#  Approximative Atomic Systems for operators in Banach spaces

Shah Jahan  Shah Jahan, Department of Mathematics, Ramjas College, University
of Delhi, Delhi-110007, India<EMAIL_ADDRESS>
###### Abstract.

$K$-frames and atomic systems for an operator $K$ in Hilbert spaces were
introduced by Gavruta [17] and further studied by Xio, Zhu and Gavruta [35].
In this paper, we have introduced the notion of an approximative atomic system
for an operator $K$ in Banach spaces and obtained interesting results. A
complete characterization of family of approximative local atoms of subspace
of Banach space has been obtained. Also, a necessary and sufficient condition
for the existence of an approximative atomic system for an operator $K$ is
given. Finally, explicit methods are given for the construction of an
approximative atomic systems for an operator $K$ from a given Bessel sequence
and approximative $\mathcal{X}_{d}$-Bessel sequence.

###### Key words and phrases:

Approximative atomic decomposition, Approximative $K$-atomic decomposition,
frames, approximative $\mathcal{X}_{d}$-frame, approximative
$\mathcal{X}_{d}$-Bessel sequence, atomic systems, local atoms.

###### 2010 Mathematics Subject Classification:

42C15; 42C30; 42C05; 46B15

##  1\. Introduction

A sequence $\\{f_{n}\\}_{n=1}^{\infty}$ in separable Hilbert space
$\mathcal{H}$ is called a frame for the Hilbert space $\mathcal{H},$ if there
exist positive constants $A,~{}B>0$ such that

$\displaystyle A\|f\|^{2}_{\mathcal{H}}\leq\sum_{n=1}^{\infty}|\langle
f,f_{n}\rangle|^{2}\leq B\|f\|^{2}_{\mathcal{H}},~{}~{}~{}\mbox{for
all}~{}~{}f\in\mathcal{H}$ (1.1)

The positive constants $A$ and $B$ are called the lower and upper frame bounds
of the frame, respectively. The inequality in (1.1) is called the frame
inequality of the frame. The frame is called a tight frame if $A=B$ and is
called normalized tight frame if $A=B=1.$ If $\\{f_{n}\\}_{n=1}^{\infty}$ is a
frame for $\mathcal{H}$ then the following operators are associated with it.

  1. (a)

Pre-frame operator $T:l^{2}(\mathbb{N})\longrightarrow\mathcal{H}$ is defined
as
$T\\{c_{n}\\}_{n=1}^{\infty}=\sum\limits_{k=1}^{\infty}c_{n}f_{n},~{}~{}\\{c_{n}\\}_{n=1}^{\infty}\in
l^{2}(\mathbb{N}).$

  2. (b)

Analysis operator $T^{*}:\mathcal{H}\longrightarrow
l^{2}(\mathbb{N}),T^{*}f=\\{\langle
f,f_{k}\rangle\\}_{k=1}^{\infty}~{}~{}~{}f\in\mathcal{H}.$

  3. (c)

Frame operator
$S=TT^{*}=:\mathcal{H}\longrightarrow\mathcal{H},~{}~{}Sf=\sum\limits_{k=1}^{\infty}\langle
f,f_{k}\rangle f_{k},~{}~{}f\in\mathcal{H}.$ The frame operator $S$ is
bounded, linear and invertible on $\mathcal{H}$. Thus, a frame for
$\mathcal{H}$ allows each vector in $\mathcal{H}$ to be written as a linear
combination of the elements in the frame, but the linear independence between
the elements is not required; i.e for each vector $f\in\mathcal{H}$ we have,

$\displaystyle f=SS^{-1}f=\sum\limits_{k=1}^{\infty}\langle f,f_{k}\rangle
f_{k}.$

Frames in Hilbert spaces were introduced by Duffin and Schaeffer [11] in 1952,
while addressing some deep problems in non-harmonic Fourier series. Frames
were reintroduced by Daubechies, Grossmann and Meyer [9] after three decades,
in 1986, frames were brought to life, and were widely studied after this nobel
work. Frames are generalization of orthonormal basis. The main property of
frames which makes them useful is their redundancy. Now, frames play an
important role not only in pure mathematics but also in applied mathematics.
Representation of signals using frames is advantageous over basis expansions
in a variety of practical applications in science and engineering. In
particular, frames are widely used in sampling theory [1, 13], wavelet theory
[10], signal processing [4], image processing [28], pseudo-differential
operators [19], filter banks [6], quantum computing [14], wireless sensor
network [21], coding theory [32], geometry[23, 24] and so on. Feichtinger and
Gröcheing [16] extended the notion of frames to Banach space and defined the
notion of atomic decomposition. Gröcheing [20] introduced a more general
concept for Banach spaces called Banach frame. Banach frames and atomic
decompositions were further studied in [7, 20, 27]. Casazza, Christensen and
Stoeva [5] studied $\mathcal{X}_{d}$-frame and $\mathcal{X}_{d}$-Bessel
sequence in Banach spaces. Shah [25] defined and studied approximative
$\mathcal{X}_{d}$-frames and approximative $\mathcal{X}_{d}$-Bessel sequences.
He gave the following definition.

######  Definition 1.1.

[25] A sequence
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subseteq\mathcal{X}^{*},$
where $\\{m_{n}\\}$ is an increasing sequence of positive integers, is called
an approximative _$\mathcal{X}_{d}$ -frame_ for $\mathcal{X}$ if

  1. (a)

$\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\in\mathcal{X}_{d}$,
for all $x\in\mathcal{X}.$

  2. (b)

There exist constants $A$ and $B$ with $0<A\leq B<\infty$ such that

$\displaystyle
A\|x\|_{\mathcal{X}}~{}\leq\|\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\|_{\mathcal{X}_{d}}~{}\leq
B\|x\|_{\mathcal{X}},~{}\text{for all}\ x\in\mathcal{X}.$ (1.2)

The constants $A$ and $B$ are called approximative _$\mathcal{X}_{d}$ -frame_
bounds. If atleast (a) and the upper bound condition in (1.2) are satisfied,
then $\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$ is
called an approximative _$\mathcal{X}_{d}$ -Bessel sequence_ for
$\mathcal{X}.$
One may note that if $\\{f_{n}\\}$ is an $\mathcal{X}_{d}$-frame for
$\mathcal{X},$ then for
$\\{h_{n,i}\\}=f_{i},~{}i=1,2,3,...,n;~{}n\in\mathbb{N}$,
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$ is an
approximative $\mathcal{X}_{d}$-frame for $\mathcal{X}.$ Also, note that if
$\\{f_{n}\\}$ is an $\mathcal{X}_{d}$-Bessel sequence for $\mathcal{X},$ then
for $\\{h_{n,i}\\}=f_{i},~{}i=1,2,3,...,n;~{}n\in\mathbb{N}$,
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$ is an
approximative $\mathcal{X}_{d}$-Bessel sequence for $\mathcal{X}.$ The bounded
linear operator $U:\mathcal{X}\rightarrow\mathcal{X}_{d}$ given by

$\displaystyle
U(x)=\\{h_{n,i}(x)\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}},x\in\mathcal{X}$
(1.3)

is called an _analysis operator_ associated to the approximative
$\mathcal{X}_{d}$-Bessel sequence
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$. If
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$ is an
approximative _$\mathcal{X}_{d}$ -frame_ for $\mathcal{X}$ and there exists a
bounded linear operator $S:\mathcal{X}_{d}\longrightarrow\mathcal{X}$ such
that $S(\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})=x,$
for all $x\in\mathcal{X}$, then
$(\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}},S)$ is called
an approximative _Banach frame_ for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$.
Atomic systems for an operator $K$ in Hilbert spaces were introduced by
Gavruta [17]. Xiao et al. [35] discussed relationship between $K$-frames and
ordinary frames in Hilbert spaces. Poumai and Jahan [30] introduced atomic
systems for operators in Banach spaces. Frames for operators in Banach spaces
were further studied in [8, 18].
Outline of the paper. $K$-frames and atomic systems for an operator $K$ in
Hilbert spaces were introduced by Gavruta [17] and further studied by Xiao,
Zhu and Gavruta [35] discussed relationship between $K$-frames and frames in
Hilbert spaces. In the present paper, we define approximative atomic system
for an operator $K$ in a Banach space and prove some results on the existence
of approximative atomic system for $K$. We also define approximative family of
local atoms for subspaces and give a characterization for the approximative
family of local atoms for subspaces. Also, we discuss methods to construct
approximative atomic system for an operator $K$ from an approximative Bessel
sequence and an approximative $\mathcal{X}_{d}$-Bessel sequence.

Throughout this paper, $\mathcal{X}$ will denote a Banach space over the
scalar field $\mathbb{K}$($\mathbb{R}$ or $\mathbb{C}$), $\mathcal{X}^{*}$ the
dual space of $\mathcal{X}$, $\mathcal{X}_{d}$ a BK-space,
$\mathcal{X}_{d}^{*}$ denotes dual of $\mathcal{X}_{d}$ and we will assume
that $\mathcal{X}_{d}$ has a sequence of canonical unit vectors as basis. By
$\\{h_{n,i}\\}$ we means a sequence of coefficient functionals of row finite
matrix. $L(\mathcal{X})$ will denote the set of all bounded linear operators
from $\mathcal{X}$ into $\mathcal{X}$. For $T\in L(\mathcal{X})$, $T^{*}$
denotes the adjoint of $T$. $\pi:\mathcal{X}\longrightarrow\mathcal{X}^{**}$
is the natural canonical projection from $\mathcal{X}$ onto
$\mathcal{X}^{**}$. A sequence space $S$ is called a _BK-space_ if it is a
Banach space and the co-ordinate functionals are continuous on $S.$ That is
the relations $x_{n}=\\{{\alpha_{j}}^{(n)}\\}$, $x=\\{\alpha_{j}\\}\in S$,
$\lim\limits_{n\longrightarrow\infty}x_{n}=x$ imply
$\lim\limits_{n\longrightarrow\infty}\alpha_{j}^{(n)}=\alpha_{j}~{}~{}~{}(j=1,2,3,...).$

##  2\. Preliminaries

Gavruta [17], introduced the notion of $K$-frame and atomic system for an
operator $K$ in a Hilbert space. She gave the following definition.

######  Definition 2.1.

[17] Let $\mathcal{H}$ be a Hilbert space, $K\in L(\mathcal{H})$ and
$\\{x_{n}\\}\subseteq\mathcal{H}$. Then
(a) $\\{x_{n}\\}$ is called a _K-frame_ for $\mathcal{H}$ if there exist
constants $A,B>0$ such that

$\displaystyle A\|K^{*}x\|^{2}\leq\sum\limits_{n=1}^{\infty}|\langle
x,x_{n}\rangle|^{2}\leq B\|x\|^{2},\ \text{for all}\ x\in\mathcal{H}.$

(b) $\\{x_{n}\\}$ is called an _atomic system for $K$_ if

  1. (i)

the series $\sum\limits_{n=1}^{\infty}c_{n}x_{n}$ converges for all
$c=\\{c_{n}\\}\in l^{2}$.

  2. (ii)

there exists $C>0$ such that for every $x\in\mathcal{H}$ there exists
$\\{a_{n}\\}\in l^{2}$ such that $\|\\{a_{n}\\}\|_{l^{2}}\leq C\|x\|$ and
$K(x)=\sum\limits_{n=1}^{\infty}a_{n}x_{n}.$

Shah [25] defined and studied approximative K-atomic decompositions and
approximative $\mathcal{X}_{d}$-frames in Banach spaces. For further studies
related to this concept one may refer [22, 26].

Next, we give some results which we will use throughout this manuscript.

######  Definition 2.2.

[29] Let $T\in L(\mathcal{X})$, we say that an operator S$\in L(\mathcal{X})$
is a pseudoinverse of $T$ if $TST=T$. Also, $S\in L(\mathcal{X})$ is the
generalized inverse of $T$ if $TST=T$ and $STS=S$.

######  Lemma 2.3.

[36] Let $\mathcal{X}$, $\mathcal{Y}$ be Banach spaces and
$T:\mathcal{X}\longrightarrow\mathcal{Y}$ be a bounded linear operator. Then
the following statements are equivalent:

  1. (1)

There exist two continuous projection operators
$P:\mathcal{X}\rightarrow\mathcal{X}$ and
$Q:\mathcal{Y}\rightarrow\mathcal{Y}$ such that

$\displaystyle P(\mathcal{X})=kerT\quad and\quad
Q(\mathcal{Y})=T(\mathcal{X}).$ (2.1)

  2. (2)

There exist closed subspaces $W$ and $Z$ such that $\mathcal{X}=KerT\oplus W$
and $\mathcal{Y}=T(\mathcal{X})\oplus Z.$

  3. (3)

$T$ has a pseudo inverse operator $T^{\dagger}$.

If two continuous projection operators $P:\mathcal{X}\rightarrow\mathcal{X}$
and $Q:\mathcal{Y}\rightarrow\mathcal{Y}$ satisfies (2.1), then there exists a
pseudo inverse operator $T^{\dagger}$ of $T$ such that

$T^{\dagger}T=I_{\mathcal{X}}-P\ and\ TT^{\dagger}=Q,$

where $I_{\mathcal{X}}$ is the identity operator on $\mathcal{X}$.

######  Lemma 2.4.

[3, 33] Let $\mathcal{X}$ be a Banach space. If $T\in L(\mathcal{X})$ has a
generalized inverse $S\in L(\mathcal{X}).$ Then $TS$,$ST$ are projections and
$TS(\mathcal{X})=T(\mathcal{X})$ and $ST(\mathcal{X})=S(\mathcal{X}).$

######  Lemma 2.5.

[15] Let $\mathcal{X}$ and $\mathcal{Y}$ be Banach spaces. Let
$L:\mathcal{X}\rightarrow\mathcal{X}$ and
$L_{1}:\mathcal{X}\rightarrow\mathcal{Y}$ be linear operators. Then the
following conditions are equivalent:

  1. (a)

$L=L_{2}L_{1}$, for some continuous linear operator
$L_{2}:L_{1}(\mathcal{X})\rightarrow\mathcal{X}$.

  2. (b)

$\|L(\mathcal{X})\|\leq k\|L_{1}(\mathcal{X})\|$, for some $k\geq 0$ and for
all $x\in\mathcal{X}$.

  3. (c)

$Range(L^{*})\subseteq Range(L_{1}^{*})$.

######  Lemma 2.6.

[2] Assume $T\in B(\mathcal{X},\mathcal{Y})$. If $S\in
B(\mathcal{X},\mathcal{Z})$ with $Range(S^{*})\subseteq Range(T^{*})$ and
$\overline{Range(T)}$ is complemented, then there exists $V\in
B(\mathcal{Y},\mathcal{Z})$ such that $S=VT.$

######  Lemma 2.7.

[5] Let $\mathcal{X}_{d}$ be a BK-space for which the canonical unit vectors
$\\{e_{n}\\}$ form a Schauder basis. Then the space
$\mathcal{Y}_{d}=\\{\\{h(e_{n})\\}|h\in\mathcal{X}_{d}^{*}\\}$ with norm
$\|\\{h(e_{n})\\}\|_{\mathcal{Y}_{d}}=\|h\|_{\mathcal{X}_{d}^{*}}$ is a BK-
space isometrically isomorphic to $\mathcal{X}_{d}^{*}$. Also, every
continuous linear functional $\Phi$ on $\mathcal{X}_{d}$ has the form
$\Phi\\{c_{n}\\}=\sum\limits_{n=1}^{\infty}c_{n}d_{n},$ where
$\\{d_{n}\\}\in\mathcal{Y}_{d}$ is uniquely determined by $d_{n}=\Phi(e_{n})$,
and $\|\Phi\|=\|\\{\Phi(e_{n})\\}\|_{\mathcal{Y}_{d}}$.

Terekhin [34] introduced and studied frames in Banach spaces.

######  Definition 2.8.

[34] Let $\mathcal{X}$ be a Banach space and $\mathcal{X}_{d}$ be a BK-space
with the sequence of canonical unit vectors $\\{e_{n}\\}$ as basis. Let
$Y_{d}$ be a sequence space mentioned in Lemma 2.7. A sequence
$\\{x_{n}\\}_{n=1}^{\infty}\subseteq\mathcal{X}$ is called a frame with
respect to $\mathcal{X}_{d}$ if

  1. (a)

$\\{f(x_{n})\\}\in Y_{d},$ for all $f\in\mathcal{X}^{*}$,

  2. (b)

there exist constants $A$ and $B$ with $0<A\leq B<\infty$ such that

$\displaystyle
A\|f\|_{\mathcal{X}^{*}}\leq\|\\{f(x_{n})\\}\|_{\mathcal{Y}_{d}}\leq
B\|f\|_{\mathcal{X}^{*}},\ \text{for all}\ f\in\mathcal{X}^{*}.$ (2.2)

We refer (2.2) as the _frame inequality_. If at least (a) and the upper bound
condition in (2.2) are satisfied, then $\\{x_{n}\\}$ is called a _Bessel
sequence_ for $\mathcal{X}$ with respect to $\mathcal{X}_{d}$. A frame
$\\{x_{n}\\}$ is called exact if on removal of its one element $x_{n}$, it is
no longer a frame for $\mathcal{X}$.

##  3\. Approximative atomic system for operators

Gavruta [17], introduced and studied atomic system for an operator $K$ in
Hilbert spaces. Poumai and Jahan [30] defined and studied atomic systems for
operators in Banach spaces as a generalization of atomic system in Hilbert
spaces. Here we have generalize this concept further and introduce the concept
of approximative atomic systems for operators in Banach spaces and obtain new
and interesting results. We starts this section with the following definition
of approximative atomic system for K:

######  Definition 3.1.

Let $\mathcal{X}$ be a Banach Space and $K$ be a bounded linear operator in
$\mathcal{X}$. A sequence $\\{x_{n}\\}\subseteq\mathcal{X}$ is called an
_approximative atomic system for K_ in $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ if

  1. (a)

$\\{x_{n}\\}$ is a Bessel sequence for $\mathcal{X}$ with respect to
$\mathcal{X}_{d},$

  2. (b)

there exists an approximative $\mathcal{X}_{d}$-Bessel Sequence
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subset\mathcal{X}^{*}$
for $\mathcal{X}$ such that

$\displaystyle
K(x)=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\
\text{for all}\ x\in\mathcal{X}.$

The sequence $\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$
is called the associated approximative $\mathcal{X}_{d}$-Bessel sequence.

Observation: If $\\{x_{n}\\}$ is an atomic system for $K$ in $\mathcal{X}$
with respect to $\mathcal{X}_{d},$ then for
$h_{n,i}=f_{i},~{}~{}i=1,2,3,...,n,~{}~{}n\in\mathbb{N},$ $\\{x_{n}\\}$ is an
approximative atomic system for $K.$

######  Definition 3.2.

Let $\mathcal{X}$ be a Banach Space and $M$ be a closed subspace of
$\mathcal{X}$. A sequence $\\{x_{n}\\}\subseteq\mathcal{X}$ is called _an
approximative family of local atoms_ for $M$ if

  1. (a)

$\\{x_{n}\\}_{n=1}^{\infty}$ be a Bessel sequence for $\mathcal{X}$ with
respect to $\mathcal{X}_{d},$

  2. (b)

there exists an approximative $\mathcal{X}_{d}$-Bessel Sequence
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subset\mathcal{X}^{*}$
for $M$ such that

$\displaystyle
x=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\
\text{for all}\ x\in M.$

The sequence $\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$
is called the associated approximative $\mathcal{X}_{d}$-Bessel sequence.

Observations (I) If $\\{x_{n}\\}$ is a family of local atoms for $M$, then for
$h_{n,i}=f_{i},~{}~{}i=1,2,3,...,n,~{}~{}n\in\mathbb{N},$ $\\{x_{n}\\}$ is an
approximative family of local atoms for $M.$

(II) Let $\mathcal{X}$ be a Banach space and $M$ be a closed subspace of
$\mathcal{X}$. Let $\\{x_{n}\\}$ be an approximative family of local atoms for
$M$ with respect to $\mathcal{X}_{d}$ and let
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$ be its
associated approximative $\mathcal{X}_{d}$-Bessel sequence with bound $B$.
Then, $(x_{n},\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}})$
is an approximative atomic decomposition for $M$ with respect to
$\mathcal{X}_{d}$ and $\\{x_{n}\\}$ is a frame for $M$. Indeed, by given
hypotheses,

$\displaystyle
x=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\text{for
all}\ x\in M.$

Let $A$ be the bound of Bessel sequence $\\{x_{n}\\}.$ Then for $x\in M$

$\displaystyle\|x\|$ $\displaystyle=$
$\displaystyle\sup\limits_{h\in\mathcal{X}^{*},\|h\|=1}|\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)h(x_{n})|\leq
A\|\\{h_{n,i}(x)\\}\|$

With Similar argument, we have

$\displaystyle 1/B\|f\|\leq\|\\{f(x_{n})\\}\|,\ \text{for all}\ f\in M^{*}.$

(III) Let $\mathcal{X}$ be a Banach space and $K\in L(\mathcal{X})$. If
$\\{x_{n}\\}$ is an approximative atomic system for $K$ in $\mathcal{X}$.
Then, there exist constants $C,D>0$ such that

$\displaystyle C\|K(x)\|\leq\|\\{h_{n,i}(x)\\}\|,\text{for all}\
x\in\mathcal{X},$

and

$\displaystyle D\|K^{*}(f)\|\leq\|\\{h(x_{n})\\}\|,\text{for all}\
h\in\mathcal{X}^{*}.$

(IV) $\\{x_{n}\\}\subseteq\mathcal{X}$ is a Bessel sequence for $\mathcal{X}$
with respect to $\mathcal{X}_{d}$ if and only if there exists a bounded linear
operator $T$ from $\mathcal{X}_{d}$ into $\mathcal{X}$ for which
$T\\{h_{n}\\}=\sum\limits_{n=1}^{\infty}h_{n}x_{n}$,
$\\{h_{n}\\}\in\mathcal{X}_{d}.$ Recall that $T$ is called the _synthesis
operator_ associated with the Bessel sequence $\\{x_{n}\\},$ and the bounded
linear operator $R:\mathcal{X}^{*}\rightarrow\mathcal{Y}_{d}$ given by

$\displaystyle R(h)=\\{h(x_{n})\\},\ \text{for}\ h\in\mathcal{X}^{*},$

is called the _analysis operator_ of Bessel sequence $\\{x_{n}\\}$. Also
observe that from Lemma 2.7, $\mathcal{Y}_{d}$ is isometrically isomorphic to
$\mathcal{X}_{d}^{*}$. Let
$J_{d}:\mathcal{X}_{d}^{*}\rightarrow\mathcal{Y}_{d}$ denote an isometric
isomorphism from $\mathcal{X}_{d}^{*}$ onto $\mathcal{Y}_{d}$. Note that the
synthesis operator $T$ need not be onto. Indeed, let
$\mathcal{X}_{d}=\mathcal{X}=l_{1}$ and let $\\{e_{n}\\}$ be a sequence of
canonical unit vectors as basis of $\mathcal{X}$. Take $x_{n}=e_{n+1}$, for
$n\in\mathbb{N}$. Let
$h=\\{h_{n}\\}_{n=1}^{\infty}\in\mathcal{X}^{*}=l^{\infty}$ then
$\\{h(x_{n})\\}\in l^{\infty}$ and $\|\\{h(x_{n})\\}\|_{l^{\infty}}\leq
A\|h\|_{l^{\infty}}$, where $A>0$ is some constant. Thus, $\\{x_{n}\\}$ is
Bessel sequence for $\mathcal{X}$. But
$T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ given by $T(e_{n})=x_{n}$, for
$n\in\mathbb{N}$ is a bounded linear operator which is not onto.
$\\{x_{n}\\}\subseteq\mathcal{X}$ is a frame if and only if there exists a
bounded linear operator $T$ from $\mathcal{X}_{d}$ onto $\mathcal{X}$ for
which $T\\{c_{n}\\}=\sum\limits_{n=1}^{\infty}c_{n}x_{n}$,
$\\{c_{n}\\}\in\mathcal{X}_{d}$.

From the frame inequality, $T^{*}$ is one-one and Range of $T^{*}$ is closed
in $\mathcal{X}_{d}^{*}$. So by [[31], page 103], $T$ is onto.
Conversely, if $T$ is onto. Then by [[31], page 103] $T^{*}$ is one-one and
range of $T^{*}$ is closed. Also, by [[12], page 487], there exists a constant
$D>0$ such that

$\displaystyle\|h\|\leq D\|T^{*}(h)\|=D\|\\{h(x_{n})\\}\|,\ \text{for all}\
h\in\mathcal{X}^{*}.$

In the following result, we construct an approximative family of local atoms
for $K(\mathcal{X})$ and an approximative atomic decomposition for
$[K(\mathcal{X})]^{*}$ from a given approximative atomic system for a bounded
linear operator $K$.

######  Theorem 3.3.

Let $\\{x_{n}\\}$ be an approximative atomic system for $K$ in $\mathcal{X}$
and $K$ has pseudo inverse $K^{\dagger}$. Then, $\\{x_{n}\\}$ is an
approximative family of local atoms for $K(\mathcal{X})$ in $\mathcal{X}$.
Moreover, if $\mathcal{X}_{d}^{*}$ has a sequence of canonical unit vectors
$\\{e_{n}^{*}\\}$ as basis, then there exists an approximative
$\mathcal{X}_{d}$-Bessel sequence
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}\subset\mathcal{X}^{*}$
for $K(\mathcal{X})$ such that $(h_{n,i},\pi(x_{n}))$ is an approximative
atomic decomposition for $[K(\mathcal{X})]^{*}$ with respect to
$\mathcal{X}_{d}^{*}$.

###### Proof.

Since $\\{x_{n}\\}$ is an approximative atomic system for $K$ in $\mathcal{X}$
$K$ has a pseudo inverse $K^{\dagger}$, $KK^{\dagger}$ is a projection from
$\mathcal{X}$ onto $K(\mathcal{X})$ and $KK^{\dagger}(x)=x,\ \text{for all}\
x\in K(\mathcal{X}).$ Thus,
$KK^{\dagger}|_{K(\mathcal{X})}=I_{K(\mathcal{X})}$. Let $\\{h_{n,i}\\}$ be
its associated approximative $\mathcal{X}_{d}$-Bessel sequence with bound C.
Take $f_{n,i}=(K^{\dagger}|_{K(\mathcal{X})})^{*}(h_{n,i})$, $n\in\mathbb{N}$
and let $x\in K(\mathcal{X})$. Then, we compute

$\displaystyle x$ $\displaystyle=$ $\displaystyle
K(K^{\dagger}|_{K(\mathcal{X})}(x))$ $\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(K^{\dagger}|_{K(\mathcal{X})}(x))x_{i}$
$\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}(K^{\dagger}|_{K(\mathcal{X})^{*}}h_{n,i}(x))x_{i}$
$\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}f_{n,i}(x)x_{i}$

Now, we will show that $\\{f_{n,i}\\}$ is an approximative Bessel sequence for
$K(\mathcal{X})$ with respect to $\mathcal{X}_{d}$. Let $x\in K(\mathcal{X}).$
Then we have

$\displaystyle\\{f_{n,i}(x)\\}=\\{h_{n,i}(K^{\dagger}|_{K(\mathcal{X})}(x)\\}\in\mathcal{X}_{d},\
\text{for all}\ x\in K(\mathcal{X})$

and

$\displaystyle\|\\{f_{n,i}(x)\\}\|$ $\displaystyle=$
$\displaystyle\|\\{h_{n,i}(K^{\dagger}|_{K(\mathcal{X})}(x))\\}\|\leq
C\|K^{\dagger}|_{K(\mathcal{X})}(x)\|$ $\displaystyle\leq$ $\displaystyle
C\|K^{\dagger}\|\|x\|,\ \text{for all}\ x\in K(\mathcal{X}).$

Also, $(K^{\dagger}|_{K(\mathcal{X})})^{*}K^{*}=I^{*}_{[K(\mathcal{X})]^{*}}$.
Let $h\in[K(\mathcal{X})]^{*}$. Then

$\displaystyle\|h\|$ $\displaystyle=$
$\displaystyle\|(K^{\dagger}|_{K(\mathcal{X})})^{*}K^{*}(h)\|\leq\|K^{\dagger}\|\|K^{*}(h)\|$
$\displaystyle=$ $\displaystyle\|K^{\dagger}\|\sup\limits_{x\in
K(\mathcal{X}),\|x\|=1}|h(\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i})|$
$\displaystyle\leq$ $\displaystyle C\|K^{\dagger}\|\|\\{h(x_{i})\\}\|.$

Let $h\in[K(\mathcal{X})]^{*}.$ Then for $N\in\mathbb{N},$ we have

$\displaystyle\|h-\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}\|$
$\displaystyle=$ $\displaystyle\sup\limits_{x\in
K(\mathcal{X}),\|x\|=1}|h(x)-\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}(x)|$
$\displaystyle=$ $\displaystyle\sup\limits_{x\in
K(\mathcal{X}),\|x\|=1}|\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}(x)|$
$\displaystyle\leq$ $\displaystyle
C\|K^{\dagger}\|\|\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})e_{i}^{*}\|$
$\displaystyle\rightarrow$ $\displaystyle 0\ \text{as N}\rightarrow\infty.$

Thus,
$f=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}$,
for all $h\in[K(\mathcal{X})]^{*}$. ∎

If $\\{x_{n}\\}$ is an approximative atomic system for $K$ and $K$ has pseudo
inverse $K^{\dagger}$, then for $h_{n,i}=f_{i}$, $\\{x_{n}\\}$ is frame as
well as an approximative family of local atoms for $K(\mathcal{X})$.
The following theorem gives a necessary condition for the existence of an
approximative atomic system for a bounded linear operator $K$.

######  Theorem 3.4.

If $\\{x_{n}\\}$ is an approximative atomic system for $K$, then there exists
a bounded linear operator $T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ such that
$T(e_{n})=x_{n}$, $n\in\mathbb{N}$ and Range$K^{**}\subseteq RangeT^{**}$,
where $\\{e_{n}\\}$ is the sequence of canonical unit vectors as a basis of
$\mathcal{X}_{d}$.

###### Proof.

Since $\\{x_{n}\\}$ is an approximative atomic system for $K$,
$T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ given
$T(\\{h_{n,i}\\})=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i}$
is a well defined bounded linear operator such that $T(e_{i})=x_{i}$,
$n\in\mathbb{N}$. Also, there exists an approximative $\mathcal{X}_{d}$-Bessel
sequence $\\{h_{n,i}\\}$ for $\mathcal{X}$ with bound $B$ such that

$\displaystyle
K(x)=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\
\text{for all}\ x\in\mathcal{X}.$

Then, for $h\in\mathcal{X}^{*}$, we have

$\displaystyle\|K^{*}(h)\|$ $\displaystyle=$
$\displaystyle\sup\limits_{x\in\mathcal{X},\|x\|=1}|K^{*}h(x)|=\sup\limits_{x\in\mathcal{X},\|x\|=1}|\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)h(x_{i})|$
$\displaystyle\leq$ $\displaystyle B\|\\{h(x_{i})\\}\|=B\|T^{*}(h)\|.$

Hence, by Lemma 2.6, we have $RangeK^{**}\subseteq RangeT^{**}$. ∎

Gavruta [17], had proved that for a separable Hilbert space $\mathcal{H}$ a
sequence $\\{x_{n}\\}\subseteq\mathcal{H}$ is an atomic system for $K$ if and
only if there exists a bounded linear operator $L:l^{2}\rightarrow\mathcal{H}$
such that $L(e_{n})=x_{n}$ and $Range(K)\subseteq Range(L)$, where
$\\{e_{n}\\}$ is an orthonormal basis for $l^{2}$. Towards the converse of the
Theorem 3.4, we have the following theorem.

######  Theorem 3.5.

Let $\mathcal{X}_{d}$ be reflexive, $\mathcal{X}_{d}^{*}$ has a sequence of
canonical unit vectors $\\{e_{n}^{*}\\}$ as basis and $\\{x_{n}\\}$ be a
sequence in $\mathcal{X}$. Let $K\in L(\mathcal{X})$,
$T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ be bounded linear operator with
$T(e_{n})=x_{n},~{}n\in\mathbb{N}$, $RangeK^{**}\subseteq RangeT^{**}$ and
$\overline{T^{*}(\mathcal{X}^{*})}$ is complemented subspace of
$\mathcal{X}_{d}^{*}$. Then $\\{x_{n}\\}$ is an approximative atomic system
for K.

###### Proof.

Since $T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ is a bounded linear operator
and is given by

$\displaystyle
T(\\{\alpha_{n}\\})=\sum\limits_{n=1}^{\infty}\alpha_{n}x_{n},~{}~{}\\{\alpha_{n}\\}\in\mathcal{X}_{d}^{*}.$

So, by Observation IV, $\\{x_{n}\\}$ is a Bessel sequence with bound say $B$.
Since $RangeK^{**}\subseteq RangeT^{**}$ and
$\overline{T^{*}(\mathcal{X}^{*})}$ is complemented subspace of
$\mathcal{X}_{d}^{*}$, by Lemma 2.6, there exists a bounded linear operator
$\theta:\mathcal{X}_{d}^{*}\rightarrow\mathcal{X}^{*}$ such that $K^{*}=\theta
T^{*}$. Take
$h_{n,i},~{}~{}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}=\theta(e_{n}^{*})$,
$n\in\mathbb{N}$. Then, for $x\in\mathcal{X}$, we get

$\displaystyle\theta^{*}(x)(e_{j}^{*})=\pi(x)\theta(e_{j}^{*})=h_{n,i}(x);i=1,2,...{m_{n}}\
\text{for}\ n\in\mathbb{N}.$

This gives $\\{h_{n,i}(x)\\}=\theta^{*}(x)\in\mathcal{X}_{d},\ \text{for all}\
x\in\mathcal{X}$ and
$\|\\{h_{n,i}(x)\\}\|=\|\theta^{*}(x)\|\leq\|\theta\|\|x\|,\ \text{for all}\
x\in\mathcal{X}.$ Also, for $h\in\mathcal{X}^{*},$ we have

$\displaystyle
K^{*}(h)=\theta(\\{h(x_{n})\\})=\theta(\sum\limits_{n=1}^{\infty}h(x_{n})e_{n}^{*})$
$\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h(x_{i})h_{n,i}.$

and for $x\in\mathcal{X}$ and $n\in\mathbb{N},$ we compute

$\displaystyle\|K(x)-\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{n}h_{n,i}(x)x_{i}\|$
$\displaystyle=$
$\displaystyle\sup\limits_{h\in\mathcal{X}^{*},\|h\|=1}|h(K(x))-\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{n}h_{n,i}(x)h(x_{i})|$
$\displaystyle=$
$\displaystyle\sup\limits_{h\in\mathcal{X}^{*},\|h\|=1}|\sum\limits_{i=n+1}^{\infty}h_{n,i}(x)h(x_{i})|$
$\displaystyle\leq$ $\displaystyle
B\|\sum\limits_{i=n+1}^{\infty}h_{n,i}(x)e_{i}\|\rightarrow 0\ \text{as}\
n\rightarrow\infty.$

Hence,
$K(x)=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$,
for all $x\in\mathcal{X}$. ∎

The following theorem gives a complete characterization for the approximative
family of local atoms for the closed subspace of a Banach space.

######  Theorem 3.6.

Let $\\{x_{n}\\}$ be a Bessel sequence for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$, $\mathcal{X}_{d}^{*}$ has a sequence of canonical unit
vectors $\\{e_{n}^{*}\\}$ as basis and $M$ be a closed subspace of
$\mathcal{X}$. Let there exists a projection P from $\mathcal{X}$ onto M.
Then, the following statements are equivalent:

  1. (a)

$\\{x_{n}\\}$ is an approximative family of local atoms for M.

  2. (b)

$\\{x_{n}\\}$ is an approximative atomic system for P.

  3. (c)

There exists a bounded linear operator $U:M\rightarrow\mathcal{X}_{d}$ such
that $TUP=P$, where $T$ is the synthesis operator of the Bessel sequence
$\\{x_{n}\\}$.

###### Proof.

$(a)\Rightarrow(b)$ Since $\\{x_{n}\\}$ is an approximative family of local
atoms for M so, there exists an approximative $\mathcal{X}_{d}$-Bessel
sequence $\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$ for
$M$ such that

$\displaystyle
x=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\
\text{for all}\ x\in M.$

Now for each $y\in\mathcal{X}$, $P(y)\in M$ and

$\displaystyle
P(y)=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(P(y))x_{i}=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}(P^{*}(h_{n,i})(y)x_{i},\
\text{for all}\ y\in\mathcal{X}.$

Clearly $\\{P^{*}(h_{n,i})\\}$ is an approximative $\mathcal{X}_{d}$-Bessel
sequence for $\mathcal{X}$. Hence, $\\{x_{n}\\}$ is an approximative atomic
system for $P$.
$(b)\Rightarrow(a)$ Since $\\{x_{n}\\}$ is an approximative atomic system for
P and $x=P(x)$, for all $x\in M$, the proof follows.
$(a)\Rightarrow(c)$ Since $\\{x_{n}\\}$ is an approximative family of local
atoms for M so there exists an approximative $\mathcal{X}_{d}$-Bessel sequence
$\\{h_{n,i}\\}\underset{n\in\mathbb{N}}{{}_{i=1,2,3,...,m_{n}}}$ for M such
that

$\displaystyle
P(x)=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(P(x))x_{i},\
\text{for all}\ x\in\mathcal{X}.$

Let $U:M\rightarrow\mathcal{X}_{d}$ be the analysis operator for
$\\{h_{n,i}\\}$ given by $U(P(x))=\\{h_{n,i}(P(x))\\},x\in\mathcal{X}.$ Also,
$\\{h_{n,i}(P(x)\\}\in\mathcal{X}_{d}$, for all $x\in\mathcal{X}$. Hence

$\displaystyle P(x)$ $\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(P(x))x_{i}=T(\\{h_{n,i}(P(x))\\})$
$\displaystyle=$ $\displaystyle TUP(x),\ \text{for all}\ x\in\mathcal{X}.$

$(c)\Rightarrow(a)$ Since $U:M\rightarrow\mathcal{X}_{d}$ is a bounded linear
operator such that $TUP=P$. Take $h_{n,i}=U^{*}(e_{n}^{*}),n\in\mathbb{N}$ and
$x\in M$. Then $h_{n,i}(x)=U^{*}(e_{n}^{*})(x)=e_{n}^{*}(U(x)).$ Therefore,
$\\{h_{n,i}(x)\\}=U(x)\in\mathcal{X}_{d}$, for all $x\in M$ and

$\displaystyle\|\\{h_{n,i}(x)\\}\|=\|U(x)\|\leq\|U\|\|x\|,\ \text{for all}\
x\in M.$

Also, for each, $x\in M,$ there exists $y\in\mathcal{X}$ such that $x=P(y)$.
Moreover, $\\{h_{n,i}(P(y))\\}\in\mathcal{X}_{d}$ for all $y\in\mathcal{X}$
and

$\displaystyle
x=TUP(y)=T(\\{h_{n,i}(P(y))\\})=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(P(y))x_{i}=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(y)x_{i}$

∎

Next, we shall show that if $\\{x_{n}\\}$ is an approximative $K$ atomic
system with $K=I_{\mathcal{X}}$, then every complemented subspace of
$\mathcal{X}$ has an approximative family of local atoms and in case
$h_{n,i}=f_{n}$ then $\\{x_{n}\\}$ is a $K$ atomic system and every
complemented subspace of $\mathcal{X}$ has a family of local atoms.

######  Theorem 3.7.

Let $\\{x_{n}\\}$ be a frame for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$. If there exists an approximative $\mathcal{X}_{d}$-Bessel
sequence $\\{h_{n,i}\\}$ such that

$\displaystyle
x=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(P(x))x_{i},\
\text{for all}\ x\in\mathcal{X},$

then every complemented subspace of $\mathcal{X}$ has an approximative family
of local atoms.

###### Proof.

Let $\\{x_{n}\\}$ be a frame for $\mathcal{X}$ and $T$ be the synthesis
operator of $\\{x_{n}\\}$. Then by given hypothesis, there exists an
approximative $\mathcal{X}_{d}$-Bessel sequence $\\{h_{n,i}\\}$ such that

$\displaystyle
x=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\
\text{for all}\ x\in\mathcal{X}.$

Let $S$ be the analysis operator of $\\{h_{n,i}\\}$. Note that
$I_{\mathcal{X}}=TS$. Let $M$ be a complemented subspace of $\mathcal{X}$ and
$P$ be the projection from $\mathcal{X}$ onto $M$. Take $y_{n}=P(x_{n})$, for
$n\in\mathbb{N}$. Define $T_{1}:\mathcal{X}_{d}\rightarrow M$ by $T_{1}=P\circ
T$. Then clearly $T_{1}$ is a bounded linear operator from $\mathcal{X}_{d}$
onto $M$ such that

$\displaystyle
T_{1}(\\{h_{n,i}\\})=P(\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i})=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)y_{i},~{}~{}\\{h_{n,i}\\}\in\mathcal{X}_{d}.$

Thus, $\\{y_{n}\\}$ is frame for $M$ and a Bessel sequence for $\mathcal{X}$
with respect to $\mathcal{X}_{d}$. Let $N=S(M)$ and let
$\alpha=\\{h_{n,i}\\}\in\mathcal{X}_{d}$. Then, $T_{1}(\alpha)=x,$ for some
$x\in M.$ Now take $\alpha_{1}=S(x).$ Then, $\alpha_{1}\in N.$ Also, if
$\alpha_{0}=\alpha-\alpha_{1},$ then

$\displaystyle T_{1}(\alpha_{0})=T_{1}(\alpha)-T_{1}(\alpha_{1})=x-P(x)=0.$

This gives $\alpha_{0}\in kerT_{1}.$ Thus $\alpha=\alpha_{0}+\alpha_{1}$. Then
$\lambda\in N\cap kerT_{1}$. Then $T_{1}(\lambda)=0$ and there exists
$x_{1}\in M$ such that $\lambda=S(x_{1})$. Therefore

$\displaystyle x_{1}=P(x_{1})=PTS(x_{1})=T_{1}(\lambda)=0.$

Thus, $\lambda=0$ and so $N\cap kerT_{1}=\\{0\\}$. Therefore
$\mathcal{X}_{d}=N\oplus kerT_{1}$. Now, by Lemma 2.3, $T_{1}$ has a pseudo
inverse $T_{1}^{\dagger}$. Moreover $T_{1}T_{1}^{\dagger}$ is a projection
from $M$ onto $T_{1}(\mathcal{X}_{d})=M$. This gives
$T_{1}T_{1}^{\dagger}=I_{M}$. Let $\\{z_{n}\\}$ be the sequence of coordinate
functionals on $\mathcal{X}_{d}$. Take $f_{n,i}=(T_{1}^{\dagger})^{*}(l_{n})$,
for $n\in\mathbb{N}$ and let $x\in M$. Then

$\displaystyle
f_{n,i}(x)=(T_{1}^{\dagger})^{*}(z_{n})(x)=z_{n}(T_{1}^{\dagger}(x)),\
\text{for}\ n\in\mathbb{N},\ x\in M.$

Therefore $\\{f_{n,i}(x)\\}=T_{1}^{\dagger}(x)\in\mathcal{X}_{d}$, for all
$x\in M$ and

$\displaystyle\|\\{f_{n,i}(x)\\}\|=\|T_{1}^{\dagger}(x)\|\leq\|T_{1}^{\dagger}\|\|x\|,\
\text{for all}\ x\in M.$

Thus $\\{f_{n,i}\\}$ is an approximative $\mathcal{X}_{d}$-Bessel sequence for
$M$. Therefore, we compute

$\displaystyle
x=T_{1}T_{1}^{\dagger}(x)=T_{1}(\\{f_{n,i}(x)\\})=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}f_{n,i}(x)y_{i},\
\text{for all}\ x\in M.$

∎

Let $\\{x_{n}\\}$ be a Bessel sequence for $\mathcal{X}$ with respect to
$\mathcal{X}_{d}$ and let T be its associated synthesis operator. Also, let
$\\{h_{n,i}\\}$ be an approximative $\mathcal{X}_{d}$-Bessel sequence for
$\mathcal{X}$ with its associated analysis operator S. Then
$TS:\mathcal{X}\rightarrow\mathcal{X}$ is an operator such that

$\displaystyle
TS(x)=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\
\text{for all}\ x\in\mathcal{X}.$

Note that $\\{x_{n}\\}$ is an approximative atomic system for $K$ in
$\mathcal{X}$ with its associated approximative $\mathcal{X}_{d}$-Bessel
sequence $\\{h_{n,i}\\}$ if and only if

$\displaystyle TS=K.$ (3.1)

Also one may observe that in case $\\{h_{n,i}\\}=\\{f_{i}\\}$ then
$\\{x_{n}\\}$ is an atomic system for $K$ in $\mathcal{X}$ with its associated
$\mathcal{X}_{d}$-Bessel sequence $\\{f_{i}\\}$
The following theorem gives a method how to construct an approximative atomic
system for $K$ from a given Bessel sequence.

######  Theorem 3.8.

Let $\mathcal{X}_{d}^{*}$ has a sequence of canonical unit vectors
$\\{e_{n}^{*}\\}$ as basis. Let $\\{x_{n}\\}$ be a Bessel sequence for
$\mathcal{X}$ with respect to $\mathcal{X}_{d}$ and
$T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ be its synthesis operator such that
T has pseudoinverse $T^{\dagger}$. Let $K\in L(\mathcal{X})$ with
$K(\mathcal{X})\subseteq T(\mathcal{X}_{d})$. Then $\mathcal{X}$ has an
approximative atomic system for $K.$

###### Proof.

Since $T$ has a pseudo inverse $T^{\dagger}$, by Lemma 2.4, $TT^{\dagger}$ is
a projection from $\mathcal{X}$ onto $T(\mathcal{X}_{d})$. Define
$S:\mathcal{X}\rightarrow\mathcal{X}_{d}$ as

$\displaystyle S=T^{\dagger}K+W-T^{\dagger}TW,$ (3.2)

where $W:\mathcal{X}\rightarrow\mathcal{X}_{d}$ is a bounded linear operator.
Then, we compute

$\displaystyle TS$ $\displaystyle=$ $\displaystyle
T(T^{\dagger}K+W-T^{\dagger}TW)$ $\displaystyle=$ $\displaystyle
TT^{\dagger}K+TW-TT^{\dagger}TW$ $\displaystyle=$ $\displaystyle
TT^{\dagger}K=K.$

Note that $T^{*}=J_{d}^{-1}\circ R$ and $\\{J_{d}e_{n}^{*}\\}$ is a basis of
$\mathcal{Y}_{d}$. Taking $h_{n,i}=S^{*}(e_{n}^{*})$,
$f_{n,i}=(T^{+})^{*}(e_{n}^{*})$ and $g_{n}=W^{*}(e_{n}^{*})$ , for
$n\in\mathcal{N},$ we compute

$\displaystyle
T^{*}(T^{\dagger})^{*}e_{n}^{*}=T^{*}(f_{n,i})=J_{d}(R(f_{n,i}))=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}f_{n,i}(x_{i})e_{i}^{*}$

and

$\displaystyle h_{n,i}$ $\displaystyle=$ $\displaystyle
K^{*}(T^{\dagger})^{*}(e_{n}^{*})+W^{*}(e_{n}^{*})-W^{*}T^{*}(T^{\dagger})^{*}(e_{n}^{*})$
$\displaystyle=$ $\displaystyle
K^{*}f_{n,i}+g_{i}-\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}f_{n,i}(x_{i})g_{i},\
\text{for all}\ n\in\mathbb{N}.$

Also, for $x\in\mathcal{X}$, we have

$\displaystyle h_{n,i}(x)=S^{*}(e_{n}^{*})(x)=e_{n}^{*}(S(x)),\ \text{for
all}\ n\in\mathbb{N}.$

Therefore $\\{h_{n,i}(x)\\}=S(x)\in\mathcal{X}_{d}$, for all
$x\in\mathcal{X}$. Also,

$\displaystyle\|\\{h_{n,i}(x)\\}\|$ $\displaystyle=$ $\displaystyle\|S(x)\|$
$\displaystyle\leq$
$\displaystyle\|T^{\dagger}K+W-T^{\dagger}TW\|\|x\|,~{}x\in\mathcal{X}.$

Thus, $\\{h_{n,i}\\}$ is an approximative $\mathcal{X}_{d}$\- Bessel sequence
and

$\displaystyle\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i}$
$\displaystyle=$ $\displaystyle
T(\\{h_{n,i}(x)\\})=T(\\{S^{*}(e_{n}^{*})(x)\\})$ $\displaystyle=$
$\displaystyle T(\\{e_{n}^{*}(S(x))\\})=TS(x)$ $\displaystyle=$ $\displaystyle
K(x),\ x\in\mathcal{X}.$

Hence, $\\{x_{n}\\}$ is an approximative atomic system for $K$. ∎

Note that (3.1) is the general formula for all $U$ satisfying the equality
(3.2). As a result of this, let $U_{\circ}$ be a linear operator such that
$TU_{\circ}=K$ and take $W=U_{\circ}$ in the right side of (3.2), then

$\displaystyle
T^{\dagger}K+U_{\circ}-T^{\dagger}TU_{\circ}=T^{\dagger}K+U_{\circ}-T^{\dagger}K=U_{\circ}$

The following theorem shows that we can construct an approximative atomic
system for $K$ from a given approximative $\mathcal{X}_{d}$-Bessel sequence.

######  Theorem 3.9.

Let $\mathcal{X}_{d}$ be a BK-space and let $\\{e_{n}\\}$ be a sequence of
canonical unit vectors as basis of $\mathcal{X}_{d}$ and $\mathcal{X}_{d}^{*}$
has a sequence of canonical unit vectors $\\{e_{n}^{*}\\}$ as basis. Let
$\\{h_{n,i}\\}$ be an approximative $\mathcal{X}_{d}$-Bessel sequence for
$\mathcal{X}$ with $S:\mathcal{X}\rightarrow\mathcal{X}_{d}$ as its analysis
operator and S has pseudo inverse $S^{\dagger}$. Let $K\in L(\mathcal{X})$ and
$K^{*}(\mathcal{X}^{*})=[h_{n,i}]$. Then, $\mathcal{X}$ has an approximative
atomic system for $K.$

###### Proof.

Since $\\{h_{n,i}\\}$ is an approximative $\mathcal{X}_{d}$-Bessel sequence
for $\mathcal{X}$ with $S:\mathcal{X}\rightarrow\mathcal{X}_{d}$ as its
analysis operator and S has pseudoinverse $S^{\dagger}.$ Define
$T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ by

$\displaystyle T=KU^{\dagger}+W(I-UU^{\dagger}),$ (3.3)

where $W:\mathcal{X}_{d}\rightarrow\mathcal{X}$ is a bounded linear operator.
Then, we compute

$\displaystyle TS$ $\displaystyle=$
$\displaystyle(KS^{\dagger}+W(I-SS^{\dagger}))S$ $\displaystyle=$
$\displaystyle KS^{\dagger}S+WS-WSS^{\dagger}S$ $\displaystyle=$
$\displaystyle KS^{\dagger}S.$

Also, for $x\in\mathcal{X}$,
$S^{*}:\mathcal{X}_{d}^{*}\rightarrow\mathcal{X}^{*}$ is given by

$\displaystyle S^{*}(e_{n}^{*})(x)=e_{n}^{*}(S(x))=h_{n,i}(x),\
n\in\mathcal{N}.$

So, we have

$\displaystyle[h_{n,i}]=[S^{*}(e_{n}^{*})]=S^{*}(\mathcal{X}_{d}^{*})=K^{*}(\mathcal{X}^{*}).$

Also $S$ has a pseudo inverse $S^{\dagger}$, $S^{*}$ has pseudo inverse
$(S^{\dagger})^{*}$. So,

$\displaystyle
S^{*}(S^{\dagger})^{*}S^{*}(\mathcal{X}_{d}^{*})=S^{*}(\mathcal{X}_{d}^{*})=K^{*}(\mathcal{X}^{*}).$

Thus, we conclude that $S^{*}(S^{\dagger})^{*}K^{*}=K^{*}$. Therefore,
$TS=KS^{\dagger}S=K$. Let $x_{n}=T(e_{n})$, $y_{n}=S^{\dagger}(e_{n})$ and
$l_{n}=W(e_{n})$, for $n\in\mathbb{N}$. Then

$\displaystyle x_{n}=KS^{\dagger}(e_{n})+W(I-SS^{\dagger})(e_{n})$
$\displaystyle=K(y_{n})+l_{n}-\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(y_{i})l_{i},~{}~{}n\in\mathcal{N}.$

Also, $T:\mathcal{X}_{d}\rightarrow\mathcal{X}$ is a bounded linear operator
given by

$\displaystyle
T(\\{h_{n,i}\\})=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}x_{i},~{}~{}\\{h_{n,i}\\}\in\mathcal{X}_{d}.$

By observation III, $\\{x_{i}\\}$ is a Bessel sequence. Thus, we get

$\displaystyle
K(x)=TS(x)=T(\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)e_{i})=\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{m_{n}}h_{n,i}(x)x_{i},\
x\in\mathcal{X}.$

Hence $\\{x_{n}\\}$ is an approximative atomic system for K. ∎

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# A Practical Two-stage Ranking Framework for Cross-market Recommendation

Zeyuan Chen1, He Wang2, Xiangyu Zhu3, Haiyan Wu1, Congcong Gu4, Shumeng Liu2,
Jinchao Huang2, Wei Zhang1∗ 1East China Normal University, 2Xiaomi Inc.,
3JD.com, 4Pingan Inc<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(2022)

###### Abstract.

Cross-market recommendation aims to recommend products to users in a resource-
scarce target market by leveraging user behaviors from similar rich-resource
markets, which is crucial for E-commerce companies but receives less research
attention. In this paper, we present our detailed solution adopted in the
cross-market recommendation contest, i.e., WSDM CUP
2022111https://xmrec.github.io/wsdmcup/. To better utilize collaborative
signals and similarities between target and source markets, we carefully
consider multiple features as well as stacking learning models consisting of
deep graph recommendation models (Graph Neural Network, DeepWalk, etc.) and
traditional recommendation models (ItemCF, UserCF, Swing, etc.). Furthermore,
We adopt tree-based ensembling methods, e.g., LightGBM, which show superior
performance in prediction task to generate final results. We conduct
comprehensive experiments on the XMRec dataset, verifying the effectiveness of
our model. The proposed solution of our team WSDM_Coggle_ is selected as the
second place submission222The source code is available at
https://github.com/loserChen/WSDM_CUP_Rec_2022.

cross-market recommendation, user behavior analysis, feature engineering

*All the corresponding to Wei Zhang

††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
15th ACM International Conference on Web Search and Data Mining; February
21–25, 2022; Phoenix, AZ, USA††booktitle: Proceedings of the 15th ACM
International Conference on Web Search and Data Mining (WSDM ’22), February
21–25, 2022, Phoenix, AZ, USA††price: 15.00††doi:
10.1145/1122445.1122456††isbn: 978-x-xxxx-xxxx-x/YY/MM††ccs: Information
systems Recommender systems††ccs: Information systems Personalization

##  1\. Introduction

Recommender systems (RS) are ubiquitous in online platforms and mobile
applications, such as e-commerce. Cross-domain recommendation is one kind of
RS that leverages the interactions of overlapping items in source domains to
benefit the recommendations in a target domain. There has been a number of
studies with various domain definitions and recommendation scenarios (Yuan et
al., 2019; Hu et al., 2018). However, few studies have been conducted for
cross-market recommendation.

The recommendation scenario of cross-market recommendation (Bonab et al.,
2021) is that the model learns from interactions of overlapping items in
different markets to improve recommendation performance in a target domain,
hoping to utilize information from rich source markets. Although it is crucial
for E-commerce companies to combine different markets of various countries to
solve the cold-start and data sparsity problems (Zhang and Wang, 2015)
occurred in a resource-scarce target market, little progress is made partly
due to the lack of publicly available experimental data.

Thanks to the cross-market recommendation contest in WSDM CUP 2022, it
provides the XMRec dataset which contains abundant user-item interaction
records in different markets for further research purpose. To address this
challenge, we introduce a practical two-stage ranking framework which contains
hybrid model ranking and GBDT learning (Wang and Zhang, 2017).

The rest of the paper is organized as follows. We first provide a brief data
analysis in Section 2. Section 3 is about hybrid models ranking. GBDT learning
is in Section 4, which includes feature engineering, models building, and
ensemble modeling. Experimental results are illustrated in Section 5. Finally,
we make a conclusion in Section 6. The overall framework of our approach is
shown in Figure 1.

##  2\. Dataset

The dataset333https://github.com/hamedrab/wsdm22_cup_xmrec provided by the
sponsor contains five folders: s1, s2, s3, t1, and t2. The folders s1, s2, and
s3 contain the data of the source markets (train.tsv, train_5core.tsv,
valid_qrel.tsv, and valid_run.tsv) for training and validating. The other
folders t1 and t2 involve the data of the target market. Inside each, there
are the training set (train.tsv and train_5core.tsv) and the public/private
test set (valid_qrel.tsv and valid_ run.tsv/test_run.tsv), respectively. For
ease of use, we simply concatenate train.tsv and train_5core.tsv by deleting
the repeated samples and generate train_merge.tsv for each folder. Concretely,
train_merge.tsv contains the training data with the fields of userId, itemId,
and rating. valid_qrel.tsv is the validation positive samples, with a data
structure similar to train.tsv. valid_run.tsv is the validation samples
wherein each row has 99 negative samples and 1 positive sample for each unique
userId. test_run.tsv is the test candidate samples with the same positive or
negative sample ratio as valid_run.tsv. The basic statistics of the datasets
are summarized in Table 1.

Table 1. Statistics of the datasets. Dataset | # Users | # Items | # Interactions
---|---|---|---
s1 | 77,776 | 11,807 | 793,300
s2 | 20,311 | 3,408 | 794,477
s3 | 8,568 | 2,332 | 379,092
t1 | 9,955 | 3,559 | 599,600
t2 | 18,504 | 8,941 | 1,216,378
Figure 1. The whole framework of our model to address the challenge of cross-
market recommendation .

##  3\. Hybrid Model Ranking

In order to improve the diversity of the model, we choose a few deep graph
recommendation models and traditional recommendation models to implement
result prediction. Since we do not significantly change these models, we just
simply introduce them used in the contest. In what follows, we mainly
introduce our ranking-augmented graph neural network based on practical
findings during this contest.

###  3.1. Ranking-augmented Graph Neural Network

Before delving into the computational formulas of this module, we need to
firstly clarify how to use the dataset for result prediction. We use the
training set to train with sampling 99 negative items for each positive
interaction, public test set to validate model performance and test candidate
samples to predict without using any data from source markets, due to no
distinct performance improvement by trying cross-market information. In order
to distinguish the degree to which a user prefers an item, we build user-item
bipartite graph $\mathcal{G}$ based on explicit feedback, i.e., rating. As
such, we form the following formula, i.e., $((rating)/10+0.5)*0.95+0.05/2$,
which decides the fine-grained edge weights in graph $\mathcal{G}$. Assume
there are $M$ users and $N$ items occurring in the target markets, then we
have a node feature matrix $\bm{X}\in\mathbb{R}^{(M+N)\times d}$ and an
adjacency matrix $\bm{A}^{(M+N)\times(M+N)}$ for the graph.

By convention, GNN performs representation along with the edges of graph
$\mathcal{G}$, which is defined as follows:

(1)
$\bm{X}_{l+1}=\bm{\hat{A}}\bm{X}_{l}+\bm{X}_{l}\odot\bm{\hat{A}}\bm{X}_{l}\,,$

where $\bm{\hat{A}}=\bm{D}^{-0.5}\bm{A}\bm{D}^{-0.5}$ denotes the normalized
adjacency matrix without self loops and $l$ is the index of the propagation
layer. We let $\bm{X}_{0}=\bm{X}$, consisting of input user and item
representations. Distinct from conventional graph convolution networks which
do not consider information transfer between the adjacent layer, we
additionally encode the similarity between $\bm{X}_{l}$ and $\bm{X}_{l+1}$ so
that more messages from similar nodes can be passed (Zhang et al., 2021). By
doing this, it can boost the ranking performance based on our extensive
experiments.

After propagation of $L$ layer, we obtain multiple layer-wise representations,
namely {$\bm{X}_{0};\bm{X}_{1};\dots;\bm{X}_{L}$}. Since the representations
from different layers emphasize different semantics, we simply perform
average-pooling on them to constitute embedding $\bm{\bar{X}}$. In order to
mitigate the popularity bias in recommendation, we adopt a simple but
effective solution, i.e., the $L_{2}$ based normalization operation, on
$\bm{\bar{X}}$ so as to obtain final embedding $\bm{\tilde{X}}$.

Based on the final embedding $\bm{\tilde{X}}$, we could have two
representations for user $u$ and item $i$, i.e., $\bm{e}_{u}$ and
$\bm{e}_{i}$. To predict the potential interaction probability for the
considered user-item pair, we proceed some changes to fuse GMF and MLP
referred to classical neural collaborative filtering method NeuMF (He et al.,
2017) and improve the ability of model prediction, which can be defined as
follows:

(2)
$\hat{y}_{ui}=\sigma(\bm{z}_{GMF}+\bm{z}_{MLP})=\sigma(\bm{h}^{T}(\bm{e}_{u}\odot\bm{e}_{i})+\mathrm{MLPs}(\bm{e}_{u}\oplus\bm{e}_{i}))\,,$

where $\bm{h}\in\mathbb{R}^{d\times 1}$ is a trainable weight vector. ReLU is
adopted as the middle-layered activation function in $\mathrm{MLPs}$. $\odot$
denotes the element-wise product of vectors and $\oplus$ means the
concatenation operation. $\sigma$ is the sigmoid function.

For training this module, we employ the cross-entropy loss, which is given by:

(3)
$\mathcal{L}=-\big{(}y_{ui}\log\hat{y}_{ui}+(1-y_{ui})\log(1-\hat{y}_{ui})\big{)}\,.$

To avoid overfitting user preferences, the label smoothing trick (Yuan et al.,
2020) is adopted based on corresponding ratings. Finally, we keep the
prediction score of public/private test set from this module for the follow-up
experiments.

###  3.2. Other Used Recommendation Models

Unlike the way of using data mentioned above, the recommendation models adopt
another way to utilize the datasets. Here we use the training set to build the
corresponding features and give predictions on the public/private test set.
Similarly, we keep the corresponding score for further research. It is worth
noting that the training set used here includes all the data from source
markets and target markets so as to learn cross-market information and improve
model performance. Unless otherwise specified, the data usage setting is used.
The used recommendation models will be introduced simply. Generally, these
models are usually applied in the recall stage in industrial scenarios.

####  3.2.1. ItemCF

The core idea behind ItemCF (Sarwar et al., 2001) is to recommend items that
is similar to items of interest to the user in the past. ItemCF focuses on
maintaining users’ historical interests to make recommendations more
personalized and reflects users’ own interest inheritance.

####  3.2.2. UserCF

Similarly, UserCF tends to recommend items that are liked by other users with
similar interests. And UserCF focuses on the hot spots of small groups similar
to users’ interests. The recommendation results are more social and reflect
the popularity of items in users’ interest groups.

####  3.2.3. Swing

Swing considers local graph structure relations such as user-item-user. For
users who click on items i and j together, the fewer items they click on, the
more similar items i and j are.

####  3.2.4. PersonalRank

PersonalRank (Yang et al., 2018) is a graph-based algorithm which utilizes
random walk iteratively and the access probability of nodes gradually
converges.

##  4\. GBDT Learning

Thanks to high efficiency and the superior ability of GBDT, we choose GBDT as
our base model which is a popular machine learning algorithm, and has quite a
few effective implementations such as XGBoost (Chen and Guestrin, 2016),
LightGBM (Ke et al., 2017), and CatBoost (Dorogush et al., 2018). By
convention, feature engineering is a pivotal process especially for these
tree-based ensembling methods. In order to improve the fitting ability of the
whole framework, we investigate novel learning strategies to get different
models. Finally, ensembling modeling is adopted which always works very well
in different contests.

###  4.1. Feature Engineering

Except for regarding the prediction results generated by the above
recommendation models as ranking features, we also construct statistical
features, embedding features, and distance features. These features could
capture useful information across different markets, so as to benefit the
contest.

####  4.1.1. Statistical Features

Taking user statistical features as an example, we generate the corresponding
features based on the userId, itemId and rating fields such as the count of
purchasing, the number of types of items purchased as well as computing
minimum, median, maximum, mean and standard deviation of ratings given by each
user. And it is analogous to item statistical features.

####  4.1.2. Embedding Features

We build user and item embedding features respectively and the following
introduces the embeddings we used in this contest:

  * •

TF-IDF (Jing et al., 2002) is always applied to NLP and information retrieval
fields, which is a statistical quantity for measuring the importance of a word
with respect to a document. As for recommendation, we can think of users and
items as words to achieve a similar goal. To avoid dimension explosion, the
SVD method (Aharon et al., 2006) is used to remove unimportant components.

  * •

Word2Vec (Goldberg and Levy, 2014) is a neural network model to generate
vector representations of words, which also can be used for generating user
and item representations.

  * •

DeepWalk (Perozzi et al., 2014) could learn network embedding by proceeding
truncated random walk based on a user-item interaction graph.

####  4.1.3. Distance Features

Based on the embedding features we have constructed, we compute cosine
distance, Manhattan distance, Jaccard coefficient, euclidean distance and
Pearson correlation coefficient to measure the correlation between the two
embeddings of the same type from users and items, and the correlation values
are used as our distance features.

###  4.2. Model Training

Here we introduce learning strategies in detail. Based on features introduced
above, we use multi-fold cross validation against the public test set of
target markets to finish predicting test candidate samples.

In essence, recommendation is the task of learning to rank, which mainly
includes three types of learning strategies, i.e., pointwise, pairwise and
listwise. The pointwise learning regards the ranking problem as a
classification or regression problem. However, the pairwise learning does not
care about the specific value but only considers the relative order. As for
the listwise learning, it tackles the ranking problem directly by optimizing
the defined loss function on a list of items.

Thanks to the effective implementation of GBDT’s pointwise and pairwise
learning versions. We use LightGBM, XGBoost, CatBoost to train by pointwise
learning and LGBMRanker (Jankiewicz et al., 2019) to train by pairwise
learning. Thus, we could obtain four final models.

###  4.3. Ensemble Modeling

In the model ensemble stage, we simply adopt the weighted average operation to
fuse these four models to get the final results. In fact, we have stacked
predictions from multiple models to build these tree-based ensembling models,
which also belongs to the model ensemble operation.

##  5\. Experiments

We evaluate our method on the XMRec dataset provided by the contest. Table 2
presents the overall performance of our model and all the adopted baselines,
from which we have the following key observations:

  * •

In the first part of the table, GNN achieves the best performance on both
datasets compared to other traditional models. It may be attributed to the
superior power of GNN and our effective module design.

  * •

LightGBM* and LGBMRanker* are the models without using ranking features
generated by the hybrid ranking models. By fine-grained feature engineering,
they could also generate comparable performance.

  * •

As for the third part of the table, it proves that the GBDT models could
obtain major improvements by stacking learning models and multiple models
ensemble.

Table 2. Main results w.r.t. NDCG@10 and HR@10 for cross-market recommendation
on target market datasets. The best and second-best performed methods in each
metric are highlighted in “bold” and underline, respectively.

Method | t1 | t2
---|---|---
NDCG@10 | HR@10 | NDCG@10 | HR@10
GNN | 0.7131 | 0.7968 | 0.6226 | 0.7255
ItemCF | 0.5955 | 0.6782 | 0.5235 | 0.6136
UserCF | 0.6136 | 0.6948 | 0.5472 | 0.6547
Swing | 0.6049 | 0.6834 | 0.5489 | 0.6350
PersonalRank | 0.5922 | 0.6756 | 0.5582 | 0.6618
LightGBM* | 0.7197 | 0.8224 | 0.6286 | 0.7421
LGBMRanker* | 0.7039 | 0.7801 | 0.6381 | 0.7506
CATBoost | 0.7310 | 0.8246 | 0.6371 | 0.7474
XGBoost | 0.7354 | 0.8283 | 0.6391 | 0.7512
LightGBM | 0.7362 | 0.8324 | 0.6399 | 0.7534
LGBMRanker | 0.7388 | 0.8356 | 0.6388 | 0.7555
Model Ensemble | 0.7393 | 0.8369 | 0.6457 | 0.7610

##  6\. Conclusion

In this paper, we have introduced our practical two-stage ranking framework
for the cross-market recommendation competition of the WSDM Cup 2022. Our team
ranks the second place on the final
leaderboard444https://competitions.codalab.org/competitions/36050#results with
an excellent performance very close to the first place. In our solution, we
first conduct various models to give predictions. After that, we train 4 GBDT
models by utilizing different learning strategies and implementations of GBDT.
Finally, we ensemble these 4 models by weighted average operation. The
comprehensive experiment results demonstrate the superiority and effectiveness
of our model.

###### Acknowledgements.

We thank everyone associated with organizing and sponsoring the WSDM Cup 2022.

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