# Learning Partially Aligned Item Representation for
Cross-Domain Sequential Recommendation

Mingjia Yin<EMAIL_ADDRESS>0009-0005-0853-1089 University of
Science and Technology of China & State Key Laboratory of Cognitive
IntelligenceHefeiChina ,  Hao Wang<EMAIL_ADDRESS>0000-0001-9921-2078
University of Science and Technology of China & State Key Laboratory of
Cognitive IntelligenceHefeiChina ,  Wei Guo<EMAIL_ADDRESS>0000-0001-8616-0221 ,  Yong Liu<EMAIL_ADDRESS>0000-0001-9031-9696
Huawei Singapore Research CenterSingapore ,  Zhi Li
<EMAIL_ADDRESS>Shenzhen International Graduate School, Tsinghua
UniversityShenzhenChina ,  Sirui Zhao<EMAIL_ADDRESS>0000-0001-8103-0321 University of Science and Technology of China & State Key
Laboratory of Cognitive IntelligenceHefeiChina ,  Defu Lian
<EMAIL_ADDRESS>0000-0002-3507-9607 University of Science and Technology
of China & State Key Laboratory of Cognitive IntelligenceHefeiChina and
Enhong Chen<EMAIL_ADDRESS>0000-0002-4835-4102 University of Science and
Technology of China & State Key Laboratory of Cognitive IntelligenceHefeiChina

(2018)

###### Abstract.

Cross-domain sequential recommendation (CDSR) aims to uncover and transfer
users’ sequential preferences across multiple recommendation domains. While
significant endeavors have been made, they primarily concentrated on
developing advanced transfer modules and aligning user representations using
self-supervised learning techniques. However, the problem of aligning item
representations has received limited attention, and misaligned item
representations can potentially lead to sub-optimal sequential modeling and
user representation alignment. To this end, we propose a model-agnostic
framework called Cross-domain item representation Alignment for Cross-Domain
Sequential Recommendation (CA-CDSR), which achieves sequence-aware generation
and adaptively partial alignment for item representations. Specifically, we
first develop a sequence-aware feature augmentation strategy, which captures
both collaborative and sequential item correlations, thus facilitating
holistic item representation generation. Next, we conduct an empirical study
to investigate the partial representation alignment problem from a spectrum
perspective. It motivates us to devise an adaptive spectrum filter, achieving
partial alignment adaptively. Furthermore, the aligned item representations
can be fed into different sequential encoders to obtain user representations.
The entire framework is optimized in a multi-task learning paradigm with an
annealing strategy. Extensive experiments have demonstrated that CA-CDSR can
surpass state-of-the-art baselines by a significant margin and can effectively
align items in representation spaces to enhance performance. The anonymous
code can be found at https://anonymous.4open.science/r/KDD2024-58E8/.

Sequential Recommendation, Cross-Domain Recommendation, Contrastive Learning,
Item Representation Alignment

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation email; June 03–05, 2018; Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06

##  1\. INTRODUCTION

Figure 1. A toy CDSR scenario in Book and Movie domains.

To alleviate the ongoing situation of information explosion, recommendation
systems play a dominant role, which aims to provide users with personalized
services by suggesting suitable items instead of leaving them to self-seeking
(Wu et al., 2023; Han et al., 2024; Wang et al., 2021b; Zhang et al., 2022a;
Wang et al., 2019). The sequential recommendation (SR) system is an important
research area within the field of recommendation systems, aiming to capture
users’ dynamic preferences from chronologically organized interaction
sequences (Wang et al., 2022; Fang et al., 2020; Wang et al., 2021a; Hidasi et
al., 2015; Kang and McAuley, 2018; Yin et al., 2023). Despite achieving
notable success, conventional methods often face challenges in effectively
modeling user behavior due to the data-sparsity issue based on a single
recommendation domain (Zhu et al., 2021; Zang et al., 2022; He et al., 2023).

To address the issue of mining and transferring users’ sequential preferences
across different domains, the Cross-Domain Sequential Recommendation (CDSR)
problem has been proposed and received lots of attention. Early works focused
on developing sophisticated transfer modules, like $\pi$-net (Ma et al., 2019)
and PSJNet (Sun et al., 2021) utilized gated transfer modules to transfer
knowledge between domains. Subsequent studies, including MIFN (Ma et al.,
2022) and DA-GCN (Guo et al., 2021), leveraged the powerful high-order
modeling ability of graph neural networks (GNNs) for cross-domain knowledge
transfer. In recent times, the rise of self-supervised learning has led to the
adoption of a contrastive learning paradigm in some works, such as C2DSR (Cao
et al., 2022) and Tri-CDR (Ma et al., 2023), to align cross-domain user
representations and facilitate effective knowledge transfer between domains.

Most existing works rely on randomly initialized item representations without
elaborate processing, leading to potential misguidance in subsequent
sequential modeling and user representation alignment. Therefore, we aim to
align item representations for CDSR in a model-agnostic manner. Our idea is to
explore the implicit item correlations across different domains. To illustrate
this concept, we provide a toy scenario in Fig. 1, highlighting two essential
processes for aligning item representations: (1) item representation
generation. To explore cross-domain item correlations, it is essential to
first model single-domain item correlations, i.e., collaborative and
sequential correlations. For instance, in Fig. 1, the movie ”Lord Rings II”
appears in both Seq.1 and Seq.2. Analyzing the neighboring items reveals a
significant connection between the movie ”Lord Rings I” in Seq.2 and the movie
”Lord Rings III” in Seq.1, representing a collaborative correlation.
Additionally, we observe a sequential correlation, indicating a preference
evolution trend among these three movies from the ”Lord Rings” series. (2)
item representation alignment. Cross-domain item representations with implicit
correlations should be partially aligned. We notice that the ”Lord Rings”
books exhibit similar single-domain correlations as the movies, making it
reasonable to assume they could be aligned accordingly. However, the content
of books and movies does not strictly correspond (inherent domain gap),
meaning they should not be strictly aligned, which we define as partial
alignment. Ideally, aligned item representations can help recommend items that
users have not interacted with in other domains. For instance, we could
recommend the movie ”Lord Rings III” to a user who recently read the book
”Lord Rings II”, even if they have not watched movies.

There are several challenges to solving the problem of item representation
alignment in CDSR to achieve this goal. First, previous works, such as (Yu et
al., 2022a; Wu et al., 2021), have employed a self-supervised learning
paradigm to model item representations and enhance their effectiveness with
feature augmentation techniques (Feng et al., 2021; Yu et al., 2022a).
However, most existing feature augmentation methods are sequence-agnostic
(Feng et al., 2021; Li et al., 2021; Yu et al., 2022a), neglecting sequential
correlations and resulting in sub-optimal item representations that can
mislead the subsequent item alignment process. How to incorporate both
collaborative and sequential information into item representation learning is
a nontrivial problem. Secondly, achieving partial item representation
alignment is a nontrivial task. The scarcity of annotated data presents
challenges in determining the specific portion of the item representation that
requires alignment. Previous studies(Cao et al., 2022; Ma et al., 2023)
commonly resorted to adjusting the weights of alignment losses to control the
degree of user alignment. However, this approach is overly simplistic and can
give rise to negative transfer issues when the alignment weight is high.
Conversely, a lower weight may lead to the omission of valuable information
during the transfer process. An ideal method should have the capability to
adaptively identify the relevant parts of item representations for alignment.
Hence, the pursuit of adaptive partial alignment across domains remains an
unresolved challenge in the field of CDSR.

To address the challenges mentioned above, we propose a model-agnostic
framework CA-CDSR named Cross-domain item representation Alignment for Cross-
Domain Sequential Recommendation problem, which aims to achieve sequence-aware
representation generation and partial alignment. To this end, we first
introduce a sequence-aware feature augmentation strategy that condenses
collaborative and sequential information into item representations, resulting
in stronger representational ability for the subsequent alignment process. We
then conduct an empirical study to reveal the partial representation alignment
problem from a spectrum perspective. Based on this analysis, we propose a
simple yet effective spectrum filter to adaptively align item representations
across different domains. Additionally, we employ a graph-based sequential
encoder to obtain user representations based on aligned item representations.
Notably, as a model-agnostic framework, CA-CDSR can be integrated with most
sequential recommenders. We optimize the framework in a multi-task learning
paradigm with an annealing strategy to facilitate mutual reinforcement of each
module. In summary, our contributions can be briefly summarized as follows:

  * •

We study a novel problem of aligning item representations for CDSR, to capture
implicit item correlations and achieve adaptive partial alignment in a model-
agnostic manner.

  * •

We propose a sequence-aware feature augmentation strategy for item
representation generation. It can effectively capture collaborative and
sequential item correlations.

  * •

We conduct an innovative empirical study to verify the necessity of partial
alignment of item representations across domains from a spectrum perspective
and propose an Adaptive Spectrum Filter to alleviate the domain gap issues
efficiently.

  * •

Extensive experiments have demonstrated that CA-CDSR can significantly surpass
competitive baselines. Furthermore, detailed analysis has revealed the
generality of CA-CDSR for alignment.

##  2\. RELATED WORKS

###  2.1. Sequential Recommendation System

As one of the most influential branches of recommendation systems, sequential
recommendation (SR) aims to discover users’ dynamic preferences in users’
interaction sequences (Wang et al., 2022; Fang et al., 2020; Wang et al.,
2021a; Yin et al., 2024; Han et al., 2023; Huang et al., 2024). With the rise
of deep learning, researchers have made efforts to incorporate deep neural
networks for modeling complex sequential preferences. These include CNN (Tang
and Wang, 2018; Yan et al., 2019), RNN (Hidasi et al., 2015; Li et al., 2017),
and GNN (Wu et al., 2019; Xu et al., 2019; Wang et al., 2020). Among all the
deep SR models, Transformer-based methods, such as SASRec (Kang and McAuley,
2018) and BERT4Rec (Sun et al., 2019), have gained dominance due to their
powerful modeling capability. To address the data sparsity issue, Self-
Supervised Learning (SSL) (Yu et al., 2022b) has been introduced in SR. S3-Rec
(Zhou et al., 2020a) proposed auxiliary pre-training tasks based on different
data views. Subsequent works, like CL4SRec (Xie et al., 2022) and ICLRec (Chen
et al., 2022), employed sequence-level contrastive learning to better capture
users’ sequential preferences. Recent works like DuoRec (Qiu et al., 2022)
constructed contrastive pairs at the model level.

###  2.2. Cross-Domain Sequential Recommendation

Despite the prosperity of recommendation systems, conventional methods often
face the challenges of data sparsity and cold-start, as they are limited to
one single recommendation domain. To address these problems, Cross-Domain
Sequential Recommendation (CDSR) methods have been proposed to transfer
knowledge from relatively richer domains to sparser domains (Zhu et al., 2021;
Zang et al., 2022; Zhang et al., 2024).

CDSR focuses on mining and transferring users’ sequential preferences across
domains. Early attempts such as $\pi$-net (Ma et al., 2019) and PSJNet (Sun et
al., 2021) proposed transferring domain knowledge using elaborate transfer
modules. Additionally, GNNs are known for their ability to capture high-order
relationships (Wu et al., 2020, 2022), which motivated the development of DA-
GCN (Guo et al., 2021) and MIFN (Ma et al., 2022) that utilize graphs to model
the complex relationships between domains. Recently, some works proposed to
learn transferable item representations with item description text, like
UniSRec(Hou et al., 2022) and VQ-Rec(Hou et al., 2023). Moreover, self-
supervised learning has been adopted to align user representations in CDSR.
For instance, C2DSR (Cao et al., 2022) proposed randomly substituting items
belonging to one domain in a sequence to generate contrastive samples. Tri-CDR
(Ma et al., 2023) instead employed sequence-level contrastive learning to
explore coarse-grained similarities and fine-grained distinctions among domain
sequences.

However, most existing CDSR works overlook the item representation alignment
problem. Misaligned item representations will potentially mislead subsequent
sequential modeling and user representation alignment processes.

##  3\. PROBLEM DEFINITION

In cross-domain sequential recommendation, the main task is to model user
preferences based on their interaction records in different domains and
recommend the next item to them in each domain. In this work, we consider a
CDSR scenario consisting of two domains, where every user has interaction
sequences in both domains. We can formally define the problem as follows:

###### Definition 3.1.

(Cross-Domain Sequential Recommendation). Denote the two domains as domain
$\mathbf{X}$ and domain $\mathbf{Y}$. We define the user set $\mathcal{U}$,
the item set $\mathcal{X}$ in domain $\mathbf{X}$, and the item set
$\mathcal{Y}$ in domain $\mathbf{Y}$. For each user $u\in\mathcal{U}$, their
interaction sequence in domain $\mathbf{X}$ is denoted as
$S_{u}^{\mathbf{X}}=[x_{1},x_{2},\dots,x_{t},\dots,x_{|S_{u}^{\mathbf{X}}|}]$,
where $x_{t}\in\mathcal{X}$ is the item interacted by user $u$ at time step
$t$, and $|S_{u}^{\mathbf{X}}|$ is the length of the sequence with a maximum
value $N$. Similarly, $S_{u}^{\mathbf{Y}}$ can be defined for domain
$\mathbf{Y}$. Furthermore, we can combine $S_{u}^{\mathbf{X}}$ and
$S_{u}^{\mathbf{Y}}$ to obtain a mixed sequence
$S_{u}=[x_{1},y_{1},y_{2},x_{2},\dots,x_{|S_{u}^{\mathbf{X}}|},\dots,y_{|S_{u}^{\mathbf{Y}}|}]$,
which is organized by timestamps. Eventually, given the sequence triplet
($S_{u}^{\mathbf{X}}$, $S_{u}^{\mathbf{Y}}$, $S_{u}$), the goal of CDSR is to
predict the next item for each user $u\in\mathcal{U}$ in domain $\mathbf{X}$
and $\mathbf{Y}$, respectively.

Building upon the behavior sequences, our objective is to model the
collaborative and sequential correlations between items. To accomplish this,
we employ graph-based self-supervised learning, which has been recognized for
its capacity to capture high-order correlations. Specifically, we construct
two single-domain graph $\mathcal{A}^{\mathbf{X}}$ and
$\mathcal{A}^{\mathbf{Y}}$ and a cross-domain graph $\mathcal{A}$ as follows:

###### Definition 3.2.

(Directed Item-Item Graph). For domain $\mathbf{X}$, we construct an item-item
graph
$\mathcal{A}^{\mathbf{X}}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}|}$,
where the matrix entry $\mathcal{A}_{ij}^{\mathbf{X}}=1$ if $x_{j}$ is next to
$x_{i}$ in any sequences in $S^{\mathbf{X}}$. Similarly,
$\mathcal{A}^{\mathbf{Y}}\in\mathbb{R}^{|\mathcal{Y}|\times|\mathcal{Y}|}$ and
$\mathcal{A}\in\mathbb{R}^{(|\mathcal{X}|+|\mathcal{Y}|)\times(|\mathcal{X}|+|\mathcal{Y}|)}$
can be constructed.

Then we will detail how to generate item representations enriched with
collaborative and sequential information and achieve adaptively partial
alignment.

##  4\. METHODOLOGY

Figure 2. The framework of the proposed CA-CDSR model. We use domain
$\mathbf{X}$ as an example for domain-specific modules.

###  4.1. Overall Framework

Despite the success of existing Cross-Domain Sequential Recommendation (CDSR)
works, few studies have focused on the partial item representation alignment
problem in CDSR, which often leads to inferior performance. To address this
issue, we propose the Cross-domain Representation Alignment for Cross-Domain
Sequential Recommendation (CA-CDSR) framework, as illustrated in Fig. 2. The
CA-CDSR framework aims to achieve sequence-aware item representation modeling
and partial item representation alignment for CDSR. First, in Sec. 4.2, we
perform sequence-aware representation generation in a self-supervised learning
(SSL) fashion. We introduce a novel sequence-aware feature augmentation
strategy (SAFA) to preserve collaborative and sequential correlations in item
representations, which serves as the foundation for the subsequent
representation alignment process. Then, in Sec. 4.3, we investigate the
partial item representation alignment problem from a spectrum perspective
through an empirical study. Motivated by this study, we propose an adaptive
spectrum filter (ASF) that achieves partial representation alignment
adaptively and efficiently. Subsequently, in Sec. 4.4, we adopt a graph-
enhanced self-attention network to obtain user representations based on the
aligned item representations. Finally, in Sec. 4.5, we elaborate on the
optimization of the CA-CDSR through a multi-task learning paradigm. Notably,
CA-CDSR can be easily integrated with different backbone models.

###  4.2. Sequence-Aware Item Representation

Previous methods have relied on data augmentation techniques in the discrete
space, such as masking an item from a sequence. However, this approach poses
challenges in generating semantically consistent augmentation samples(Qiu et
al., 2022).

To begin with, we take the item representation generation of the domain
$\mathbf{X}$ as an example. The modeling process for representations in domain
$\mathbf{Y}$ follows a similar approach. However, due to the data sparsity
issue, obtaining ideal item representations without sufficient supervised
signals is challenging. Therefore, we aim to model item representations in a
contrastive learning fashion. Previous methods have relied on data
augmentation techniques in the discrete space, such as masking an item from a
sequence. However, this approach poses challenges in generating semantically
consistent augmentation samples(Qiu et al., 2022). Therefore, we propose a
sequence-aware representation augmentation (SAFA) strategy, which operates in
the continuous space, to construct augmented data that preserves both
collaborative and sequential correlations.

Collaborative correlations. Motivated by the powerful ability of GNNs to
capture high-order relationships(Wu et al., 2020), we aim to capture
collaborative correlations using single-domain graphs
$\mathcal{A}^{\mathbf{X}}\in\mathbb{R}^{|\mathcal{X}|\times|\mathcal{X}|}$ and
$\mathcal{A}^{\mathbf{Y}}\in\mathbb{R}^{|\mathcal{Y}|\times|\mathcal{Y}|}$
defined in Sec. 3. Given $\mathcal{A}^{\mathbf{X}}$, we condense neighborhood
information into item representations using a simplified graph neural network
LightGCN(He et al., 2020):

(1)
$\mathbf{E}^{\mathbf{X}}_{(l)}=\text{Norm}(\mathcal{A}^{\mathbf{X}})\mathbf{E}^{\mathbf{X}}_{(l-1)},$
(2)
$\hat{\mathbf{E}}^{\mathbf{X}}=\frac{(\mathbf{E}^{\mathbf{X}}_{(0)}+\mathbf{E}^{\mathbf{X}}_{(1)}+\dots+\mathbf{E}^{\mathbf{X}}_{(L)})}{L+1},$

where $\mathbf{E}^{\mathbf{X}}_{(0)}$ is a randomly initialized embedding
table $\mathbf{E}^{\mathbf{X}}\in\mathbb{R}^{|\mathcal{X}|\times d}$, d is the
embedding size, and $L$ is the number of layers in the graph convolution. With
the graph-enhanced item representations $\hat{\mathbf{E}}^{\mathbf{X}}$, we
can now incorporate collaborative correlation information into item
representations through contrastive learning. Specifically, we first augment
the item representations in the representation space:

(3)
$\tilde{\mathbf{E}}^{\mathbf{X}}=\hat{\mathbf{E}}^{\mathbf{X}}+\alpha\mathbf{\beta},$

where $\alpha$ is a constant scalar controlling the magnitude of augmentation,
and $\mathbf{\beta}\in\mathbb{R}^{|\mathcal{X}|\times d}$ is a noise variable
sampled from distribution $\mathcal{D}$, which will be detailed later. We then
use InfoNCE(Oord et al., 2018) to model item representations with
collaborative correlations:

(4)
$\mathcal{L}^{\mathbf{X}}=-\sum_{i=1}^{|B|}\log{\frac{e^{cos(\hat{\mathbf{e}}_{i}^{\mathbf{X}},\tilde{\mathbf{e}}_{i}^{\mathbf{X}})/\tau}}{\sum_{j=1}^{|B|}e^{cos(\hat{\mathbf{e}}_{i}^{\mathbf{X}},\tilde{\mathbf{e}}_{j}^{\mathbf{X}})/\tau}}},$

where $|B|$ is the batch size, $cos(\cdot,\cdot)$ denotes cosine similarity,
and $\tau$ is a temperature hyper-parameter. By utilizing Equation 4, we can
effectively capture the intra-domain collaborative item correlations.

Sequential correlations. Considering the collaborative correlations alone are
inadequate to achieve holistic representation modeling, we further explore the
sequential item correlations, which are crucial for capturing users’ evolving
preferences in subsequent sequence modeling. Previous feature augmentation
methods(Yu et al., 2022a) sample noise from a data-independent distribution.
However, item representations exhibit a trend of maintaining sequential
correlations with other items in the sequences. It is crucial for augmented
item representations to preserve these correlations. Therefore, we introduce a
learnable noise generator that produces sequence-aware noise, facilitating the
modeling of sequential correlations.

First, we utilize an encoder to extract sequential information. Given an
interaction sequence
$S^{\mathbf{X}}=[x_{1},x_{2},\dots,x_{t},\dots,x_{|S^{\mathbf{X}}|}]$, we
generate its corresponding sequential representation:

(5) $\hat{\mathbf{h}}^{\mathbf{X}}=\text{Encoder}(S^{\mathbf{X}}),$

where Encoder is a sequential encoder, and we instantiate it as a gated
recurrent network for its effectiveness in capturing sequential signals. To
capture sequential information, we define the noise distribution $\mathcal{D}$
as a learnable Multivariate Normal distribution
$\mathcal{N}(\mathbf{\mu}^{\mathbf{X}}(\hat{\mathbf{h}}^{\mathbf{X}}),(\mathbf{\sigma}^{\mathbf{X}}(\hat{\mathbf{h}}^{\mathbf{X}}))^{2})$.
Specifically, $\mathbf{\mu}(\cdot)$ and $\mathbf{\sigma}(\cdot)$ are
implemented as MLP networks, enabling the noise to be sampled from a sequence-
aware distribution. Since the sampling procedure is non-differentiable, we
utilize the reparameterization(Kingma and Welling, 2013) trick:

(6)
$\mathbf{\beta}={\mathbf{\mu}}^{\mathbf{X}}(\hat{\mathbf{h}}^{\mathbf{X}})+\epsilon\mathbf{\sigma}^{\mathbf{X}}(\hat{\mathbf{h}}^{\mathbf{X}}),$

where $\epsilon\in\mathbb{R}^{d}$ is randomly sampled from
$\mathcal{N}(\mathbf{0},\mathbf{I})$.

By modeling both collaborative and sequential item correlations, we obtain
item representations enriched with holistic knowledge, which lay a solid
foundation for the subsequent alignment process.

###  4.3. Partial Item Representation Alignment

Given the item representations with well-preserved collaborative and
sequential correlations, we can now start to perform representation alignment
across domains. However, we are still in the dilemma of lacking explicit data
to guide the alignment process. Therefore, we instead perform item
representation alignment based on common users in both two domains.

To leverage the information of these common users, we first concatenate the
two embedding matrices
$\mathbf{E}^{\mathbf{X}}\in\mathbb{R}^{|\mathcal{X}|\times d}$ and
$\mathbf{E}^{\mathbf{Y}}\in\mathbb{R}^{|\mathcal{Y}|\times d}$ into
$\mathbf{E}\in\mathbb{R}^{(|\mathcal{X}|+|\mathcal{Y}|)\times d}$. Then, given
the mixed item-item graph
$\mathcal{A}\in\mathbb{R}^{(|\mathcal{X}|+|\mathcal{Y}|)\times(|\mathcal{X}|+|\mathcal{Y}|)}$
in Sec. 3, which is constructed based on common users, we can obtain the
cross-domain global item representation $\hat{\mathbf{E}}$ as in Eq.( 1) and
Eq.( 2). With the cross-domain global item representation $\hat{\mathbf{E}}$,
and the augmented local item representations $\tilde{\mathbf{E}}^{\mathbf{X}}$
and $\tilde{\mathbf{E}}^{\mathbf{Y}}$, we can now perform representation
alignment by pulling in local and global item representations:

(7)
$\mathcal{L}_{\text{align}}^{\text{inter}}=-\sum_{i=1}^{|B|}\log{\frac{e^{\cos(\hat{\mathbf{e}}_{i},\tilde{\mathbf{e}}_{i}^{\mathbf{X}})/\tau}}{\sum_{j=1}^{|B|}e^{\cos(\hat{\mathbf{e}}_{i},\tilde{\mathbf{e}}_{j}^{\mathbf{X}})/\tau}}}-\sum_{i=1}^{|B|}\log{\frac{e^{\cos(\hat{\mathbf{e}}_{i},\tilde{\mathbf{e}}_{i}^{\mathbf{Y}})/\tau}}{\sum_{j=1}^{|B|}e^{\cos(\hat{\mathbf{e}}_{i},\tilde{\mathbf{e}}_{j}^{\mathbf{Y}})/\tau}}}.$

Since cross-domain information is condensed in $\hat{\mathbf{E}}$, aligning
$\hat{\mathbf{E}}$ with $\tilde{\mathbf{E}}^{\mathbf{X}}$ or
$\tilde{\mathbf{E}}^{\mathbf{Y}}$ will facilitate the implicit alignment
between item representations of domain $\mathbf{X}$ and domain $\mathbf{Y}$.

However, perfect representation alignment will eliminate the inherent domain
gap between domains, leading to the negative transfer problem (Zhang et al.,
2022b; Zhu et al., 2021). Therefore, we question whether the two domains
should be partially aligned, i.e., only a portion of the global representation
$\hat{\mathbf{e}}_{i}$ should be aligned. Since the representation spectrum
(singular values) can reflect the information carried by representations, we
investigate the problem of partial item representation alignment from a
spectrum perspective. Furthermore, as different singular values carry distinct
information, we conduct an empirical study to examine how singular values
contribute to representation alignment, as illustrated in Fig. 3. From the
figure, we draw the following conclusions: (1) The optimal filtering strengths
for the two groups of singular values differ, indicating that different
singular values exhibit varying contributions to the representation alignment
process. (2) Similarly, the optimal filtering strengths differ for distinct
domains, indicating that different domains require different filtering
strengths. Both of the two observations suggest the need for adaptive
manipulation of the spectrum to adaptively achieve partial alignment for each
domain.

(a) Food

(b) Kitchen

Figure 3. The empirical study on manipulating the spectrum of global item
representation $\hat{\mathbf{e}}_{i}$ on the Food-Kitchen dataset.
Specifically, we focused on the top 10 largest singular values, dividing them
into two groups: top 1-5 and top 6-10. We reduced the singular values in each
group by multiplying a coefficient. This can be viewed as a manual filtering
operation that selectively removes a portion of the information.

Although manual manipulation of the representation spectrum can partially
mitigate the issue stemming from perfect representation alignment, two
challenges remain unresolved. (1) The SVD decomposition is not easily
parallelizable, resulting in inefficient training(Yu et al., 2020). (2) The
search space for manual filtering is intractable, making it impractical to
identify the optimal filter for every singular value and domain. Building on
empirical findings, we introduce an adaptive spectrum filter to tackle these
challenges.

Regarding the first challenge, the rank-1 update method(Yu et al., 2020)
enables efficient spectrum modification. However, it lacks sensitivity to
domain information, and its filtering procedure is non-learnable, which fails
to fulfill the aforementioned requirement. Consequently, we introduce a
straightforward yet highly effective module called the adaptive spectrum
filter (ASF). This enables adaptive and efficient spectrum filtering, leading
to partial representation alignment. Subsequently, we provide a detailed
description of the procedure, using domain $\mathbf{X}$ as an illustrative
example.

Specifically, our objective is to derive the filtered global representations
$\bar{\mathbf{E}}^{\mathbf{X}}$ by utilizing the cross-domain item
representations $\hat{\mathbf{E}}$. Initially, we randomly sample a spectrum
filter $\mathbf{v}$ from a filter distribution
$\mathcal{N}(\mu^{\mathbf{X}},1)$. Here, $\mu^{\mathbf{X}}$ represents a
learnable embedding that can be optimized using a technique akin to the
reparameterization trick(Kingma and Welling, 2013). Subsequently, the filtered
representation can be obtained by improving the rank-1 update process in (Yu
et al., 2020) as follows:

(8)
$\begin{gathered}\hat{\mathbf{v}}=\mathbf{\hat{E}}^{T}\mathbf{\hat{E}}\mathbf{v},\\\
\bar{\mathbf{E}}^{\mathbf{X}}=\bar{\mathbf{E}}-\bar{\mathbf{E}}\frac{\hat{\mathbf{v}}\hat{\mathbf{v}}^{T}}{\parallel\hat{\mathbf{v}}\parallel_{2}^{2}}.\end{gathered}$

Similarly, we can obtain the filtered global representations
$\bar{\mathbf{E}}^{\mathbf{Y}}$ for domain $\mathbf{Y}$. Now Eq. 7 can be re-
written as:

(9)
$\mathcal{L}^{\text{inter}}_{\text{align}}=-\sum_{i=1}^{|B|}\log{\frac{e^{cos(\bar{\mathbf{e}}_{i}^{\mathbf{X}},\tilde{\mathbf{e}}_{i}^{\mathbf{X}})/\tau}}{\sum_{j=1}^{|B|}e^{cos(\bar{\mathbf{e}}_{i}^{\mathbf{X}},\tilde{\mathbf{e}}_{j}^{\mathbf{X}})/\tau}}}-\sum_{i=1}^{|B|}\log{\frac{e^{cos(\bar{\mathbf{e}}_{i}^{\mathbf{Y}},\tilde{\mathbf{e}}_{i}^{\mathbf{Y}})/\tau}}{\sum_{j=1}^{|B|}e^{cos(\bar{\mathbf{e}}_{i}^{\mathbf{Y}},\tilde{\mathbf{e}}_{j}^{\mathbf{Y}})/\tau}}}.$

As the filter is learnable, we can attain adaptive filtering magnitudes for
various singular values. Additionally, distinct filter distributions are
employed for each domain, ensuring adaptiveness to specific domains. Thus, by
utilizing the adaptive spectrum filter, the representation alignment process
can be appropriately guided.

###  4.4. Sequential Modeling

Once we have acquired the appropriately aligned item representations, we
employ a sequential encoder to generate user representations. Given the
impressive advancements of GNN-based sequential models and the self-attention
mechanism(Vaswani et al., 2017), we employ a GNN-enhanced attentive network
(GNN-Att). Importantly, as the two proposed modules are model-agnostic,
alternative sequential encoders are also viable, as demonstrated in Sec.
5.4.1.

Specifically, using the aligned item representations
$\mathbf{E}^{\mathbf{X}}$, $\mathbf{E}^{\mathbf{Y}}$, and $\mathbf{E}$, we
leverage LightGCN(He et al., 2020) to generate graph-enhanced representations
$\hat{\mathbf{E}}^{\mathbf{X}}$, $\hat{\mathbf{E}}^{\mathbf{Y}}$, and
$\hat{\mathbf{E}}$ following Eq. 1 and Eq. 2. We project an interaction
sequence
$S_{u}=[x_{1},y_{1},y_{2},x_{2},\dots,x_{|S_{u}^{\mathbf{X}}|},\dots,y_{|S_{u}^{\mathbf{Y}}|}]$
into a sequence of representations
$[\hat{\mathbf{e}}^{\mathbf{X}}_{1},\hat{\mathbf{e}}^{\mathbf{Y}}_{1},\hat{\mathbf{e}}^{\mathbf{Y}}_{2},\hat{\mathbf{e}}^{\mathbf{X}}_{2},\dots,\hat{\mathbf{e}}^{\mathbf{X}}_{|S_{u}^{\mathbf{X}}|},\dots,\hat{\mathbf{e}}^{\mathbf{Y}}_{|S_{u}^{\mathbf{Y}}|}]$,
which is enhanced with graph information. We then apply positional encoding to
the sequence representations, obtaining the initial input $\mathbf{H}^{(0)}$.
Similar to SASRec(Kang and McAuley, 2018), the attentive network comprises
multiple multi-head self-attention (MHSA) layers and feed-forward layers
(FFN), which can be represented as:

(10) $\mathbf{H}^{(l)}=FFN(MHSA(\mathbf{H}^{(l-1)})),$

where $\mathbf{H}^{(l)}$ is the representation in the $l$-th layer, and we
denote $\mathbf{H}$ as the output user representation. Besides, we can pad
$S_{u}$ to yield single-domain sequences
$S_{u}^{\mathbf{X}}=[x_{1},\text{[P]},\text{[P]},x_{2},\dots,x_{|S_{u}^{\mathbf{X}}|},\dots,\text{[P]}]$
and
$S_{u}^{\mathbf{Y}}=[\text{[P]},y_{1},y_{2},\text{[P]},\dots,\text{[P]},\dots,y_{|S_{u}^{\mathbf{Y}}|}]$,
where [P] will be projected into a zero constant vector. In this way, we can
obtain single-domain user representations $\mathbf{H}^{\mathbf{X}}$ and
$\mathbf{H}^{\mathbf{Y}}$ similarly as in Eq.(10).

###  4.5. Model Optimization

####  4.5.1. Next-item prediction objective

With the generated user representations $\mathbf{H}^{\mathbf{X}}$,
$\mathbf{H}^{\mathbf{Y}}$ and $\mathbf{H}$, the entire model can be optimized
by the next-item prediction losses. In this paper, we adopt two recommendation
objectives, whose effectiveness has been verified(Cao et al., 2022).

Single-domain next-item prediction. Denote the padded sequences in domain
$\mathbf{X}$ as $S_{u}^{\mathbf{X}}=[x_{1},\text{[P]},x_{2},\dots,x_{t}]$,
where [P] is a padding, and $x_{t+1}$ is the next item. As illustrated in Fig.
2(f), the single-domain prediction loss of domain $\mathbf{X}$ is defined as:

(11) $\begin{gathered}\mathcal{L}_{\text{single-
rec}}^{\mathbf{X}}=\sum_{S_{u}^{\mathbf{X}}\in
S^{\mathbf{X}}}\sum_{t}\mathcal{L}_{\text{single}}^{\mathbf{X}}(S_{u}^{\mathbf{X}},t),\\\
\mathcal{L}_{\text{single}}^{\mathbf{X}}(S_{u}^{\mathbf{X}},t)=-\log\left(\text{Softmax}((\mathbf{h}_{t}^{\mathbf{X}}+\mathbf{h}_{t})\mathbf{W}^{\mathbf{X}})_{x_{t+1}}\right),\end{gathered}$

where $\mathbf{h}_{t}^{\mathbf{X}}$ and $\mathbf{h}_{t}$ are the user
representations of user $u$ at timestamp $t$, and
$\mathbf{W}^{\mathbf{X}}\in\mathbb{R}^{d\times|\mathcal{X}|}$ is a learnable
parameter matrix for prediction. The purpose of Eq.(11) is to maximize the
probability of recommending the expected next item $x_{t+1}$ based on the user
representations obtained from the single-domain sequence and mixed sequence.
$\mathcal{L}_{\text{single-rec}}^{\mathbf{Y}}$ can be defined similarly.

Cross-domain next-item prediction. In the mixed sequences, users’ sequential
behaviors in one domain may also lead to an action in the other domain
(Someone who has recently purchased food requiring special handling may lead
to the corresponding kitchen utensils). Therefore, as depicted in Fig. 2(f), a
cross-domain next-item prediction loss is defined as follows:

(12) $\begin{gathered}\mathcal{L}_{\text{cross-rec}}=\sum_{S_{u}\in
S}\sum_{t}\mathcal{L}_{\text{cross}}(S_{u},t),\\\
\mathcal{L}_{\text{cross}}(S_{u},t)=\begin{cases}-\log\left(\text{Softmax}(\mathbf{h}_{t}\mathbf{W}^{\mathbf{X}})_{x_{t+1}}\right)\\\
-\log\left(\text{Softmax}(\mathbf{h}_{t}\mathbf{W}^{\mathbf{Y}})_{y_{t+1}}\right)\end{cases}.\end{gathered}$

Therefore, the mixed user representations $H$ can maintain recommendation
power for both domains, rather than be dominated by only one single domain.

####  4.5.2. Overall learning objective

We now introduce the overall learning objective by combining the previous
losses from Eq.(4), Eq.(9), Eq.(11), and Eq.(12). Intuitively, during the
initial training phase, the model should focus primarily on intra-domain
modeling. As the representations become enriched with sufficient single-domain
knowledge, the framework can progressively shift its focus toward capturing
more intricate cross-domain relationships. Based on this rationale, we present
an annealing strategy to achieve a balance between single-domain and cross-
domain training as follows:

(13) $\begin{gathered}\mathcal{L}_{rec}=\eta(\mathcal{L}_{\text{intra-
domain}})+(1-\eta)\mathcal{L}_{\text{inter-domain}},\\\
\mathcal{L}_{\text{intra-domain}}=\mathcal{L}_{\text{single-
rec}}^{\mathbf{X}}+\mathcal{L}_{\text{single-
rec}}^{\mathbf{Y}}+\lambda_{1}(\mathcal{L}^{\mathbf{X}}+\mathcal{L}^{\mathbf{Y}}),\\\
\mathcal{L}_{\text{inter-domain}}=\mathcal{L}_{\text{cross-
rec}}+\lambda_{2}\mathcal{L}_{\text{align}}^{\text{inter}},\end{gathered}$

where $\eta$ is the annealing factor, which is initialized as 1 and linearly
decreased to 0.5 with $N_{anneal}$ annealing steps, $\lambda_{1}$ is a hyper-
parameter controlling sequence-aware representation modeling, and
$\lambda_{2}$ is a hyper-parameter controlling partial representation
alignment. The entire framework will be optimized in a multi-task learning
paradigm, facilitating mutual reinforcement of each module.

####  4.5.3. Computational complexity

The computational complexity of CA-CDSR primarily arises from the graph
convolution operation in Eq.(1). To maintain brevity, we will focus solely on
discussing its complexity. By utilizing sparse matrix multiplication
operations, the complexity of Eq.(1) is $O(|\mathcal{E}|d)$, where
$|\mathcal{E}|$ represents the total number of interaction records. By
selecting $d<<|\mathcal{E}|$, CA-CDSR achieves linear complexity relative to
the number of interaction records within the datasets.

##  5\. EXPERIMENTAL EVALUATION

###  5.1. Experimental Settings

####  5.1.1. Datasets

Table 1. Statistics of the datasets.

Scenario | #Item | #Train | #Val | #Test | Avg. Length
---|---|---|---|---|---
Food | 29,207 | 34,117 | 2,722 | 2,747 | 9.91
Kitchen | 34,886 | 5,451 | 5,659
Movie | 36,845 | 58,515 | 2,032 | 1,978 | 11.98
Book | 63,937 | 5,612 | 5,730
Entertain | 8,367 | 120,635 | 4,525 | 4,485 | 29.94
Education | 11,404 | 2,404 | 2,300

To verify the superiority of the proposed CA-CDSR, we follow C2DSR(Cao et al.,
2022) to construct CDSR scenarios based on two widely-adopted public datasets
Amazon111http://jmcauley.ucsd.edu/data/amazon/index_2014.html and
HVIDEO222https://bitbucket.org/Catherine_Ma/pinet_sigir2019/src/master/HVIDEO/.
Specifically, we build three CDSR scenarios ”Food-Kitchen” (Amazon), ”Movie-
Book” (Amazon), and ”Entertain-Education” (HVIDEO). As for preprocessing the
datasets, we first extract common users belonging to both domains and filter
out users and items whose number of interactions is fewer than 10. Then we
ensure the interaction sequences for each domain contain at least 3 items. The
statistics of the preprocessed datasets are summarized in Table 1.

####  5.1.2. Compared Baselines

Table 2. The overall performance of all methods. The best results are bolded
while the second-best results are underlined in each row. * means improvements
are statistically significant with p¡0.05, and ** means p¡0.01.

Datasets | Metrics | BPR | ItemKNN | NCF | CoNet | GRU4Rec | SASRec | SR-GNN |  $\pi$-net | MIFN | C2DSR | Tri-CDR | CA-CDSR | Improv
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Food | MRR | 0.0101 | 0.0097 | 0.0111 | 0.0102 | 0.0143 | 0.0176 | 0.0193 | 0.0189 | 0.0210 | 0.0247 | 0.0254 | 0.0271∗∗ | 6.69%
Recall@10 | 0.0174 | 0.0171 | 0.0195 | 0.0182 | 0.0265 | 0.0337 | 0.0361 | 0.0352 | 0.0385 | 0.0461 | 0.0468 | 0.0506∗∗ | 8.12%
Recall@20 | 0.0252 | 0.0251 | 0.0285 | 0.0256 | 0.0382 | 0.0475 | 0.0523 | 0.0497 | 0.0551 | 0.0626 | 0.0639 | 0.0716∗∗ | 12.1%
NDCG@10 | 0.0124 | 0.0115 | 0.0134 | 0.0125 | 0.0172 | 0.0221 | 0.0235 | 0.0220 | 0.0244 | 0.0273 | 0.0284 | 0.0303∗∗ | 6.69%
NDCG@20 | 0.0143 | 0.0135 | 0.0159 | 0.0148 | 0.0205 | 0.0263 | 0.0279 | 0.0257 | 0.0286 | 0.0316 | 0.0327 | 0.0356∗∗ | 8.87%
Kitchen | MRR | 0.0040 | 0.0037 | 0.0045 | 0.0041 | 0.0064 | 0.0072 | 0.0075 | 0.0069 | 0.0082 | 0.0093 | 0.0098 | 0.0105∗∗ | 7.14%
Recall@10 | 0.0052 | 0.0050 | 0.0058 | 0.0060 | 0.0088 | 0.0112 | 0.0117 | 0.0106 | 0.0126 | 0.0145 | 0.0150 | 0.0164∗∗ | 9.33%
Recall@20 | 0.0075 | 0.0071 | 0.0082 | 0.0085 | 0.0125 | 0.0158 | 0.0166 | 0.0150 | 0.0173 | 0.0209 | 0.0212 | 0.0233∗ | 9.91%
NDCG@10 | 0.0039 | 0.0037 | 0.0041 | 0.0042 | 0.0060 | 0.0074 | 0.0078 | 0.0076 | 0.0081 | 0.0097 | 0.0100 | 0.0107∗∗ | 7.00%
NDCG@20 | 0.0046 | 0.0043 | 0.0049 | 0.0049 | 0.0070 | 0.0088 | 0.0091 | 0.0089 | 0.0094 | 0.0114 | 0.0117 | 0.0125∗∗ | 6.84%
Movie | MRR | 0.0069 | 0.0068 | 0.0074 | 0.0073 | 0.0088 | 0.0091 | 0.0093 | 0.0100 | 0.0118 | 0.0171 | 0.0179 | 0.0197∗∗ | 10.1%
Recall@10 | 0.0091 | 0.0095 | 0.0098 | 0.0099 | 0.0126 | 0.0124 | 0.0123 | 0.0138 | 0.0171 | 0.0258 | 0.0265 | 0.0296∗ | 11.7%
Recall@20 | 0.0115 | 0.0120 | 0.0123 | 0.0125 | 0.0155 | 0.0152 | 0.0157 | 0.0170 | 0.0221 | 0.0309 | 0.0314 | 0.0372∗∗ | 18.5%
NDCG@10 | 0.0070 | 0.0071 | 0.0075 | 0.0077 | 0.0078 | 0.0075 | 0.0081 | 0.0089 | 0.0117 | 0.0185 | 0.0192 | 0.0207∗ | 7.81%
NDCG@20 | 0.0077 | 0.0079 | 0.0083 | 0.0082 | 0.0084 | 0.0082 | 0.0090 | 0.0093 | 0.0126 | 0.0199 | 0.0203 | 0.0226∗∗ | 11.3%
Book | MRR | 0.0037 | 0.0036 | 0.0041 | 0.0041 | 0.0048 | 0.0053 | 0.0050 | 0.0063 | 0.0073 | 0.0095 | 0.0098 | 0.0104∗∗ | 6.12%
Recall@10 | 0.0055 | 0.0052 | 0.0061 | 0.0057 | 0.0072 | 0.0081 | 0.0078 | 0.0096 | 0.0111 | 0.0162 | 0.0163 | 0.0173∗∗ | 6.13%
Recall@20 | 0.0067 | 0.0062 | 0.0070 | 0.0068 | 0.0084 | 0.0095 | 0.0095 | 0.0116 | 0.0128 | 0.0179 | 0.0186 | 0.0206∗∗ | 10.8%
NDCG@10 | 0.0043 | 0.0040 | 0.0044 | 0.0045 | 0.0047 | 0.0055 | 0.0056 | 0.0057 | 0.0062 | 0.0105 | 0.0109 | 0.0116∗ | 6.42%
NDCG@20 | 0.0055 | 0.0047 | 0.0053 | 0.0054 | 0.0058 | 0.0067 | 0.0064 | 0.0069 | 0.0074 | 0.0112 | 0.0116 | 0.0123∗ | 6.03%
Entertainment | MRR | 0.3244 | 0.3340 | 0.3204 | 0.3229 | 0.3252 | 0.3523 | 0.3539 | 0.3718 | - | 0.4521 | 0.4594 | 0.4724∗∗ | 2.83%
Recall@10 | 0.4550 | 0.4581 | 0.4458 | 0.4641 | 0.4673 | 0.5094 | 0.5025 | 0.5128 | - | 0.6193 | 0.6217 | 0.6320∗∗ | 1.66%
Recall@20 | 0.5009 | 0.5199 | 0.5059 | 0.5109 | 0.5041 | 0.5495 | 0.5588 | 0.5645 | - | 0.6810 | 0.6840 | 0.6957∗∗ | 1.71%
NDCG@10 | 0.3940 | 0.4008 | 0.4238 | 0.4514 | 0.4419 | 0.4586 | 0.4601 | 0.4432 | - | 0.4889 | 0.4927 | 0.5062∗∗ | 2.74%
NDCG@20 | 0.4170 | 0.4330 | 0.4485 | 0.4540 | 0.4535 | 0.4903 | 0.4919 | 0.4740 | - | 0.4984 | 0.5012 | 0.5195∗∗ | 3.65%
Education | MRR | 0.2890 | 0.3024 | 0.2904 | 0.3129 | 0.3225 | 0.3407 | 0.3438 | 0.3457 | - | 0.4012 | 0.4058 | 0.4174∗∗ | 2.86%
Recall@10 | 0.4374 | 0.4239 | 0.4453 | 0.4497 | 0.4940 | 0.5206 | 0.5246 | 0.5003 | - | 0.5834 | 0.5861 | 0.5957∗∗ | 1.64%
Recall@20 | 0.4651 | 0.4694 | 0.4684 | 0.4980 | 0.5470 | 0.5593 | 0.5751 | 0.5542 | - | 0.6422 | 0.6451 | 0.6562∗∗ | 1.72%
NDCG@10 | 0.3493 | 0.3539 | 0.3697 | 0.3747 | 0.4244 | 0.4216 | 0.4236 | 0.4155 | - | 0.4411 | 0.4449 | 0.4547∗ | 2.20%
NDCG@20 | 0.3587 | 0.3633 | 0.3835 | 0.3847 | 0.4270 | 0.4420 | 0.4380 | 0.4354 | - | 0.4558 | 0.4593 | 0.4696∗∗ | 2.24%

We compare the performance of our method with four groups of state-of-the-art
baselines:

  * •

Traditional Methods. (1) BPR(Rendle et al., 2012) is a classic matrix
factorization method with a pair-wise ranking loss function. (2)
ItemKNN(Sarwar et al., 2001) models item relations and use these relations to
recommendations.

  * •

Cross-Domain Methods. We select two classic CDR baselines: (1) NCF(He et al.,
2017) is a neural CF model that adopts multiple hidden layers to capture
collaborative signals. We adapt it to the CDR problem by sharing user
embeddings and setting domain-specific MLP networks for each domain. (2)
CoNet(Hu et al., 2018) first models domain-specific interactions with MLP
networks and then adopts cross-connection networks to transfer knowledge
between domains.

  * •

Sequential Methods. We include three classic deep sequential baselines based
on network structures: RNN-based method GRU4Rec(Hidasi et al., 2015),
Attention-based method SASRec(Kang and McAuley, 2018), and GNN-based method
SR-GNN(Wu et al., 2019).

  * •

Cross-Domain Sequential Methods. As the main competitors of the proposed
method, we select CDSR baselines from two perspectives. Network-based: (1)
$\pi$-net(Ma et al., 2019) proposes a gating mechanism to transfer sequential
patterns between domains. (2) MIFN(Ma et al., 2022) is a graph-based CDSR
baseline, which utilizes knowledge graphs to guide the transfer process. SSL-
based: (1) C2DSR(Cao et al., 2022) proposes a contrastive infomax objective to
capture and transfer user’s cross-domain sequential preferences. (2) Tri-
CDR(Ma et al., 2023) instead performs sequence-level contrastive learning to
explore coarse-grained similarities and fine-grained distinctions among domain
sequences.

####  5.1.3. Evaluation Protocols

We adopt the commonly used leave-one-out evaluation strategy to evaluate the
top-K recommendation performance as in (Kang and McAuley, 2018; Sun et al.,
2019; Zhou et al., 2020b; Cao et al., 2022). Specifically, we equally divide
users’ latest interaction sequences into validation and test sets, and all the
remaining interactions are treated as the training set. As for evaluation
metrics, we employ MRR(Voorhees et al., 1999), NDCG@10(Järvelin and
Kekäläinen, 2002), and HR@10. Besides, as suggested by Krichene and
Rendle(Krichene and Rendle, 2020), we adopt the whole item set as the
candidate item set during evaluation.

####  5.1.4. Implementation Details

We implement CA-CDSR based on PyTorch(Paszke et al., 2019). The maximum number
of training epochs is set to 100. The training will be stopped when MRR on the
validation data doesn’t increase for 5 epochs. For CA-CDSR and all baselines,
we adopt the following settings to perform a fair comparison: the training
batch size $B$ is fixed to 256, Adam(Kingma and Ba, 2014) with a learning rate
{1e-4, 5e-4, 1e-3} is used as the optimizer, the embedding size $d$ is set to
256 as previous works(Ma et al., 2019; Cao et al., 2022), and the maximum
sequence length $N$ is set to 30 for all datasets. Besides, the non-CDR
methods are trained upon the mixed datasets.

Considering particular hyper-parameters of CA-CDSR, the number of GNN layers
$L$ is fixed to 2, the strength of the two contrastive learning tasks
$\lambda_{1}$ and $\lambda_{2}$ are searched from 0.1 to 0.9 with a step 0.1,
the temperature $\tau$ of InfoNCE is fixed to 0.2, the magnitude of feature
augmentation $\alpha$ is fixed to 0.1, the annealing step $N_{\text{anneal}}$
is searched among {25, 50, 75, 100, 200}.

###  5.2. Overall Performance

In this subsection, we compare the overall performance of all methods, which
are presented in Table 2. Notably, the results of MIFN on “Entertain-
Education” are missing, which is because we could not get the item knowledge
graph. From the table, We can draw the following conclusions from the results:
(1) Traditional methods yield inferior results compared to cross-domain
methods, indicating the superiority of CDR methods in addressing the data
sparsity issue. (2) CDSR methods, which consider both sequential
characteristics and cross-domain modeling, achieve the best results across the
three datasets. Additionally, methods based on self-supervised learning (SSL)
demonstrate better performance, indicating the potential of contrastive
learning in enhancing representation learning and knowledge transfer in CDSR.
(3) Among the SSL-based CDSR methods, CA-CDSR consistently outperforms other
approaches, particularly in the sparser datasets Food-Kitchen and Movie-Book.
This can be attributed to two factors: Firstly, CA-CDSR adopts an SSL paradigm
based on feature augmentation, which proves to be a more effective approach.
Secondly, CA-CDSR incorporates representation alignment using carefully
designed item representations, which guides the modeling of sequential user
preferences and leads to improved performance compared to C2DSR and Tri-CDR.
(4) CA-CDSR exhibits superior performance on sparser datasets. We can observe
more significant improvement in the two sparser datasets. This can be
attributed to the fact that the limitations of previous self-supervised
strategies have been mitigated by an ample supply of supervised signals
present in the data.

###  5.3. Ablation Studies

Table 3. Abalation studies of CA-CDSR on MRR.

Datasets | Food | Kitchen | Book | Movie | Entertain | Education
---|---|---|---|---|---|---
(A) CA-CDSR | 0.0271 | 0.0105 | 0.0197 | 0.0104 | 0.4724 | 0.4714
(B) -all | 0.0239 | 0.0092 | 0.0163 | 0.0090 | 0.4566 | 0.4554
(C) -generation | 0.0264 | 0.0102 | 0.0186 | 0.0099 | 0.4661 | 0.4664
(D) -seq noise | 0.0264 | 0.0103 | 0.0191 | 0.0101 | 0.4682 | 0.4695
(E) -alignment | 0.0259 | 0.0096 | 0.0182 | 0.0098 | 0.4625 | 0.4633
(F) -ASF | 0.0261 | 0.0098 | 0.0178 | 0.0098 | 0.4639 | 0.4621
(G) -annealing | 0.0266 | 0.0103 | 0.0191 | 0.0103 | 0.4699 | 0.4693

In this subsection, we aim to validate the effectiveness of essential modules
in CA-CDSR. The corresponding results are presented in Table 3, where several
variations are examined: (B) removes both the representation generation and
representation alignment by setting $\lambda_{1}$ and $\lambda_{2}$ to 0, (C)
removes the representation generation by setting $\lambda_{1}$ to 0, (D)
changes the sequence-aware noise distribution to the standard normal
distribution $\mathcal{N}(\textbf{0},\textbf{I})$, (E) removes the
representation alignment by setting $\lambda_{2}$ to 0, (F) removes the ASF
module, and (G) removes the annealing strategy by fixing $\eta$ to 0.5.

Comparing (A) to (B) shows significant performance degradation, highlighting
the need to address partial item representation alignment in CDSR.
Furthermore, comparing (A) to (C) and (E) reveals the contributions of
representation modeling and alignment modules to overall performance.
Comparing (A) to (D) shows performance drops due to inadequate feature
augmentation based on random noise capturing essential sequential item
correlations. Additionally, comparing (E) and (F) suggests that unfiltered
full cross-domain alignment may even deteriorate performance. Comparing (A) to
(F) demonstrates the significant performance enhancement from the ASF module,
as it can filter out non-transferable knowledge during the alignment process.
Lastly, comparing (A) to (G) shows slight improvement in recommendation
performance with the annealing strategy for a more appropriate training
process. Our empirical studies validate the effectiveness of each component in
CA-CDSR.

###  5.4. Further Experiments

In this section, we aim to verify the generality of CA-CDSR, compare different
augmentation strategies, and demonstrate the robustness of the hyper-
parameters of CA-CDSR.

####  5.4.1. Generality of CA-CDSR

Table 4. Analysis of generality of CA-CDSR on MRR (%).

Datasets | Food | Kitchen | Movie | Book | Entertain | Education
---|---|---|---|---|---|---
GRU4Rec | 0.0143 | 0.0064 | 0.0088 | 0.0048 | 0.3252 | 0.3225
GRU4Rec+ | 0.0194 | 0.0076 | 0.0115 | 0.007 | 0.3564 | 0.3877
SASRec | 0.0176 | 0.0072 | 0.0091 | 0.0053 | 0.3523 | 0.3407
SASRec+ | 0.0213 | 0.0076 | 0.0131 | 0.0081 | 0.3866 | 0.4091
GNN-Attention | 0.0239 | 0.0092 | 0.0163 | 0.009 | 0.4566 | 0.4554
CA-CDSR | 0.0271 | 0.0105 | 0.0197 | 0.0104 | 0.4724 | 0.4714

Since CA-CDSR is model-agnostic and can be easily combined with other
sequential encoders, we have applied the proposed modules to some
representative sequential encoders such as GRU4Rec (Hidasi et al., 2015) and
SASRec (Kang and McAuley, 2018). The corresponding results are summarized in
Table 4, where ’+’ indicates the encoder enhanced by the proposed modules, and
GNN-Attention refers to the graph-enhanced sequential encoder introduced in
Sec. 4.4. From the table, it can be observed that the performance of all
encoders is greatly improved when combined with CA-CDSR, which further
confirms the generality of CA-CDSR. Additionally, the results highlight the
significance of item representation alignment in CDSR.

####  5.4.2. Comparison of different augmentation strategy

To verify the superiority of the proposed SAFA strategy, we compared it with
other augmentation methods, including Edge Drop (Wu et al., 2021) (graph
structure augmentation), SimGCL (Yu et al., 2022a) (feature augmentation), and
Tri-CDR (Ma et al., 2023) (sequence-aware augmentation). The same sequential
encoder is adopted for all methods. The results are presented in Table 5, and
we can draw these conclusions: (1) Comparing (A) with (B) and (C) with (D),
the feature augmentation methods (B and D) outperform the non-sequential
competitors (A and C), demonstrating the effectiveness of feature
augmentation. (2) Comparing (A) with (C) and (B) with (D), the sequence-aware
augmentation strategies (C and D) outperform the non-sequential competitors (A
and B). This indicates that sequence-aware strategies can capture essential
sequential information, leading to improved performance.

Table 5. Results of augmentation strategies on MRR (%). Datasets | Food | Kitchen | Movie | Book
---|---|---|---|---
Sequence Encoder | 0.0239 | 0.0092 | 0.0163 | 0.0090
(A) Edge Drop | 0.0232 | 0.0086 | 0.0168 | 0.0085
(B) SimGCL | 0.0244 | 0.0094 | 0.0168 | 0.009
(C) Tri-CDR | 0.0254 | 0.0098 | 0.0179 | 0.0098
(D) CA-CDSR | 0.0271 | 0.0105 | 0.0197 | 0.0104

####  5.4.3. Hyper-Parameter Sensitivity

In this section, we investigate the sensitivity of hyper-parameters in CA-
CDSR, specifically the strength of representation modeling ($\lambda_{1}$) and
representation alignment ($\lambda_{2}$) in Eq. 13. Since the trend is similar
for the three datasets, we only report the results for the Food-Kitchen
dataset.

(a) Food

(b) Kitchen

Figure 4. Recommendation performance (%) w.r.t $\lambda_{1}$.

We set $\lambda_{1}$ in the range of $[0,0.3,0.5,0.7,1,10]$ and the results
are depicted in Fig. 4. In the figure, the performance exhibits an increasing
trend as $\lambda_{1}$ increases, as modeling collaborative and sequential
item correlations can facilitate the representation alignment process.
However, the performance decreases when $\lambda_{1}$ exceeds a threshold, as
too large $\lambda_{1}$ will instead suppress the representation alignment.

(a) Food

(b) Kitchen

Figure 5. Recommendation performance (%) w.r.t $\lambda_{2}$.

We set $\lambda_{2}$ in the range of $[0,0.3,0.5,0.7,1,10]$, and the results
are illustrated in Fig. 5. The performance initially increases and then
decreases. This observation suggests that the domain gap should be narrowed to
an optimal extent. Insufficient representation alignment may not fully exploit
the cross-domain knowledge, while excessive representation alignment can
impair the inherent gap between domains, resulting in the negative transfer
problem.

##  6\. CONCLUSIONS

In this paper, we proposed a model-agnostic framework named CA-CDSR, which
studies cross-domain sequential recommendation from a partial item
representation alignment perspective. Particularly, we proposed a sequence-
aware feature augmentation strategy and an adaptive spectrum filter, which
respectively achieved sequence-aware generation and adaptive partial alignment
for item representations. Extensive experiments demonstrated that CA-CDSR not
only outperformed other baselines with significant margins but also exhibited
promising potential in aligning items in the representation space. In the
future, we will explore the partial alignment problem on datasets with
accessible item properties, which can provide explicit guidance for the
alignment process.

###### Acknowledgements.

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