al. (2020) constructed a global graph using session data and
modifies the model structure to introduce the global information learned from
the global graph.

  * •

Disen-GNNLi et al. (2022) proposes a disentangled graph neural network to
capture the session purpose.

  * •

$S^{2}$-DHCNXia et al. (2021) constructs each session as a hypergraph and
utilizes contractive learning methods.

###  4.4 Implementation Details

To align with the previous work, we set the hidden size $d=100$, while model
parameters are initialized using Gaussian distribution with a mean of 0 and
deviation of 0.1. The mini-batch Adam optimizer is utilized to optimize model
parameters. The initial learning rate is set to 1e-3 and decays by 0.1 every 3
epochs. Dropout probabilities for all dropout layers are set to 0.1. For the
contrastive module, the temperature parameter $\tau$ is set to 12, and $\beta$
is set to 0.1 for Yoochoose 1/64 and 1 for Diginetica. All the hyperparameters
are tuned on the validation set.

###  4.5 Experiment Results

Table 2: Recommendation performance on two datasets. The best performing
method in each column is boldfaced, and the second best method is underlined.

Method | Yoochoose 1/64 | Diginetica
---|---|---
Recall@20 | MRR@20 | Recall@20 | MRR@20
POP | 6.71 | 1.65 | 0.89 | 0.20
S-POP | 30.44 | 18.35 | 21.06 | 13.68
Item-KNN | 51.60 | 21.81 | 35.75 | 11.57
BPR | 31.31 | 12.08 | 5.24 | 1.98
FPMC | 45.62 | 15.01 | 26.53 | 6.95
GRU4REC | 60.64 | 22.89 | 29.45 | 8.33
NARM | 68.32 | 28.63 | 49.70 | 16.17
STAMP | 68.74 | 29.67 | 45.64 | 14.32
SR-GNN | 70.57 | 30.94 | 50.73 | 17.59
FGNN | 71.75 |  31.71 | 51.36 | 18.47
GCE-GNN | 70.91 | 30.63 | 54.22 | 19.04
Disen-GNN | 71.46 | 31.36 | 53.79 | 18.99
$S^{2}$-DHCN | 70.39 | 29.92 | 53.66 | 18.51
SimCGNN |  71.61 | 31.99 |  54.01 | 19.04

To further demonstrate the overall performance of our proposed SimCGNN, we
compare it with the selected baselines described above. The experimental
results are shown in Table 2. From Table 2, we can say that our SimCGNN
achieves the best performance on all two datasets, especially in MRR@20, which
illustrates the superior ranking capability compared with other baseline
methods.

Among traditional methods, the Item-KNN achieves the best performance,
although the overall performance of all traditional methods is relatively
poor. To our surprise, the simple yet effective S-POP shows better performance
than those of BPR and FPMC. Notably, the S-POP takes only item popularity into
consideration, which means both the Yoochoose 1/64 dataset and the Diginetica
dataset are suffered from popularity bias. It’s also vital to point out that,
the Item-KNN utilizes pairwise item similarities only and performs better than
FPMC, which is an MC-based approach with the assumption that only the last
interacted items are needed to perform sequential recommendation. This
phenomenon indicates that the simple MC assumption is not suitable for such
complex sessions.

As for Deep learning-based methods, all DL-based methods consistently
outperform traditional methods. GRU4REC and NARM are both based on RNN
structure, and they together achieved decent performances. However, since NARM
adds an attention mechanism to the original RNN to give different weights to
items at different positions in the session, the performance of NARM has a
significant improvement over GRU4REC. STAMP, which replaces RNN with
attentional MLPs, shows comparative performance over NARM. At last, all DL-
base methods share better performance over FPMC, which shows the importance of
modeling the whole interaction sequence instead of considering merely the last
click. However, both RNNs and MLPs are not suitable for capturing complex
transitions among sessions. This may be the reason why they perform worse than
graph-based methods.

Graph-based methods outperform all other baselines by a large margin. More
specifically, GCE-GNN outperforms SR-GNN as it effectively leverages global
information in different ways. FGNN also shows competitive results by
rethinking item order and replacing the assembling method by a well-designed
readout function. Disen-GNN gains decent performance on both datasets, showing
the importance of disentangled session graphs. The $S^{2}-DHCN$, however,
performs the worst on Yoochoose 1/64, which does not match the capacity of
hypergraph neural networks. Compared with these state-of-the-art methods, our
methods achieve comparative performances or even better performances without
explicitly applying global information. Expanding on this, our method has
optimal MRR@20 on both datasets. The Recall@20 of our method is higher than
GCE-GNN and lower than FGNN on the Yoochoose 1/64 dataset and the opposite on
the Diginetica dataset. We must also point out that the number of parameters
used in our SimCGNN is consistent with the SR-GNN. This means that we have
achieved a top-level performance using the least parameters among all graph-
based methods, which strongly demonstrates the superiority of our network
structure. In conclusion, graph-based methods have an inherent advantage over
traditional methods and DL-based methods in modeling complex sessions.

###  4.6 Ablation Study

Table 3: Performance of variants of SimCGNN.

Method | Yoochoose 1/64 | Diginetica
---|---|---
Recall@20 | MRR@20 | Recall@20 | MRR@20
SR-GNN | 70.57 | 30.94 | 50.73 | 17.59
SimCGNN |  71.61 | 31.99 | 54.01 | 19.04
-Contrast | 71.31 |  31.80 | 53.49 |  19.01
-WeakNeg | 71.65 | 31.16 |  53.99 | 18.92
-Norm | 71.01 | 31.55 | 51.77 | 17.58
-PE | 71.39 | 30.92 | 53.80 | 18.94

To further validate the effectiveness of each module in our SimCGNN, we
compare our SimCGNN with the following four variants.

  * •

SimCGNN-Contrast. We removed the contrastive module of SimCGNN to prove its
effectiveness.

  * •

SimCGNN-WeakNeg. We randomly sample negative sessions instead of choosing
sessions with the same last item.

  * •

SimCGNN-Norm. We removed all normalizations to prove the effectiveness of
normalized session embeddings.

  * •

SimCGNN-PE. We removed positional embeddings to validate whether the
positional information is useful for session graphs.

The results of the proposed ablation studies are shown in Table 3.

As the most essential component of our approach, we first verified the
effectiveness of the proposed contrastive module. It is not difficult to see
from the experimental results that SimCGNN-Contrast performs weaker than the
original SimCGNN on both datasets. This demonstrates the effectiveness of
using the contrastive related approach to enhance the representation ability
of session embeddings.

In SimCGNN-WeakNeg, we did not emphasize the importance of negative sessions
with the same last-item, which greatly affected the ranking performance of the
model on both datasets. In terms of recall metrics, the model was not affected
too much, and there was even a marginal increase on the Yoochoose 1/64
dataset. This illustrates that our sampling method can give more favorable
rankings to items that are more suitable for the target session in a
relatively similar set of candidate items.

As shown in Table 3, the overall performance of SimCGNN-Norm is severely
damaged. This is not only because normalization is effective in suppressing
popularity bias, but also because removing normalization makes it more
difficult for the contrastive module to learn the intrinsic discrepancies
between sessions.

We finally verified the effectiveness of introducing positional information
into the session graph. Experimental results demonstrate that the ranking
performance of the model, especially on the Yoochoose 1/64 dataset, is greatly
affected after the removal of the positional embedding.

###  4.7 Case Study

####  4.7.1 Effect on Solving Same Last-Item Confusion

Consistent with the Introduction section, we still take sessions with the same
last-item (item 33889) and utilize the trained SR-GNN and SimCGNN to provide
predictions. For each session, 20 items are recommended and counted. The
relationship between the number of occurrences of an item in the prediction
candidate set and its corresponding occurrence ranking is shown in Fig.3. From
the figure we can see that SimCGNN predicts a lower curve of item occurrences
compared to SR-GNN, which shows that SimCGNN can provide different
recommendations for sessions with the same last-item. At the same time,
SimCGNN recommended a total of 142 kinds of items, 20 more than the 122 kinds
of items in SR-GNN, which also demonstrates the effectiveness of our method in
solving the same last-item confusion.

Figure 3: Predicted results of SimCGNN and SR-GNN on sessions with the same
last-item (item 33889).

####  4.7.2 Effect on Solving Popularity Bias

To measure the popularity of recommended items, we calculated Average
Recommendation Popularity (ARP) on SR-GNN and SimCGNN to compare the
difference in popularity of recommended items between the two methods. The ARP
can be calculated as follows,

$ARP=\frac{1}{|S|}\sum_{s\in S}\frac{\sum_{i\in L_{s}}\phi(i)}{K},$ (17)

where $\phi(i)$ is the number of times that item $i$ appears in the training
dataset, $L_{s}$ is the recommended item list for session $s$, and $S$ is the
set of sessions in the test dataset.

From Table 4, it is easy to see that our SimCGNN method has a huge difference
in the popularity of recommended items on both datasets compared to SR-GNN.
This proves that our method can indeed solve the problem of popularity bias to
some extent.

Table 4: SimCGNN versus SR-GNN in terms of Average Recommendation Popularity (ARP). Lower ARP value indicates lower popularity bias. Method | Yoochoose 1/64 | Diginetica
---|---|---
SR-GNN | 4128.54 | 495.25
SimCGNN | 2678.64 | 285.85

##  5 Conclusion

In this paper, to address the session-based recommendation problem, we
proposed a novel Simple Contrastive Graph Neural Network (SimCGNN), which
introduces a contrastive module to deal with the ”same last-item confusion”
problem and normalized item and session embeddings to cope with popularity
bias. Experiments on two real-world datasets validate that our SimCGNN
outperforms the state-of-the-art approaches with a significant margin in terms
of Recall@20 and MRR@20. In future work, we aim to propose a new combination
approach that can eliminate the impact of the last item to a greater extent.

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