# Be Causal: De-biasing Social Network Confounding in Recommendation

Qian Li<EMAIL_ADDRESS>University of Technology, SydneyP.O. Box
123BroadwayNSW 2007Australia ,  Xiangmeng Wang
<EMAIL_ADDRESS>University of Technology, SydneyP.O. Box
123BroadwayNSW 2007Australia and  Guandong Xu<EMAIL_ADDRESS>University of Technology, SydneyP.O. Box 123BroadwayNSW 2007Australia

(2018)

###### Abstract.

In recommendation systems, the existence of the missing-not-at-random (MNAR)
problem results in the selection bias issue, degrading the recommendation
performance ultimately. A common practice to address MNAR is to treat missing
entries from the so-called “exposure” perspective, i.e., modeling how an item
is exposed (provided) to a user. Most of the existing approaches use heuristic
models or re-weighting strategy on observed ratings to mimic the missing-at-
random setting. However, little research has been done to reveal how the
ratings are missing from a causal perspective. To bridge the gap, we propose
an unbiased and robust method called DENC (_De-bias Network Confounding in
Recommendation_) inspired by confounder analysis in causal inference. In
general, DENC provides a causal analysis on MNAR from both the inherent
factors (e.g., latent user or item factors) and auxiliary network’s
perspective. Particularly, the proposed exposure model in DENC can control the
social network confounder meanwhile preserves the observed exposure
information. We also develop a deconfounding model through the balanced
representation learning to retain the primary user and item features, which
enables DENC generalize well on the rating prediction. Extensive experiments
on three datasets validate that our proposed model outperforms the state-of-
the-art baselines.

Recommendation; Missing-Not-At-Random; Causal Inference; Bias; Propensity

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††conference: Woodstock ’18: ACM Symposium on Neural
Gaze Detection; June 03–05, 2018; Woodstock, NY††booktitle: Woodstock ’18: ACM
Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Information systems††ccs:
Collaborative filtering††ccs: Computer systems organization Robotics††ccs:
Networks Network

##  1\. Introduction

Recommender systems aim to handle information explosion meanwhile to meet
users’ personalized interests, which have received extensive attention from
both research communities and industries. The power of a Recommender system
highly relies on whether the observed user feedback on items “correctly”
reflects the users’ preference or not. However, such feedback data often
contains only a small portion of observed feedback (e.g., explicit ratings),
leaving a large number of missing ratings to be predicted. To handle the
partially observed feedback, a common assumption for model building is that
the feedback is missing at random (MAR), i.e., the probability of a rating to
be missing is independent of the value. When the observed data follows the
MAR, using only the observed data via statistical analysis methods can yield
“correct” prediction without introducing bias (Marlin and Zemel, 2009; Lim et
al., 2015). However, this MAR assumption usually does not hold in reality and
the missing pattern exhibits _missing not at random_ (MNAR) phenomenon.
Generally, MNAR is related to selection bias. For instance, in movie
recommendation, instead of randomly choosing movies to watch, users are prone
to those that are highly recommended, while in advertisement recommendation,
whether an advertisement is presented to a user is purely subject to the
advertiser’s provision, rather than at random. In these scenarios, the missing
pattern of data mainly depends on whether the users are exposed to the items,
and consequently, the ratings in fact are _missing not at random_ (MNAR) (He
et al., 2016). These findings shed light on the origination of selection bias
from MNAR (Sportisse et al., 2020). Therefore the selection bias cannot be
ignored in practice and it has to be modeled properly in order for reliable
recommendation prediction. How to model the missing data mechanism and debias
the rating performance forms up the main motivation of this research.

Existing MNAR-aware Methods

There are abundant methods for addressing the MNAR problem on the implicit or
explicit feedback. For implicit feedback, traditional methods (Hu et al.,
2008) take the uniformity assumption that assigns a uniform weight to down-
weight the missing data, assuming that each missing entry is equally likely to
be negative feedback. This is a strong assumption and limits models’
flexibility for real applications. Recently, researchers tackle MNAR data
directly through simulating the generation of the missing pattern under
different heuristics (Hernández-Lobato et al., 2014). Of these works,
probabilistic models are presented as a proxy to relate missing feedback to
various factors, e.g., item features. For explicit feedback, a widely adopted
mechanism is to exploit the dependencies between rating missingness and the
potential ratings (e.g., 1-5 star ratings) (Koren and Bell, [n. d.]). That is,
high ratings are less likely to be missing compared to items with low ratings.
However, these paradigm methods involve heuristic alterations to the data,
which are neither empirically verified nor theoretically proven (Saito, 2020).

A couple of methods have recently been studied for addressing MNAR (Hernández-
Lobato et al., 2014; Liang et al., 2016; Schnabel et al., 2016) by treating
missing entries from the so-called “exposure” perspective, i.e., indicating
whether or not an item is exposed (provided) to a user. For example, ExpoMF
resorts modeling the probability of _exposure_ (Hernández-Lobato et al.,
2014), and up-weighting the loss of rating prediction with high _exposure_
probability. However, ExpoMF can lead to a poor prediction accuracy for rare
items when compared with popular items. Likewise, recent works (Liang et al.,
2016; Schnabel et al., 2016) resort to _propensity score_ to model _exposure_.
The _propensity score_ introduced in causal inference indicates _the
probability that a subject receiving the treatment or action_. Exposing a user
to an item in a recommendation system is analogous to exposing a subject to a
treatment. Accordingly, they adopt _propensity score_ to model the _exposure_
probability and re-weight the prediction error for each observed rating with
the inverse _propensity score_. The ultimate goal is to calibrate the MNAR
feedbacks into missing-at-random ones that can be used to guide unbiased
rating prediction.

Whilst the state-of-the-art propensity-based methods are validated to
alleviate the MNAR problem for recommendation somehow, they still suffer from
several major drawbacks: 1) they merely exploit the user/item latent vectors
from the ratings for mitigating MNAR, but fail to disentangle different causes
for MNAR from a causal perspective; 2) technically, they largely rely on
propensity score estimation to mitigate MNAR problem; the performance is
sensitive to the choice of propensity estimator (Wang et al., 2019), which is
notoriously difficult to tune.

Figure 1. The causal view for MNAR problem: _treatment_ and _outcome_ are
terms in the theory of causal inference, which denote an action taken
(e.g.,_exposure_) and its result (e.g., _rating_), respectively. The
_confounder_ (e.g., _social network_) is the common cause of treatment and
outcome.

The proposed approach

To overcome these obstacles, in contrast, we aim to address the fundamental
MNAR issue in recommendation from a novel causal inference perspective, to
attain a robust and unbiased rating prediction model. From a causal
perspective, we argue that the selection bias (i.e., MNAR) in the
recommendation system is attributed to the presence of _confounders_. As
explained in Figure 1, _confounders_ are factors (or variables) that affect
both the treatment assignments (exposure) and the outcomes (rating). For
example, friendships (or social network) can influence both users’ choice of
movie watching and their further ratings. Users who choose to watch the movie
are more likely to rate than those who do not. So, _the social network is
indeed a confounding factor that affects which movie the user is exposed to
and how the user rates the movie_. The confounding factor results in a
_distribution discrepancy between the partially observed ratings and the
complete ratings_ as shown in Figure 2. Without considering the distribution
discrepancy, the rating model trained on the observed ratings fails to
generalize well on the unobserved ratings. With this fact in mind, our idea is
to analyze the confounder effect of social networks on rating and exposure,
and in turn, fundamentally alleviate the MNAR problem to predict valid
ratings.

Figure 2. The training space of conventional recommendation models is the
observed rating space $\mathcal{O}$, whereas the inference space is the entire
exposure space $\mathcal{D}$. The discrepancy of data distribution between
$\mathcal{O}$ and $\mathcal{D}$ leads to selection bias in conventional
recommendation models.

In particular, we attempt to study the MNAR problem in recommendation from a
causal view and propose an unbiased and robust method called DENC (_De-bias
Network Confounding in Recommendation_). To sufficiently consider the
selection bias in MNAR, we model the underlying factors (i.e., inherent user-
item information and social network) that can generate observed ratings. In
light of this, as shown in Figure 4, we construct a causal graph based
recommendation framework by disentangling three determinants for the ratings,
i.e., _inherent factors_ , _confounder_ and _exposure_. Each determinant
accordingly corresponds to one of three specific components in DENC:
_deconfonder model_ , _social network confounder_ and _exposure model_ , all
of which jointly determine the rating outcome.

In summary, the key contributions of this research are as follows:

  * •

Fundamentally different from previous works, DENC is the first method for the
unbiased rating prediction through disentangling determinants of selection
bias from a causal view.

  * •

The proposed _exposure model_ is capable of revealing the exposure assignment
and accounting for the confounder factors derived from the _social network
confounder_ , which thus remedies selection bias in a principled manner.

  * •

We develop a _deconfonder model_ via the balanced representation learning that
embeds inherent factors independent of the exposure, therefore mitigating the
distribution discrepancy between the observed rating and inference space.

  * •

We conduct extensive experiments to show that our DENC method outperforms
state-of-the-art methods. The generalization ability of our DENC is also
validated by verifying different degrees of confounders.

##  2\. Related Work

###  2.1. MNAR-aware Methods

####  2.1.1. Traditional Heuristic Models

Early works on explicit feedback formulate a recommendation a rating
prediction problem in which the large volume of unobserved ratings (i.e.,
missing data) is assumed to be extraneous for user preference (Hu et al.,
2008). Following this unreliable assumption, numerous recommenders have been
developed including basic matrix factorization based-recommenders (Rendle and
Schmidt-Thieme, 2008) and sophisticated ones such as SVD++ (Koren and Bell,
[n. d.]). As statistical analysis with missing data techniques, especially
MNAR proposition, find widespread applications, there is much interest in
understanding its impacts on the recommendation system. Previous research has
shown that for explicit-feedback recommenders, users’ ratings are MNAR (Marlin
and Zemel, 2009). Marlin and Zemel (Marlin and Zemel, 2009) first study the
effect of violating MNAR assumption in recommendation methods; they propose
statistical models to address MNAR problem of missing ratings based on the
heuristic that users are more likely to supply ratings for items that they do
like. Another work of (Hernández-Lobato et al., 2014) also has focused on
addressing MNAR problem; they propose a probabilistic matrix factorization
model (ExpoMF) for collaborative filtering that learns from observation data.
However, these heuristic paradigm methods are neither empirically verified nor
theoretically proven (Schnabel et al., 2016; Liang et al., 2016).

####  2.1.2. Propensity-based Model

The basic idea of propensity scoring methods is to turn the outcomes of an
observational study into a pseudo-randomized trial by re-weighting samples,
similarly to importance sampling. Typically, using Inverse Propensity
Weighting (IPW) estimator, Liang (Liang et al., 2016) proposes a framework
consisted of one exposure model and one preference model. The exposure model
is estimated by a Poisson factorization, then preference model is fit with
weighted click data, where each click is weighted by the inverse of exposure
and be used to alleviate popularity bias. Based on Self Normalized Inverse
Propensity Scoring (SNIPS) estimator, the model in (Schnabel et al., 2016) are
developed either directly through observed ratings of a missing-completely-at-
random sample estimated by SNIPS or indirectly through user and item
covariates. These works re-weight the observational click data as though it
came from an “experiment” where users are randomly shown items. Thus, the
measurement is still adopting re-weighting strategies to mimic the missing-
completely-at-random like most of the heuristic models do (Yang et al., 2018).
Besides, these works are sensitive to the choice of propensity score
estimators (Wang et al., 2019). In contrast, our work relies solely on the
observed ratings: we do not require ratings from a gold-standard randomized
exposure estimation and nor do we use external covariates; moreover, we
consider another important bias in the recommendation scenario, namely, social
counfounding bias.

###  2.2. Social Network-based Methods

The effectiveness of social networks has been proved by a vast amount of
social recommenders. Purushotham (Purushotham et al., 2012) has explored how
traditional factorization methods can exploit network connections; this brings
the latent preferences of connected users closer to each other, reflecting
that friends have similar tastes. Other research has included social
information directly into various collaborative filtering methods. TrustMF
(Yang et al., 2016) adopts collaborative filtering to map users into low-
dimensional latent feature spaces in terms of their trust relationship; the
remarkable performance of the proposed model reflects individuals among a
social network will affect each other while reviewing items. SocialMF (Jamali
and Ester, 2010) incorporates trust propagation into the matrix factorization
model, which assumes the factors of every user are dependent on the factor
vectors of his/her direct neighbors in the social network. However, despite
the remarkable contribution of social network information in various
recommendation methods, it has not been utilized in controlling for
confounding bias of causal inference-based recommenders yet.

##  3\. DENC Method

###  3.1. Notations

We first give some preliminaries of our method and used notation. Suppose we
have $m\times n$ rating matrix $Y\in\mathbb{R}^{m\times n}=[\dot{y}_{ui}]$
describing the numerical rating of $m$ users on $n$ items. Let
$U=\\{u_{1},u_{2},...,u_{n}\\}$ and $I=\\{i_{1},i_{2},...,i_{m}\\}$ be the set
of users and items respectively. For each user-item pair, we use $a_{ui}$ to
indicate whether user $u$ has been exposed to item $i$ and
$a_{ui}\in\\{0,1\\}$. We use $y_{ui}$ to represent the rating given by $u$ to
item $i$.

###  3.2. A Causal Inference Perspective

Viewing recommendation from a causal inference perspective, we argue that
exposing a user to an item in recommendation is an intervention analogous to
exposing a patient to a treatment in a medical study. Following the potential
outcome framework in causal inference (Rubin, 1974), we reformulate the rating
prediction as follows.

###### Problem 1 (Causal View for Recommendation).

For every user-item pair with a binary exposure $a_{ui}\in\\{0,1\\}$, there
are two potential rating outcomes $y_{ui}(0)$ and $y_{ui}(1)$. We aim to
_estimating the ratings had all movies been seen by all users_ , i.e.,
estimate $y_{ui}(1)$ (i.e., $y_{ui}$) for all $u$ and $i$.

Figure 3. The causal graph for recommendation.

As we can only observe the outcome $y_{ui}(1)$ when the user $u$ is exposed by
the item $i$, i.e., $a_{ui}=1$, we target at the problem that _what would
happen to the unobserved rating $y_{ui}$ if we set exposure variable by
setting $a_{ui}=1$_. In our settings, the confounder derived from the social
network among users are denoted as a common cause that affects the exposure
$a_{ui}$ and outcome $y_{ui}$. We aim to disentangle the underlying factors in
observation ratings and social networks as shown in Figure 3. The intuition
behind Figure 3 is that the observed rating outcomes are generated as a result
of both inherent and confounding factors. The inherent factors refer to the
user preferences and inherent item properties, and auxiliary factors are the
confounding factors from the social network. By disentangling determinants
that cause the observed ratings, we can account for effects separately from
the selection bias of confounders and the exposure, which ensures to attain an
unbiased rating estimator with superior generalization ability.

Followed the causal graph in Figure 3, we now design our DENC method
incorporates three determinants in Figure 4. Each component accordingly
corresponds to one of three specific determinants: _social network confounder_
, _exposure model_ and _deconfonder model_ , which jointly determine the
rating outcome.

Figure 4. Our DENC method consists of four components: _Social network
confounder_ , _exposure model_ , _deconfonder model_ and _rating model_.

###  3.3. Exposure Model

To cope with the selection bias caused by users or the external social
relations, we build on the causal inference theory and propose an effective
exposure model. Guided by the treatment assignment mechanism in causal
inference, we propose a novel exposure model that computes the probability of
exposure variable specific to the user-item pair. This model is beneficial to
understand the generation of the _Missing Not At Random_ (MNAR) patterns in
ratings, which thus remedies selection biases in a principled manner. For
example, user A goes to watch the movie because of his friend’s strong
recommendation. Thus, we propose to mitigate the selection bias by exploiting
the network connectivity information that indicating _to which extent the
exposure for a user will be affected by its neighbors_.

####  3.3.1. Social Network Confounder

To control the selection bias arisen from the external social network, we
propose a confounder representation model that quantifies the common biased
factors affecting both the exposure and rating outcome.

We now discuss the method of choosing and learning exposure. Let $G$ present
the social relationships among users $U$, where an edge denotes there is a
friend relationship between users. We resort to node2vec (Grover and Leskovec,
2016) method and learn network embedding from diverse connectivity provided by
the social network. More details about node2vec method can be found in Section
A.4 in the appendix. To mine the deep social structure from $G$, for every
source user $u$, node2vec generates the network neighborhoods $N_{s}(u)\subset
G$ of node $u$ through a sampling strategy to explore its neighborhoods in a
breadth-first sampling as well as a depth-first sampling manner. The
representation $Z_{u}$ for user $u$ can be learned by minimizing the negative
likelihood of preserving network neighborhoods $N_{s}(u)$:

(1) $\begin{split}\mathcal{L}_{z}&=-{\sum_{u\in G}{\log P(N_{s}(u)|Z_{u})}}\\\
&=\sum_{u\in G}\left[\log\sum_{v\in G}\exp(Z_{v}\cdot Z_{u})-\sum_{u_{i}\in
N_{s}(u)}Z_{u_{i}}\cdot Z_{u}\right]\end{split}$

The final output $Z_{u}\in\mathbb{R}^{d}$ sufficiently explores diverse
neighborhoods of each user, which thus represents to what extent the exposure
for a user is influenced by his friends in graph $G$.

####  3.3.2. Exposure Assignment Learning

The exposure under the recommendation scenario is not randomly assigned. Users
in social networks often express their own preferences over the social
network, which therefore will affect their friends’ exposure policies. In this
section, to characterize the _Missing Not At Random_ (MNAR) pattern in
ratings, we resort to causal inference (Pearl, 2009) to build the exposure
mechanism influenced by social networks.

To begin with, we are interested in the binary exposure $a_{ui}$ that defines
whether the item $i$ is exposed ($a_{ui}=1$) or unexposed ($a_{ui}=1$) to user
$u$, i.e., $a_{ui}=1$. Based on the informative confounder learned from social
network, we propose the notation of _propensity_ to capture the exposure from
the causal inference language.

###### Definition 0 (Propensity).

Given an observed rating $y_{ui}\in\text{rating}$ and confounder $Z_{u}$ in
(1), the propensity of the corresponding exposure for user–item pair $(u,i)$
is defined as

(2) $\pi(a_{ui};Z_{u})=P(a_{ui}=1|y_{ui}\in\text{rating};Z_{u})$

In view of the foregoing, we model the exposure mechanism by the probability
of $a_{ui}$ being assigned to 0 or 1.

(3) $\displaystyle P(a_{ui})$
$\displaystyle=\prod_{u,i}P\left(a_{ui}\right)=\prod_{(u,i)\in\mathcal{O}}P\left(a_{ui}=1\right)\prod_{(u,i)\notin\mathcal{O}}P\left(a_{ui}=?\right)$

where $\mathcal{O}$ is an index set for the observed ratings. The case of
$a_{ui}=1$ can result in an observed rating or unobserved rating: 1) for the
observed rating represented by $y_{ui}\in\text{rating}$, we definitely know
the item $i$ is exposed, i.e., $a_{ui}=1$; 2) an unobserved rating
$y_{ui}\notin\text{rating}$ may represent a negative feedback (i.e., the user
is not reluctant to rating the item) on the exposed item $a_{ui}=1$. In light
of this, based on (2), we have

(4)
$\begin{split}P(a_{ui}=1)&=P(a_{ui}=1,y_{ui}\in\text{rating})+P(a_{ui}=1,y_{ui}\notin\text{rating})\\\
&=\pi(a_{ui};Z_{u})P(y_{ui}\in\text{rating})+W_{ui}P(y_{ui}\notin\text{rating})\end{split}$

where $W_{ui}=P(a_{ui}=1|y_{ui}\notin\text{rating})$. The exposure $a_{ui}$
that is unknown follows the distributions as

(5) $\begin{split}P(a_{ui}=?)=1-P(a_{ui}=1)\end{split}$

By substituting Eq. (4) and Eq. (5) for Eq. (3), we attain the exposure
assignment for the overall rating data as

(6)
$P(a_{ui})=\prod_{(u,i)\in\mathcal{O}}\pi(a_{ui};Z_{u})\prod_{(u,i)\notin\mathcal{O}}\left(1-W_{ui}\right)$

Inspired by (Pan et al., 2008), we assume uniform scheme for $W_{ui}$ when no
side information is available. According to most causal inference methods
(Shalit et al., 2017; Pearl, 2009), a widely-adopted parameterization for
$\pi(a_{ui};Z_{u})$ is a logistic regression network parameterized by
$\Theta=\\{W_{0},b_{0}\\}$, i.e.,

(7)
$\pi(a_{ui};Z_{u},\Theta)=\mathbb{I}_{y\in\text{rating}}\cdot\left[1+e^{-(2a_{ui}-1)\left(Z_{u}^{\top}\cdot
W_{0}+b_{0}\right)}\right]^{-1}$

Based on Eq. (7), the overall exposure $P(a_{ui})$ in Eq. (6) can be written
as the function of parameters $\Theta=\\{W_{0},b_{0}\\}$ and $Z_{u}$, i.e.,

(8) $\begin{split}\mathcal{L}_{a}=\sum_{u,i}-\log
P(a_{ui};Z_{u},\Theta)\end{split}$

where social network confounder $Z_{u}$ is learned by the pre-trained node2vec
algorithm. Similar to supervised learning, $\Theta$ can be optimized through
minimization of the negative log-likelihood.

###  3.4. Deconfounder Model

Traditional recommendation learns the latent factor representations for user
and item by minimizing errors on the observed ratings, e.g., matrix
factorization. Due to the existence of selection bias, such a learned
representation may not necessarily minimize the errors on the unobserved
rating prediction. Inspired by (Shalit et al., 2017), we propose to learn a
balanced representation that is independent of exposure assignment such that
it represents inherent or invariant features in terms of users and items. The
invariant features must also lie in the inference space shown in Figure 2,
which can be used to consistently infer unknown ratings using observed
ratings. This makes sense in theory: if the learned representation is hard to
distinguish across different exposure settings, it represents invariant
features related to users and items.

According to Figure 3, we can define two latent vectors
$U\in\mathbb{R}^{k_{d}}$ and $I\in\mathbb{R}^{k_{d}}$ to represent the
inherent factor of a user and a item, respectively. Recall that different
values for $W_{ui}$ in Eq. (6) can generate different exposure assignments for
the observed rating data. Following this intuition, we construct two different
exposure assignments $a$ and $\hat{a}$ corresponding two settings of $W_{ui}$.
Accordingly, $\Phi_{(a)}$ and $\Phi_{(\hat{a})}$ are defined to include
inherent factors of users and items, i.e.,
$\Phi_{(a)}=\left[U_{1}^{(a)},\cdots,U_{M}^{(a)},I_{1}^{(a)},\cdots,I_{M}^{(a)}\right]\in\mathbb{R}^{k_{d}\times
2M}$,
$\Phi_{(\hat{a})}=\left[U_{1}^{(\hat{a})},\cdots,U_{M}^{(\hat{a})},I_{1}^{(\hat{a})},\cdots,I_{M}^{(\hat{a})}\right]\in\mathbb{R}^{k_{d}\times
2M}$. Figure 3 also indicates that the inherent factors of user and item would
keep unchanged even if the exposure variable is altered from 0 to 1, and vice
versa. That means $U\in\mathbb{R}^{k_{d}}$ and $I\in\mathbb{R}^{k_{d}}$ should
be independent of the exposure assignment, i.e., $U^{(a)}\perp\\!\\!\\!\perp
U^{(\hat{a})}$ or $I^{(a)}\perp\\!\\!\\!\perp I^{(\hat{a})}$. Accordingly,
minimizing the discrepancy between $\Phi_{(a)}$ and $\Phi_{(\hat{a})}$ ensures
that the learned factors embeds no information about the exposure variable and
thus reduce selection bias. The penalty term for such a discrepancy is defined
as

(9)
$\begin{split}\mathcal{L}_{d}=\text{disc}\left(\Phi_{(\hat{a})},\Phi_{(a)}\right)\\\
\end{split}$

Inspired by (Müller, 1997), we employ _Integral Probability Metric_ (IPM) to
estimate the discrepancy between $\Phi_{(\hat{a})}$ and $\Phi_{(a)}$.
$\text{IPM}_{\mathcal{F}}(\cdot,\cdot)$ is the (empirical) integral
probability metric defined by the function family $\mathcal{F}$. Define two
probability distributions $\mathbb{P}=P(\Phi_{(\hat{a})})$ and
$\mathbb{Q}=P(\Phi_{(a)})$, the corresponding IPM is denoted as

(10)
$\text{IPM}_{\mathcal{F}}(\mathbb{P},\mathbb{Q})=\sup_{f\in\mathcal{F}}\left|\int_{S}fd\mathbb{P}-\int_{S}fd\mathbb{Q}\right|$

where $\mathcal{F}:S\rightarrow\mathbb{R}$ is a class of real-valued bounded
measurable functions. We adopt $\mathcal{F}$ as 1-Lipschitz functions that
lead IPM to the Wasserstein-1 distance, i.e.,

(11)
$Wass(\mathbb{P},\mathbb{Q})=\inf_{f\in\mathcal{F}}\sum_{\mathbf{v}\in\text{col}_{i}(\Phi_{(\hat{a})})}\|f(\mathbf{v})-\mathbf{v}\|\mathbb{P}(\mathbf{v})d\mathbf{v}$

where $\mathbf{v}$ is the $i$-th column of $\Phi_{(\hat{a})}$ and the set of
push-forward functions $\mathcal{F}=\left\\{f\mid
f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\text{ s.t.
}\mathbb{Q}(f(\mathbf{v}))=\mathbb{P}(\mathbf{v})\right\\}$ can transform the
representation distribution of the exposed $\Phi_{(\hat{a})}$ to that of the
unexposed $\Phi_{(a)}$. Thus, $\|f(\mathbf{v})-\mathbf{v}\|$ is a pairwise
distance matrix between the exposed and unexposed user-item pairs. Based on
the discrepancy defined in (12), we define
$C(\Phi)=\|f(\mathbf{v})-\mathbf{v}\|$ and reformulate penalty term in (9) as

(12)
$\mathcal{L}_{d}=\inf_{\gamma\in\Pi\left(\mathbb{P},\mathbb{Q}\right)}\mathbb{E}_{(\mathbf{v},f(\mathbf{v}))\sim\gamma}C(\Phi)$

We adopt the efficient approximation algorithm proposed by (Shalit et al.,
2017) to compute the gradient of (12) for training the deconfounder model. In
particular, a mini-batch with $l$ exposed and $l$ unexposed user-item pairs is
sampled from $\Phi_{(\hat{a})}$ and $\Phi_{(a)}$, respectively. The element of
distance matrix $C(\Phi)$ is calculated as
$C_{ij}=\|\text{col}_{i}(\Phi_{(\hat{a})})-\text{col}_{j}(\Phi_{(a)})\|$.
After computing $C(\Phi)$, we can approximate $f$ and the gradient against the
model parameters 111For a more detailed calculation, refer to Algorithm 2 in
the appendix of prior work (Shalit et al., 2017). In conclusion, the learned
latent factors generated by the deconfounder model embed no information about
exposure variable. That means all the confounding factors are retained in
social network confounder $Z_{u}$.

###  3.5. Learning

####  3.5.1. Rating prediction

Having obtained the final representations $U$ and $I$ by the deconfounder
model, we use an inner product of $U^{\top}I$ as the inherent factors to
estimate the rating. As shown in the causal structure in Figure 4, another
component affecting the rating prediction is the social network confounder. A
simple way to incorporate these components into recommender systems is through
a linear model as follows.

(13)
$\hat{y}_{ui}=\sum_{u,i\in\mathcal{O}}U^{\top}I+{W_{u}}^{\top}Z_{u}+\epsilon_{ui},\quad\epsilon_{ui}\sim\mathcal{N}(0,1)$

where $W_{u}$ is coefficient that describes how much the confounder $Z_{u}$
contributes to the rating. To define the unbiased loss function for the biased
observations $y_{ui}$, we leverage the IPS strategy (Schnabel et al., 2016) to
weight each observation with _Propensity_. By Definition 1, the intuition of
the inverse propensity is to down-weight the commonly observed ratings while
up-weighting the rare ones.

(14)
$\mathcal{L}_{y}=\frac{1}{|\mathcal{O}|}\sum_{u,i\in\mathcal{O}}\frac{\left(y_{ui}-\hat{y}_{ui}\right)^{2}}{\pi(a_{ui};Z_{u})}$

####  3.5.2. Optimization

To this end, the objective function of our DENC method to predict ratings
could be derived as:

(15)
$\displaystyle\mathcal{L}=\mathcal{L}_{y}+\lambda_{a}\mathcal{L}_{a}+\lambda_{z}\mathcal{L}_{z}+\lambda_{d}\mathcal{L}_{d}+\mathcal{R}(\Omega)$

where $\Omega$ represents the trainable parameters and $\mathcal{R}(\cdot)$ is
a squared $l_{2}$ norm regularization term on $\Omega$ to alleviate the
overfitting problem. $\lambda_{a}$, $\lambda_{z}$ and $\lambda_{d}$ are trade-
off hyper-parameters. To optimize the objective function, we adopt Stochastic
Gradient Descent(SGD) (Bottou, 2010) as the optimizer due to its efficiency.

##  4\. Experiments

To more thoroughly understand the nature of MNAR issue and the proposed
unbiased DENC, experiments are conducted to answer the following research
questions:

  * •

(RQ1) How confounder bias caused by the social network is manifested in real-
world recommendation datasets?

  * •

(RQ2) Does our DENC method achieve the state-of-the-art performance in
debiasing recommendation task?

  * •

(RQ3) How does the embedding size of each component (e.g., social network
confounder and deconfounder model) in our DENC method impact the debiasing
performance?

  * •

(RQ4) How do the missing social relations impact the debiasing performance of
our DENC method?

###  4.1. Setup

####  4.1.1. Evaluation Metrics

We adopt two popular metrics including Mean Absolute Error (MAE) and Root Mean
Square Error (RMSE) to evaluate the performance. Since improvements in MAE or
RMSE terms can have a significant impact on the quality of the Top-$K$
recommendations (Koren, 2008), we also evaluate our DENC with Precision@K and
Recall@K for the ranking performance222We consider items with a rating greater
than or equal to 3.5 as relevant.

####  4.1.2. Datasets

We conduct experiments on three datasets including one semi-synthetic dataset
and two benchmark datasets Epinions 333http://www.cse.msu.edu/
tangjili/trust.html and Ciao (Tang et al., 2012) 444http://www.cse.msu.edu/
tangjili/trust.html. We maintain all the user-item interaction records in the
original datasets instead of discard items that have sparse interactions with
users.555Models can benefit from the preprocessed datasets in which all items
interact with at least a certain amount of users, for such preprocessing will
reduce the dataset sparsity.  The semi-synthetic dataset is generated by
incorporating the social network into
MovieLens666https://grouplens.org/datasets/movielens dataset. The details of
these datasets are given in Section A.1 in the appendix.

####  4.1.3. Baselines

We compare our DENC against three groups of methods for rating prediction: (1)
Traditional methods, including NRT (Li et al., 2017b) and PMF (Mnih and
Salakhutdinov, 2008). (2) Social network-based methods, including GraphRec
(Fan et al., 2019), DeepFM+ (Guo et al., [n. d.]), SocialMF (Jamali and Ester,
2010), SREE (Li et al., 2017a) and SoReg (Ma et al., 2011). (3) Propensity-
based methods, including CausE (Bonner and Vasile, 2018) and D-WMF (Wang et
al., 2018). More implementation details of baselines and parameter settings
are included in Section A.2 in the appendix.

Table 1. Statistics of Datasets. Density for rating (density-R) is
$\\#ratings/(\\#users\cdot\\#items)$, Density for social relations (density-
SR) is $\\#relations/(\\#users\cdot\\#users)$.

| Epinions | Ciao | MovieLens-1M
---|---|---|---
# users | 22,164 | 7,317 | 6,040
# items | 296,277 | 104,975 | 3,706
# ratings | 922,267 | 283,319 | 1000,209
density-R (%) | 0.0140 | 0.0368 | 4.4683
# relations | 355,754 | 111,781 | 9,606
density-SR (%) | 0.0724 | 0.2087 | 0.0263

####  4.1.4. Parameter Settings

We implement all baseline models on a Linux server with Tesla P100 PCI-E 16GB
GPU. 777Our code is currently shared on Github, however, due to the double-
blind submission policy requirement, we leave the link void now but promise to
activate it after paper acceptance. Datasets for all models except CausE 888
As in CausE, we sample 10% of the training set to build an additional debiased
dataset (mandatory in model training), where items are sampled to be uniformly
exposed to users.  are split as training/test sets with a proportion of 80/20,
and 20% of the training set are validation set.

We optimize all models with Stochastic Gradient Descent(SGD) (Bottou, 2010).
For fair comparison, a grid search is conducted to choose the optimal
parameter settings, e.g., dimension of user/item latent vector $k_{MF}$ for
matrix factorization-based models and dimension of embedding vector $d$ for
neural network-based models. The embedding size is initialized with the Xavier
(Glorot and Bengio, 2010) and searched in $[8,16,32,64,128,256]$. The batch
size and learning rate are searched in $[32,64,128,512,1024]$ and
$[0.0005,0.001,0.005,0.01,0.05,0.1]$, respectively. The maximum epoch
$N_{epoch}$ is set as 2000, an early stopping strategy is performed. Moreover,
we employ three hidden layers for the neural components of NRT, GraphRec and
DeepFM+. Like our DENC method, DeepFM+ uses node2vec to train the social
network embeddings. Hence, the embedding size of its node2vec is set as the
same as in our DENC for a fair comparison.

Without specification, unique hyperparameters of DENC are set as: three
coefficients $\lambda_{n}$, $\lambda_{u}$ and $\lambda_{i}$ are tuned in
$\\{10^{-3},10^{-4},10^{-5}\\}$. The dimension of node2vec embedding size
$k_{a}$ and the dimension of inherent factor $k_{d}$ are tuned in
$[8,16,32,64,128,256]$, and their influences are reported in Section 4.4.

Table 2.  Performance comparison: bold numbers are the best results. Strongest
baselines are highlighted with underlines.

|  | Traditional | Social network-based | Propensity-based | Ours
---|---|---|---|---|---
Dataset | Metrics | PMF | NRT | SocialMF | SoReg | SREE | GraphRec | DeepFM+ | CausE | D-WMF | DENC
Epinions | MAE | 0.9505 | 0.9294 | 0.8722 | 0.8851 | 0.8193 | 0.7309 | 0.5782 | 0.5321 | 0.3710 | 0.2684
| RMSE | 1.2169 | 1.1934 | 1.1655 | 1.1775 | 1.1247 | 0.9394 | 0.6728 | 0.7352 | 0.6299 | 0.5826
Ciao | MAE | 0.8868 | 0.8444 | 0.7614 | 0.7784 | 0.7286 | 0.6972 | 0.3641 | 0.4209 | 0.2808 | 0.2487
| RMSE | 1.1501 | 1.1495 | 1.0151 | 1.0167 | 0.9690 | 0.9021 | 0.5886 | 0.8850 | 0.5822 | 0.5592
MovieLens-1M | MAE | 0.8551 | 0.8959 | 0.8674 | 0.9255 | 0.8408 | 0.7727 | 0.5786 | 0.4683 | 0.3751 | 0.2972
$\Delta(Z_{u})=-0.35$ | RMSE | 1.0894 | 1.1603 | 1.1161 | 1.1916 | 1.0748 | 0.9582 | 0.6730 | 0.8920 | 0.6387 | 0.5263
MovieLens-1M | MAE | 0.8086 | 0.8801 | 0.8182 | 0.8599 | 0.7737 | 0.7539 | 0.5281 | 0.4221 | 0.3562 | 0.2883
$\Delta(Z_{u})=0$ | RMSE | 1.0034 | 1.1518 | 1.0382 | 1.1005 | 0.9772 | 0.9454 | 0.6477 | 0.8333 | 0.6152 | 0.5560
MovieLens-1M | MAE | 0.7789 | 0.7771 | 0.7969 | 0.8428 | 0.7657 | 0.7423 | 0.3672 | 0.4042 | 0.3151 | 0.2836
$\Delta(Z_{u})=0.35$ | RMSE | 0.9854 | 0.9779 | 1.0115 | 1.0792 | 0.9746 | 0.9344 | 0.5854 | 0.8173 | 0.5962 | 0.5342

###  4.2. Understanding Social Confounder (RQ1)

We initially conduct an experiment to understand to what extent the
confounding bias caused by social networks is manifested in real-world
recommendation datasets. The social network as a confounder will bias the
interactions between the user and items. We aim to verify two kinds of
scenarios: (1) User in the social network interacts with more items than users
outside the social network. (2) The pair of user-neighbor in the social
network has more common interacted items than the pair of user-neighbor
outside the social network. Intuitively, an unbiased platform should expect
users to interact with items broadly, which indicates that interactions are
likely to be evenly distributed. Thus, we investigate the social confounder
bias by analyzing the statistics of interactions in these two scenarios in
Epinions and Ciao dataset.

(a) (a) Distribution on Ciao.

(b) (b) Distribution on Epinions.

Figure 5. Scenario (1): the distribution of $x$ (the number of items
interacted by a user). The smooth probability curves visualize how the number
of items is distributed.

(a) (a) Distribution on Ciao.

(b) (b) Distribution on Epinions.

Figure 6. Scenario (2): the distribution of $x$ (the number of items commonly
interacted by a user-pair).

For the first scenario, we construct two user sets within or outside the
social network, i.e., $\mathcal{U}_{G}$ and $\mathcal{U}_{\backslash G}$.
Specially, $\mathcal{U}_{G}$ is constructed by randomly sampling a set of
users in social network $G$, and $\mathcal{U}_{\backslash G}$ is randomly
sampled out of $G$. The size of $\mathcal{U}_{G}$ and $\mathcal{U}_{\backslash
G}$ is the same and defined as $n$. Following the above guidelines, we sample
$n=70$ users for $\mathcal{U}_{G}$ and $\mathcal{U}_{\backslash G}$. Figure 5
depicts the distributions of the interacted items by users in
$\mathcal{U}_{G}$ and $\mathcal{U}_{\backslash G}$. The smooth curves are
continuous distribution estimates produced by the kernel density estimation.
Apparently, the distribution for $\mathcal{U}_{\backslash G}$ is significantly
skewed: Most of the users interact with few items. For example, on Ciao, more
than 90% of users interact with fewer than 50 items. By contrast, most users
in the social network tend to interact with items more frequently, which is
also confirmed by the even distribution. In general, the distribution curve of
$\mathcal{U}_{G}$ is quite different from $\mathcal{U}_{\backslash G}$, which
reflects that the social network influences the interactions between users and
items. In addition, the degree of bias varies across datasets: Epinions is
less biased than Ciao.

For the second scenario, based on $\mathcal{U}_{G}$ and
$\mathcal{U}_{\backslash G}$, we further analyze the number of commonly
interacted items by the user-pair. Particularly, we randomly sample four one-
hop neighbours for each user in $\mathcal{U}_{G}$ to construct user-pairs.
Since users in $\mathcal{U}_{\backslash G}$ have no neighbours, for each of
them, we randomly select another four users999According to the statistics, we
discover that 90% of users have at least four one-hop neighbours in Ciao and
Epinions in $\mathcal{U}_{\backslash G}$ to construct four user-pairs. Recall
that $\mathcal{U}_{G}$ and $\mathcal{U}_{\backslash G}$ both have 70 users,
then we totally have $4\times 70$ user-pairs for $\mathcal{U}_{\backslash G}$
and $\mathcal{U}_{\backslash G}$, respectively. Figure 6 represents the
distribution of how many items are commonly interacted by the users in each
pair.101010For example, $\\{user1,user2,user3,user4\\}$ are one-hop neighbours
of $user5$. If the number of commonly items interacted by $user1$ and $user5$
is 3, then $x=3$ in the $x$-axis of Figure 6 is nonzero. Figure 6 indicates
most user-neighbour pairs in the social network have fewer than 10 items in
common. However the user-pairs outside the social network nearly have no items
in common, i.e., less than 1. We can conclude that social networks can
encourage users to share more items with their neighbours, compared with users
who are not connected by any social networks.

###  4.3. Performance Comparison (RQ2)

We compare the rating prediction of DENC with nine recommendation baselines on
three datasets including Epinions, Ciao and MovieLens-1M. Table 2 demonstrates
the performance comparison, where the confounder $\Delta{(Z_{u})}$ in
MovieLens-1M is assigned with three different settings, i.e., -0.35, 0 and
0.35. Analyzing Table 2, we have the following observations.

  * •

Overall, our DENC consistently yields the best performance among all methods
on five datasets. For instance, DENC improves over the best baseline model
w.r.t. MAE/RMSE by 10.26/4.73%, 3.21/2.3%, and 7.79/11.24% on Epinions, Ciao
and MovieLens-1M ($\Delta{(Z_{u})}$=-0.35) datasets, respectively. The results
indicate the effectiveness of DENC on the task of rating prediction, which has
adopted a principled causal inference way to leverage both the inherent
factors and auxiliary social network information for improving recommendation
performance.

  * •

Among the three kinds of baselines, propensity-based methods serves as the
strongest baselines in most cases. This justifies the effectiveness of
exploring the missing pattern in rating data by estimating the propensity
score, which offers better guidelines to identify the unobserved confounder
effect from ratings. However, propensity-based methods perform worse than our
DENC, as they ignore the social network information. It is reasonable that
exploiting the social network is useful to alleviate the confounder bias to
rating outcome. The importance of social networks can be further verified by
the fact that most of the social network-based methods consistently outperform
PMF on all datasets.

  * •

All baseline methods perform better on Ciao than on Epinions, because Epinions
is significantly sparser than Ciao with 0.0140% and 0.0368% density of
ratings. Besides this, DENC still achieves satisfying performance on Epinions
and its performance is competitive with the counterparts on Ciao. This
demonstrates that its exposure model of DENC has an outstanding capability of
identifying the missing pattern in rating prediction, in which biased user-
item pairs in Epinions can be captured and then alleviated. In addition, the
performance of DENC on three Movielens-1M datasets is stable w.r.t. different
levels of confounder bias, which verifies the robust debiasing capability of
DENC.

###  4.4. Ablation Study (RQ3)

In this section, we conduct experiments to evaluate the parameter sensitivity
of our DENC method. We have two important hyperparameters $k_{a}$ and $k_{d}$
that correspond to the embedding size in loss function $\mathcal{L}_{a}$ and
$\mathcal{L}_{d}$, respectively. Based on the hyperparameter setup in Section
4.1.4, we vary the value of one hyperparameter while keeping the others
unchanged.

(a) (a) MAE on Ciao.

(b) (b) RMSE on Ciao.

(c) (a) MAE on Epinions.

(d) (b) RMSE on Epinions.

Figure 7. Our DENC: parameter sensitivity of $k_{a}$ and $k_{d}$ against (a)
MAE (b) RMSE on Ciao and Epinions dataset.

Figure 7 lays out the performance of DENC with different parameter settings.
For both datasets, the performance of our DENC is stable under different
hyperparameters $k_{a}$ and $k_{d}$. The performance of DENC increases while
the embedding size increase from approximately 0-15 for $k_{d}$; afterwards,
its performance decreases. It is clear that when the embedding size is set to
approximately $k_{a}$=45 and $k_{d}$=15, our DENC method achieves the optimal
performance. Our DENC is less sensitive to the change of $k_{a}$ than $k_{d}$,
since MAE/RMSE values change with a obvious concave curve along $k_{d}$=0 to
50 in Figure 7, while MAE/RMSE values only change gently with a downwards
trend along $k_{a}$=0 to 50. It is reasonable since $k_{d}$ controls the
embedding size of disentangled user-item representation attained by the
deconfounder model, i.e., the inherent factors, while social network embedding
size $k_{a}$ serves as the controller for auxiliary social information, the
former can influence the essential user-item interaction while the latter
affects the auxiliary information.

###  4.5. Case Study (RQ4)

We first investigate how the missing social relations affect the performance
of DENC. We randomly mask a percentage of social relations to simulate the
missing connections in social networks. For Epinions, Ciao and MovieLens
dataset, we fix the social network confounder as $\Delta(Z_{u})=0$. Meanwhile,
we exploit different percentages of missing social relations including {20%,
50%, 80%}. Note that we do not consider the missing percentage of $100\%$,
i.e., the social network information is completely unobserved. Considering
that the social network is viewed as a proxy variable of the confounder, the
social network should provide partially known information. Following this
guideline, we firstly investigate how the debias capability of our DENC method
varies under the different missing percentages. Secondly, we also report the
ranking performance of DENC (percentages of missing social relations is set to
$0\%$) under Precision@K and Recall@K with $K=\\{10,15,20,25,30,35,40\\}$ to
evaluate our model thoroughly.

(a) (a) MAE of DENC.

(b) (b) RMSE of DENC.

(c) (c) Precision@20 of DENC.

(d) (d) Recall@20 of DENC.

Figure 8. Our DENC: debias performance w.r.t. different missing percentages of
social relation.

Figure 8 illustrates our debias performance w.r.t. different missing
percentages of social relations on three datasets. As shown in Figure 8, the
missing social relations can obviously degrade the debias performance of DENC
method. The performance evaluated by four metrics in Figure 8 consistently
degrades when the missing percentage increases from 0% to 80%, which is
consistent with the common observation. This indicates that the underlying
social network can play a significant role in a recommendation, we consider
the because it can capture the preference correlations between users and their
neighbours.

(a) (a) Epinions.

(b) (b) Ciao.

(c) (c) MovieLens-1M
($\Delta(Z_{u})=0$).

Figure 9. Performance of DENC in terms of Precision@K and RecallG@K under
difference $K$

Based on the evaluation on Precision@K and Recall@K, Figure 9 show that DENC
achieves stable performance on Top-$K$ recommendation when $K$ (i.e., the
length of ranking list) varies from 10 to 40. Our DENC can recommend more
relevant items within top $K$ positions when the ranking list length
increases.

##  5\. Conclusion and Future Work

In this paper, we have researched the missing-not-at-random problem in the
recommendation and addressed the confounding bias from a causal perspective.
Instead of merely relying on inherent information to account for selection
bias, we developed a novel social network embedding based de-bias recommender
for unbiased rating, through correcting the confounder effect arising from
social networks. We evaluate our DENC method on two real-world and one semi-
synthetic recommendation datasets, with extensive experiments demonstrating
the superiority of DENC in comparison to state-of-the-arts. In future work, we
will explore the effect of different exposure policies on the recommendation
system using the intervention analysis in causal inference. In addition,
another promising further work is to explore the selection bias arisen from
other confounder factors, e.g., user demographic features. This can be
explained that a user’s nationality affects which restaurant he is more likely
to visit (i.e., exposure) and meanwhile affects how he will rate the
restaurant (i.e., outcome).

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##  Appendix A APPENDIX

###  A.1. Datasets

The statistics of baseline datasets are given in Table 1. In Epinions and
Ciao, the rating values are integers from 1 (like least) to 5 (like most).
Since observed ratings are very sparse (rating density 0.0140% for Epinions
and 0.0368% for Ciao), thus the rating prediction on these two datasets is
challenging.

In addition, we also simulate a semi-synthetic dataset based on MovieLens. It
is well-known that MovieLens is a benchmark dataset of user-movie ratings
without social network information. For MovieLens-1M, we first need to
construct a social network $G$ by placing an edge between each pair of users
independently with a probability 0.5 depending on whether the nodes belong to
$G$. Recall that the social network is viewed as the confounder (common cause)
which affects both exposure variables and ratings. We generate the exposure
assignment by the confounder $Z_{u}$ of three levels
$\Delta(Z_{u})\in\\{-0.35,0,0.35\\}$. Then, the exposure $a_{ui}$ and rating
outcome $y_{ui}$ are simulated as follows.

$\displaystyle a_{ui}$
$\displaystyle\sim\operatorname{Bern}\left(\Delta\left(Z_{u}\right)\right)$
$\displaystyle y_{ui}$
$\displaystyle=a_{ui}\cdot(y_{ui}^{\text{mov}}+\beta_{u}\Delta\left(Z_{u}\right)+\varepsilon)$
$\displaystyle\varepsilon\sim N(0,1),\quad u\in G$ $\displaystyle y_{ui}$
$\displaystyle=y_{ui}^{\text{mov}}$ $\displaystyle u\notin G$

where $y_{ui}^{\text{mov}}$ is the original rating in MovieLens and the
parameter $\beta_{u}$ controls the amount of social network confounder. The
exposure $a_{ui}$ indicating whether item $i$ being exposed to user $u$ is
given by a Bernoulli distribution parameterized by the confounder $Z_{u}$. The
non-zero $a_{ui}$ is used to simulate the semi-synthetic rating $y_{ui}$ by
the second equation. The third equation indicates that the ratings of user
will keep unchanged if s/he is not connected by $G$.

###  A.2. Baselines

We compare our DENC against three groups of methods, covering matrix
factorization method, social network-based method, and propensity-based
method. For each group, we select its representative baselines with details as
follows.

  * •

PMF (Mnih and Salakhutdinov, 2008): The method utilizes user-item rating
matrix and models latent factors of users and items by Gaussian distributions;

  * •

NRT (Li et al., 2017b): A deep-learning method that adopts multi-layer
perceptron network to model user-item interactions for rating predictions.

  * •

SocialMF (Jamali and Ester, 2010): It considers the social information by
adding the propagation of social relation into the matrix factorization model.

  * •

SoReg (Ma et al., 2011): It models social information as regularization terms
to constrain the Matrix Factorization framework.

  * •

SREE (Li et al., 2017a): It models users and items embeddings into a Euclidean
space as well as users’ social relations.

  * •

GraphRec (Fan et al., 2019): This is a state-of-the-art social recommender
that models social information with Graph Neural Network, it organizes user
behaviors as a user-item interaction graph.

  * •

DeepFM (Guo et al., [n. d.])+: DeepFM is a state-of-the-art recommender that
integrates Deep Neural Networks and Factorization Machine (FM). To incorporate
the social information into DeepFM, we change the output of FM in DeepFM+ to
the linear combination of the original FM function in (Guo et al., [n. d.])
and the pre-trained node2vec user embeddings. We also change the task of
DeepMF from click-through rate (CTR) to rating prediction.

  * •

CausE (Bonner and Vasile, 2018): It firstly fits exposure variable embedding
with Poisson factorization, then integrates the embedding into PMF for rating
prediction.

  * •

D-WMF (Wang et al., 2018): A propensity-based model which uses Poisson
Factorization to infer latent confounders then augments Weighted Matrix
Factorization to correct for potential confounding bias.

###  A.3. Model Variants Configuration

To get a better understanding of our DENC method, we further evaluate the key
components of DENC including Exposure model and Social network confounder. We
evaluate the performance of DENC on the condition that if a specific component
is removed, and then compare the performance of the intact DENC method. In the
following, we define two variants of DENC as (1) DENC-$\alpha$ that removes
Exposure model; (2) DENC-$\beta$ that removes Social network confounder. Note
that we do not consider the evaluation of removing Deconfounder in DENC, since
Deconfounder models the inherent factors of user-item information, removing
user-item information in a recommender can result in poor performance. We
record evaluation results in Table 3 and have the following findings:

  * •

By comparing DENC with DENC-$\alpha$, we find that Exposure model is important
for capturing missing patterns and thus boosting the recommendation
performance. Removing Exposure model can lead a drastic degradation of
MAE/RMSE by 20.41%/24.08% on Epinions and 18.93%/24.34% on Ciao, respectively.

  * •

We observe that without Social network confounder, the performance of
DENC-$\beta$ is deteriorated significantly, with the degradation of MAE/RMSE
by 16.10%/20.50% on Epinions and 13.83%/11.31% on Ciao, respectively.

  * •

Exposure model has a greater impact on DENC compared with Social network
confounder. It is reasonable since Exposure model simulates the missing
patterns, then Social network confounder can consequently debias the potential
confounding bias under the guidance of missing patterns.

Table 3.  Experimental results of DENC-$\alpha$ and DENC-$\beta$.  Dataset | Models | MAE | RMSE
---|---|---|---
Epinions | DENC-$\alpha$ | 0.4725 | 0.8234
| DENC-$\beta$ | 0.4294 | 0.7876
| DENC | 0.2684 | 0.5826
Ciao | DENC-$\alpha$ | 0.4380 | 0.8026
| DENC-$\beta$ | 0.3870 | 0.6723
| DENC | 0.2487 | 0.5592

###  A.4. Investigation on Different Network Embedding Methods

We construct network embedding with node2vec (Grover and Leskovec, 2016) that
has the capacity of learning richer representations by adding flexibility in
exploring neighborhoods of nodes. Besides, by adjusting the weight of the
random walk between breadth-first and depth-first sampling, embeddings
generated by node2vec can balance the trade-off between homophily and
structural equivalence (Henderson et al., 2012), both of which are essential
feature expressions in recommendation systems. The key characteristic of
node2vec is its scalability and efficiency as it scales to networks of
millions of nodes.

By comparison, we further investigate how different network embedding methods
impact the performance of DENC, i.e., LINE (Tang et al., 2015), SDNE (Wang et
al., 2016).

  * •

LINE (Tang et al., 2015) preserves both first-order and second-order
proximities, it suits arbitrary types of information networks and can easily
scale to millions of nodes.

  * •

SDNE (Wang et al., 2016) is a Deep Learning-based network embedding method,
like LINE, it exploits the first-order and second-order proximity jointly to
preserve the network structure.

We train the three embedding methods with embedding size $d$=10 while the
batch size and epochs are set to 1024 and 50, respectively. The experimental
results are given in Table 4.

Table 4.  Experimental results of DENC under node2vec, LINE, SDNE.

Dataset | Embedding | MAE | RMSE | Precision@20 | Recall@20
---|---|---|---|---|---
Epinions | node2vec | 0.2684 | 0.5826 | 0.2832 | 0.2501
| LINE | 0.4241 | 0.6307 | 0.1736 | 0.1534
| SDNE | 0.4021 | 0.6137 | 0.1928 | 0.1837
Ciao | node2vec | 0.2487 | 0.5592 | 0.2703 | 0.2212
| LINE | 0.5218 | 0.7605 | 0.1504 | 0.1209
| SDNE | 0.4538 | 0.6274 | 0.2082 | 0.1594

The results show that under the same experimental settings, DENC performs
worse with embeddings trained by LINE and SDNE compared with node2vec on both
datasets. Although LINE considers the higher-order proximity, unlike node2vec,
it still cannot balance the representation between homophily and structural
equivalence (Henderson et al., 2012), in which connectivity information and
network structure information can be captured jointly. The results show that
our DENC benefits more from the balanced representation that can learn both
the connectivity information and network structure information. Based on
higher-order proximity, SDNE develops a deep-learning representation method.
However, compared with node2vec, SDNE suffers from higher time complexity. The
deep architecture of SDNE framework mainly causes the high time complexity of
SDNE, the input vector dimension can expand to millions for the auto-encoder
in SDNE (Cui et al., 2018). Thus, we consider it reasonable that our DENC with
SDNE embedding cannot outperform the counterpart with node2vec embedding under
the same training epochs, since it requires more iterations for SDNE to get
finer representation.


# Logarithmic uncertainty principle for Clifford-wavelet transform

Sabrine Arfaoui  Sabrine Arfaoui, 1Laboratory of Algebra, Number Theory and
Nonlinear Analysis LR18ES15, Department of Mathematics, Faculty of Sciences,
University of Monastir, Monastir 5019, Tunisia.
2Higher Institute of Computer Science El-Kef, University of Jendouba, 5 Street
of Saleh Ayech - Kef 7100, Tunisia<EMAIL_ADDRESS>Hicham
Banouh  Departement of Mathematics, Faculty of Sciences and Applied Sciences,
University of Akli Mohand Oulhadj of Bouira, Bouira 10000, Algeria.
<EMAIL_ADDRESS>Anouar Ben Mabrouk  1Higher Institute of Applied
Mathematics and Computer Science, University of Kairaouan, Street of Assad Ibn
Alfourat, Kairaouan 3100, Tunisia.
2Laboratory of Algebra, Number Theory and Nonlinear Analysis LR18ES15,
Department of Mathematics, Faculty of Sciences, Monastir 5019, Tunisia.
3Department of Mathematics, Faculty of Sciences, University of Tabuk, King
Faisal Rd, Tabuk 47512, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

The uncertainty principle constitutes one of the famous physical concepts
which continues to attract researchers from different related fields since its
discovery due to its utility in many applications. Among the classical
(Fourier-based) forms, modified variants such as wavelet-based and logarithmic
versions have been investigated by researchers. In the present work we are
concerned with the development of a new logarithmic uncertainty principle
combining both wavelets and logarithmic variants in the Clifford wavelet
transform framework.

###### Key words and phrases:

Harmonic analysis, Clifford algebra, Clifford analysis, Continuous wavelet
transform, Clifford Fourier transform, Clifford wavelet transform, Uncertainty
principle.

###### 1991 Mathematics Subject Classification:

30G35, 42C40, 42B10, 15A66.

##  1\. Introduction

In the last decade(s) a coming back to Clifford algebras and analysis has
taken place in many fields such as mathematics, physics and computer science.
This may be due to the fact that Clifford algebras incorporate inside one
single structure the geometrical and algebraic properties of Euclidean space.
Incite of a Clifford algebra geometric entities are treated according to their
dimension such as scalars, vectors, bivectors and volume elements. This
allowed researchers especially in mathematical physics to develop harmonic
analysis on Clifford algebras as extension of complex one. Famous operators
and related inequalities such as Cauchy-Riemann operator and uncertainty
principle have been extended to the Clifford framework.

Recently extensions of wavelet theory to the Clifford framework have been
developed and applied in many fields, and consequently connection between
wavelet theory and Clifford algebra/analysis is now growing up, and
applications are widespread such as in signal/image processing in physics,
nuclear magnetics, electric engineering, colour images. See [10, 17, 18, 24].

Wavelet analysis of functions starts by checking that the mother wavelet is
admissible to guarantee the reconstruction, Fourier-Plancherel and Parseval
rules.

The present work is concerned with the development of a sharp uncertainty
principle in the Clifford framework based on Clifford wavelets and thus
extending the one developed in [2, 4]. We propose a new Heisenberg uncertainty
principle applied to continuous Clifford wavelet transform in the settings of
the non commutative Clifford algebras. In section 2 some preliminaries on
Clifford toolkit and wavelets are presented. We also recall briefly the most
recent existing works on the uncertainty principle in both its classical forms
and Clifford one. Section 4 is devoted to the development of our main results.

##  2\. Clifford wavelets toolkit

In this section, we review briefly the basic concepts of Clifford analysis and
the continuous wavelet transform on the real Clifford algebra as well as the
uncertainty principle .

The Clifford algebra $\mathbb{R}_{n}$ associated to $\mathbb{R}^{n}$ is the
$2^{n}$-dimensional non commutative associative algebra generated by the
canonical basis $\left\\{e_{1},e_{2},\dots,e_{n}\right\\}$ relatively to the
non commutative product

$e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij},$

where $\delta$ is the Kronecker symbol. Any element $a\in\mathbb{R}_{n}$ has a
unique representation

$a=\sum_{A}a_{A}e_{A}=\sum_{k=0}^{n}\sum_{\left|A\right|=k}a_{A}e_{A},\;a_{A}\in\mathbb{R},$

where $|A|$ is the length of the multi-index $A$. We define here the module
(or the norm) of $a$ in the Clifford algebra $\mathbb{R}_{n}$ by

$|a|=(\sum_{A}|a_{A}|^{2})^{1/2},$

where for $x\in\mathbb{R}$, $|x|$ stands for its absolute value.

On the algebra $\mathbb{R}_{n}$ there are some types of involutive operators
defined on the basis elements. A main-involution satisfying

$\widetilde{e_{A}}=(-1)^{\left|A\right|}e_{A}.$

A second type called reversion stating that

$e_{A}^{\ast}=(-1)^{\frac{\left|A\right|(\left|A\right|-1)}{2}}e_{A}$

and finally a conjugation operator defined by

$\overline{e_{A}}=(-1)^{\frac{|A|(|A|+1)}{2}}e_{A}.$

In the case of the complex extension
$\mathbb{C}_{n}=\mathbb{R}_{n}+i\mathbb{R}_{n}$ the last operator has to be
extended to the so-called Hermitian conjugation defined by

$\lambda^{\dagger}=\overline{a}-i\overline{b},$

for $\lambda=a+ib\in\mathbb{C}_{n}$ with $a,b\in\mathbb{R}_{n}$. The module
(or the norm) of $\lambda$ in the Clifford algebra, will be

$|\lambda|^{2}=\lambda^{\dagger}\lambda=|a|^{2}+|b|^{2},$

As $\mathbb{R}^{n}$ is a subspace of $\mathbb{R}_{n}$ any element
$x=(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}$ is identified to
$\underline{x}={\displaystyle\sum_{j=1}^{n}e_{j}x_{j}}$. This allows to
introduce the Clifford product of vectors by

$\underline{x}\underline{y}=-<\underline{x},\underline{y}>+\underline{x}\wedge\underline{y},$

where $<\underline{x},\underline{y}>$ is the usual inner product and
$\underline{x}\wedge\underline{y}$ is the exterior product on
$\mathbb{R}^{n}$.

Finally we recall that for two functions
$f,g:\mathbb{R}^{n}\longrightarrow\mathbb{R}_{n}$ (the same apply for
functions with values in $\mathbb{C}_{n}$) the inner product is defined by

$<f,g>_{L^{2}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))}=\int_{\mathbb{R}^{n}}\left[f(\underline{x})\right]^{\dagger}g(\underline{x})dV(\underline{x})$
(2.1)

and the associated norm by

$\|f\|_{L^{2}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))}=<f,f>_{L^{2}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))}^{\frac{1}{2}}.$

We denote also

$\left\|f\right\|_{L^{1}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))}=\int_{\mathbb{R}^{n}}|f(\underline{x})|dV(\underline{x})$

where $dV(\underline{x})$ stands for the Lebesgue measure, and
$|f(x)|^{2}=\left[f(\underline{x})\right]^{\dagger}f(\underline{x})$

As for multidimensional Euclidean spaces the Fourier and wavelet transforms
have been extended to the case of Clifford framework. The Clifford-Fourier
transform of a Clifford-valued function $f\in L^{1}\cap
L^{2}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))$ is defined by

$\mathcal{F}\left[f\right](\underline{\xi})=\widehat{f}(\underline{\xi})=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}e^{-i<\underline{x},\underline{\xi}>}f(\underline{x})dV(\underline{x}).$

In the present paper we use spin theory as in [1, 2, 4]. Let $\psi\in
L^{1}\cap L^{2}(\mathbb{R}^{n},dV(\underline{x}))$ be such that

  * •

${\psi}$ is a Clifford-valued function.

  * •

$\widehat{\psi}(\underline{\xi})\left[\widehat{\psi}(\underline{\xi})\right]^{\dagger}$
is scalar.

  * •

$\mathcal{A}_{\psi}={\displaystyle(2\pi)^{n}\int_{\mathbb{R}^{n}}\frac{\widehat{\psi}(\underline{\xi})\left[\widehat{\psi}(\underline{\xi})\right]^{\dagger}}{|\underline{\xi}|^{n}}dV(\underline{\xi})<\infty.}$

$\psi$ is called a Clifford mother wavelet. The spin group of order $n$ is
defined by

$S_{n}=\left\\{s\in\mathbb{R}_{n};\;s={\displaystyle\prod\limits_{j=1}^{2l}}\underline{\omega}_{j},\;\underline{\omega}_{j}^{2}=-1,1\leq
j\leq 2l\right\\}.$

For $(a,\underline{b},s)\in\mathbb{R}_{+}\times\mathbb{R}^{n}\times S_{n}$, we
denote

$\psi^{a,\underline{b},s}(\underline{x})=\frac{1}{a^{\frac{n}{2}}}s\psi(\frac{\overline{s}(\underline{x}-\underline{b})s}{a})\overline{s}.$

In [1, 2, 4] the authors showed that the copies ${\psi^{a,\underline{b},s}}$
constitute a dense set in
$L^{2}(\mathbb{R}^{n},\mathbb{R}^{n},dV(\underline{x}))$. The Clifford-wavelet
transform of an analyzed function $f\in
L^{2}(\mathbb{R}^{n},\mathbb{R}^{n},dV(\underline{x}))$ is defined by

$T_{\psi}\left[f\right](a,\underline{b},s)=\int_{\mathbb{R}^{n}}\left[\psi^{a,\underline{b},s}(\underline{x})\right]^{\dagger}f(\underline{x})dV(\underline{x}).$

Let
$\mathcal{H}_{\psi}=T_{\psi}\left(L^{2}(\mathbb{R}^{n},dV(\underline{x}))\right)$.
We define an inner product on $\mathcal{H}_{\psi}$ by

$\left[T_{\psi}\left[f\right],T_{\psi}\left[g\right]\right]=\frac{1}{\mathcal{A}_{\psi}}\int\limits_{S_{n}}\int\limits_{\mathbb{R}^{n}}\int\limits_{\mathbb{R}_{+}}\mathcal{K}_{\psi}\left[f,g\right](a,\underline{b},s)d\mu(a,\underline{b},s),$

where

$\mathcal{K}_{\psi}\left[f,g\right](a,\underline{b},s)=(T_{\psi}\left[f\right](a,\underline{b},s))^{\dagger}T_{\psi}\left[g\right](a,\underline{b},s),$
$d\mu(a,\underline{b},s)=\frac{da}{a^{n+1}}dV(\underline{b})ds,$

and where $ds$ is the Haar measure on $S_{n}$.

It is straightforward that the operator $T_{\psi}$ is an isometry from
$L^{2}(\mathbb{R}^{n},dV(\underline{x}))$ to $\mathcal{H}_{\psi}$ ,and we have

$\int\limits_{Spin(n)}\int\limits_{\mathbb{R}^{n}}\int\limits_{\mathbb{R}^{+}}\left|T_{\psi}\left[f\right](a,\underline{b},s)\right|{}^{2}\frac{da}{a^{n+1}}dV(\underline{b})ds=\mathcal{A}_{\psi}\left\|f\right\|_{2}^{2}.$
(2.2)

See [1, 2, 4]. As a result any analyzed function in
$L^{2}(\mathbb{R}^{n},dV(\underline{x}))$ may be reconstructed in the
$L^{2}$-sense by means of its Clifford-wavelet transform which constitutes the
Clifford-wavelet Plancherel-Parseval formulas.

Our purpose in the present work is to establish a logarithmic uncertainty
principle in the Clifford framework by using the Clifford wavelet continuous
transform.

The uncertainty principle originally due to Heisenberg constitutes since its
discovery an interesting concept in quantum mechanics. Mathematically, it is
resumed by the fact that a non-zero function and its Fourier transform cannot
both be sharply localized. In the last decades many studies have been
developed on the uncertainty principle using pure Fourier transform, pure
wavelets, Clifford Fourier transform and recently Clifford wavelet transform.
We will not review here in details these works. However the readers may refer
to [30, 31, 12, 17, 18, 22, 23, 24, 25, 26, 27, 28, 19].

##  3\. Logarithmic uncertainty for Clifford Wavelet Transform

Based on the classical Pitt’s inequality, Beckner [5] proved the logarithmic
version of Heisenberg’s uncertainty principle. Recently, this principle has
been carried out for different two-dimensional time-frequency domain
transforms [2,7,19]. Here, we derive the logarithmic inequality in Clifford
wavelet transform domains [6, 7, 8, 16, 5, 15, 20, 23, 11, 13].

In [1, 2, 3, 4], the authors have established the two following results on the
uncertainty principle applied to the continuous Clifford wavelet transform.

######  Theorem 3.1.

[1, 2, 3, 4] Let $\psi\in L^{2}(\mathbb{R}_{n},dV(\underline{x}))$ be an
admissible Clifford mother wavelet. Then, for $f\in
L^{2}(\mathbb{R}_{n},dV(\underline{x}))$ the following inequality holds

$\displaystyle\left(\displaystyle\int_{Spin(n)}\displaystyle\int_{\mathbb{R}^{+}}\left\|b_{k}T\left[f\right](a,\cdot,s)\right\|_{2}^{2}\frac{da}{a^{n+1}}ds\right)^{\frac{1}{2}}\left\|\xi_{k}\widehat{f}\right\|_{2}\geq\frac{(2\pi)^{\frac{n}{2}}}{2}\sqrt{A_{\psi}}\left\|f\right\|_{2}^{2},$
(3.1)

where $k=1,2,\cdots,n$.

######  Theorem 3.2.

[2, 3, 4] Let $\psi\in L^{2}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))$
be an admissible Clifford mother wavelet. Then for $f\in
L^{2}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))$ the following
inequality holds

$(\int_{Spin(n)}\int_{\mathbb{R}^{+}}\left\|b_{k}T_{\psi}\left[f\right](a,\bullet,s)\right\|_{2}^{2}\frac{da}{a^{n+1}}ds)^{\frac{1}{2}}\left\|\xi_{k}\widehat{f}\right\|_{2}\geq\sqrt{2^{n+1}\pi{}^{n}A_{\psi}}\left\\{\left\|f\right\|_{2}^{2}+2\left|\left\langle
f_{1},f_{2}\right\rangle\right|\right\\}$

where

$\begin{cases}f_{1}(\underline{x})&=\dfrac{1}{A_{\psi}}\displaystyle\int_{Spin(n)}\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{+}}\psi^{a,\underline{b},s}(\underline{x})\partial_{b_{k}}T_{\psi}\left[f\right](a,\underline{b},s)\frac{da}{a^{n+1}}dV(\underline{b})ds\\\
f_{2}(\underline{x})&=\dfrac{1}{A_{\psi}}\displaystyle\int_{Spin(n)}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{+}}\psi^{a,\underline{b},s}(\underline{x})b_{k}T_{\psi}\left[f\right](a,\underline{b},s)\frac{da}{a^{n+1}}dV(\underline{b})ds\end{cases}$

Now we present the main result of this work. First we give

######  Theorem 3.3.

[16, 20, 5, 15] For
$f\in\mathcal{S}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))$

$\int_{\mathbb{R}^{n}}\ln\left|\underline{x}\right|\left|f(\underline{x})\right|^{2}dV(\underline{x})+\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{f}(\underline{\xi})\right|^{2}dV(\underline{\xi})\geq(\varphi(\frac{n}{4})+\ln(2))\left\|f\right\|_{2}^{2}$

where
$\varphi(t)=\frac{d}{dt}\ln\Gamma(t)=\frac{\Gamma^{\prime}(t)}{\Gamma(t)}$ is
the Digamma function.

######  Theorem 3.4.

Let $\psi\in L^{2}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))$ be an
admissible Clifford mother wavelet. Then for
$f\in\mathcal{S}(\mathbb{R}^{n},\mathbb{R}_{n},dV(\underline{x}))$

$\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{b}\right|\left|T_{\psi}\left[f\right](a,\underline{b},s)\right|^{2}dV(\underline{b})\frac{da}{a^{n+1}}ds$
$+\frac{A_{\psi}}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{f}(\underline{\xi})\right|^{2}dV(\underline{\xi})\geq(\frac{\Gamma^{\prime}(\frac{n}{4})}{\Gamma(\frac{n}{4})}+\ln(2))A_{\psi}\left\|f\right\|_{2}^{2}.$

To prove this result we need the following lemma.

######  Lemma 3.5.

$\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{T_{\psi}\left[f\right]}(a,\underline{\xi},s)\right|^{2}dV(\underline{\xi})\frac{da}{a^{n+1}}ds$
$=\frac{A_{\psi}}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{f}(\underline{\xi})\right|^{2}dV(\underline{\xi})$

###### Proof.

We have

$\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{T_{\psi}\left[f\right]}(a,\underline{\xi},s)\right|^{2}dV(\underline{\xi})\frac{da}{a^{n+1}}ds$
$=\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|a^{\frac{n}{2}}\overline{s}\left[\widehat{\psi}(a\overline{s}\underline{\xi}s)\right]^{\dagger}s\widehat{f}(\underline{\xi})\right|^{2}dV(\underline{\xi})\frac{da}{a^{n+1}}ds$
$=\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|a^{\frac{n}{2}}\overline{s}\left[\widehat{\psi}(a\overline{s}\underline{\xi}s)\right]^{\dagger}s\widehat{f}(\underline{\xi})$
$\left[a^{\frac{n}{2}}\overline{s}\left[\widehat{\psi}(a\overline{s}\underline{\xi}s)\right]^{\dagger}s\widehat{f}(\underline{\xi})\right]^{\dagger}dV(\underline{\xi})\frac{da}{a^{n+1}}ds$
$=\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|a^{\frac{n}{2}}\overline{s}\left[\widehat{\psi}(a\overline{s}\underline{\xi}s)\right]^{\dagger}$
$s\widehat{f}(\underline{\xi})\left[\widehat{f}(\underline{\xi})\right]^{\dagger}\overline{s}\widehat{\psi}(a\overline{s}\underline{\xi}s)sa^{\frac{n}{2}}dV(\underline{\xi})\frac{da}{a^{n+1}}ds.$

As $s\overline{s}=1$, we obtain

$\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{T_{\psi}\left[f\right]}(a,\underline{\xi},s)\right|^{2}dV(\underline{\xi})\frac{da}{a^{n+1}}ds$
$=\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left[\widehat{\psi}(a\overline{s}\underline{\xi}s)\right]^{\dagger}\widehat{f}(\underline{\xi})\left[\widehat{f}(\underline{\xi})\right]^{\dagger}\widehat{\psi}(a\overline{s}\underline{\xi}s)dV(\underline{\xi})\frac{da}{a}ds.$

we know from [2, 4, equation 3.5] that

$\int_{Spin(n)}\int_{\mathbb{R}^{+}}\left[\widehat{\psi}(a\overline{s}\underline{\xi}s)\right]^{\dagger}\widehat{\psi}(a\overline{s}\underline{\xi}s)\frac{da}{a}ds=\frac{A_{\psi}}{(2\pi)^{n}}.$

So, we get

$\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{T_{\psi}\left[f\right]}(a,\underline{\xi},s)\right|^{2}dV(\underline{\xi})\frac{da}{a^{n+1}}ds$
(3.2)
$=\frac{A_{\psi}}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{f}(\underline{\xi})\right|^{2}dV(\underline{\xi}.$

∎

###### Proof of theorem 3.4.

First we apply theorem 3.3 to the function $\underline{b}\longmapsto
T_{\psi}\left[f\right](a,\underline{b},s)$. So we obtain

$\int_{\mathbb{R}^{n}}\ln\left|\underline{b}\right|\left|T_{\psi}\left[f\right](a,\underline{b},s)\right|^{2}dV(\underline{b})+\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{T_{\psi}\left[f\right]}(a,\underline{\xi},s)\right|^{2}dV(\underline{\xi})$
$\geq(\frac{\Gamma^{\prime}(\frac{n}{4})}{\Gamma(\frac{n}{4})}+\ln(2))\left\|T_{\psi}\left[f\right]\right\|_{2}^{2}.$

Integrating by respect to the measure $\frac{da}{a^{n+1}}ds$ and taking in
account Lemma 3 and Formula (2.2) we obtain

$\int_{Spin(n)}\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{n}}\ln\left|\underline{b}\right|\left|T_{\psi}\left[f\right](a,\underline{b},s)\right|^{2}dV(\underline{b})\frac{da}{a^{n+1}}ds$
$+\frac{A_{\psi}}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\ln\left|\underline{\xi}\right|\left|\widehat{f}(\underline{\xi})\right|^{2}dV(\underline{\xi})\geq(\frac{\Gamma^{\prime}(\frac{n}{4})}{\Gamma(\frac{n}{4})}+\ln(2))A_{\psi}\left\|f\right\|_{2}^{2}$

∎

##  4\. Conclusion

In this paper, a logarithmic uncertainty principle based on the continuous
wavelet transform in the Clifford algebra’s settings has been formulated and
proved. Compared to existing formulations in Clifford framework, the proposed
uncertainty principle here is sharper.

##  5\. Data availability statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

##  6\. Conflict of interest statement

All authors declare that they have no conflicts of interest regarding the
present work.

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