# Deep Clustering based Fair Outlier Detection

Hanyu Song, Peizhao Li, Hongfu Liu  Michtom School of Computer Science,
Brandeis UniversityWalthamMAUSA<EMAIL_ADDRESS>
(2021)

###### Abstract.

In this paper, we focus on the fairness issues regarding unsupervised outlier
detection. Traditional algorithms, without a specific design for algorithmic
fairness, could implicitly encode and propagate statistical bias in data and
raise societal concerns. To correct such unfairness and deliver a fair set of
potential outlier candidates, we propose Deep Clustering based Fair Outlier
Detection (DCFOD) that learns a good representation for utility maximization
while enforcing the learnable representation to be subgroup-invariant on the
sensitive attribute. Considering the coupled and reciprocal nature between
clustering and outlier detection, we leverage deep clustering to discover the
intrinsic cluster structure and out-of-structure instances. Meanwhile, an
adversarial training erases the sensitive pattern for instances for fairness
adaptation. Technically, we propose an instance-level weighted representation
learning strategy to enhance the joint deep clustering and outlier detection,
where the dynamic weight module re-emphasizes contributions of likely-inliers
while mitigating the negative impact from outliers. Demonstrated by
experiments on eight datasets comparing to 17 outlier detection algorithms,
our DCFOD method consistently achieves superior performance on both the
outlier detection validity and two types of fairness notions in outlier
detection.

Outlier Detection; Fair Representation Learning; Deep Clustering on Outlier
Detection

††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the
27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining; August
14–19, 2021; Virtual Event, Singapore.††booktitle: Proceedings of the 27th ACM
SIGKDD Conference on Knowledge Discovery and Data Mining (KDD ’21), August
14–19, 2021, Virtual Event, Singapore††price: 15.00††isbn:
978-1-4503-8332-5/21/08††doi: 10.1145/3447548.3467225††ccs: Computing
methodologies Anomaly detection††ccs: Applied computing Law, social and
behavioral sciences

##  1\. Introduction

Utility-oriented machine learning systems have trickled down to the real-world
in aid of high-stake decision making in various fields. The inherent bias
brought by unbalanced data and embedded in those models, if not controlled
with specific algorithmic design, will inevitably backfire and deteriorate the
existing social barricades. Due to the rising societal concerns, fairness in
machine learning has received increasing attention in recent years
(Chouldechova and Roth, 2018; Hashimoto et al., 2018; Kearns et al., 2019; Li
et al., 2020b). Outlier detection is one of the most vulnerable domains in
face of the lurking algorithmic unfairness. The task is to find rare or
suspicious individuals that deviate from the majority. Due to the strong
correlation between the minority and outliers, an outlier detector might
easily suffer from discrimination against certain sensitive attributes like
ethnicity or gender. For instance, in credit risk assessment, credit card
applicants may suffer from a biased credit scoring algorithm against certain
sexuality (Evelina Nedlund, 2019); racial equality will deteriorate when
criminality detection heavily hinges on the object’s appearance (NEWS, 2020).
To recognize the significance of equality preservation, we need to address the
urgent demands of fairness-aware outlier detection methods. However, this
recently-emerged task has received little attention before 2020 and has not
been fully addressed yet.

On a colloquial formulation, unsupervised fair outlier detection aims to find
potential outlier candidates that substantially differ from the majority
instances while maintaining insignificant to sensitive attribute
subgroups111In the paper we use ”sensitive attribute subgroup”, ”sensitive
subgroup”, and ”protected subgroup” interchangeably. (i.e., gender or
ethnicity) (Davidson and Ravi, 2020). To our best knowledge, only two methods
exist along this direction and are both proposed in 2020. Deepak and Abraham
(2020) propose the FairLOF algorithm, which improves classical LOF (Breunig et
al., 2000) in regards to three heuristic principles: neighborhood diversity,
apriori distribution, and attribute asymmetry, to prevent an unfair sensitive
subgroup distribution among top-ranked outlier candidates. Shekhar et al.
(2020) design FairOD, a deep model using a standard autoencoder component for
self-reconstruction with the statistical parity and group fidelity as fairness
constraints to ensure the majority protected subgroup share the same outlier
rate as the minority one.

Although the above attempts have made pioneering progress in addressing this
newly emerging yet crucial problem, we see some limitations in the existing
work that could be further improved. FairLOF operates on the original feature
space, which limits its capacity towards detection accuracy and fairness
degree. Since the original feature space often does not expose much
information on the underlying data pattern, it adds difficulty to separate
anomalies from inliers. FairOD employs the classic autoencoder for
representation learning via minimizing the self-reconstruction loss and uses
it as the outlier detection criteria. However, such a deep embedding treats
all samples indifferently. It does not filter the noise and disruptions that
outliers may impose on the feature space, thus preventing itself from pursuing
high performance. Moreover, existing methods evaluate fairness degree only in
terms of the outlier rates in different groups, while ignoring the disparate
predictive validity between groups, which could also lead to unjust results.
Consider the case where an outlier detector returns a consistent $5\%$ outlier
rate for two sensitive subgroups, but the true positive rates for two groups
are $5\%$ and $10\%$, respectively. The detector yields a huge performance
penalty on the second subgroup, yet such a performance gap is unrecognizable
under the existing evaluative phase. Finally, existing methods mostly perform
evaluations on a limited number of benchmarks, including synthetic datasets.
An extensive exploration with more real-world datasets is heavily needed.

To solve the above challenges, we propose a novel Deep Clustering based Fair
Outlier Detection (DCFOD) method. Inspired by the reciprocal relationship
between outlier detection and clustering (Chawla and Gionis, 2013; Liu et al.,
2019), we employ a joint deep clustering/outlier detection framework and
propose weighted representation learning towards each instance. The degree of
outlierness for each individual is associated with the distance to its nearest
cluster centroid. The dynamic weight module enhances outlier detection
validity by re-emphasizing likely-inliers’ contributions while mitigating
outliers’ negative impacts. To simultaneously ensure fairness adaptation, we
utilize fairness-adversarial training with a min-max strategy to conceal
sensitive information and achieve group fairness, while preserving the feature
validity on the embedded representation. In terms of fairness evaluation, we
propose two metrics to measure the degree of fairness, based on the diagnostic
ability gap among subgroups and the subgroup distribution drift among top-
ranked outliers. To sum up, we underline our contributions as follows:

  * •

We address the newly-emerged fair outlier detection problem and propose a
novel Deep Clustering based Fair Outlier Detection (DCFOD) framework.

  * •

Our DCFOD adopts representation learning and fairness-adversarial training,
with a novel dynamic weight in regulation of negative impacts from outlier
points, to obtain a downstream task-favorable representation while
simultaneously ensuring improvement in fairness degree.

  * •

We strengthen the fairness measurements in the context of outlier detection by
proposing two fairness metrics that test subgroup-wise agnostic ability gap
and subgroup distribution drift in detected outliers.

  * •

Extensive experiments on eight real-world public datasets demonstrate evident
edge of DFCOD on all metrics in competitions with 17 unsupervised outlier
detection methods, including recently-proposed fair outlier detection methods
and other conventional outlier detection algorithms.

##  2\. Related Work

In this section, we illustrate the related work in terms of unsupervised
outlier detection, deep outlier detection, fair machine learning, and recent
advances in fair outlier detection.

Unsupervised Outlier Detection. Typical unsupervised outlier detection
algorithms calculate a continuous score for each data point to quantify its
outlier degree. Based on diverse assumptions, a number of outlier detection
methods have been proposed, including linear models (Shyu et al., 2003;
Schölkopf et al., 2001), proximity-based models (Breunig et al., 2000; Tang et
al., 2002; He et al., 2003; Goldstein and Dengel, 2012), and probability-based
models (Kriegel et al., 2008; Li et al., 2020c). Moreover, some studies purse
outlier detection by subspace learning (Kriegel et al., 2009), low-rank (Zhao
and Fu, 2015), matrix-completion (Kannan et al., 2017), and random walk (Pang
et al., 2016). Since most above-mentioned methods rely heavily on various
assumptions, ensemble-based outlier detectors are brought up to alleviate
assumption dependence, including iForest (Liu et al., 2008), bi-sampling
outlier detection (Liu et al., 2016), feature bagging (Lazarevic and Kumar,
2005), lightweight online detector of anomalies (Pevnỳ, 2016), and clustering
with outlier removal (Liu et al., 2019).

Deep Outlier Detection. Recent advances in representation learning (He et al.,
2016; Krizhevsky et al., 2017) have demonstrated that deep neural networks are
capable of extracting effective features for downstream tasks. The key in deep
outlier detection is to seek a self-supervised signal for representation
learning. Autoencoder (An and Cho, 2015) and variational autoencoder (Kingma
and Welling, 2013) are two widely used frameworks that optimize the self-
reconstruction loss. Their variants include robust autoencoder with $L_{1}$
sparse constraint (Zhou and Paffenroth, 2017), autoencoder ensemble (Chen et
al., 2017) and stacked autoencoder (Wan et al., 2019). Other state-of-the-art
deep approaches include random mapping (Wang et al., 2020), which predicts
proximity information in a randomly projected space; generative adversarial
networks (Zenati et al., 2018; Li et al., 2018), which focuses on generating
fake outliers by min-max training; predictability models (Liu et al., 2018),
which predicts the current data instances using the representations of the
previous instances within a temporal window. More details on deep outlier
detection can be found in this recent review (Pang et al., 2021).

Figure 1. Overview of Deep Clustering based Fair Outlier Detection. The model
conducts weighted representation learning with min-max training between the
feature extractor and fairness-adversarial subgroup discriminator to conceal
the protected subgroups on the learned representation and eventually achieve
subgroup-invariant outlier detection. Distance-based outlier scoring then
propagates weight adjustment to guide learning on the next iteration.

Fair Machine Learning and Fair Outlier Detection. According to various
application scenarios, abundant fairness notions have been proposed, including
individual fairness (Dwork et al., 2012), where similar individuals are
expected to be treated in a similar fashion; group fairness (Li et al., 2021;
Dwork et al., 2012; Kleinberg et al., 2016), where samples in different groups
ought to be treated equally; subgroup fairness (Kearns et al., 2018), which
extends group fairness by demanding fairness on combinations of groups (i.e.
subgroups); causality-based fairness (Kusner et al., 2017; Chiappa, 2019),
where causal inference on multiple features is conducted to track the
relationship between sensitive attributes and samples while mitigating
historical bias. Aside from fairness definitions, debiasing techniques can be
categorized into three main approaches: pre-processing (Feldman et al., 2015),
in-processing (Zemel et al., 2013; Madras et al., 2018) and post-processing
(Lohia et al., 2019). The deep representation learning is a typical in-
processing method that incorporates fairness constraints into the model
optimization objectives to obtain an ideal model parameterization that
maximizes performance and fairness (Caton and Haas, 2020). In particular,
representation learning with adversary has become a widely-used method in
recent years and has demonstrated effectiveness on multiple tasks, including
anonymization (Edwards and Storkey, 2015), clustering (Li et al., 2020b),
classification (Madras et al., 2018), transfer learning (Madras et al., 2018),
and domain adaptation (Tzeng et al., 2017; Li et al., 2020a).

Existing two algorithms in pursuit of fair outlier detection typically target
group fairness and contain a post-processing technique that seeks an outlier
ranking that is consistent with the baseline result. FairLOF (Deepak and
Abraham, 2020) is the first paper to address the fair outlier detection
problem, which incorporates a corrective term on the baseline LOF algorithm,
in regards to local sensitive subgroup diversity and global outlier alignment
with the baseline. FairOD (Shekhar et al., 2020) is another pioneering work,
which has recently been released on arXiv. This method targets an equal
outlier rate on the majority and minority sensitive subgroups. Specifically,
it employs an autoencoder (An and Cho, 2015) as the base outlier detector and
performs subgroup debiasing with statistical parity fairness constraint, while
maintaining fidelity to within-group rankings with respect to the baseline.

However, one concern the post-processing approach may have is the baseline
restricting the fair model’s detection validity, due to their aligned outlier
ranking. Therefore, we set to tackle the fair outlier detection problem with
an in-processing deep representation learning methodology, where we ensure
detecting validity on the learned task-friendly feature space while conducting
fairness-adversarial training to concurrently maintain group fairness.

##  3\. Fair Outlier Detection

Given a set of data instances with categorical sensitive attributes,
unsupervised fair outlier detection aims to seek top outlier candidates in a
sensitive attribute-invariant fashion. Specifically, we define the
unsupervised fair outlier detection problem as follows:

Problem Formulation (Unsupervised fair outlier detection). Given data points
$X=\\{x_{i}\\}_{i=1}^{N}$ in space $\mathcal{X}$ sampled i.i.d. from an
underlying distribution $\mathcal{D}$, with sensitive attribute
$S=\\{s_{i}\\}_{i=1}^{N}$ ranging in $M$ subgroup classes
$C=\\{c_{i}\\}_{i=1}^{M}$ and inaccessible ground truth outlier label $Y$, the
problem of unsupervised fair outlier detection aims to seek an outlier score
assignment $O=\\{o_{i}\\}_{i=1}^{N}$ that faithfully distinguishes the
underlying outliers from the majority inliers, while maintaining statistical
independence to sensitive subgroups,

(1) $~{}\Pr[o_{i}=o|s_{i}=c]=\Pr[o_{i}=o],\forall o\in\mathbb{R},c\in
C,i\in[1,N],$

and holding an accuracy parity between sensitive subgroups,

(2)
$~{}\mathbb{E}_{\mathcal{D}_{i}}[\left|Y-O\right||S=c_{i}]=\mathbb{E}_{\mathcal{D}_{j}}[\left|Y-O\right||S=c_{j}],\forall
i,j\in[1,M],$

Eq. (1) and (2) both reflect the targeted group fairness (Dwork et al., 2012;
Kleinberg et al., 2016). In Eq. (1), the probability distribution of $O$ is
invariant to sensitive subgroups; In Eq. (2), where $\mathcal{D}_{i}$ denotes
the underlying data distribution for subgroup $i$, the detection discrepancy
between the true labels and outlier scores ought to be the same for all
subgroups.

Existing methods (Deepak and Abraham, 2020; Shekhar et al., 2020) on fair
outlier detection achieve group fairness by balancing out the protected
subgroup diversity, which adds difficulty to outlier uncovering as it blurs
the underlying data pattern. To alleviate that downside, they preserve a
consistent outlier ranking with respect to a pre-obtained baseline result, yet
this may also impose restrictions on detection validity. Motivated by the
above limitations, we adopt the representation learning with adversarial
training framework, which guarantees the detection accuracy on a task-
favorable feature space, while achieving the expected group fairness by hiding
sensitive subgroup information on the learned representation.

##  4\. Methodology

We elaborate our proposed Deep Clustering based Fair Outlier Detection
(DCFOD), using symbols summarized in Table 1.

Table 1. Main notations and descriptions Notation | Type | Description
---|---|---
$N$ | Input | Number of instances
$M$ | Input | Number of subgroup classes
$D$ | Input | Dimension of embedded feature space
$K$ | Input | Number of clusters
$x_{i}$ | Input |  Vector of $i$-th data point
$s_{i}$ | Input |  Sensitive subgroup of $i$-th data point
$c_{i}$ | Learnable |  Cluster membership of $i$-th data point
$o_{i}$ | Learnable |  Outlier score of $i$-th data point
$w_{i}$ | Learnable |  Weight of $i$-th data point
$\mu_{k}$ | Learnable |  Vector of $k$-th centroid

###  4.1. Framework Overview

Unsupervised fair outlier detection can be decomposed into unsupervised
outlier detection and fair learning. Inspired by the huge progress in deep
neural networks, we tackle the above two problems under a same umbrella with
representation learning. Figure 1 shows our proposed DCFOD framework.
Administered by a dynamic weight, the representation learning of DCFOD
consists of a feature extractor and a fairness-adversarial counterpart. For
fairness-adaptation, we conduct a min-max training, where a sensitive subgroup
discriminator strives to predict instances’ subgroup category, while the
feature extractor undermining that prediction. Expected group fairness is
achieved when the prediction ceases to improve, due to the neutralization of
the subgroup information on the learned feature embedding. The dynamic weight
module secures the outlier detection validity amidst fair learning.
Specifically, we perform a joint clustering and outlier detection on the
embedded space. Given the strong coupling between clustering and outlier
detection, the well-discovered cluster structure fosters outlier uncovering,
while assisting the weight calculation. Data points close to their associated
centroids are assigned higher weights while those far away from centroids,
identified as potential outlier candidates, are assigned lower weights.
Therefore, inlier points with dominant weights capture the intrinsic cluster
structure, which, in turn, brings benefits to outlier detection. By this
means, we diminish outliers’ disturbance on representation learning, thus
guaranteeing the outlier diagnostic quality on the obtained representation.

###  4.2. Weighted Representation Learning

We denote the feature extractor and subgroup discriminator as $f$ and $g$,
where $f$ seeks a latent representation for samples $X$ and $g$ predicts its
sensitive subgroup membership, i.e.,
$f:\mathcal{X}~{}\rightarrow\mathcal{Z}\subseteq\mathbb{R}^{D}$,
$g:\mathcal{Z}~{}\rightarrow\mathbb{R}^{M}$. Furthermore, we formulate $K$
cluster centroids on the latent space: $\mu\subseteq\mathbb{R}^{K\times D}$,
and calculate outlier score $o_{i}$ with dynamic weight $w_{i}$ for each
instance $x_{i}$. Based on the above components, we establish the training
objective as three loss functions: self-reconstruction loss $\mathcal{L}_{s}$,
fairness-adversarial loss $\mathcal{L}_{f}$, and clustering regularization
$\mathcal{L}_{r}$. In what follows, we elaborate the formulations of each
component and the calculation of $o_{i}$ and $w_{i}$.

Self-reconstruction Loss. We employ a weighted autoencoder as the feature
extractor, which contains an encoder $f$ that seeks a feature representation
and a decoder $h:\mathcal{Z}~{}\rightarrow\mathcal{X}$ to reconstruct back to
the original space. The weight module $w_{i}$, which will be elaborated in
detail in Eq. (9), mitigates the negative impact from outliers’ abnormal
feature representations on model parameter training. $\mathcal{L}_{s}$, with a
L2 distance, is expressed as:

(3) $\mathcal{L}_{s}\coloneqq\sum_{i}^{N}w_{i}\cdot\|x_{i}-h\circ
f(x_{i})\|_{2}^{2},$

where $\circ$ denotes the function composition.

Fairness-adversarial Loss. Furthermore, the learned representation is also
expected to be sensitive subgroup-invariant. To achieve this, we employ a
sensitive attribute subgroup discriminator $g$ predicting each data point’s
likelihood of $M$ subgroup categories in the latent space. $\mathcal{L}_{f}$
records the predicting loss as follows:

(4) $\mathcal{L}_{f}\coloneqq\sum_{i}^{N}w_{i}\cdot\ell(g\circ f(x_{i}),\
s_{i}),$

where $\ell$ is the cross-entropy loss and $s_{i}$ is the true subgroup class
for $x_{i}$. During the fairness min-max training, $g$ and $f$ are trained in
opposite directions. Specifically, $g$ aims to accurately predict protected
subgroups via minimizing $\mathcal{L}_{f}$, while $f$, in adversary, maximizes
the loss by hiding protected information on the feature embedding. As training
progresses, the obtained feature representation becomes subgroup-invariant for
fair outlier detection.

Clustering Assignment and Regularization. To perform joint clustering and
outlier detection, we first obtain a soft clustering assignment $P$ on the
feature embedding. Specifically, we calculate the probability of assigning
each $f(x_{i})\in\mathcal{Z}$ to each of the $K$ clusters based on its
distance to the centroids with Student’s t-distribution:

(5)
$~{}p_{ik}\coloneqq\frac{(1+\|f(x_{i})-\mu_{k}\|_{2}^{2})^{-1}}{\sum_{j}^{K}(1+\|f(x_{i})-\mu_{j}\|_{2}^{2})^{-1}},$

where $p_{ik}$ is the probability of assigning the $i$-th data point to the
$k$-th cluster, and $\mu_{k}$ denotes the $k$-th cluster centroid. The degree
of freedom for the t-distribution is 1.

To strengthen the cluster predictive confidence, we follow Xie et al. (2016)
and employ a clustering regularizer, which computes an auxiliary distribution
$Q$ based on the soft clustering assignment $P$ as follows:

(6)
$~{}q_{ik}\coloneqq\frac{p_{ik}^{2}/\sum_{i}^{N}p_{ik}}{\sum_{j}^{K}p_{ij}^{2}/\sum_{i}^{N}p_{ij}}.$

In this way, the auxiliary distribution enlarges the distinctions between
predictions on different clusters. The clustering regularizer then pulls $P$
toward $Q$ by minimizing their KL divergence. Similar to the above losses,
$\mathcal{L}_{r}$ is also regulated by the weight module:

(7)
$\mathcal{L}_{r}\coloneqq\sum_{i}^{N}w_{i}\cdot\textup{D}_{\textup{KL}}(P\|Q)=\sum_{i}^{N}\sum_{k}^{K}w_{i}\cdot
p_{ik}\cdot\log\frac{p_{ik}}{q_{ik}}.$

Algorithm 1 DCFOD

0: Data point $x_{i}$ with subgroup membership $s_{i}$, $i=1$ to $N$; Cluster
number $K$. Learning rate $\eta$.

0: Continuous outlier score $o_{i}$, $i=1$ to $N$.

1: Initialize parameters for $f,g$, and $h$;

2: $\mu\leftarrow Kmeans([f(x_{1}),\cdots,f(x_{N})],K)$;

3: while model not converged do

4: Calculate the soft assignment $p_{ik}$ and the auxiliary distribution
$q_{ik}$ by Eq. (5) and (6);

5: Calculate the outlier score of each data point $o_{i}$ by Eq. (8);

6: Calculate the weight of each data point $w_{i}$ by Eq. (9);

7: Calculate $\bigtriangledown\mathcal{L}_{f}$ with $w_{i},s_{i}$, $f$, and
$g$;

8: Calculate
$\bigtriangledown(\mathcal{L}_{s}+\mathcal{L}_{r}-\mathcal{L}_{f})$ with
$w_{i}$, $p_{ik}$, $q_{ik}$, $s_{i}$, $f$, $g$, $h$ and $\mu$;

9: Update $g\leftarrow g-\eta\bigtriangledown\mathcal{L}_{f}$;

10: Update
$(f,h,\mu)\leftarrow(f,h,\mu)-\eta\bigtriangledown(\mathcal{L}_{s}+\mathcal{L}_{r}-\mathcal{L}_{f})$;

11: end while

12: return Outlier score $o_{i}$ for $i=1$ to $N$.

Outlier Score. Outliers are conceptually defined by clusters as data points
that deviate from the majority cluster structure. Based on that, we define
outlierness as the within-cluster distance ratio to the centroid.
Specifically, $o_{i}$ is computed by the Euclidean distance between $f(x_{i})$
and its closest centroid, normalized by that of the furthest point belonging
to the same centroid:

(8)
$~{}o_{i}\coloneqq\frac{\min_{k\in[1,K]}\|f(x_{i})-\mu_{k}\|_{2}}{\max_{j\in[1,N]}\mathbb{I}_{m_{j}=k}\cdot\|f(x_{j})-\mu_{k}\|_{2}},$

where $\mathbb{I}$ is an indicator function, and $m_{j}$ is the most probable
cluster membership for the $j$-th point, based on clustering distribution $P$.

Weight Module. To mitigate negative impacts from outliers, we guide the
representation learning with an instance-level weight module $w_{i}$, where we
deemphasize outliers with lower weights compared to inliers. Specifically,
$w_{i}$ in $\mathcal{L}_{s}$ alleviates the disruption in feature extractor’s
parameter updating due to outliers’ abnormal feature representations; $w_{i}$
in $\mathcal{L}_{r}$ prevents strengthening the predictive confidence for
outliers, as conceptually, they do not belong to any cluster; For $w_{i}$ in
$\mathcal{L}_{f}$, the idea is to increase focus in subgroup debiasing on
potential outlier candidates.

Therefore, with the above $o_{i}$, we compute the dynamic weight mechanism
that governs the designated components in our objective as follows:

(9) $~{}w_{i}\coloneqq\frac{e^{-o_{i}}}{\sum_{j}^{N}e^{-o_{j}}},$

which is essentially a Softmax function imposed on the negative values of
outlier scores. We design $w_{i}$ such that it is in negative correlation with
$o_{i}$, while keeping $\sum_{i}^{N}w_{i}=1$, a global balance that avoids
weights excessively overvalued.

In conclusion, the overall objective function of our framework can be written
as the following minimax optimization:

(10) $\begin{split}\max_{f,\ h,\
\mu}\quad\beta\mathcal{L}_{f}-\alpha\mathcal{L}_{s}-\mathcal{L}_{r},\quad\min_{g}\quad\mathcal{L}_{f},\end{split}$

where $\alpha$ and $\beta$ are hyperparameters that control the balance in the
objective function. In the end, our method returns a set of outlier scores
that maintains detection validity and group fairness. The complete process of
DCFOD is shown in Algorithm 1.

##  5\. Experiment

In this section, we first introduce the experimental settings, datasets and
evaluative metrics for outlier validation and degree of fairness. Then we
compete our model with 17 unsupervised outlier detection methods on eight
public datasets. Finally, we conduct a factor exploration on our model for an
in-depth analysis.

###  5.1. Experimental Setup

Datasets. We choose eight UCI 222https://archive.ics.uci.edu/ml/datasets.php
datasets in diverse fields, including credit rating, personal income, academic
performance, drug review, and clinical screening. Table 2 shows detailed
characteristics of these datasets. All datasets except german contain either
Gender or Race as the sensitive attribute, while that of german is provided as
a combination of gender and marital status. The Outlier Definition column
shows the benchmark outlier criteria for each dataset, which is used as the
ground truth for outlier validation. The numbers of instances in these
datasets vary from 704 to nearly 300,000, and the outlier percentage ranges
from 6.2% to 30%.

Competitive Methods. We compare our model with the fair outlier detectors
FairLOF (Deepak and Abraham, 2020), FairOD (Shekhar et al., 2020), and 15
conventional unsupervised outlier detection methods, including linear models:
Principal Component Analysis (PCA) (Shyu et al., 2003), One-class Support
Vector Machine (OCSVM) (Schölkopf et al., 2001); proximity-based models: Local
Outlier Factor (LOF) (Breunig et al., 2000), Connectivity-Based Outlier Factor
(COF) (Tang et al., 2002), Clustering Based Local Outlier Factor (CBLOF) (He
et al., 2003); probability-based models: Fast angle-based Outlier Detector
(FABOD) (Kriegel et al., 2008), Copula Based Outlier Detector (COPOD) (Li et
al., 2020c); ensemble-based models: Feature Bagging (FB) (Lazarevic and Kumar,
2005), iForest (Liu et al., 2008), Lightweight On-line Detector of Anomalies
(LODA) (Pevnỳ, 2016), Clustering with Outlier Removal (COR) (Liu et al.,
2019); neural networks: AutoEncoder (AE) (An and Cho, 2015), Variational Auto
Encoder (VAE) (Kingma and Welling, 2013), Random Distance Prediction (RDP)
(Wang et al., 2020).

Implementation. We implement FairLOF, FairOD in PyTorch, build RDP, COR based
on their public source code, and other classical outlier detection methods are
implemented with PyOD (Zhao et al., 2019). In practice, we use the following
default settings. The hyperparameters for FairLOF and FairOD strictly follow
their literature. The number of nearest neighbors for LOF, COF, and FABOD are
set to 20; the number of clusters for CBLOF and COR are set to 10; the number
of bins and random cuts for LODA is 10 and 100, respectively; the number of
base estimators and sub-sampling size for iForest are 100 and 256
respectively; the base estimator for feature bagging is LOF, its number of
estimators is 10 and its sub-sampling size equals to the sample size. For AE
and VAE, the hidden neuron structures are 64-32-32-64 and 128-64-32-32-64-128.
In cases when the feature dimension is smaller than 64, we modify their
structures into 16-8-8-16 and 16-8-4-4-8-16, accordingly. Both of them run for
100 epochs with a batch size of 32.

Table 2. Characteristics of datasets

Dataset | #Instances | #Features | Domain | Sensitive Attribute | Outlier Definition | %Outlier
---|---|---|---|---|---|---
student | 1045 | 33 | Academic | Gender | Final grade $\leq$ 6 | 9.58%
asd | 704 | 21 | Clinical | Gender | Should pursue diagnosis | 26.84%
obesity | 2111 | 17 | Clinical | Gender | Insufficient weight | 13.60%
cc | 30000 | 24 | Credit | Gender | Credit default | 22.12%
german | 1000 | 21 | Credit | Marital & Gender | Good credit | 30.00%
drug | 1885 | 13 | Drug | Gender | Used within last week | 8.44%
adult | 48842 | 14 | Income | Race | Income ¿ 50K | 23.93%
kdd | 299285 | 40 | Income | Race | Income ¿ 50K | 6.2%

Table 3. Outlier validation comparison of 18 algorithms on 8 benchmark
datasets by $AUC$

Dataset | Fair Model | Linear Model | Proximity Based | Probability Based | Ensemble Based | Neural Networks | Ours
---|---|---|---|---|---|---|---
FairLOF | FairOD | PCA | OCSVM | LOF | COF | CBLOF | FABOD | COPOD | FB | iForest | LODA | COR | AE | VAE | RDP | DCOD | DCFOD
student | 0.662 | 0.717$\pm$0.123 | 0.718 | 0.848 | 0.777 | 0.710 | 0.761 | 0.834 | 0.750 | 0.804 | 0.679 | 0.794$\pm$0.054 | 0.568 | 0.728 | 0.719 | 0.834$\pm$0.021 |  0.912$\pm$0.011 |  0.921$\pm$0.013
asd | 0.543 | 0.642$\pm$0.214 | 0.630 | 0.707 | 0.523 | 0.491 | 0.387 | 0.469 | 0.811 | 0.521 | 0.542 | 0.524$\pm$0.152 | 0.857 | 0.627 | 0.629 | 0.695$\pm$0.037 |  0.939$\pm$0.038 |  0.945$\pm$0.032
obesity | 0.428 | 0.514$\pm$0.167 | 0.720 | 0.675 | 0.355 | 0.377 | 0.602 | 0.540 | 0.742 | 0.383 | 0.757 | 0.572$\pm$0.134 | 0.552 | 0.719 | 0.719 | 0.612$\pm$0.021 |  0.800$\pm$0.029 |  0.825$\pm$0.017
cc | 0.465 | 0.554$\pm$0.016 | 0.604 | 0.621 | 0.451 | 0.465 | 0.579 | 0.531 | 0.557 | 0.451 | 0.594 |  0.605$\pm$0.001 | 0.546 | 0.604 | 0.604 | 0.597$\pm$0.011 | 0.571$\pm$0.018 | 0.566$\pm$0.024
german | 0.569 | 0.570$\pm$0.021 | 0.546 | 0.584 | 0.589 | 0.568 | 0.581 | 0.589 | 0.547 | 0.582 | 0.546 | 0.506$\pm$0.034 | 0.475 | 0.546 | 0.546 | 0.581$\pm$0.030 | 0.558$\pm$0.016 | 0.559$\pm$0.013
drug | 0.543 | 0.551$\pm$0.035 | 0.562 | 0.562 | 0.546 | 0.534 | 0.555 | 0.562 | 0.556 | 0.544 | 0.570 | 0.521$\pm$0.055 | 0.522 | 0.560 | 0.562 | 0.567$\pm$0.013 |  0.588$\pm$0.031 |  0.586$\pm$0.021
adult | 0.486 |  0.605$\pm$0.022 | 0.473 | 0.525 | 0.488 | 0.479 | 0.474 | 0.471 | 0.488 | 0.438 | 0.426 | 0.504$\pm$0.095 | 0.442 | 0.473 | 0.473 | 0.541$\pm$0.014 | 0.487$\pm$0.072 |  0.592$\pm$0.051
kdd | N/A∗ | 0.706$\pm$0.003 | 0.661 | 0.630 | 0.549 | N/A | 0.687 | 0.519 | 0.686 | 0.566 | 0.589 | 0.472$\pm$0.114 | 0.424 | 0.661 | 0.661 | 0.675$\pm$0.000 |  0.752$\pm$0.007 |  0.747$\pm$0.011
$Score_{AUC}$ | 0.751 | 0.852 | 0.854 | 0.892 | 0.752 | 0.737 | 0.816 | 0.792 | 0.882 | 0.751 | 0.786 | 0.751 | 0.821 | 0.855 | 0.854 | 0.886 | 0.953 | 0.979
∗N/A means out-of-memory error. For this method, the according dataset is not
included in the $Score$ measurement calculation.

Table 4. Degree of fairness comparison of 18 algorithms on 8 benchmarks by
$F_{Gap}$

Dataset | Fair Model | Linear Model | Proximity Based | Probability Based | Ensemble Based | Neural Networks | Ours
---|---|---|---|---|---|---|---
FairLOF | FairOD | PCA | OCSVM | LOF | COF | CBLOF | FABOD | COPOD | FB | iForest | LODA | COR | AE | VAE | RDP | DCOD | DCFOD
student | 0.008 | 0.043$\pm$0.027 | 0.053 | 0.042 | 0.001 | 0.069 | 0.100 | 0.011 | 0.049 | 0.009 | 0.073 | 0.060$\pm$0.040 | 0.003 | 0.049 | 0.053 | 0.028$\pm$0.006 | 0.020$\pm$0.013 | 0.012$\pm$0.011
asd | 0.013 | 0.024$\pm$0.019 | 0.034 | 0.040 | 0.032 | 0.016 | 0.068 | 0.004 | 0.016 | 0.004 | 0.013 | 0.055$\pm$0.036 | 0.030 | 0.034 | 0.034 | 0.019$\pm$0.010 | 0.034$\pm$0.013 | 0.031$\pm$0.011
obesity | 0.024 | 0.013$\pm$0.009 | 0.184 | 0.064 | 0.069 | 0.094 | 0.114 | 0.136 | 0.239 | 0.035 | 0.163 | 0.150$\pm$0.125 | 0.362 | 0.181 | 0.184 | 0.047$\pm$0.008 |  0.007$\pm$0.006 |  0.004$\pm$0.004
cc | 0.001 | 0.004$\pm$0.003 | 0.005 | 0.012 | 0.010 | 0.016 | 0.006 | 0.007 | 0.005 | 0.004 | 0.001 | 0.005$\pm$0.000 | 0.003 | 0.005 | 0.005 |  0.001$\pm$0.001 | 0.006$\pm$0.003 | 0.005$\pm$0.004
german | 0.076 | 0.098$\pm$0.041 | 0.126 | 0.157 | 0.118 | 0.151 | 0.109 | 0.134 | 0.103 | 0.098 | 0.071 | 0.255$\pm$0.081 | 0.186 | 0.123 | 0.126 | 0.069$\pm$0.021 |  0.068$\pm$0.029 |  0.067$\pm$0.030
drug | 0.051 | 0.049$\pm$0.027 | 0.036 | 0.059 | 0.059 | 0.030 | 0.211 | 0.114 | 0.076 | 0.101 | 0.078 | 0.046$\pm$0.042 | 0.110 | 0.034 | 0.036 | 0.025$\pm$0.011 |  0.014$\pm$0.010 |  0.009$\pm$0.008
adult | 0.151 | 0.059$\pm$0.003 | 0.103 | 0.040 | 0.214 | 0.203 | 0.071 | 0.081 | 0.058 | 0.212 | 0.061 | 0.126$\pm$0.050 | 0.167 | 0.103 | 0.103 | 0.051$\pm$0.012 |  0.048$\pm$0.020 | 0.049$\pm$0.015
kdd | N/A |  0.042$\pm$0.004 | 0.152 | 0.078 | 0.136 | N/A | 0.052 | 0.243 | 0.047 | 0.146 | 0.152 | 0.102$\pm$0.061 | 0.087 | 0.152 | 0.152 | 0.015$\pm$0.003 | 0.054$\pm$0.005 |  0.032$\pm$0.009
$Score_{F}$ | 0.328 | 0.180 | 0.123 | 0.100 | 0.227 | 0.116 | 0.099 | 0.210 | 0.140 | 0.251 | 0.092 | 0.251 | 0.254 | 0.126 | 0.123 | 0.300 | 0.277 | 0.402

We implement our DCFOD in PyTorch333Source code: https://github.com/brandeis-
machine-learning/FairOutlierDetection. In order to showcase the fairness
adaptation effect from the adversarial training, we also compare DCFOD with
its independent derivative, DCOD. Other than the absence of subgroup
discriminator on DCOD, the two versions share the same implementation.
Specifically, we set the number of clusters to 10 and dimension of embedded
space to 64. For datasets with sample size greater than 10,000, we set the
epoch to 40 and minibatch size to 256 to accelerate training process,
otherwise to 90 and 64 respectively. Following Xie et al. (2016), we set the
network structure as drop-$n$-500-500-2000-64 and 64-2000-500-500-$n$ for
encoder and decoder, where drop is a dropout layer with rate 0.1. We set the
structure for the discriminator to be 64-500-500-2000-$m$, and the cluster
centroid layer $\mu$ to be of size (10 * 64). We perform weight initialization
on all linear layers with Xavier uniform initialization (Glorot and Bengio,
2010). Hyperparameter $\alpha$ and $\beta$ are set to 8 and 100 for all
datasets. The starting learning rate for encoder, decoder and discriminating
layer is set to $1e^{-5}$, that of the centroid layer is set to $1e^{-4}$.
Learning rates are multiplied by a factor of 0.1 for every 30 epochs. In the
embedded space, we initialize the cluster centroids with MiniBatchKmeans
(Sculley, 2010) implemented in sklearn.

During evaluation process, we report the mean and standard deviation for non-
deterministic methods by repeating their experiments on 20 random seeds. AE
and VAE are not reported as such, since their fluctuations are small enough to
be neglected.

Metrics. Since all the outlier detectors return one continuous set of score
and all associated datasets provide the ground truth outlier criteria, we use
Area Under Receiver Operating Characteristic Curve (AUC) to evaluate the
outlier diagnostic ability. To measure degree of group fairness, we propose
$F_{Gap}$ and $F_{Rank}$, based on the subgroup-wise diagnostic ability gap
and the subgroup distribution drift among top-ranked outliers, respectively.
Moreover, to thoroughly compare the performance of different methods across
datasets, we also propose $Score_{AUC}$ and $Score_{F}$ functions that
aggregate the $AUC$ score and fairness measurements on all datasets.

In specific, AUC evaluates the degree of alignment between the outlier score
and the ground truth label under varying thresholds:

(11)
$\begin{split}AUC\coloneqq&1-\frac{1}{|\mathcal{T}^{+}||\mathcal{T}^{-}|}\sum_{x^{+}\in\mathcal{T}^{+}}\sum_{x^{-}\in\mathcal{T}^{-}}\\\
&\left(\mathbb{I}(O(x^{+})<O(x^{-}))+\frac{1}{2}\mathbb{I}(O(x^{+})=O(x^{-}))\right),\end{split}$

where $\mathcal{T},\mathcal{T}^{+}$, and $\mathcal{T}^{-}$ represent the total
sample, true inliers and outliers, respectively, $\mathbb{I}$ is an indicator
function and $O(x)$ is the outlier score of data point $x$ given by one
outlier detector. In practice, we use $roc\\_auc\\_score$ in
sklearn.metrics444https://scikit-learn.org to calculate AUC.

For degree of fairness, $F_{Gap}$ evaluates the diagnostic validity gap
between different protected subgroups, where a smaller gap indicates less
group bias. Specifically, we calculate the $AUC$ score for each subgroup, and
derive $F_{Gap}$ with the difference between the highest $AUC$ and the lowest:

(12) $\displaystyle F_{Gap}\coloneqq\max_{s\in
S}AUC(\mathcal{T}_{s})-\min_{s^{\prime}\in S}AUC(\mathcal{T}_{s^{\prime}}),$

where $S$ is the sensitive subgroup set, and $\mathcal{T}_{s}$ represents the
sample in subgroup $s$. $AUC(\mathcal{T}_{s})$ denotes the $AUC$ score of
group $s$.

Figure 2. Execution time comparison on 8 benchmarks.

Beyond the performance-related fairness, it is crucial to measure whether
sensitive subgroups are distributed fairly among detected outliers. $F_{Rank}$
achieves that goal by calculating the KL-divergence between the subgroup
distribution in top outlier candidates in the output and that of the dataset,
which is a reference distribution:

(13) $\displaystyle F_{Rank}\coloneqq\max_{r\in
R}\textup{D}_{\textup{KL}}(d_{r}\|\tilde{d}),\quad R=\\{5,6,7,...,20\\},$

where we rank all output scores from the highest to lowest, with $d_{r}$
representing the subgroup distribution among the top $r\%$ outlier candidates
derived by the outlier detector, and $\tilde{d}$ being the fixed reference
distribution of the dataset. Specifically, $F_{Rank}$ records the most
significant distribution drift in a range of top $5\%$ to $20\%$ outliers.
Both $F_{Gap}$ and $F_{Rank}$ are negative measurements, where the smaller
value indicates better fairness degree.

To better compare model performance under the variability on different
datasets, we propose two scoring functions to produce one score for each
algorithm on each metric, indicating its aggregated performance across all
datasets. We denote the collection of algorithms and datasets with $A,T$, and
$A^{(i)},T^{(j)}$ represent the $i$-th algorithm and $j$-th dataset,
respectively. For the diagnostic validity metric $AUC$, we propose
$Score_{AUC}$ as the following:

(14) $\displaystyle
Score_{AUC}(A^{(i)})\coloneqq\frac{1}{J}\sum_{j}\frac{AUC(A^{(i)},T^{(j)})}{\max_{i}AUC(A^{(i)},T^{(j)})},$

where $AUC(A^{(i)},T^{(j)})$ is the $AUC$ of algorithm $A^{(i)}$ on dataset
$T^{(j)}$, and $J$ is the total number of datasets. We then propose
$Score_{F}$ for the two fairness metrics $F_{Gap}$ and $F_{Rank}$, both
denoted as $F$:

(15) $\displaystyle
Score_{F}(A^{(i)})\coloneqq\frac{1}{J}\sum_{j}\frac{\min_{i}F(A^{(i)},T^{(j)})+\epsilon}{F(A^{(i)},T^{(j)})+\epsilon},$

where $\epsilon=1e^{-5}$. In essence, $Score$ functions rescale numerical
results by normalizing each evaluative value using the best result among all
algorithms on the same dataset, then average over the scaled value on all
datasets. In the end, higher values of $Score_{AUC}$ and $Score_{F}$ indicate
better diagnostic validity and more fairness.

Table 5. Degree of fairness comparison of 18 algorithms on 8 benchmarks by
$F_{Rank}$

Dataset | Fair Model | Linear Model | Proximity Based | Probability Based | Ensemble Based | Neural Networks | Ours
---|---|---|---|---|---|---|---
FairLOF | FairOD | PCA | OCSVM | LOF | COF | CBLOF | FABOD | COPOD | FB | iForest | LODA | COR | AE | VAE | RDP | DCOD | DCFOD
student | 0.003 | 0.016$\pm$0.010 | 0.009 | 0.067 | 0.004 | 0.030 | 0.007 | 0.008 | 0.057 | 0.008 | 0.006 | 0.066$\pm$0.070 | 0.206 | 0.007 | 0.009 | 0.010$\pm$0.005 | 0.011$\pm$0.011 | 0.007$\pm$0.003
asd | 0.012 | 0.030$\pm$0.026 | 0.023 | 0.013 | 0.013 | 0.005 | 0.023 | 0.019 | 0.030 | 0.007 | 0.006 | 0.087$\pm$0.236 | 0.030 | 0.023 | 0.023 | 0.016$\pm$0.011 | 0.013$\pm$0.012 | 0.011$\pm$0.010
obesity | 0.031 | 0.008$\pm$0.007 | 0.013 | 0.030 | 0.015 | 0.018 | 0.073 | 0.009 | 0.067 | 0.009 | 0.011 | 0.057$\pm$0.187 | 0.669 | 0.012 | 0.013 |  0.003$\pm$0.001 | 0.008$\pm$0.007 |  0.005$\pm$0.003
cc | 0.001 |  0.001$\pm$0.000 | 0.010 | 0.010 | 0.004 | 0.003 | 0.014 | 0.012 | 0.026 | 0.002 | 0.024 | 0.049$\pm$0.001 | 0.112 | 0.010 | 0.010 |  0.002$\pm$0.000 |  0.001$\pm$0.000 |  0.001$\pm$0.000
german | 0.101 | 0.038$\pm$0.017 | 0.014 | 0.105 | 0.102 | 0.046 | 0.061 | 0.054 | 0.024 | 0.148 | 0.042 | 0.029$\pm$0.560 | 0.255 | 0.014 | 0.015 | 0.016$\pm$0.002 | 0.048$\pm$0.018 | 0.044$\pm$0.022
drug | 0.003 | 0.007$\pm$0.006 | 0.003 | 0.009 | 0.010 | 0.001 | 0.002 | 0.005 | 0.009 | 0.004 | 0.009 | 0.093$\pm$0.324 | 0.091 | 0.003 | 0.003 | 0.005$\pm$0.002 | 0.008$\pm$0.006 | 0.008$\pm$0.008
adult | 0.131 |  0.001$\pm$0.000 | 0.483 | 0.009 | 0.154 | 0.153 | 0.071 | 0.244 | 0.662 | 0.045 | 0.562 | 0.073$\pm$0.344 | 0.014 | 0.483 | 0.483 |  0.001$\pm$0.000 |  0.001$\pm$0.000 |  0.001$\pm$0.000
kdd | N/A |  0.001$\pm$0.000 | 0.185 | 0.022 | 0.023 | N/A | 0.016 | 0.003 | 0.235 | 0.057 | 0.140 | 0.095$\pm$0.112 | 0.047 | 0.185 | 0.185 |  0.000$\pm$0.000 |  0.000$\pm$0.000 |  0.000$\pm$0.000
$Score_{F}$ | 0.370 | 0.236 | 0.279 | 0.107 | 0.215 | 0.391 | 0.179 | 0.191 | 0.126 | 0.246 | 0.084 | 0.246 | 0.276 | 0.292 | 0.279 | 0.325 | 0.346 | 0.442

Environment. Experiments were run on a 12 core processor Linux server with
Intel Core<EMAIL_ADDRESS>Ghz and 64GB RAM.

(a) asd

(b) german

Figure 3. Weight and score trends of inliers and outliers during iterations on
asd and german.

###  5.2. Performance

Table 3, 4 and 5 show the performance comparison of 18 algorithms on 8
benchmarks in terms of $AUC$, $F_{Gap}$ and $F_{Rank}$, where the best and
second best results are highlighted in red bold and blue italic.

For outlier validation validity, different algorithms achieve the best results
on different datasets. OCSVM gets 0.621 on cc, LOF and FABOD achieve 0.589 on
german. Although these algorithms are hardly considered state-of-the-art, they
are still effective in some scenarios. FairOD gets a 0.605 on adult, which
shows its fairness adaptation can also bring high detection accuracy in
certain cases. Our baseline method DCOD delivers two best results out of eight
datasets, while our proposed DCFOD provides three best results and excels
other competitive methods by a large margin on the comprehensive measurement
$Score_{AUC}$. The demonstrated superior performance on DCFOD comes from two
main aspects: (1) our model adopts a well-generalized clustering based
framework, which holds a strong coupling with the unsupervised outlier
detection task. (2) The weighted representation learning is capable of
generating a feature embedding that captures the unique intrinsic cluster
structure for each dataset. In particular, the dynamic weight module mitigates
the influences from outliers’ abnormal representations and fosters a task-
favorable feature space.

For fairness measurements, DCFOD has the highest $Score_{F}$ values on both
$F_{Gap}$ and $F_{Rank}$. Specifically, our model delivers the best results on
obesity, german, drug and kdd by $F_{Gap}$, where the adversarial training
shrinks the detection validity gap between subgroups by an average of 30$\%$
compared to DCOD. Other competitive methods, including FairLOF, OCVSM, LOF,
FB, iForest, and RDP, share some best results on other datasets. Similar
phenomena occurs on $F_{Rank}$, as shown in Table 5, where DCFOD achieves the
most fair performance on cc, adult, and kdd. On the rest datasets, the
adversarial training also generates a fairer subgroup distribution compared to
DCOD and improves $F_{Rank}$ by an average of $10\%$.

(a) AUC

(b) $F_{Gap}$

Figure 4. Effectiveness of weight adjustment on AUC and fairness measurement
$F_{Gap}$.

(a) student

(b) asd

(c) obesity

(d) drug

Figure 5. Hyperparameter analysis for $\alpha$ and $\beta$. (a) & (b) The
trends of AUC and $F_{Gap}$ on student and asd, while $\alpha$ ranging from
[2, 4, 6, 8, 10, 15, 20, 25, 30, 35, 40, 45, 50] and $\beta$ pinned at 100.
(c) & (d) Pareto diagram of AUC and $F_{Gap}$ for all competitive methods on
obesity and drug. $\beta$ ranges from [0, 1, 10, 20, 50, 80, 100, 150, 200,
250, 300], while $\alpha$ pinned at 8. Points with $\beta=0$ and $\beta=100$
are annotated with arrows, which are the model w/o adversarial training and
the pareto front, respectively.

We then discuss FairLOF and FairOD, two recently proposed fair outlier
detectors. They set out to sacrifice certain degrees of detection validity in
exchange for more fairness compared to their base methods LOF and autoencoder,
which has been demonstrated in our experiment. Compared to LOF, FairLOF has a
lower AUC with improved degrees of fairness on both $F_{Gap}$ and $F_{Rank}$,
while FairOD has a slightly lower AUC score compared to that of AE, with a
similar $F_{Rank}$ and improved $F_{Gap}$. The above models both apply a post-
processing technique that seeks a consistent outlier ranking with a baseline
detector, whose capacity might restrict the fair model from pursuing a higher
detection accuracy. In contrast, DCFOD adopts a baseline-free, in-processing
procedure to learn a fair feature representation where the detection accuracy
is simultaneously optimizable. As shown in Table 3, the fairness-adapted DCFOD
surpasses DCOD on $AUC$, which demonstrates that the goals of accurate
diagnosis and fairness adaptation may be beneficial to each other in a proper
feature space. Based on the above illustrations, with the representation
learning framework and fairness-adversarial training, the legitimacy of
DCFOD’s superior performance on both the detection validity and group fairness
is well demonstrated.

We compare the execution time of all methods in Figure 2. On datasets with
less than 50,000 instances, all algorithms except COF have fast performances.
Several maintain a low running time across all datasets, shown at the bottom
lines in the figure. As sample size increases, FairLOF and COF encounter an
out-of-memory error, and several other methods dramatically increase their
running time. Meanwhile, DCFOD remains stable and keeps a flat increase rate
due to its linear time complexity to the sample size $N$ and parameters $D,K$.

###  5.3. Factor Exploration

We hereby explore the impact of critical factors in DCFOD. We look into the
weight module for its quantitative effects on model training and performance,
then test model volatility on hyperparameters. We use $AUC$ and $F_{Gap}$
performance for visualization due to the space limitation.

Weight and Outlier Score. The proposed DCFOD model assigns a dynamic weight
for each instance during training. The likely-outlier candidates that are far
away from the cluster centroids have higher outlier scores while assigned with
lower weights. With this approach, we alleviate outliers’ disturbance toward
the overall cluster structure and improve the model performance. Figure 3
shows the dynamic relationship between outlier scores and weight adjustments
on asd and german. For each dataset, we choose three inliers and two outliers
for demonstration, where consistent trends are exhibited. After several
iterations, the weights and scores stabilize into two distinct groups. For
outliers, scores converge to high values, and weights go to the lowest value
of 0.008, while inliers have more dominant weights at around 0.02 and low
outlier scores. The demonstration uses a minibatch of size 64, where one data
point’s original contribution is $1/64\approx 0.016$. Our adaptive weight
module reinforces inliers’ contributions by approximately $25\%$ and decreases
that of outliers by $50\%$, respectively.

Figure 4 compares AUC and $F_{Gap}$ on all datasets with respect to the weight
module’s presence, whose beneficial effects towards both the detection
validity and fairness degree are clearly exhibited. Particularly, the weight
module brings an average of $46\%$ increase in AUC on student, asd, and kdd
datasets, while decreasing the detection validity gap between subgroups by an
average of $55\%$ on student, german, and drug. The weight module’s
enhancement in outlier validation invariably applies to all subgroups, thus
closing their validity gaps and making the model fairer.

(a) AUC

(b) $F_{Gap}$

Figure 6. Performance of DCFOD with different cluster numbers on asd, cc and
kdd by $AUC$ and $F_{Gap}$.

Hyperparameter Analysis. In our objective function, $\alpha$ and $\beta$
govern the balance between the self-reconstruction loss and fairness-
adversarial loss. We hereby explore the volatility of the model with respect
to changes on hyperparameters and verify the legitimacy of the default values
(i.e., $\alpha=8,\beta=100$). As shown in Figure 5(a) and 5(b), the model
reaches the best combination between $AUC$ and $F_{Gap}$ when $\alpha=8$. As
$\alpha$ further increases, detection validity decreases and the subgroup
validity gap subsequently enlarges. Figure 5(c) and 5(d) are the pareto
diagrams of $AUC$ and $F_{Gap}$ on dataset obesity and drug. DCFOD is observed
to consistently yield results that are both fair and of high detection
validity under varying $\beta$, compared to all other competitive methods.
$\beta=100$ takes the pareto front on both datasets, with clear improvements
on fairness degree $F_{Gap}$ compared to $\beta=0$ (w/o adversarial training).

Another hyperparameter in our model is the number of clusters $K$. In a
typical Clustering based algorithm, it is set as the number of classes.
However, in real-world unsupervised outlier detection tasks, we are often not
provided with the actual number of classes. In our experiment, we uniformly
set the number of clusters $K=10$ for all datasets. Figure 6 shows the $AUC$
and $F_{Gap}$ on cc, kdd, and asd with different cluster numbers. The
consistent performance shows that our DCFOD is not sensitive to the cluster
number.

##  6\. Conclusion

We studied the fairness issues in the task of unsupervised outlier detection.
We proposed a Deep Clustering based Fair Outlier Detection framework that
conducts downstream task-favorable representation learning with adversarial
training to optimize the detection validity concurrently with group fairness.
Specifically, we designed an adaptive weight module to administer model
training by reinforcing the likely-inliers’ contributions while alleviating
the outliers’ negative effects. We extended the existing fairness evaluative
criteria in the context of outlier detection and proposed two metrics
regarding group-wise diagnostic accuracy gap and subgroup distribution among
likely-outliers. Experiments showcased our model’s superiority over the recent
fair models and conventional outlier detection methods in both outlier
validation and degree of fairness.

##  7\. Acknowledgment

This work was supported in part by NSF OAC 1920147.

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# Novel quantum phase of the chromium spinel oxide HgCr2O4
in high magnetic fields

Shojiro Kimura1, Shusaku Imajo2, Masaki Gen2, Tsutomu Momoi3, Masayuki
Hagiwara4, Hiroaki Ueda5, Yoshimitsu Kohama2 1Institute for Materials
Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan
2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8531,
Japan
3Condense Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
4Center for Advanced High Magnetic Field Science, Graduate School of Science,
Osaka University, Toyonaka Osaka 560-0043, Japan
5Department of Chemistry, Graduate School of Science, Kyoto University,
Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

###### Abstract

In this study, we have performed the magnetocaloric effect and the specific
heat measurements of chromium spinel oxide HgCr2O4, wherein the magnetic Cr3+
ions form a highly frustrated pyrochlore lattice with significant spin-lattice
coupling. In addition to the known magnetic-field-induced phases, our
thermodynamic measurements detect a novel quantum phase just before the
saturation of the magnetization, which has not been expected from the
classical theories of the pyrochlore lattice antiferromagnet with spin-lattice
coupling. Based on recent theoretical model calculation, we discuss the
possibility of a spin nematic state appearing for this quantum phase.

††preprint: APS/123-QED

The spin nematic state, which has long been discussed for magnets with
frustrated interactions, is a spin analog of the nematic state in a liquid
crystalBlume ; Andreev ; Chubukov ; Shannon ; Lacroix . In this unconventional
magnetic state, the fluctuations of spins, behaving like rod- or disk-shaped
molecules, spontaneously select a preferred direction without conventional
magnetic orderLacroix ; Lauchli ; Smerald . Under an external magnetic field,
the spin nematic state is defined by finite values of transverse spin
quadrupolar order parameters ${\langle}S_{i}^{-}S_{j}^{-}+h.c.{\rangle}$ with
no transverse dipolar order, i.e.
${\langle}S_{i}^{x}{\rangle}={\langle}S_{i}^{y}{\rangle}=0$Shannon ; Orlova .
A promising route to obtain the spin nematic state is the condensation of
magnon bound pairs, which leads to the finite quadrupolar order
parametersShannon ; Momoi . As a realistic system, which is able to stabilize
the two-magnon bound state, a square lattice and one-dimensional chain magnets
with competing ferromagnetic and antiferromagnetic exchange interactions was
first suggestedChubukov ; Shannon ; Hikihara ; Zhitomirsky ; Ueda . The
competition between the exchange interactions suppresses the hopping of
individual magnons, while the ferromagnetic interaction, which acts as an
attractive force between magnons, stabilizes the bound states of multiple
magnons, causing the spin nematic ground state. Based on this prediction, the
searches for the spin nematic state have been intensively conducted in several
frustrated magnets with spin $S$ = 1/2, such as LiCuVO4Svistov ; Orlova ;
Mourigal ; Gen , Cu3V2O7(OH)2$\cdot$2H2O (volborthite)Yoshida ; Kohama , and
BaCdVO(PO4)2Skoulatos . Moreover, a new method to realize the spin nematic
order by condensation of two-magnon bound states was recently proposed for an
antiferromagnet on the highly frustrated pyrochlore lattice with spin-lattice
coupling, which is fulfilled in the chromium spinel oxides $A$Cr2O4 ($A=$ Mg,
Zn, Cd, and Hg)Takata . It was demonstrated theoretically that the effective
biquadratic spin interaction, which has been known to stabilize the spin
nematic stateLacroix ; Lauchli ; Smerald ; Tsunetsugu , is derived from the
spin-lattice couplingpenc1 ; penc2 ; Motome . Takata $\it{et.al.}$
demonstrated that this biquadratic interaction provides a significant energy
gain for the two-magnon bound state on the pyrochlore lattice to lead the spin
nematic ground state just below the saturation fieldTakata .

In $A$Cr2O4, which belongs to the cubic $Fd\bar{3}m$ space group, magnetic
Cr3+ ions with $S$ = 3/2 form the pyrochlore lattice, which is composed of a
three-dimensional arrangement of corner-sharing tetrahedra. Because the strong
spin frustration from the lattice geometry causes vast degeneracy in the
ground state, both classical $S=\infty$ and quantum $S=1/2$ pyrochlore
antiferromagnets with the nearest neighbor Heisenberg interaction are believed
to show no long range order at any temperatureMoessner ; Canals . However, in
$A$Cr2O4, the spin ordering is allowed to occur due to the weak further
neighbor exchange interaction and the spin-lattice coupling, causing lattice
distortion at the ordering temperature to relieve the ground state
degeneracyKino ; Plumier ; Rovers ; Yamashita ; Tchernyshyov1 ; Tchernyshyov2
. Under the external magnetic fields $H$, $A$Cr2O4 shows successive field-
induced phase transitions, including a distinctive half-magnetization plateau
phase with three up and one down spin configuration in a tetrahedronueda1 ;
ueda2 ; Mitamura ; Kojima ; miyata1 ; miyata2 ; miyata3 ; miyata4 ; miyata5 .
The magnetization curve in HgCr2O4, of which the exchange interaction is the
weakest among $A$Cr2O4, is shown in Fig. 1ueda2 ; Kimura1 ; Kimura2 . The
magnetization $M$, which gradually increases in the low field region, jumps to
the plateau with half of the saturation magnetization, at $H_{c1}$ = 10 T.
Then, $M$ starts to increase again above $H_{c2}$ = 27 T and shows a kink at
$H_{c3}$ = 36 T with a hysteresis, refrecting the first order nature of the
transition. This magnetization process is accompanied by deformations of the
crystal lattice, which cause large changes in the nearest neighbor exchange
interactions. Via the transition to the plateau phase above $H_{c1}$, the
crystal symmetry of HgCr2O4 changes to the distorted cubic $P4_{3}32$Matsuda ,
causing an imbalance in the exchange interactions in the tetrahedron, which
stabilizes the magnetization plateauKimura1 ; Kimura2 . Above $H_{c2}$, where
the canted 3:1 magnetic structure results in the gradual increase of $M$, this
imbalance is gradually diminished as the magnetic field $H$ is increased. It
then disappears at $H_{c3}$, indicating restoration of the high symmetry
$Fd\bar{3}m$ lattice with no distortionKimura1 ; Kimura2 . The interplay
between the lattice and the magnetism in $A$Cr2O4 is accounted for the
effective spin Hamiltonian, derived by integrating out the lattice degrees of
freedompenc1 ; penc2 ; Motome . Its most simple form is the one with the
bilinear and the aforementioned biquadratic interactions with an constant $b$:

$\displaystyle{\mathcal{H}}=J\sum_{i,j}\left\\{\bm{S}_{i}\bm{S}_{j}-b(\bm{S}_{i}\bm{S}_{j})^{2}\right\\}-g\mu_{B}\bm{H}\sum_{i}\bm{S}_{i}.$
(1)

Our previous work demonstrated that classical mean field calculations, based
on the magnetoelastic Hamiltonian with several spin-lattice coupling constants
for each normal tetrahedral distortion mode, reproduce the overall field
dependencies of $M$ and the exchange interactions, deduced from the ESR
measurementsKimura2 . However, there is an unresolved discrepancy between the
experiments and the mean field calculations. While the classical calculation
indicates the saturation of the $M$ at $H_{c3}$, the experimentally observed
$M$ still gradually increases above $H_{c3}$. It is noteworthy that there is a
tiny peak in the $dM/dH$ curve at $H_{c4}$ = 39 T as shown in Fig. 1. This
indicates the existence of a magnetic phase, which is not anticipated from the
classical model, for $H_{c3}<H<H_{c4}$. In this unanticipated phase, the
crystal lattice has no distortion while the magnetization does not saturate
but still increases as the magnetic field is increasedKimura1 . We consider
that this phase is an unconventional state, because, even in the magnetic
field, a pyrochlore antiferromagnet with no lattice distortion possesses vast
degeneracy in the ground statepenc1 , which is expected to strongly disturb
conventional magnetic order. As a reasonable candidate for this phase, the
spin nematic state has been recently proposedTakata . To search for the
possible spin nematic phase, we have performed thermodynamic experiments in
high magnetic fields. The thermodynamic methods can detect phase transitions
characterized by any type of order parameter, including the long range order
of the spin quadrupole moment i.e., spin nematic order. In this study, we have
measured the high-field specific heat and magnetocaloric effect of HgCr2O4 in
pulsed magnetic fields of up to 42 T to gain thermodynamic evidence for the
existence of ordering in the field region between $H_{c3}$ and $H_{c4}$.

Polycrystalline samples of HgCr2O4 were prepared by thermal decomposition of
Hg2CrO4ueda2 . We have measured the temperature $T$ dependence of the specific
heat $C_{p}$ in HgCr2O4 at the high magnetic fields of 15.8, 26.9, 34.8, and
38.5 T using the quasi-adiabatic method under a flat-top pulsed fieldImajo ;
Kohama1 . The $C_{p}$ at 0 T and 8.0 T was measured using the conventional
thermal relaxation method by utilizing a commercial measurement system
(Quantum Design, PPMS). To measure the field dependence of specific heat under
the quasi-isothermal condition, the ac-calorimetry of up to 40 T was performed
at 250 HzKohama2 . The magnetocaloric effect (MCE) measurements were also
performed with high magnetic fields of up to 42 T using the same setup without
applying AC heat. By adjusting the amount of helium gas in the sample space,
the MCE data were obtained in different thermal environments: quasi-adiabatic
or nearly isothermal conditions.

Figure 2(a) shows the MCE curve $T(H)$ started at 0.8 K under quasi-adiabatic
condition in the up-sweep of the field pulse. The $T$($H$) exhibits a sharp
dip at $H_{c1}$, and shows a maximum around 17 T in the plateau phase, which
roughly corresponds to the center of the magnetization plateau.Then, the
$T$($H$) decreases as the magnetic field is increased and exhibits a minimum
at $H_{c2}$. Above $H_{c2}$, a kink of $T$($H$) is observed at $H_{c3}$.
Between $H_{c2}$ and $H_{c3}$, a small dip of $T$($H$) is found at arond
$H_{c2^{\prime}}$ = 31 T. An anomaly around 31 T in the ${dM/dH}$ curve, which
has been overlooked in previous papersueda2 ; Kimura1 ; Kimura2 , is also
shown in Fig. 1. This anomaly suggests some weak but sudden change in the spin
structure. However its detail is not clear at present. In ref. 47, the
positive MCE by entering the plateau phase has been predicted. According to
the thermodynamic equations, the field derivative of the $T$($H$), observed
under an adiabatic condition, is given asRossi ; Schmidt :

$\displaystyle\biggl{(}\frac{\partial T}{\partial
H}\biggr{)}_{S}=-\frac{T}{C_{p}}\biggl{(}\frac{\partial M}{\partial
T}\biggr{)}_{H}.$ (2)

Thus, the sign of ($\partial T$/$\partial H$)S is determined by that of
($\partial M$/$\partial T$)H because both $C_{p}$ and $T$ are positive. The
sharp dip at $H_{c1}$ and the subsequent increase in $T$($H$), observed in our
experiments, are consistent with the ($\partial M$/$\partial T$)H obtained
from the classical Monte Carlo simulations and analytical calculations based
on the magnetoelastic Hamiltonian with the weak ferromagnetic third nearest
neighbor interaction $J_{3}$ in the supplementary materials for ref. 47.
However, while the calculated ($\partial M$/$\partial T$)H suggests a
monotonic increase in $T$($H$) in the entire plateau region, the experimental
$T$($H$) starts to decrease of above 17 T. The calculation indicates that the
change in the sign of the ($\partial M$/$\partial T$)H from negative to
positive occurs at $H_{c2}$, namely, just at the end of the plateau phase. In
contrast, the temperature dependence of the experimental $M$, reported in ref.
33, displayed the sign change of the ($\partial M$/$\partial T$)H within the
plateau phase, therefore agreeing with our MCE result. The classical Monte
Carlo simulation with $J_{3}$ = 0 by Shannon $et$ $at$.Shannon2  also
suggested that the sign change occurs at the magnetic field below $H_{c2}$. We
suspect that the discrepancy between the calculation in the supplementary
materials for ref. 47 and the experimental results may likely comes from the
assumed ferromagnetic $J_{3}$, which is incompatible with the magnetic
structure of the $P4_{3}32$ symmetry in the plateau phase, and a possible
quantum effect. A more apparent discrepancy between the calculations and the
experimental result is observed around $H_{c3}$. The calculation suggested a
deep and sharp minimum of $T$($H$) at $H_{c3}$ and also an increase of
$T$($H$) above $H_{c3}$, whereas the experimental $T$($H$) is almost constant
above $H_{c3}$. The strong anomaly, derived from the classical calculation, is
associated to the energy-gap-closing of the localized spin wave modeRossi ,
which is characteristic of pyrochlore lattice magnets, at the saturation field
and the subsequent gap-reopening. The gap-closing at $H_{c3}$ was detected
from the previous high-field ESR measurements in HgCr2O4Kimura1 . However,
when the spin nematic state is realized, the system still remains to be
gapless above $H_{c3}$ because of an emergence of the Goldstone mode for the
quadrupolar order parameter by breaking of rotational symmetry inherent in the
spin nematic orderLauchli ; Smerald . This gapless nature of the spin nematic
state may suppress the strong anomaly of $T$($H$) at $H_{c3}$ as
experimentally observed.

To clarify the temperature dependence of the magnetic-field-induced
transitions, we performed the MCE measurements at several temperatures in a
nearly isothermal condition (Fig. 2(b)). Note that, these MCE curves were
taken under the nearly isothermal condition with a small amount of 3He gas.
The anomalies observed in this condition are more smeared out than those in
quasi-adiabatic condition due to the heat leak via the helium gas around the
sample. When the amount of gas is large, the measurement conditions are closer
to the isothermal limit, and the transitions become less visible, as shown in
the MCE data obtained at 1.7 K in Fig. 2(b). As shown in the inset in Fig.
2(a), the MCE result under a condition intermediate between adiabatic and
nearly-isothermal at around 2 K clearly exhibits sharp dips both at $H_{c3}$
$\simeq$ 37 T and $H_{c4}$ $\simeq$ 40 T.This results unambiguously indicates
an existence of the new high field phase between $H_{c3}$ and $H_{c4}$.
Together with the MCE data obtained in the quasi-adiabatic and intermediate
conditions and the $C_{p}(T,H)$ data shown below, the MCE data in the nearly
isothermal condition were used to construct the $(B,T)$ phase diagram in Fig.
4.

Figure 3(a) shows the temperature dependencies of the specific heat $C_{p}$.
At zero magnetic field, a peak of the $C_{p}$, which is indicative of a
magnetic long-range order, is observed around the reported N$\rm{\acute{e}}$el
temperature of 5.8 K. This peak slightly shifts to lower temperature in the
magnetic field at 8 T. In the plateau phase at 15.8 T, a sharp peak of $C_{p}$
is observed at a higher temperature. The peak broadens above $H_{c2}$, and
then the broad peak is still observed at 38.5 T, which is well above $H_{c3}$.
It is important to note that the peak in $C_{p}$ could not fit to the
Schottky-type heat anomaly related to a non-cooperative thermal excitation in
a gapped system. This suggests that the peak is attributed to the critical
behavior in a phase transition. Figure 3(b) shows the field dependence of
$C_{\rm{p}}$ observed at 1.3 K. The anomalies are observed at $H_{c1}$,
$H_{c2}$ and $H_{c3}$. In contrast to the fact that the specific heat of an
antiferromagnet often shows a pronounced peak at a saturation field because of
the subsequent reopening of the spin wave gap as reported in ref. Kohama3 ,
$C_{\rm{p}}$ in HgCr2O4 exhibits only the tiny anomaly at $H_{c3}$ and the
gradual increase above $H_{c3}$, implying the existence of the subsequent
phase transition at the higher magnetic field. These behaviors are also in
contrast to the sharp anomaly in $dM/dH$ at $H_{c3}$. The observed
$C_{\rm{p}}$ suggests that the $H_{c3}$ is not the saturation field, and a
gapless ordered phase exists between $H_{c3}$ and $H_{c4}$. The field-
temperature phase diagram of HgCr2O4, obtained from this study, is shown in
Fig. 4. As deduced from previous magnetization measurements in pulsed magnetic
fields ueda2 , the plateau phase is expanded to a higher temperature region
than the antiferromagnetic phase below $H_{c1}$ because of the stabilization
by thermal fluctuation. The phase boundary between the paramagnetic and the
plateau phases is suggested to be located at higher temperatures compared to
the temperature reported from the magnetization measurements. This is possibly
because the magnetization measurement does not consider the magnetocaloric
effect but assumes isothermal conditions.

As mentioned above, the spin nematic state has been theoretically proposed for
the magnetic phase just below $H_{c4}$Takata . This spin nematic state is
stabilized in the pyrochlore lattice antiferromagnet owing to the interplay of
the spin-lattice coupling, quantum fluctuation, and geometrical frustration.
In the Heisenberg-type pyrochlore lattice magnet, the lowest-energy magnon
states in a fully spin-polarized region above the saturation field are
localized with flat dispersions owing to the geometrical configuration of the
latticeZhitomirsky ; Zhitomirsky2 . Therefore, the hopping of the single
lowest-energy magnon is prohibited, thereby preventing kinetic energy gain.
Meanwhile the spin-lattice coupling in $A$Cr2O4 brings the effective
biquadratic interaction in Eq. (1). This biquadratic interaction contains a
term $(S_{i}^{-})^{2}(S_{j}^{-})^{2}+h.c.$, which induces hopping of a two-
spin flipped state on a Cr3+ site to the nearest neighbor sites. Because of
this hopping term, the two-magnon bound state becomes stable and has a lower
energy than two independent localized magnons in the fully polarized spin
stateTakata . Decreasing the magnetic field in the fully polarized spin state
causes softening of the two-magnon bound state. Then, the bound state touches
to the ground state at the saturation field, resulting in its condensation,
which leads to the spin nematic state. Because the dispersion curve of the
two-magnon bound state has a minimum at $\bm{k}=$ (0 0 0) in the reciprocal
lattice space, where $\bm{k}$ is the center-of-mass wave vector, the resulting
spin nematic state is characterized by the ferro-quadrupolar order. Thus, the
spin nematic order causes no lattice distortion via the spin-lattice coupling.
This is consistent with the absence of the symmetry lowering in the lattice
from the $Fd\bar{3}m$ symmetry above $H_{c3}$ in HgCr2O4, as suggested from
the previous ESR experimentsKimura1 . From the theoretical calculation, Takata
$et$ $al.$ found that the spin nematic order can appear with relatively weak
$b$ $>$ 0.043Takata . This is also consistent with the reported value of
$b_{A}$ = 0.05 in HgCr2O4Kimura2 , which is a biquadratic interaction constant
for the fully symmetric $A_{1}$ mode. According to the calculation, the spin-
component distribution, of which the projection to the plane perpendicular to
the magnetic field has an ellipsoid shape, is aligned owing to the spin
nematic order, thereby breaking the rotational symmetry around fieldTakata .
From our experimental results that suggest the existence of the gapless
magnetic ordered phase between $H_{c3}$ and $H_{c4}$, we consider that such a
spin nematic order most probably appears in HgCr2O4. The existence of an
unknown magnetic phase between the 3:1 canted and the fully polarized spin
states was also reported for other chromium spinel oxides, CdCr2O4 and
ZnCr2O4, from high-field experimentsmiyata2 ; miyata3 ; miyata4 ; Nakamura ;
Zherlitsyn ; Kimura3 . It is possible that the appearance of the spin nematic
phase is of universal nature in $A$Cr2O4. To gain more detailed microscopic
information of the phase between $H_{c3}$ and $H_{c4}$, nuclear magnetic
resonance measurements in high magnetic fields are desirable.

In summary, we have performed the high-field thermodynamic measurements of the
chromium spinel oxide HgCr2O4 in pulsed magnetic fields of up to 40 T. From
the temperature and field dependences of the specific heat, we observed the
appearance of the new magnetic phase between the 3:1 canted and the fully
polarized spin states. We consider that the spin nematic state, which is
stabilized due to the interplay of spin-lattice coupling, quantum fluctuation,
and geometrical frustration in the pyrochlore lattice antiferromagnet, is
realized in this magnetic phase.

The experimental work was performed under the Visiting Researcher’s Program of
the Institute for Solid State Physics, The University of Tokyo. This work was
in part supported by the Grant-in-Aid for Scientific Research (Grant No.
121H01026 and No. 22H00101) from MEXT Japan.

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Figure 1: High field magnetization curve of a polycrystalline sample of
HgCr2O4, observed at 1.3 K. Solid bold curves are the experimental
magnetization ($M$) curve, and a dashed curve is the theoretical curve
obtained from the classical mean field calculation based on the magnetoelestic
Hamiltonian. Thin solid curves are the differential magnetization ( $dM$/$dH$)
curve. The experimental and theoretical data are from Kimura1  and Kimura2 ,
respectively. Figure 2: Magnetocaloric effect of HgCr2O4, observed under (a)
quasi-adiabatic and (b) quasi-isothermal conditions. The inset shows the
result of the magnetocaloric effect measurement under a condition intermediate
between quasi-adiabatic and quasi-isothermal.

Figure 3: (a) Temperature dependence of the specific heat $C_{p}$ decided by
temperature $T$. The data at 26.9 T are multiplied by 1/2. (b) Magnetic field
dependence of $C_{p}$ of HgCr2O4, observed at 1.3 K. Thin solid curves are the
differential magnetization ( $dM$/$dH$) curve.  Figure 4: Temperature to
magnetic field phase diagram of HgCr2O4, obtained from thermodynamic
experiments. Squares, circles, and triangles are the transition points
obtained from the magnetocaloric effect, temperature, and field dependenceis
of the specific heat measurements, respectively. Dashed curves are drawn as
guides for the eyes.