# A Look Into News Avoidance Through AWRS: An Avoidance-Aware Recommender
System

Igor L.R. Azevedo  0000-0001-5144-825X The University of TokyoTokyoJapan
<EMAIL_ADDRESS>,  Toyotaro Suzumura  0000-0001-6412-8386 The
University of TokyoTokyoJapan<EMAIL_ADDRESS>and  Yuichiro Yasui
0000-0002-4175-9318 Nikkei Inc.TokyoJapan<EMAIL_ADDRESS>
###### Abstract.

In recent years, journalists have expressed concerns about the increasing
trend of news article avoidance, especially within specific domains. This
issue has been exacerbated by the rise of recommender systems. Our research
indicates that recommender systems should consider avoidance as a fundamental
factor. We argue that news articles can be characterized by three principal
elements: exposure, relevance, and avoidance, all of which are closely
interconnected. To address these challenges, we introduce AWRS, an Avoidance-
Aware Recommender System. This framework incorporates avoidance awareness when
recommending news, based on the premise that news article avoidance conveys
significant information about user preferences. Evaluation results on three
news datasets in different languages (English, Norwegian, and Japanese)
demonstrate that our method outperforms existing approaches.

Recommender Systems, News Modeling, News Avoidance

##  1\. Introduction

Recommender systems are extensively used to present users with options that
align closely with their interests. In the news domain, these systems face
unique challenges not typically encountered in other areas, such as the
importance of timeliness, novelty, and relevance. These factors create a
rapidly changing environment where news consumption patterns shift quickly.

Traditional methods relying solely on user click history and basic profiling
have given way to sophisticated techniques that delve deeper into
understanding user preferences and behavioral patterns. For instance, the NRMS
model (Wu et al., 2019d) uses multi-head self-attention to learn news
representations from news titles by modeling the interactions between words.
In the user encoder, they learn representations of users from their browsed
news and use multi-head self-attention to capture the relatedness between the
news. Moreover, the NAML (Wu et al., 2019b) model proposes a neural news
recommendation approach which can learn informative representations of users
and news by exploiting different kinds of news information such as titles,
bodies, and topic categories. Additionally, the LSTUR model (An et al., 2019)
learns representations of news from their titles and topic categories and uses
an attention network to select important words. In the user encoder, they
propose to learn long-term user representations from the embeddings of their
IDs.

More recently, other models have tried to increase their performance by
focusing on different paradigms. For instance, the GLORY model (Yang et al.,
2023) integrates global information with local user interactions to optimize
content personalization. Moreover, LANCER (Bae et al., 2023) incorporates the
concept of news lifetime to strategically enhance the negative sample space by
perceiving that news articles have a finite influence period. Addressing
challenges such as the cold-start problem and popularity bias, the PP-Rec
model (Qi et al., 2021) utilizes news popularity metrics to enhance
recommendation accuracy. In their method, the ranking score for recommending a
candidate news to a target user is the combination of a personalized matching
score and a news popularity score. Meanwhile, the CAUM model (Qi et al., 2022)
leverages candidate-aware self-attention networks to capture global user
interests based on candidate news items. They propose a candidate-aware CNN
network to incorporate candidate news into local behavior context modeling and
learn candidate-aware short-term user interest. Moreover, MANNeR (Iana et al.,
2024), a modular framework for flexible multi-aspect (neural) news
recommendation that supports ad-hoc customization over individual aspects at
inference time, focuses on balancing recommendation performance with diversity
across various metrics.

However, one factor that has become increasingly common in today’s digital
landscape is the phenomenon of news avoidance. This behavior reflects a
deliberate rejection or unintentional neglect of traditional news consumption,
often influenced by a preference for alternative media sources (Schrøder,
2019; Villi et al., 2022; Fitzpatrick, 2022). This trend, which can be
temporary or selective, underscores a nuanced understanding of how consumers
interact with news, shaped by their interests and a growing skepticism towards
certain topics. As the demand rises for news that aligns with personal values,
especially in a competitive and diverse media landscape, the concept of
relevance becomes paramount. Challenges exacerbated by the pandemic, such as
news fatigue and distrust in mainstream media, emphasize the importance of
recommender systems capable of recognizing and integrating avoidance
behaviors.

Despite the aforementioned advancements, the integration of news avoidance
behaviors into recommender systems has been overlooked, at least to the best
of our knowledge. By engaging domain experts such as journalists and
addressing their concerns (Schrøder, 2019; Villi et al., 2022; Fitzpatrick,
2022; Heitz et al., 2022), we explored how to perceive and integrate avoidance
strategies within these news recommendation systems. This approach not only
enhances the relevance and effectiveness of news recommendations but also
acknowledges the evolving dynamics of user engagement with news content in
today’s digital landscape.

In summary, our contributions are as follows: (1) Our model presents a novel
concept, and to the best of our knowledge, we are the first to explore the
perspective of avoidance in recommender systems. (2) We introduce the AWRS
framework (Avoidance-Aware Recommender System), which incorporates avoidance
awareness, including time and relevance modules, to improve performance in
user matching recommendations. (3) Extensive experiments on three diverse
real-world datasets show that AWRS consistently delivers superior performance
across a wide range of metrics.

##  2\. Related Work

###  2.1. Personalized News Recommender Systems

Neural content-based models have become the leading method for personalized
news recommendation, surpassing traditional systems that relied on manual
feature engineering (Wu et al., 2023). These Neural News Recommender (NNR)
models typically include a news encoder that transforms input features into
embeddings and a user encoder that creates user-level representations from
clicked news (Wu et al., 2023, 2019d, 2019c; Okura et al., 2017; An et al.,
2019; Wu et al., 2022b). The recommendation score is computed by comparing the
candidate news embedding against the user embedding (Wang et al., 2018; Wu et
al., 2019b). These models are trained using point-wise classification
objectives with negative sampling (Huang et al., 2013; Wu et al., 2019d).

Personalized NNR systems focus on matching news content to user preferences,
often limiting exposure to diverse viewpoints and creating ”filter bubbles”
(Iana et al., 2024; Heitz et al., 2022; Pariser, 2011; Li and Wang, 2019).
This leads to homogeneous news consumption, reinforcing users’ initial
stances. To address these issues, researchers have explored various methods to
diversify recommendations. One approach involves re-ranking personalized
recommendation sets (Gharahighehi and Vens, 2021) to increase diversity.
Another approach involves multi-task training of NNRs (Moreira et al., 2019;
Qi et al., 2022; Wu et al., 2022a; Choi et al., 2022), where the primary
personalization objective is coupled with auxiliary objectives that promote
aspect-based diversification.

Recent advancements in news recommenders aim to enhance content-based
personalization by considering additional aspects: (1) Sentiment \- SentiRec
(Wu et al., 2020b) recommends news with diverse sentiments. (2) Popularity \-
PP-Rec (Qi et al., 2021) incorporates news popularity to address cold-start
and diversity issues. (3) Global Representations \- GLORY (Yang et al., 2023)
uses a global news graph and gated graph neural networks to enrich news
representations. (4) Recency \- LANCER (Bae et al., 2023) introduces the
concept of lifetime to improve news recommendation models.

###  2.2. Open Possibilities and Current Limitations

As previously mentioned, news avoidance has become a growing concern for news
platforms, particularly those utilizing recommender systems due to the
creation of ”filter bubbles” (Pariser, 2011). We decided to investigate how
avoidance affects the behavior of news article clicks. Subsequent sections
will illustrate the intrinsic relationship between avoidance, exposure, and
the number of clicks a news article receives. This aspect, overlooked in prior
studies to the best of our knowledge, can enhance news recommendations by
providing insights into the interaction between exposure and avoidance,
especially regarding popular items.

In the news domain, the item with the most clicks is often highly exposed and
less avoided. We argue that news article avoidance conveys significant
information about user preferences. For instance, during an election,
election-related articles may be popular but often avoided by those fatigued
by political news. If a user reads a political article that is largely avoided
by others, it suggests a strong interest in that topic. Conversely, clicking
on a popular football article, one that is little avoided, might indicate less
about the user’s preference and more about its current popularity. In other
words, largely avoided news articles provide more information about user
tastes than less avoided ones. This understanding allows us to incorporate
such contextual information into recommender systems, improving the
understanding of user preferences. The following sections will formally define
each of the concepts discussed above.

##  3\. Principal Elements

We identified three principal elements for characterizing a news article:
exposure, avoidance, and relevance. To explain these concepts please refer to
the Figure 1. In the figure, the black dashed line represents the behaviors
dataset, listing news articles (impressions) at each time point $t$. For
simplicity, assume each positive news candidate (clicked article) has a fixed
number of negative candidates. The off-white square labeled ”exposed news
articles” shows all news articles viewed by users at time $t_{1}$. Imagine a
news platform where at any time $t$, a certain number of news articles
$\mathcal{N}_{t}=\\{n_{1},n_{2},...,n_{k}\\}$ and users
$\mathcal{U}_{t}=\\{u_{1},u_{2},...,u_{j}\\}$ interact. When user $u_{1}$
views article $n_{1}$, they are ”exposed” to it and can choose to click or
not. The box contains all $k$ articles viewed by all $j$ users at time $t$.

Figure 1. A schematic for the explanation of avoidance and exposure.

The light pink rectangle, called the ”impression list,” represents all news
articles viewed by user $u$, both positive and negative. An impression log
records the articles displayed to a user at a specific time and their click
behaviors (Wu et al., 2020a). The set of impressions at time $t$ is termed
exposure. Exposure is contingent upon the publication date (time) of news
articles before they can be displayed to users. Therefore, at a specific time
$t$, certain news articles may be exposed while others are not, simply because
they have not yet been published. This concept is illustrated by the arrow at
the bottom of the figure, depicting how the impression lists change over time.
Finally, we use dark blue and dark pink to indicate whether articles have been
clicked or not, respectively.

###  3.1. Exposure

###### Definition 3.1 (Number of Exposures - $n_{E}(n,t)$).

The total number of times a specific news article $n$ is presented to users on
the platform up to a defined time $t$. This metric quantifies all instances
where the article $n$ was displayed to active users, thus providing
opportunities for being clicked.

###### Definition 3.2 (Number of Impressions - $n_{I}(t)$).

An Impression is defined as an individual data record in a news dataset in a
given time $t$.

###### Definition 3.3 (Exposure Per Impression - $EPI(n,t)$).

The Exposure Per Impression ($EPI$) for a news article $n$ at time $t$ is
calculated as follows:

(1) $EPI(n,t)=\frac{n_{E}(n,t)}{n_{I}(t)}$

This ratio provides a relative measure of how extensively a news article $n$
has been exposed in relation to the total number of impressions recorded.

Example: At time $t_{1}$, we know that we have 100 impression lists, and the
news article $n_{174}$ has been displayed in 50 of them. Thus, we have
$n_{E}(n_{174},t_{1})=50$ and $n_{I}(t_{1})=100$. Hence, the exposure per
impression of this news article $n_{174}$ is
$EPI(n_{174},t_{1})=\frac{50}{100}=0.5$.

Figure 2. $Av(n,t)\text{ vs. }EPI(n,t)$ for MIND-small.

###  3.2. Avoidance

Avoidance reflects the deliberate non-engagement with specific news content by
readers. This phenomenon may arise from various factors, such as a lack of
interest in the topic, discomfort with the content, or distrust towards the
source. Understanding the patterns of avoidance is crucial for analyzing user
behavior and optimizing content delivery to enhance reader engagement.

###### Definition 3.4 (Avoidance - $Av(n,t)$).

Avoidance for a news article $n$ over a timeframe $t$ quantifies the
proportion of potential exposures that did not result in interaction. This is
mathematically defined as follows:

(2) $Av(n,t)=1-\frac{n_{clk}(n,t)}{n_{E}(n,t)}$

here, $n_{clk}(n,t)$ represents the number of clicks received by the article
$n$ up to the specified time $t$, and $n_{E}(n,t)$ denotes the total number of
exposures of the article $n$ within the same timeframe. The value of $Av(n,t)$
ranges from 0 to 1, where 1 indicates total avoidance (no clicks relative to
exposures) and 0 signifies complete engagement (every exposure resulted in a
click).

Example: Given that the news article $n_{174}$ at time $t_{1}$ received 20
clicks out of 50 exposures, its avoidance is calculated as
$Av(n_{174},t_{1})=1-\frac{n_{clk}(n_{174},t_{1})}{n_{E}(n_{174},t_{1})}=1-\frac{20}{50}=0.6$.

####  3.2.1. Visualization of Avoidance

We analyzed the correlation among article avoidance, exposure, and the number
of clicks for news articles. We plotted $Av(n,t)$ versus $EPI(n,t)$ for the
MIND-small dataset on November 9, 2019, at 16:00 and 22:00, as shown in Figure
2. Additionally, we incorporated a color scale to indicate the normalized
number of clicks and represented the radius of each news article based on its
number of clicks. The graph demonstrates that articles maintaining a balanced
level of exposure and low avoidance tend to attract more clicks over time.
Articles that show an increase in $EPI$ and a decrease in $Av$ (as indicated
by the red arrow) receive the highest number of clicks. For example, article
N41881 maintains a stable balance between $EPI$ and $Av$ from 16:00 to 22:00,
making it the most clicked news article. In contrast, article N1034 shows a
shift in its $EPI$ and $Av$ balance during the same time window. It
demonstrates an upward trend in exposure ($\uparrow EPI$) and an increase in
avoidance ($\rightarrow Av$) over time, impacting its click rate, which
indicates a gradual increase in clicks (larger circle radius). This analysis
underscores how avoidance and exposure dynamics influence click rates,
offering insights to enhance recommender systems.

(a) MIND-small \- English

(b) Adressa one-week \- Norwegian

(c) Nikkei \- Japanese

Figure 3. The graph for $Av(n,t)$ vs $EPI(n,t)$ is computed for the MIND-small
(a), Adressa one-week (b), and Nikkei (c) datasets using $D=5$, resulting in
25 distinct regions.

###  3.3. Relevance

Relevance indicates how well a news article aligns with readers’ interests,
but measuring it is complex and requires specific data. Key factors in
measuring relevance include: (1) time since publication, (2) news text
embedding, (3) exposure per impression $EPI(n,t)$, (4) avoidance $Av(n,t)$,
and (5) number of clicks $clk(n,t)$. Each factor is weighted within the model
to evaluate relevance, offering a robust framework for assessing the alignment
of a news article with user interests. More details on the calculation of
relevance can be found in Section 5.3.

##  4\. User Engagement Embedding

To measure the impact of avoidance and exposure on user interaction with a
news article, we calculate a metric called user engagement ($\bm{ue}$)—how
users interact with a specific news article based on its avoidance and
exposure per impression. We divide our graph of $Av(n,t)\text{ vs. }EPI(n,t)$
into $D\times D$ distinct regions. Depending on the values of $Av(n,t)$ and
$EPI(n,t)$, a news article falls into different regions. Formally, the user
engagement indices are computed as:

(3) $i_{\text{ue}}=D\cdot epi_{\text{idx}}+av_{\text{idx}}$

where $epi_{\text{idx}}$ is the vector of exposure per impression indices for
historical clicks, $av_{\text{idx}}$ is the vector of avoidance indices for
historical clicks, and $D$ is a constant representing the number of distinct
values. The user engagement embedding is then obtained using an embedding
layer:

(4) $\bm{ue}=\mathbf{W}_{\text{ue}}(i_{\text{ue}})$

where $\mathbf{W}_{\text{ue}}$ is an embedding layer that maps each index to
an embedding vector, and $i_{\text{ue}}$ are the computed indices from the
above equation 3. The user engagement embedding ($\bm{ue}$) captures the
spatial representation based on the quantized values of avoidance and exposure
per impression. This embedding is used as input for our models to capture the
information conveyed when a specific news article is largely avoided by many
users but still clicked by a particular user. The concept behind the user
engagement embedding ($\mathbf{ue}$) is to integrate avoidance-awareness into
recommender models. Figure 3 illustrates this division for $D=5$, resulting in
$D\times D=25$ distinct regions.

##  5\. Methodology

As highlighted in section 2.1, different models focus on various aspects of
news recommendations. We introduce a new aspect: Avoidance. Our method, AWRS,
improves recommendations by incorporating avoidance-awareness. Unlike
traditional methods based only on clicks and popularity, AWRS integrates
avoidance and relevance. The conceptual framework is shown in Figure 4.

###  5.1. Problem Formulation

Given a user $u$ and a candidate news article $n_{c}$, our goal is to compute
an interest score $Int_{s}$ that quantifies user $u$’s potential engagement
with $n_{c}$. We evaluate a collection of candidate news articles
$\mathbf{N}_{c}=[n_{c1},n_{c2},\ldots,n_{cL}]$ and recommend the highest-
ranking articles to user $u$. User $u$ has a history of clicked news articles
$\mathbf{H}_{u}=[h_{1},h_{2},\ldots,h_{M}]$. Each news article is
characterized by its title ($\mathcal{T}$), abstract ($\mathcal{A}$), category
($\mathcal{C}_{cat}$), and associated entities
($\mathbf{E}_{i}=[e_{1},e_{2},\ldots,e_{k}]$).

Figure 4. Overview of our proposed approach for AWRS.

###  5.2. Document Encoder

Our news encoder, inspired by (Qi et al., 2021, 2022; Wu et al., 2019b),
extracts detailed information from text and entities within a news article
using a multi-head attention architecture. It incorporates embeddings from the
title, category, and entities, generating embeddings with Pretrained Language
Models (PLMs) following (Iana et al., 2024). We used RoBERTa Base (Liu et al.,
2019) for the MIND-small dataset, NB-BERT Base (Kummervold et al., 2021) for
the Adressa one-week dataset, and a finetuned version of the DeBERTa model (He
et al., 2021) called Japanese DeBERTa V2111https://huggingface.co/ku-
nlp/deberta-v2-base-japanese. For all models, only the last four PLM layers
were fine-tuned. The final news embedding n combines $\textbf{n}_{t}$ (title),
$\textbf{n}_{c}$ (category), and $\textbf{n}_{e}$ (entity). For the Nikkei and
Adressa one-week datasets, only title and category information were used.

###  5.3. Avoidance-aware Relevance Predictor

Figure 5. (a) Avoidance-aware Relevance Predictor and (b) Avoidance-aware User
Encoder Schematics.

As discussed in Section 3.3, measuring relevance at a given time $t$ involves
considering several factors: the content of the article, the time elapsed
since publication, avoidance and exposure per impression values, and the
number of clicks. Therefore, the Avoidance-aware Relevance Predictor,
illustrated in Figure 5 (a), evaluates the relevance of each previously
clicked news article $n$. This assessment incorporates the time elapsed,
encoded using the Time2Vec (Kazemi et al., 2019) model, resulting in the time
elapsed embedding $\bm{t}_{el}$. The news embedding $\bm{n}$, obtained from
the news encoder described in Section 5.2, along with the number of clicks at
time $t$ denoted as $n_{clk}(n,t)$, and the impact of avoidance and exposure
per impression are integrated into the user engagement embedding $\bm{ue}$, as
detailed in Section 4. Consequently, the avoidance-aware relevance scores
($\textbf{r}_{aw}$) for each news article are computed as follows:

(5) $\textbf{r}_{aw}=\sigma\left(n_{\textit{clk}}(n,t)\cdot
w_{\text{ctr}}+\hat{r}\cdot w_{\hat{r}}\right)$

here, $\hat{r}$ is the weighted sum of the relevance scores influenced by the
news article conveyed information, defined as

(6) $\hat{r}=W\cdot\hat{r}_{\text{ic}}+(1-W)\cdot\hat{r}_{\text{tue}}$

here the weight $W$ is computed as
$W=\sigma(\Psi_{1}([\textbf{n},\bm{ue},\bm{t}_{el}]))$, where
$[\cdot,\cdot,\cdot]$ denotes vector concatenation, $\sigma$ is the sigmoid
function, and $\Psi_{1}$ is a dense layer. The term $\hat{r}_{\text{ic}}$
represents the relevance score influenced by the news article embedding (n):

(7)
$\hat{r}_{\text{ic}}=\Psi_{2}(\textbf{n}),\quad\hat{r}_{\text{tue}}=\Psi_{3}([\bm{ue},\textbf{t}_{el}])$

here the term $\hat{r}_{\text{tue}}$ represents the relevance score influenced
by the combination of the user engagement embedding ($\bm{ue}$) and the time
elapsed embedding ($\bm{t}_{el}$). Finally, the overall relevance score
($\bm{r}_{aw}$) is obtained by considering both the number of clicks up to
time $t$ ($n_{clk}(n,t)$) and the computed relevance score ($\hat{r}$),
adjusted by their respective weights ($w_{n_{clk}}$ and $w_{\hat{r}}$) and
passed through a sigmoid activation function ($\sigma$) for normalization, as
indicated by Equation 5.

###  5.4. Avoidance-aware User Encoder

The core idea of AWRS is to integrate avoidance-awareness. Leveraging the CAUM
model’s architecture (Qi et al., 2022), which facilitates the extraction of
how closely candidate news articles relate to previously clicked ones, AWRS
enhances the candidate-aware attention network with user engagement
embeddings. This enhancement refines the comprehension of user preferences by
examining the categories of news articles users interact with, as well as
their levels of exposure and avoidance. This approach yields a nuanced
depiction of user interests.

####  5.4.1. Historical and Candidate News Vector Concatenation

For each historical and candidate news item, we enhance its representation by
appending the corresponding user engagement embeddings. This generates
enriched feature vectors for a more tailored recommendation approach. Given a
set $\mathbf{N}_{c}$ of candidate news articles and a set $\mathbf{H}_{u}$ of
historical clicks for a user $u$, we concatenate the user engagement embedding
$\bm{ue}$ for each item. Let $\mathscr{H}$ and $\mathscr{N}$ be the sets with
concatenated user engagement embeddings. Formally, for each $i$-th item
$\bm{h}_{i}=[\textbf{h}_{u_{i}},\bm{ue}_{h_{i}}]$ and
$\bm{n}_{i}=[\mathbf{n}_{c_{i}},\bm{ue}_{c_{i}}]$. Where $\bm{ue}_{h_{i}}$ and
$\bm{ue}_{c_{i}}$ are the user engagement embeddings for the $i$-th historical
and candidate news items, respectively, and $[\cdot,\cdot]$ denotes vector
concatenation. Thus, we have the sets:

(8)
$\mathscr{H}=[\bm{h}_{1},...,\bm{h}_{M}],\quad\mathscr{N}=[\bm{n}_{c1},...,\bm{n}_{cL}]$

####  5.4.2. Avoidance-aware Self-Attention Layer

Given the history of clicked news articles $\mathscr{H}_{u}^{M}$, where $M$
represents the total number of articles previously clicked by the user $u$, we
use multiple self-attention heads to assess similarities and extract
relatedness information between the $i$-th and $j$-th historical clicks,
$\textbf{h}_{i}$ and $\textbf{h}_{j}$, respectively:

(9)
$\hat{r}^{k}_{i,j}=\textbf{q}^{T}_{i}\textbf{W}^{r}_{k}\bm{h}_{j},\quad\textbf{q}^{T}_{i}=\textbf{Q}_{u}\bm{h}_{i}$

here, $\hat{r}^{k}_{i,j}$ is the attention score from the $k$-th head, with
$\textbf{Q}_{u}$ as the projection matrix and $\textbf{W}^{r}_{k}$ as the
parameters of the $k$-th attention head. We adaptively select significant
long-range relatedness metrics to model user interest in the candidate news
$\mathscr{N}$ based on their contextual relevance

(10)
$r^{k}_{i,j}=\hat{r}^{k}_{i,j}+\textbf{q}^{T}_{c}\textbf{W}^{r}_{k}\bm{h}_{j}$

where $\textbf{q}^{T}_{c}=\textbf{Q}_{c}\mathscr{N}$. This enhances the
attention score with candidate-specific adjustments. The augmented
representation $\textbf{l}^{k}_{i}$ for each click is formulated through
attention weights:

(11)
$\textbf{l}^{k}_{i}=\textbf{W}^{k}_{o}\sum_{j=1}^{N}\gamma^{k}_{j}\bm{h}_{j},\quad\gamma^{k}_{j}=\frac{\exp(r^{k}_{i,j})}{\sum_{p=1}^{N}\exp(r^{k}_{i,p})}$

$\textbf{W}^{k}_{o}$ is the projection matrix for the $k$-th head. The
comprehensive contextual representation $\textbf{l}_{i}$ for each click is
derived by merging outputs from all $K$ attention heads. Note that by
incorporating user engagement embeddings, our relatedness information now
considers the influence of both avoidance and exposure, providing more context
to user click patterns.

####  5.4.3. Avoidance-aware CNN Layer

The historical news vectors undergo processing through a convolutional neural
network (CNN) equipped with self-attention mechanisms. According to (Qi et
al., 2022), this involves applying multiple filters to capture patterns among
local contexts of adjacent clicks and candidate news, formulated as

(12)
$\textbf{s}_{i}=\textbf{W}_{awc}[\bm{h}_{i-h};\dots;\bm{h}_{i};\dots;\bm{h}_{i+h};\mathscr{N}]$

here, $\textbf{s}_{i}$ denotes the local contextual representation of the
$i$-th click, $2h+1$ represents the CNN window size, and $\textbf{W}_{awc}$
refers to the parameters of the avoidance-aware filters. These local
contextual representations $[s_{1},s_{2},...,s_{N}]$ of all clicked news
encode candidate-aware short-term user interests. A unified contextual
representation $\textbf{m}_{i}$ for each $i$-th click is then formed by
aggregating $\textbf{l}_{i}$ and $\textbf{s}_{i}$, expressed as
$\textbf{m}_{i}=\Phi_{1}[\textbf{s}_{i},\textbf{l}_{i}]$, where $\Phi_{1}$
denotes a dense layer.

####  5.4.4. Avoidance-aware Final Attention Layer

We use a candidate-aware attention network to model the importance of clicked
news based on their relevance to avoidance-aware candidate news $\mathscr{N}$.
This builds the avoidance-aware user embedding representation
$\textbf{u}_{aw}$, given by
$\textbf{u}_{aw}=\sum^{N}_{i=1}\alpha_{i}\textbf{m}_{i}$ where $\alpha_{i}$ is
the weight of the $i$-th click:

(13)
$\alpha_{i}=\frac{\exp(\Phi_{2}(\textbf{m}_{i},\mathscr{N}))}{\sum^{N}_{j=1}\exp(\Phi_{2}(\textbf{m}_{j},\mathscr{N}))}$

here, $\Phi_{2}$ is a dense layer. This approach encodes avoidance-aware user
interests relevant to the candidate news into $\textbf{u}_{aw}$, enhancing
interest matching accuracy. The general overview of the Avoidance-aware User
Encoder is shown in Figure 5 (b).

####  5.4.5. Final Relevance Scores

The relevance scores combine the news candidate’s avoidance-awareness
$\mathscr{N}$ with the score from the avoidance-aware user vector
$\textbf{u}_{aw}$. Specifically, the preliminary interest scores are
calculated as $Int_{s}^{\prime}=\mathscr{N}^{T}\cdot\textbf{u}_{aw}$. However,
to obtain the final scores, we apply our Avoidance-aware Relevance Predictor,
which considers the time-varying nature of avoidance, as shown in Figure 4.
Thus,

(14) $Int_{s}=(1-\eta)\cdot\mathbf{r}_{aw}+\eta\cdot Int_{s}^{\prime}$

where $\eta=\sigma(\Phi_{3}(\mathbf{u}_{aw}))$. Here, $\Phi_{3}$ is a dense
layer, $\mathbf{r}_{aw}$ is the relevance avoidance-aware score as defined
earlier by equation 5, and $\eta$ is computed from the user representation
$\mathbf{u}_{aw}$ using a dense network with sigmoid activation. This approach
allows our model to better capture patterns related to the information
conveyed by theavoidance and exposure values of a news article.

###  5.5. Loss Function

Inspired by (Wu et al., 2019d), we use negative sampling during model
training. For each clicked news item (positive sample), we randomly sample $K$
non-clicked items (negative samples) from the same impression and shuffle
their order. The click probability score of the positive item is
$\hat{y}^{+}$, and the scores for the $K$ negative items are
$[\hat{y}^{-}_{1},\hat{y}^{-}_{2},\ldots,\hat{y}^{-}_{K}]$. Then, these scores
are normalized using the softmax function given by

(15)
$p_{i}=\frac{\exp(\hat{y}^{+}_{i})}{\exp(\hat{y}^{+}_{i})+\sum_{j=1}^{K}\exp(\hat{y}^{-}_{i,j})}$

We frame the click prediction task as a pseudo $(K+1)$-way classification
problem. The loss function is the negative log-likelihood of all positive
samples $S$: $\mathcal{L}=-\sum_{i\in S}\log(p_{i})$. This method trains the
model to distinguish between clicked and non-clicked news items.

##  6\. Experiment

###  6.1. Experimental Settings

####  6.1.1. Datasets

We evaluated AWRS on three datasets: MIND-small (English) (Wu et al., 2020a),
Adressa one-week (Norwegian) (Gulla et al., 2017), and Nikkei (Japanese).
Nikkei Inc.222https://www.nikkei.com/, based in Japan, is a leading media
corporation renowned for its comprehensive coverage of business, economic, and
financial news. Table 1 summarizes the train, validation, and test splits used
in our study. For Nikkei, we analyzed 15,803 news articles, dividing user
click behavior into training (Jan 16-20, 2023), validation (Jan 21, 2023), and
testing (Jan 22, 2023) sets. The Adressa one-week dataset includes 22,136 news
articles, with user click behaviors split into training (Jan 1-5, 2017),
validation (Jan 6, 2017), and testing (Jan 7, 2017) sets. And the MIND-small
dataset comprises 93,698 news articles, with user click behaviors segmented
into training (Nov 9-13, 2019), validation (Nov 14, 2019), and testing (Nov
15, 2019) sets.

Table 1. Dataset Statistics Dataset | Train | Validation | Test
---|---|---|---
| Impr./Users | Impr./Users | Impr./Users
Adressa | 181,279 / 83,599 | 36,412 / 27,943 | 145,626 / 68,565
MIND-small | 124,229 / 45,214 | 29,498 / 19,703 | 70,938 / 48,593
Nikkei | 137,142 / 23,139 | 10,560 / 6,201 | 9,695 / 5,805

####  6.1.2. Baselines

We conducted a comparative analysis of the AWRS model against several state-
of-the-art (SOTA) baseline models to evaluate its performance. The models
compared include: (1) NRMS (Wu et al., 2019d), (2) NAML (Wu et al., 2019b),
(3) LSTUR (An et al., 2019), (4) TANR (Wu et al., 2019a), (5) SentiRec (Wu et
al., 2020b), (6) MINER (Li et al., 2022), (7) MINS (Wang and Lu, 2022), (8)
CenNewsRec (Qi et al., 2020), (9) MANNeR-CR333For the Nikkei dataset and
MANNeR-CR baseline model, we had to reduce the maximum number of tokens due to
memory constraints. (Iana et al., 2024), (10) PP-REC (Qi et al., 2021), and
(11) CAUM (Qi et al., 2022).

####  6.1.3. Evaluation Metrics

In accordance with prior research (Qi et al., 2022; Yang et al., 2023), we
evaluate model performance using Area under the ROC Curve (AUC), Mean
Reciprocal Rank (MRR), and normalized discounted cumulative gain (nDCG@5 and
nDCG@10).

Table 2. Performance comparisson for the Nikkei, MIND-small, and Adressa one-week datasets. The best results are bolded, and the second best are underlined. All the results reported here used $D=5$.  | Nikkei | MIND-small | Adressa one-week
---|---|---|---
MODEL | AUC | MRR | nDCG@5 | nDCG@10 | AUC | MRR | nDCG@5 | nDCG@10 | AUC | MRR | nDCG@5 | nDCG@10
NRMS | 0.5140 | 0.3110 | 0.2117 | 0.3069 | 0.5000 | 0.2731 | 0.2532 | 0.3198 | 0.5847 | 0.2102 | 0.1824 | 0.2894
NAML | 0.5011 | 0.3323 | 0.2324 | 0.3322 | 0.5000 | 0.2806 | 0.2606 | 0.3246 | 0.5000 | 0.2267 | 0.1984 | 0.2945
LSTUR | 0.4953 | 0.2878 | 0.1929 | 0.2883 | 0.5000 | 0.3000 | 0.2830 | 0.3468 | 0.5358 | 0.3091 | 0.2931 | 0.3576
TANR | 0.5007 | 0.3005 | 0.2059 | 0.3011 | 0.5224 | 0.2964 | 0.2773 | 0.3425 | 0.5359 | 0.2622 | 0.2491 | 0.3136
SentiRec | 0.5000 | 0.3051 | 0.2042 | 0.2991 | 0.5289 | 0.2773 | 0.2642 | 0.3281 | 0.5313 | 0.2087 | 0.1754 | 0.2831
MINER | 0.5439 | 0.3296 | 0.2342 | 0.3613 | 0.6141 | 0.2722 | 0.2591 | 0.2385 | 0.6012 | 0.2165 | 0.1908 | 0.3054
MINS | 0.5007 | 0.2723 | 0.1674 | 0.2677 | 0.4266 | 0.1995 | 0.1756 | 0.2385 | 0.7018 | 0.3511 | 0.3837 | 0.4703
CenNewsRec | 0.5249 | 0.2873 | 0.1679 | 0.2521 | 0.4837 | 0.1992 | 0.1770 | 0.2384 | 0.5980 | 0.3211 | 0.3194 | 0.3781
MANNeR-CR | 0.5000 | 0.3422 | 0.2422 | 0.3451 | 0.5917 | 0.3018 | 0.2826 | 0.3469 | 0.5000 | 0.2248 | 0.1741 | 0.2481
PP-REC | 0.4970 | 0.2731 | 0.1869 | 0.2808 | 0.5686 | 0.2912 | 0.2695 | 0.3301 | 0.6729 | 0.4335 | 0.4129 | 0.4714
CAUM | 0.5606 | 0.3278 | 0.2391 | 0.3411 | 0.5528 | 0.3133 | 0.2972 | 0.3615 | 0.6300 | 0.2476 | 0.2434 | 0.3014
AWRS | 0.6723 | 0.4973 | 0.4028 | 0.4781 | 0.5978 | 0.3267 | 0.3105 | 0.3722 | 0.7670 | 0.4826 | 0.5134 | 0.5627

####  6.1.4. Environment Configuration

During training, we used mixed precision and the Adam optimizer (Kingma and
Ba, 2017). The learning rates were 1e-5 for MIND-small and Nikkei, and 1e-6
for Adressa one-week. We used $k=4$ negative samples for all datasets. Each
model for the MIND-small was trained for 10 epochs and for Adressa one-week
and Nikkei 3 epochs, using a single NVIDIA A100 GPU, with implementations in
the newsreclib framework444Upon publication, we plan to release parts of the
code to the research community for further development and replication of our
results. (Iana et al., 2023). When calculating avoidance and user engagement
embedding, we measured it every 1 hour for the MIND-small dataset, every 5
hours for the Adressa one-week dataset, and every 2 hours for the Nikkei
dataset.

##  7\. Results

With the results at hand we comprehensively evaluated AWRS by answering the
following research questions:

  * •

RQ1 (Accuracy): Does the proposed AWRS method outperform existing baselines in
terms of accurately predicting user-clicked news articles?

  * •

RQ2 (Effect of Each Component): How does each component of our proposed
architecture contribute to the accuracy scores? Specifically, what is the
impact of adding the Avoidance-aware User Encoder and Avoidance-aware
Relevance Predictor components?

  * •

RQ3 (Effect of Varying $D$): To compute the user engagement embedding
($\bm{ue}$), we divided our avoidance region space ($av_{idx}$) and exposure
per impression ($epi_{idx}$) into distinct regions determined by the constant
$D$. How does accuracy vary when changing this constant?

  * •

RQ4 (Performance without PLMs): How does our model perform in the absence of
pretrained language models (PLMs)?

  * •

RQ5 (Limitations): What are some of the limitations of our model compared to
existing baselines?

###  7.1. RQ1 (Accuracy)

Table 2 presents a performance comparison between the baselines and our
proposed AWRS model. As shown in the table, our model demonstrates
improvements in nearly all key metrics, attributed to its enhanced awareness
of avoidance and exposure. For instance, if a user opts to read an article
about chess, a typically avoided topic, it suggests a strong interest in this
subject. Conversely, an article about the Oscars, which is generally less
avoided, might be clicked due to its current popularity rather than genuine
interest. It is crucial to note that such patterns may change over time. For
example, during a specific period involving controversial news about chess,
articles on chess might become less avoided, whereas Oscars articles might be
more avoided if they are far from the premiere date. Therefore, we argue that
avoidance and time are closely related. This understanding allows us to delve
deeper into our discussion for each dataset.

####  7.1.1. Nikkei

Our proposed AWRS model is the best-performing model for this dataset, with
the second-best performance varying across different metrics. The second
highest AUC was achieved by CAUM (Qi et al., 2022), the second best MRR and
nDCG@5 by MANNeR (Iana et al., 2024), and the second highest nDCG@10 by MINER
(Li et al., 2022). All these baselines have complex architectures for encoding
user behaviors, demonstrating their efficacy in predicting user preferences.
However, by incorporating user engagement embeddings with our Avoidance-aware
User Encoder alongside our Avoidance-aware Relevance Predictor as indicated in
figures 4 and 5, our model achieves a better understanding of user patterns.
This leads to an avoidance-aware contextualized assessment of each article,
enabling our model to extract more information by understanding the avoidance
of news articles.

####  7.1.2. MIND-small

For this dataset, our model outperforms others in terms of MRR, nDCG@5, and
nDCG@10. However, it performs slightly worse than MINER (Li et al., 2022) for
AUC. Generally, the second-best model was CAUM (Qi et al., 2022), for reasons
similar to those elaborated for Nikkei. We argue that by fully capturing the
avoidance behavior of news articles, our model better understands user
patterns.

####  7.1.3. Adressa one-week

We employed various baselines, each with different advantages and
disadvantages. The Adressa dataset contains click patterns for news articles
published long ago. Despite encompassing the initial weeks of 2017, the
dataset includes user interactions with very old news articles (e.g., a user
clicking on a news article from the 2000s during the early weeks of 2017).
Therefore, baselines such as PP-REC (Qi et al., 2021), which consider time as
a key metric, perform very well on this dataset. However, our AWRS model not
only incorporates time as a factor through the Time2Vec (Kazemi et al., 2019)
module, but also considers avoidance, enhancing the contextualization for the
model’s decision-making process.

###  7.2. RQ2, RQ3 and RQ4 (Ablation Studies)

We aim to address research questions RQ2, RQ3, and RQ4 by exposing an ablation
study, as demonstrated in the following sections.

Table 3. Impact of the Avoidance-aware User Encoder and the Avoidance-aware Relevance Predictor modules on AWRS. Dataset | AWRS Model | AUC | MRR | nDCG@5 | nDCG@10
---|---|---|---|---|---
MIND-small | only rel | 0.5818 | 0.3244 | 0.3083 | 0.3690
only avoid | 0.5838 | 0.3077 | 0.2884 | 0.3511
avoid + rel | 0.5978 | 0.3267 | 0.3105 | 0.3722
Adressa one-week | only rel | 0.6354 | 0.3621 | 0.3948 | 0.4526
only avoid | 0.5423 | 0.2564 | 0.2204 | 0.2499
avoid + rel | 0.7670 | 0.4826 | 0.5134 | 0.5627
Nikkei | only rel | 0.6362 | 0.4638 | 0.3701 | 0.4534
only avoid | 0.6522 | 0.4889 | 0.3784 | 0.4568
avoid + rel | 0.6723 | 0.4973 | 0.3784 | 0.4781
Table 4. AWRS scores for different datasets with varying $D$ values. Dataset | Metrics | D = 5 | D = 7 | D = 10 | D = 15 | D = 20
---|---|---|---|---|---|---
MIND-small | AUC | 0.5978 | 0.6213 | 0.6430 | 0.6479 | 0.6494
MRR | 0.3267 | 0.3214 |  0.3499 | 0.3481 | 0.3409
nDCG@5 | 0.3105 | 0.3049 | 0.3285 | 0.3289 | 0.3226
nDCG@10 | 0.3722 | 0.3684 | 0.3914 | 0.3923 | 0.3864
Adressa one-week | AUC | 0.7670 | 0.6561 | 0.6997 | 0.6443 | 0.7652
MRR | 0.4826 | 0.4310 | 0.4598 | 0.4109 | 0.5055
nDCG@5 | 0.5134 | 0.4172 | 0.4799 | 0.3869 | 0.5300
nDCG@10 | 0.5627 | 0.4670 | 0.5062 | 0.4459 | 0.6033
Nikkei | AUC | 0.6723 | 0.6466 | 0.6252 | 0.6283 | 0.6424
MRR | 0.4973 | 0.4804 | 0.4168 | 0.4497 | 0.4494
nDCG@5 | 0.4028 | 0.3732 | 0.3178 | 0.3405 | 0.3435
nDCG@10 | 0.4781 | 0.4531 | 0.4096 | 0.4279 | 0.4330
Table 5. Performance comparison for the MIND-small dataset. All the results reported here used $D=5$ and GloVe. Model | AUC | MRR | nDCG@5 | nDCG@10
---|---|---|---|---
SentiRec | 0.5425 | 0.3060 | 0.2826 | 0.3486
TANR | 0.5389 | 0.3240 | 0.3039 | 0.3683
LSTURINI | 0.5482 | 0.3194 | 0.2995 | 0.3644
NAML | 0.5010 | 0.3350 | 0.3176 | 0.3802
NRMS | 0.5571 | 0.2924 | 0.2730 | 0.3370
GLORY | 0.6603 | 0.2978 | 0.3218 | 0.3825
CAUM | 0.6200 | 0.3414 | 0.3207 | 0.3854
AWRS | 0.6613 | 0.3425 | 0.3249 | 0.3896

####  7.2.1. RQ2 (Effect of Each Component)

To assess the influence of each component in our architecture, specifically
the Avoidance-aware User Encoder and the Avoidance-aware Relevance Predictor,
we conducted experiments by removing each component separately and calculating
the MRR, nDCG@5, and nDCG@10 scores for the MIND-small, Adressa one-week, and
Nikkei datasets. As shown in Table 3, it is evident that each component
individually underperforms compared to when both components are used together.
We argue that this is because the Avoidance-aware User Encoder is responsible
for understanding how user behavior historically patterns in terms of
avoidance. It analyzes how the news articles that the user has clicked on were
avoided. For example, is the user more interested in less avoided or more
avoided news articles? Conversely, the Avoidance-aware Relevance Predictor
gains a better understanding of the temporal and global influence of
avoidance. Thus the impact of avoidance on the decision of which news article
is more relevant can vary depending on the timing of the recommendation. This
dual approach enhances the model’s ability to accurately predict user
preferences.

####  7.2.2. RQ3 (Effect of Varying $D$)

To investigate the impact of adjusting the parameter $D$ on the AWRS model, we
conducted experiments with values $D=5,7,10,15,$ and $20$. As detailed in
Table 4, varying $D$ has a notable effect on model performance, albeit with
nuanced outcomes. Notably, optimal scores for the Nikkei dataset are observed
at $D=5$ and $D=7$, whereas for the Adressa one-week dataset, peak performance
is achieved at $D=20$. In the case of the MIND-small dataset, the highest AUC
score emerges at $D=20$, with MRR peaking at $D=10$, and both nDCG@5 and
nDCG@10 showing optimal results at $D=15$. Our findings suggest that adjusting
$D$ influences the embedding space’s complexity, potentially leading to
overfitting or underfitting of avoidance behavior in news article
recommendations. Thus, our grid search aimed to pinpoint the most effective
$D$ values tailored to each dataset and metric.

####  7.2.3. RQ4 (Performance without PLMs)

We investigated how excluding pretrained language models (PLMs) in favor of
GloVe embeddings (Pennington et al., 2014) would affect our model’s
performance. For evaluation, we utilized the same datasplit from the MIND-
small dataset described in Section 6.1.1. In this comparison, we included
several baseline models for a more comprehensive evaluation: we employed GLORY
(Yang et al., 2023), SentiRec (Wu et al., 2020b), CAUM (Qi et al., 2022), TANR
(Wu et al., 2019a), NRMS (Wu et al., 2019d), NAML (Wu et al., 2019b), and
LSTUR (An et al., 2019). Table 5 demonstrates that our model, AWRS, exhibits
superior performance compared to such models. The reduced performance gains
compared to PLMs are due to PLMs’ advanced capability in capturing more
detailed contextual information from news articles.

###  7.3. RQ5 (Limitations)

From our exploration of current methodologies in news article recommendation
systems, we identify two notable approaches. Firstly, there are methods like
PP-REC (Qi et al., 2021) that leverage contextual information such as time to
enhance recommendations. Secondly, methods such as those discussed in GLORY
(Yang et al., 2023) utilize global information derived from static global
graphs. Our proposed model aligns with the former category, leveraging time-
varying avoidance data of news articles to enhance user recommendations.
However, like all models, our approach has its drawbacks. For instance, one
limitation of our model is its increased memory requirements due to the need
for extensive per-unit-time information such as avoidance ($Av$) and Exposure
Per Impression ($EPI$). Despite this increased memory demand, we did not
observe a corresponding increase in computational training time for our
models.

##  8\. Conclusion

In this study, we emphasize the importance of avoidance awareness. Our
research highlights that integrating avoidance behaviors can significantly
enhance the predictive accuracy of recommender systems. Essentially, analyzing
the timely avoidance behavior of news articles provides deeper insights into
user preferences. This understanding enables us to incorporate contextual
information into recommender systems, thereby refining our understanding of
user preferences. By integrating our Avoidance-aware Relevance Predictor and
Avoidance-aware User Encoder modules, we have developed a novel model named
AWRS. This model demonstrates robust performance across three real-world
datasets, showing improvements in various benchmark metrics. Our approach not
only shows enhanced accuracy but also establishes a foundation for future
research into understanding user behavior through avoidance.

###### Acknowledgements.

This work is partially supported by ”Joint Usage/Research Center for
Interdisciplinary Large-scale Information Infrastructures (JHPCN)” in Japan
(Project ID: jh241004), JSPS KAKENHI Grant Number 23K28098, and the
Monbukagakusho: MEXT (Ministry of Education, Culture, Sports, Science and
Technology - Japan) scholarship.

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$\displaystyle\sum_{\lambda\in\Lambda^{\perp}}\widehat{B_{N}}(\xi+\lambda)\overline{\widehat{B^{\prime}_{N}}(\xi+\lambda)}=\sum_{\lambda\in\Lambda^{\perp}}\Big{(}\prod{j=1}^{N}\widehat{(\varphi_{j}\chi_{\mho})}(\xi+\lambda)\overline{\widehat{(\psi_{j}\chi_{\mho})}(\xi+\lambda)}\Big{)}.$

By the Corollary 3.8, the subspace orthogonality and duality for the system
$\mathcal{E}^{\Lambda}(B^{\prime}_{N})$ associated with
$\mathcal{E}^{\Lambda}(B_{N})$ can be described by assigning the values on the
above expression either 0 or 1, respectively. Due to the importance of splines
in numerous applications, both the Euclidean and the LCA group setups conduct
in-depth research on them for the Gabor and Wavelet systems [10, 25].

###  6.4. For other setup

In the scenario of $p$-adic numbers $\mathbb{Q}_{p}$ and Heisenberg groups
(semidirect product LCA groups), we may examine our results for the subspace
orthogonality and duality of Bessel families. Considering that the Zak
transform plays a significant role in describing our results, such as,
Theorems 3.3, 3.5, 3.10, and 5.1, Corollary 3.8, we describe below the Zak
transform in these setups, see [3, 22] and references therein.

####  6.4.1. $p$-adic numbers $\mathbb{Q}_{p}$

For a prime number $p,$ the locally compact field of $p$-adic numbers
$\mathbb{Q}_{p}$ is
$\\{\sum_{j=m}^{\infty}c_{j}p^{j}:m\in\mathbb{Z},c_{j}\in\\{0,1,\dots,p-1\\}\\}$
in which the associated the $p$-adic norm is $|x|_{p}=p^{-m}$ for
$x=\sum_{j=m}^{\infty}c_{j}p^{j},c_{m}\neq 0.$ Indeed, it is an LCA group. The
$p$-adic integers $\mathbb{Z}_{p}$ defined by
$\\{\mathbb{Z}_{p}:=\\{x\in\mathbb{Q}_{p}:|x|_{p}\leq 1\\}\\}$ is a compact
open subgroup of $\mathbb{Q}_{p}.$ In this setup, the Zak transform is given
by $\mathcal{Z}f(x,y)=\int_{\mathbb{Z}_{p}}f(y+\xi)e^{-2\pi ix\xi}\
d{\mu_{\mathbb{Z}_{p}}}(\xi)$ for $f\in L^{1}(\mathbb{Q}_{p})\cap
L^{2}(\mathbb{Q}_{p}),$ and $x,y\in\Omega$ which can be extended from
$L^{2}(\mathbb{Q}_{p})$ to $\ell^{2}(\Omega\times\Omega),$ where $\Omega$ is
the fundamental domain.

####  6.4.2. Semidirect product of LCA groups

For LCA groups $\Gamma_{1}$ and $\Gamma_{2},$ let us consider a locally
compact group $\mathscr{G}_{\tau}=\Gamma_{1}\rtimes_{\tau}\Gamma_{2}$ by the
semidirect product of $\Gamma_{1}$ and $\Gamma_{2}$ with the binary operation
$(\gamma_{1},\gamma_{2}).(\gamma_{1}^{\prime},\gamma_{2}^{\prime})=(\gamma_{1}\gamma_{1}^{\prime},\gamma_{2}\tau_{\gamma_{1}}(\gamma_{2})),$
where $\gamma_{1}\mapsto\tau_{\gamma_{1}}$ is a group homomorphism from
$\Gamma_{1}$ to the set of all automorphisms on $\Gamma_{2},$ such that
$(\gamma_{1},\gamma_{2})\mapsto\tau_{\gamma_{1}}(\gamma_{2})$ from
$\Gamma_{1}\times\Gamma_{2}\mapsto\Gamma_{2}$ is continuous. Then the Zak
transform $\mathcal{Z}$ is defined by
$\mathcal{Z}f(\gamma_{1},w)=\int_{\Gamma_{2}}f(\gamma_{1},\gamma_{2})\overline{w(\gamma_{2})}\delta(\gamma_{1})\
d{\mu_{\Gamma_{2}}}(\gamma_{2})$ for $f\in
L^{1}(\Gamma_{1}\times_{\tau}\Gamma_{2}),$
$(\gamma_{1},\omega)\in\Gamma_{1}\rtimes_{\widehat{\tau}}\widehat{\Gamma_{2}},$
where $\delta$ is a positive homeomorphism on $\Gamma_{1}$ given by
$d{\mu_{\Gamma_{2}}}(\gamma_{2})=\delta(\gamma_{1})d{\mu_{\Gamma_{2}}}({\tau_{\gamma_{1}}}(\gamma_{2})).$
It can be extended from $L^{2}(\Gamma_{1}\times_{\tau}\Gamma_{2})$ to
$L^{2}(\Gamma_{1}\rtimes_{\widehat{\tau}}\widehat{\Gamma_{2}}).$

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