# What is Wrong with
One-Class Anomaly Detection?

Junekyu Park, Jeong-Hyeon Moon, Namhyuk Ahn & Kyung-Ah Sohn
Department of Artificial Intelligence, Ajou University
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

From a safety perspective, a machine learning method embedded in real-world
applications is required to distinguish irregular situations. For this reason,
there has been a growing interest in the anomaly detection (AD) task. Since we
cannot observe abnormal samples for most of the cases, recent AD methods
attempt to formulate it as a task of classifying whether the sample is normal
or not. However, they potentially fail when the given normal samples are
inherited from diverse semantic labels. To tackle this problem, we introduce a
latent class-condition-based AD scenario. In addition, we propose a
confidence-based self-labeling AD framework tailored to our proposed scenario.
Since our method leverages the hidden class information, it successfully
avoids generating the undesirable loose decision region that one-class methods
suffer. Our proposed framework outperforms the recent one-class AD methods in
the latent multi-class scenarios.

##  1 Introduction

With the rapid increase in the performance of deep learning-based methods, the
demands for applying this technology are emerging in recent. However, simply
adopting it may not be ideal due to the mismatched label information between
the training and test set. A self-driving system, for example, should make the
best decision on the abnormal scene or status even though it never observed
this condition before. Under this scenario, it is required to detect whether
the given data is unseen (abnormal) or not, to build a reliable and secure
machine learning system.

The anomaly detection (AD) task focuses on to identify suspicions or abnormal
events. We can categorize this task into supervised (liang2017enhancing) or
unsupervised (chalapathy2017robust; oza2018one) by the training strategy. The
former train the model with both normal and abnormal samples, while the latter
use normal data only. Since collecting the abnormal cases is time-consuming or
impossible in the real-world, unsupervised learning gets more attention
despite the superior performance of the supervised scheme. In an unsupervised
approach, most of the methods formulate the task as an one-class
classification problem (i.e. classify as normal or not). While such simplicity
works on the well-refined scenarios, they potentially fail when the given
normal dataset is composed of samples from diverse classes (Figure
LABEL:fig:overview_prev). Since conventional methods ignore the latent class
information, they tend to draw a single binary decision boundary that loosely
covering a wide range. This could be problematic when the normal dataset
contains multiple latent class information. Can we leverage this semantic
knowledge to guide the AD methods to judge the anomalies more accurately? We
believe a different approach is needed.

Base on this assumption, we introduce a latent class-condition-based AD
scenario and its benchmark datasets. This simulates the circumstance where
both normal and abnormal data have multi-class samples (Figure
LABEL:fig:overview_ours). We would like to emphasize that our proposed
scenario is realistic; in the real-world, various (object) classes could be
normal, and the abnormal cases could consist of diverse classes as well (e.g.
street signs of different countries). Because of the nature of our scenario,
the optimal solution needs to generate semantic-aware tight decision
boundaries (as in Figure LABEL:fig:overview_ours). However, neither supervised
nor unsupervised approaches can handle the scenario properly. For example,
supervised models cannot be applied since no label is given. Conventional
unsupervised AD methods create a single and loose decision region (since they
ignore latent class information). At this point, one natural question can
arise: How can we use latent semantic information? Since the latent label is
accessible to the oracle (or human) only, we may detour this limitation, for
example, by utilizing the pseudo-latent class label.

To tackle this, we propose a Confidence-based self-Labeling Anomaly Detection
(CLAD) framework to bridge the gap between the supervised and unsupervised
approaches. We model the latent class information using self-labeling so that
supervised learning can be adapted (Figure 2). To do this, we first train the
feature extraction network following xie2016unsupervised. Then, we cluster the
training samples using the extracted latent feature and allocate pseudo labels
via self-labeling (lee2013pseudo). Now the classification network can learn to
predict pseudo labels by standard supervised learning. At the inference, we
decide whether the test sample is abnormal or not by the confidence-based AD
method (liang2017enhancing). Since our method leverages the hidden class
information, it successfully avoids the generation of undesirable loose
decision regions typically suffered by one-class methods. Several experiments
on latent multi-class scenarios demonstrate that the proposed method
substantially outperforms recent one-class AD methods.111Our code is available
at https://github.com/JuneKyu/CLAD

##  2 Latent Class-condition Anomaly Detection Scenario

Traditional one-class AD methods learn a decision boundary based on the given
normal samples in the training dataset (Figure LABEL:fig:overview_prev). Such
one-class strategy is effective when a single class label is treated as normal
alone (and the rest of them are abnormal). However, this assumption may not
hold in a real environment since the normal category can consist of
heterogeneous semantic labels. With this circumstance, conventional one-class
AD generates a loose decision boundary to cover all samples drawn from diverse
classes and thus vulnerable to a false negative.

To reduce the gap between the real-world and the one-class AD scenarios, we
simulate the scenario environment where the latent sub-classes exist
implicitly (Figure LABEL:fig:overview_ours). With this environment, it is
crucial to learn a decision boundary by seeing not only the normality of the
data samples but also its semantics. Note that such class information is not
observable, thus the AD framework may require learning the semantic
representation in an unsupervised or self-supervised manner.

|
---|---

(a)

(b)

Figure 1: Comparison of two AD scenarios. (a) One-class AD scenario where the
model creates a single decision boundary to cover all normal data. Some
abnormal samples are misclassified due to the loose boundary. (b) Our proposed
AD scenario. An optimal solution can model latent class-condition information
and draws tight class-wise decision boundaries. For normal samples, each class
is denoted as different shapes. Note that the latent class information are not
observable to the models; only the oracle is able to access this.

##  3 Confidence-based Self-labeling Anomaly Detection

Before going into the details, let us define the problem statement. We have a
training set $X_{tr}=\\{\mathbf{x}_{i}\\}^{N}_{i=1}$ and a test set
$(X_{te},Y_{te})$. $X_{te}$ contains both normal/abnormal samples with
corresponding label $Y_{te}$; $0$ if normal and $1$ otherwise. The core idea
of our framework is to consider the concealed semantic information. To do
that, we generate the pseudo-label $y^{*}\in\mathcal{Y}^{*}=\\{1,...,L\\}$,
where the cardinality of $\mathcal{Y}^{*}$ is assumed pseudo-class counts.
Then, our AD framework inferences a true $y\in Y_{te}$ based on the classifier
$F$ that is trained with generated pairs $(X_{tr},Y^{*}_{tr})$, where
$Y^{*}_{tr}=\\{\mathbf{y}^{*}_{i}\\}^{N}_{i=1}$.

(1) Feature extraction. First, we train the autoencoder ($E$ and $D$) with
reconstruction loss on training dataset $X_{tr}$ and calculate the latent
feature $\mathbf{z}$ using the trained encoder $E$.

(2) Self-labeling via clustering. We further fine-tune the encoder $E$ to
increase the possibility of assigning a sample $\mathbf{x}$ to the given
cluster $C_{i}$ by the Kullback-Leibler divergence between the latent feature
$\mathbf{z}$ and soft-alignment of $C_{i}$ (xie2016unsupervised). We
simultaneously update the encoder and cluster assignment and this makes the
clustering algorithm to be robust. Then, we assign a label $y^{*}$ by
referring to the allocated cluster $C_{i}$. Now the $X_{tr}$ have associated
pseudo-labels ${Y}_{tr}^{*}$.

(3) Supervised classification. With (pseudo) latent class labels $Y^{*}_{tr}$
and the normal sample $X_{tr}$, a classifier $F$ is trained with a standard
supervised learning scheme. By referring to the class information that is
intrinsic in the latent labels, our framework can generate the semantic-aware
tight decision boundaries that envelop each relevant class sample only.

(4) Confidence-based AD. Because of the tight decision boundary by the pairs
$(X_{tr},Y^{*}_{tr})$, we argue that the AD can be viewed as the out-of-
distribution (OOD) task. In detail, the OOD sample is defined as the data
where the class label is not included in the training dataset. We assume the
label set $\mathcal{Y}^{*}$ is a multi-classes label set from $Y^{*}_{tr}$.
Then, we treat $(\mathbf{x},y^{*})$ as an OOD sample when the
$y^{*}\not\in\mathcal{Y}^{*}_{tr}$. Because the confidence-based algorithms
predict the samples as OOD when the classifier outputs a small probability for
all classes, we can safely convert anomaly detection into an OOD task. (i.e.,
it is an abnormal sample when $y^{*}\not\in\mathcal{Y}^{*}_{tr}$).

Under these assumptions, we measure anomaly detection scores by adapting the
scoring scheme in ODIN (liang2017enhancing). We define the score
$s(\mathbf{x};T,\delta)=max_{i}p(\tilde{\mathbf{x}};T)_{(i)}$, where
$p(\tilde{\mathbf{x}};T)_{(i)}$ is output of the classifier $F$ in each class
$i$. $T$ is the parameter for the temperature scaling and $\tilde{\mathbf{x}}$
is the input term $\tilde{\mathbf{x}}=\mathbf{x}-\epsilon(\mathbf{x})$
perturbed by the reverse FGSM method (goodfellow2014explaining) that makes the
OOD samples more separable. If the score $s(x;T,\delta)$ is greater than a
given threshold $\delta$, then we predict the $\hat{y}$ as normal.

Figure 2:  Overview of our proposed confidence-based self-labeling anomaly
detection framework. (1) Feature extraction by training the convolutional
autoencoder to extract the latent feature $\mathbf{z}$. (2) Self-labeling via
clustering to assign latent class ${y}^{*}$ to sample $\mathbf{x}$. (3) Train
a classifier $F$ using $(X_{tr},Y^{*}_{tr})$ pairs in supervised learning
manner. $L_{cls}$ denotes a classification loss. (4) We perform anomaly
detection by the confidence score of classifying normal sample $\mathbf{x}$ to
$y^{*}$.

##  4 Experiment

Baselines. We compare with one-class AD methods: OCSVM
(scholkopf2001estimating), OCNN (chalapathy2017robust), OCCNN (oza2018one),
SVDD (tax2004support), and DeepSVDD (ruff2018deep). We follow the
implementation setups based on the official codes.

Datasets. We use following datasets: MNIST (lecun-mnisthandwrittendigit-2010),
CIFAR-10 (krizhevsky2009learning), GTSRB (Stallkamp2012), and Tiny-ImageNet
(ILSVRC15).

We devise the super-categories by merging the semantic labels to simulate our
AD scenario. For example, MNIST is as {Curly, Straight, Mix} and GTSRB based
on the semantic meanings of traffic signs. Note that both datasets share a
similar domain prior which represents the text or symbols. For the natural
scene datasets such as CIFAR-10 and Tiny-ImageNet, we set as {Thing, Living}
and {Animal, Insect, Instrument, Structure, Vehicle}, respectively. When we
evaluate the models, we pick one super-category for training and the rest of
the subsets as a test dataset. Please see Appendix D for the detailed settings
of the scenario.

Results. Table 1 shows the AUROC of the MNIST and GTSRB datasets. Our
framework surpasses all the one-class AD methods in MNIST. Among them, it is
notable that CLAD outperforms others on Mixed in a huge margin. This scenario
is very challenging since it carries complex information due to the mixed
shape of digits such as ‘2’ or ‘6’. For the GTSRB dataset, our CLAD reaches
the best performance for SPDL, DIRC, REGN and second-best for the rest. The
only method comparable to ours is DeepSVDD. However, it suffers the
inconsistent performance (worst score in SPEC) while our framework shows
stable and high performances for all the cases.

To demonstrate the superior performance of CLAD in the complex image domain,
we evaluate it on the CIFAR-10 and Tiny-ImageNet datasets (Table 2). Our
framework achieves the best performance for all the scenarios in CIFAR-10 and
four of five cases in Tiny-ImageNet. Similar to GTSRB, the scores of DeepSVDD
are inconsistent; we claim that DeepSVDD is sensitive to the latent class
labels. It maps all the (normal) samples into a single-modal hypersphere
(Figure LABEL:fig:overview_prev), making it vulnerable to the anomalies that
close to the normals in terms of the class information.

Table 1: Performance comparison on the newly proposed AD scenario with MNIST and GTSRB. Method | MNIST | GTSRB
---|---|---
CUR | STR | MIX | SPDL | INST | WARN | DIRC | SPEC | REGN
OCSVM | 69.9 | 87.4 | 54.8 | 66.3 | 60.9 | 52.7 | 60.7 | 50.5 | 77.2
OCNN | 81.4 | 76.2 | 60.2 | 65.7 | 65.7 | 51.8 | 62.0 | 52.7 | 78.3
OCCNN | 77.5 | 78.3 | 51.5 | 59.3 | 57.5 | 60.4 | 56.4 | 59.1 | 64.3
SVDD | 58.8 | 72.4 | 52.6 | 58.2 | 55.1 | 51.1 | 50.5 | 56.9 | 67.4
DeepSVDD | 81.7 | 82.2 | 69.7 | 57.7 | 69.7 | 73.4 | 59.3 | 70.2 | 77.8
CLAD (ours) | 94.0 | 96.1 | 92.6 | 66.7 | 66.5 | 64.9 | 67.4 | 62.2 | 79.1
Table 2: Performance comparison on our AD scenario with CIFAR-10 and Tiny-ImageNet. Method | CIFAR-10 | Tiny-ImageNet
---|---|---
THG | LIV | ANML | ISCT | ISTM | STRT | VHCL
OCSVM | 51.9 | 67.7 | 63.5 | 60.9 | 50.2 | 56.7 | 55.5
OCNN | 58.5 | 67.2 | 58.2 | 57.1 | 50.3 | 51.0 | 55.1
OCCNN | 59.8 | 62.5 | 59.7 | 62.4 | 51.5 | 54.2 | 66.4
SVDD | 50.8 | 61.4 | 51.5 | 51.8 | 51.5 | 50.6 | 51.9
DeepSVDD | 65.0 | 52.7 | 59.0 | 53.5 | 53.4 | 55.4 | 53.5
CLAD (ours) | 74.9 | 72.8 | 65.9 | 66.2 | 55.6 | 62.0 | 64.7

##  5 Discussion & Conclusion

We introduced a new aspect of the AD task and proposed a confidence-based AD
framework. We assume that (ab)normal categories can have (unobservable) multi-
class samples in contrast to the one-class AD scenarios. We believe that our
scenario and method are practical in real world. One possible usage is for the
industrial environment. When the normal sensor signals with various range can
be altered by the surroundings, conventional one-class AD methods suffer
spurious detection.

Since our framework trains a classifier using self-labeling, extracting the
right representation of the data sample is crucial. However, most of the
feature extraction methods have difficulty with handling a dataset bias
(bahng2020learning). As future work, we will focus on the disentanglement of
the scene context with a key concept of the object discovery
(burgess2019monet).

## Acknowledgements

This research was supported by the National Research Foundation of Korea grant
funded by the Korea government (MSIT) (No. NRF-2019R1A2C1006608), and also
under the ITRC (Information Technology Research Center) support program
(IITP-2020-2018-0-01431) supervised by the IITP (Institute for Information &
Communications Technology Planning & Evaluation).

##  Appendix A Related Work

One-class anomaly detection. Most of the anomaly detection (AD) methods treat
a task as a binary classification: normal and abnormal. Under this assumption,
OCSVM (scholkopf2001estimating) learns a decision boundary with normal samples
only. OC-NN and OC-CNN (chalapathy2017robust; oza2018one) are the earlier
attempts on using a deep learning-based approach with an unsupervised learning
regime. However, unlike OCSVM, these works have the potential risk to be a
trivial solution due to the insufficient theoretical analysis. That is, the
learnable parameters may become a trivial solution when the given data samples
are all normal. On the other hand, SVDD (tax2004support) and DeepSVDD
(ruff2018deep) successfully avoid such trivialness by the theoretical basis.
Although the aforementioned methods have shown promising results, they are
limited to the one-class scenario which is not suitable for real-world
applications. In contrast, our proposed task assumes that the data points
could be allocated into the various classes, which is more realistic.

Self-labeling. The self-labeling is one of the most rapidly developing
approaches in the self-supervised learning context
(dosovitskiy2015discriminative; doersch2015unsupervised;
noroozi2016unsupervised; noroozi2017representation; doersch2017multi;
gidaris2018unsupervised). Base on the success of self-supervised learning,
recent works attempt to create a self-label using traditional clustering
methods. For example, DeepCluster (caron2018deep) first makes initial self-
labels using a convolutional network and iteratively updates the network
parameters with the re-assigning process by the clustering algorithms.
asano2019self extended DeepCluster making it to learn visual representation
and clustering simultaneously based on the information theory as such
maximizing the information between the labels and the input data. Although
self-labeling has shown outstanding performance in the computer vision field,
to the best of our knowledge, none of the studies exists to apply this
strategy to the anomaly detection field.

Out-of-distribution detection. Out-of-distribution (OOD) detection is the task
to discriminate the samples whether they are from the training distribution or
not. Because the deep learning-based classification model tends to predict as
a wrong class with high-confidence when the given test samples are the class
not in the training set (i.e. high-confidence problem). To tackle this issue,
many works have been studied on this problem. ODIN (liang2017enhancing)
addressed the high-confidence problem using temperature scaling and an input-
preprocessing method from the FGSM (goodfellow2014explaining). Their method
pursues better separating the out-of- and in-distribution samples.
papadopoulos2019outlier proposed an additional loss function from ODIN and
applied other machine learning tasks such as natural language problems. The
aforementioned methods rely on the multi-class labels when training, i.e.
supervised approach. This limits the OOD detection methods difficult to apply
to real-world anomaly detection problems.

##  Appendix B Model Analysis

In this section, we analyze our proposed method. First, we evaluate CLAD on
the previous one-class AD scenario. Second, we show how our method is robust
to the hyper-parameter settings such as the number of the clusters or the
hidden dimension size.

One-class AD. We can view this task as a simplification of our proposed
scenario. In one-class AD, the normal category has a single semantic label
only, in contrast to ours which sets the normal condition to contain multiple
class information. We evaluate the methods on MNIST and CIFAR-10 datasets as
shown in Table 3 and 4. Our CLAD shows comparable results on MNIST to the
other one-class-based AD methods and competes on par with DeepSVDD on the
CIFAR-10 dataset.

We would like to note that CLAD is not designed for the one-class AD task.
Because of the feature clustering and label assignment, our method could
create fragmented decision boundaries in this conventional scenario. In
contrast, DeepSVDD learns to generate a spherical decision boundary that
tightly wraps the single-class normal samples. However, on CIFAR-10 which may
have various latent semantics within the single class (e.g. different pose,
intra-class diversity), DeepSVDD fails to detect abnormal in some cases (e.g.
bird, deer) while CLAD shows consistent scores.

Table 3: One-class AD performance on MNIST dataset. Method | MNIST
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
OCSVM | 98.3 | 99.5 | 82.0 | 88.5 | 91.6 | 79.7 | 93.1 | 93.4 | 83.9 | 91.3
OCNN | 97.6 | 99.5 | 87.3 | 86.5 | 93.3 | 86.5 | 97.1 | 93.6 | 88.5 | 93.5
OCCNN | 91.8 | 98.7 | 74.9 | 78.2 | 88.3 | 72.7 | 75.6 | 85.6 | 69.1 | 78.5
SVDD | 98.6 | 99.5 | 82.5 | 88.1 | 94.9 | 77.1 | 96.5 | 93.7 | 88.9 | 93.1
DeepSVDD | 98.0 | 99.7 | 91.7 | 91.9 | 94.9 | 88.5 | 98.3 | 94.6 | 93.9 | 96.5
CLAD (ours) | 96.3 | 97.9 | 89.8 | 87.2 | 92.2 | 90.7 | 92.5 | 91.5 | 80.0 | 92.0
Table 4: One-class AD performance on CIFAR-10 dataset. Method | CIFAR-10
---|---
airplane | automobile | bird | cat | deer | dog | frog | horse | ship | truck
OCSVM | 65.1 | 59.0 | 65.2 | 50.1 | 75.1 | 51.3 | 71.7 | 51.2 | 67.6 | 51.0
OCNN | 60.4 | 62.0 | 63.7 | 53.6 | 67.4 | 56.1 | 63.3 | 60.1 | 64.7 | 60.3
OCCNN | 65.0 | 65.6 | 62.8 | 51.2 | 72.7 | 50.9 | 64.7 | 52.2 | 66.5 | 66.9
SVDD | 61.6 | 63.8 | 50.0 | 55.9 | 66.0 | 62.4 | 74.7 | 62.6 | 74.9 | 75.9
DeepSVDD | 61.7 | 65.9 | 50.8 | 59.1 | 60.9 | 65.7 | 67.7 | 67.3 | 75.9 | 73.1
CLAD (ours) | 73.0 | 64.4 | 58.3 | 60.1 | 73.1 | 63.9 | 76.0 | 60.1 | 70.7 | 69.8

(a) The effect of # clusters of our scenarios.

(b) The effect of different hidden dimension sizes.

Figure 3: Ablation study results. The red dash-line denotes the average scores
of DeepSVDD on our three MNIST scenarios. (a) With more than four clusters,
our method achieves better performance compared to DeepSVDD. (b) We vary the
hidden dimension sizes from 10 to 100. Our method shows consistent performance
for all the hidden dimension sizes.

The effects of the hyper-parameters. In our framework, we use feature
extraction and clustering modules. Since these modules are the basis of the
self-labeling procedure, we analyze how the hyper-parameters of such modules
affect the AD performance. Figure 3 shows the performance tendency when we
change the number of the clusters or hidden dimension of feature $z$.

We vary the number of clusters from 2 to 20 and hidden dimension size from 10
to 100. The red dashed line indicates the average scores of DeepSVDD in three
scenarios. If we set the number of cluster sizes to more than four, our method
surpasses DeepSVDD by a huge margin. This result implies that the robustness
of CLAD to the rough self-labeling. Our framework also shows the robustness
with the change of the hidden dimension size $\mathbf{z}$; only marginal
fluctuations are observed.

##  Appendix C Implementation Detail

Latent feature extraction. We use autoencoder-based architecture for this
network. Both encoder and decoder have five convolutional layers with
increasing channel size followed by two linear layers. We additionally apply
dropout (srivastava2014dropout) to avoid overfitting. The model training is
done for 100 epochs using Adam (kingma2014adam) with a learning rate of 0.01.
Based on the model analysis, we set the number of clusters as 10 and the size
of the hidden dimension as 100.

Self-labeling via clustering. We adopt DEC (xie2016unsupervised) to self-
assign labels to data samples. In detail, we minimize KL divergence between
the embedded data samples from encoder $E$ and the soft-alignments. When
training this module, an SGD optimizer was used with a momentum of 0.9 and a
learning rate of 0.01 for 100 epochs.

Classifier for confidence-based AD. We use ResNet-18 (he2016deep) as a
classifier $F$. We train this network using Adam (kingma2014adam) with a
learning rate of 0.0001 for 100 epochs from scratch. At the inference phase,
we adopted the temperature scaling and input perturbation following ODIN
(liang2017enhancing).

Dataset | Scenario | Class labels
---|---|---
MNIST | CUR (Curly) | 0, 3, 8
STR (Straight) | 1, 4, 7
MIX (Mixed) | 2, 5, 6, 9
GTSRB |  SPDL (Speed Limit) | Speed limit (20km/h), Speed limit (30km/h),
Speed limit (50km/h), Speed limit (60km/h),
Speed limit (70km/h), Speed limit (80km/h),
Speed limit (100km/h), Speed limit (120km/h)
INST (Driving Instruction) | No passing, No passing for vehicles over 3.5 metric tons,
No vehicles, Vehicles over 3.5 metric tons prohibited
WARN (Warning) | Right-of-way at the next intersection, General caution,
Dangerous curve to the left, Dangerous curve to the right,
Double curve, Bumpy road, Slippery road,
Road narrows on the right, Road work, Traffic signals,
Pedestrians, Children crossing,
Bicycles crossing, Beware of ice/snow, Wild animals crossing
DIRC (Direction) | Turn right ahead, Turn left ahead, Ahead only,
Go straight or right, Go straight or left,
Keep right, Keep left, Roundabout mandatory
SPEC (Special Sign) | Priority Road, Yield, Stop, No entry
REGN (Regulation) | End of speed limit (80km/h), End of no passing,
End of all speed and passing limits,
End of no passing by vehicles over 3.5 metrics tons
CIFAR-10 | THG (Thing) | Airplane, Automobile, Ship, Truck
LIV (Living) | Bird, Cat, Deer, Dog, Frog, Horse
Tiny- ImageNet |  ANML (Animal) | Golden retriever, Chihuahua, German shepherd, Labrador retriever,
Standard poodle, Yorkshire terrier, Cougar/Puma, Persian cat
ISCT (Insect) | Dragonfly, Roach, Bee, Grasshopper,
Fly, Mantis, Monarch butterfly, Sulphur butterfly
ISTM (Instrument) | Water jug, Beer bottle, Tea pot, Pop bottle/Soda bottle,
Beaker, Rugby ball, Volley ball, Pill bottle
STRT (Structure) | Triumphal arch, Suspension bridge, Fountain, Viaduct,
Bannister, Steel arch bridge, Obelisk, Beacon
VHCL (Vehicle) | School bus, Trolly bus, Sports car, bullet train,
Convertible, Tractor, Police van, beach wagon
Table 5: Descriptions of sub-class scenario selection for each dataset.

##  Appendix D Scenario Setting

MNIST. We categorized the class labels into {Curly, Straight, Mixed} by the
shape of the digit. When training, we choose a single scenario as normal and
the rest of the others as abnormal. Note that the case where Mixed as normal
is the most challenging since this category has similar features with other
scenarios (e.g. ‘6’ versus ‘8’).

GTSRB. This dataset is originally proposed for the traffic sign recognition
task. We choose the scenarios by following the subset as introduced in
Stallkamp2012. With these subsets, we can simulate the abnormal cases from the
driver or self-driving car perspective on various traffic signs. Table 5 shows
the overall scenarios and its containing class labels.

CIFAR-10. We divided the conditions into two simple scenarios as {Thing,
Living} to mimic the real-world anomaly detection that can be used in general
object recognition applications.

Tiny-ImageNet. We first categorized class labels with representative subsets
as {Animal, Insect, Instrument, Structure, Vehicle}. Since the class labels of
this dataset are annotated based on the WordNet (miller1995wordnet), we
selected the equal number of the classes for each scenario by referring to the
same hierarchy as shown in Table 5 and the representative images for each
scenario are shown in Figure 4.

|  |
---|---|---

(a)

(b)

(c)

|
---|---

(d)

(e)

---

(f)

|  |
---|---|---

(g)

(h)

(i)

|
---|---

(j)

(k)

|  |
---|---|---

(l)

(m)

(n)

|
---|---

(o)

(p)

Figure 4: Representative images of the super-categories of each benchmark
datasets: (a-c) MNIST. (d-i) GTSRB. (j-k) CIFAR-10. (l-p) Tiny-ImageNet.


A.B.K and C.B ] contributed equally.

# Localized-delocalized crossover of spin-carriers and magnetization reversal
in Co2VO4

Abhijit Bhat Kademane  Department of Mathematics and Physics, Universitetet i
Stavanger, 4036 Stavanger, Norway  Churna Bhandari  Ames Laboratory, Iowa
State University, Ames, Iowa 50011, USA  [  Durga Paudyal  Ames Laboratory,
Iowa State University, Ames, Iowa 50011, USA  Department of Electrical and
Computer Engineering, Iowa State University, Ames, Iowa 50011, USA  Stephen
Cottrell  ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Science
and Innovation Campus, Chilton, Didcot, Oxon OX11 0QX, UK  Pinaki Das  Yong
Liu  Yuen Yiu  Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA
C. M. Naveen Kumar  Institute of Solid State Physics, Vienna University of
Technology, Wiedner Hauptstrae 810, 1040 Vienna, Austria  AGH University of
Science and Technology, Faculty of Physics and Applied Computer Science,
30-059 Kraków, Poland  Konrad Siemensmeyer  Andreas Hoser  Helmholtz Zentrum
Berlin für Materialien und Energie, D-14109 Berlin, Germany  Diana Lucia
Quintero-Castro  Department of Mathematics and Physics, Universitetet i
Stavanger, 4036 Stavanger, Norway  David Vaknin  Ames Laboratory, Iowa State
University, Ames, Iowa 50011, USA  Department of Physics and Astronomy, Iowa
State University, Ames, Iowa 50011, USA  Rasmus Toft-Petersen  Department of
Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

###### Abstract

Neutron diffraction, magnetization and muon spin relaxation measurements,
supplemented by density functional theory (DFT) calculations are employed to
unravel temperature-driven magnetization reversal (MR) in inverse spinel
Co2VO4. All measurements show a second order magnetic phase transition at
$T_{\rm C}=168$ K to a collinear ferrimagnetic phase. The DFT results suggest
the moments in the ferrimagnetic phase are delocalized and undergo gradual
localization as the temperature is lowered below $T_{\rm C}$. The delocalized-
localized crossover gives rise to a maximum magnetization at $T_{\rm NC}=138$
K and the continuous decrease in magnetization produces sign-change at $T_{\rm
MR}\sim 65$ K. Muon spectroscopy results support the DFT, as a strong
$T_{1}$-relaxation is observed around $T_{\rm NC}$, indicating highly
delocalized spin-carriers gradually tend to localization upon cooling. The
magnetization reversal determined at zero field is found to be highly
sensitive to applied magnetic field, such that above $B\sim 0.25$ T instead of
a reversal a broad minimum in the magnetization is apparent at $T_{\rm MR}$.
Analysis of the neutron diffraction measurements shows two antiparallel
magnetic sub-lattice-structure, each belonging to magnetic ions on two
distinct crystal lattice sites. The relative balance of these two structural
components in essence determines the magnetization. Indeed, the order
parameter of the magnetic phase on one site develops moderately more than that
on the other site. Unusual tipping of the magnetic balance, caused by such
site-specific magnetic fluctuation, gives rise to a spontaneous flipping of
the magnetization as the temperature is lowered.

##  I Introduction

The $AB_{2}X_{4}$-type spinel compounds, with tetrahedral $A$-site and
octahedral $B$-site, have been an anchoring point for some of the earliest
work on magnetism that led to the seminal Néel Néel (1948) and the Yafet-
Kittel models Yafet and Kittel (1952). The spinel (and the inverse-spinel)
compounds display a complex interplay among lattice, spin, charge, and orbital
degrees of freedom Tsurkan  _et al._ (2021), where novel phenomena emerge
Yamasaki  _et al._ (2006); Ruff  _et al._ (2019). Indeed, interest in these
systems persists as they exhibit exotic magnetic phenomena, such as
multiferroic effects and the spin liquid state that result from spin-orbit
coupling and geometric frustration associated with the pyrochlore sub-lattice
of the $B$-site Balents (2010); Bergman  _et al._ (2007). A prototypical class
of spinel compounds is the vanadate of type $A$V2O4 and its inverse-spinel
counterpart $A_{2}$VO4. At high temperatures, these systems form a cubic
structure with space-group $Fd\bar{3}m$. For $A$V2O4 with a non-magnetic
transition metal ion, such as Mg2+ or Zn2+ in the $A$-site, a cubic-to-
tetragonal phase transition accompanied by an antiferromagnetic phase have
been reported Niitaka  _et al._ (2013); Lee  _et al._ (2004). On the other
hand, magnetic ions on the $A$-site, such as in MnV2O4, FeV2O4, CoV2O4, show a
multitude of structural and magnetic transitions by virtue of coupled spin-
and orbital- degrees of freedom Garlea  _et al._ (2008); MacDougall  _et al._
(2012); Zhang  _et al._ (2014); Ishibashi  _et al._ (2017); Nii  _et al._
(2012). Specifically, for CoV2O4, orbital glass-like behavior has been
reported Koborinai  _et al._ (2016); Reig-i Plessis  _et al._ (2016) and spin-
ice rules have been outlined in studies of MnV2O4 and FeV2O4 Garlea  _et al._
(2008); MacDougall  _et al._ (2012). The inverse-spinel vanadate counterparts
(of type $A_{2}$VO4), which are more complex due to random-cation
distribution, have yet to be thoroughly explored. Indeed, the intriguing
magnetization reversal (MR) Kumar and Yusuf (2015) has been observed in
inverse spinel Co2VO4 Menyuk  _et al._ (1960). It has been proposed that the
cation distribution of Co2VO4 is
Co${}_{A}^{2+}$(Co${}^{2+}_{\nu}$V${}^{4+}_{1-\nu})_{B}$O${}_{4}^{2-}$ Rogers
_et al._ (1963). Additionally, magnetic interaction mechanisms in Co2VO4 have
been proposed using semi-empirical rules for magnetic interactions in spinels
Menyuk  _et al._ (1960); Wickham and Goodenough (1959). While temperature-
driven magnetization reversal for Co2VO4, Co2TiO4, and Sr2DyRuO6 has been
experimentally explored Menyuk  _et al._ (1960); Sakamoto (1962); Thota  _et
al._ (2017); Nayak  _et al._ (2015); Adroja  _et al._ (2020), the fundamental
driving mechanism behind MR in the inverse spinels remains to be understood.

Here, we examine the temperature evolution of magnetic structures using
neutron diffraction, magnetization, polarized muon spectroscopy, and
electronic structure calculations of Co2VO4. We establish multiple phase
transitions in the magnetically ordered phase, including temperature-driven
MR. We find the structure to be composed of two nearly balanced antiparallel
ferromagnetic components, on the $A$\- and $B$-site, giving rise to a net
ferrimagnetic phase evolving with temperature. Performing density functional
theory (DFT), localized calculation representing low-temperatures and
delocalized calculations for high-temperature (within the ordered region), we
confirm that the MR in Co2VO4 is consistent with such a balance. Polarized
muon spectroscopy results show dynamic $T_{1}$-relaxation at high-temperature
(within the ordered region) supporting electron delocalization predicted by
the theory.

##  II Experimental and Computational Details

The polycrystalline sample of Co2VO4 was prepared using a solid-state reaction
as described in Ref. (Rüdorff  _et al._ , 1956). The X-ray diffraction (XRD)
measurement confirms the formation of Co2VO4 spinel. Rietveld refinement of
neutron diffraction and XRD data (at 300 K) confirm the cubic $Fd\bar{3}m$
symmetry. Due to $A/B$ site inversion ($A$ – $8b$-tetrahedral and $B$ –
$16c$-octahedral) the formal chemical formula of Co2VO4 is Co${}^{2+}_{\nu}$
V${}^{4+}_{1-\nu}$[V${}^{4+}_{\nu}$ Co${}^{2+}_{2-\nu}$]O4, where $\nu$ is the
inversion parameter, and the $B$-site cations are enclosed in brackets. The
refinement yields that the degree of inversion $\nu\simeq 1$, i.e., the actual
Co${}^{2+}_{A}$[V4+Co2+]BO4, where Co2+ occupies the $A$ \- site and the
remaining Co2+ and V4+ are distributed evenly in the $B$ site. Unlike the
common spinel counterparts Garlea  _et al._ (2008); Ishibashi  _et al._
(2017), we find no structural transitions in Co2VO4 at all measured
temperatures. Nonetheless, we observed an insignificant shrinking of the unit-
cell that appears to saturate at $\approx$ 65 K.

Figure 1: Temperature dependence of ZFC (black circle) and FC (red triangle)
magnetization measurements performed for applied magnetic fields 0.03 T, 0.25
T and 5 T. $T_{\rm{C}}$ and $T_{\rm{NC}}$ represent the transition
temperatures to collinear and non-collinear magnetic structures respectively,
as explained in text. Magnetization reversal temperature is represented by
$T_{\rm{MR}}$ for $B=0.03$ T. Figure 2: Neutron-diffraction patterns (black
open circles), fits from the Rietveld refinement (red solid lines), and their
differences (blue solid lines). The vertical bars are the expected Bragg peak
positions as mentioned in the panels. The asterisk indicate impurity peaks.
NPD observed (measured on POWGEN) (a) $T$ = 140 K and (b) $T$ = 25 K
corresponding to the data collected with center wavelengths 1.333 Å. In the
inset of (b) one can see the emergence of (200) between 140 K and 25 K. (c)
Magnetic structure of Co2VO4 at 140 K. (d) Magnetic structure at 25 K.
$A$-site is occupied by Co(red), whereas $B$-site occupancy is shared between
Co and V(purple). Figure 3: Temperature dependencies of integrated intensities
of the Bragg peaks (111),(220), and (200) reflections for $B=0$ T. The solid
lines represents respective Power-law fits. $T_{\rm{C}}$ features can be
identified from the ordering of (111) and (220) while $T_{\rm{NC}}$ from
ordering of (200).

Magnetization measurements were performed on a pellet of mass 6.15 mg and
carried out using a superconducting quantum interference device (Quantum
Design MPMS, SQUID) magnetometer for various applied magnetic fields in the
range of $0.03-7$ T and $2-350$ K, performing both zero-field cooled (ZFC) and
field cooled (FC) measurements. Neutron powder diffraction (NPD) was performed
on a 2 g polycrystalline sample on the time-of-flight (TOF) powder
diffractometer, POWGEN Huq  _et al._ (2011), located at the Spallation Neutron
Source at Oak Ridge National Laboratory. The measurements were carried out
with neutrons of central wavelengths 1.333 and 3.731 Å. The sample was loaded
in a vanadium can which was attached to the cold end of a cryostat to reach
temperatures in the range $2\leq T\leq 300$ K. Neutron powder diffraction in
applied magnetic fields was performed using E6 Buchsteiner and Stüßer (2009)
and E9 Franz and Hoser (2017) diffractometers at the BER II reactor at the
Helmholtz-Zentrum Berlin. The 3.5 g powder sample was dispersed in deuterated
methanol-ethanol mixture, which freezes into an amorphous glass upon cooling,
and thereby fixes the powder orientation while applying a magnetic field. The
E6 diffractometer was used to perform the temperature scans ranging from
15-200 K with an applied magnetic field of 0.25 T and 2 T with wavelength 2.44
Å. Refinements of the diffraction data were carried out using FullProf
Rodriguez-Carvajal (1993, 2001).

The EMU spectrometer Giblin  _et al._ (2014) at the ISIS Neutron and muon
source was used to perform zero applied field muon spin relaxation ( ZF
-$\mu^{+}$SR) and longitudinal field muon spin relaxation (LF-$\mu^{+}$SR)
measurements where the applied magnetic field and muon spin are collinear on
the polycrystalline sample. The polycrystalline Co2VO4 sample amounting to 2.5
g was placed in a titanium holder with 24 mm diameter window. Muons falling
outside the sample window were stopped in a silver mask, giving a non-relaxing
background. 100% spin-polarized muons are produced by the source, implanted in
the sample and their time-evolution measured. The time evolution of muon
polarization is given by $P_{z}(t)$. A normalized sample polarization
immediately after muon implantation, $P_{s}(0)$, was obtained after background
correction. The ZF-$\mu^{+}$SR scans were performed at various temperatures in
the 10-250 K range. In-order to elucidate dynamic aspects, LF-$\mu^{+}$SR
measurements were performed at 250, 125, 75, and 10 K with applied fields of
0.005, 0.05, 0.2, and 0.3 T. The data were analyzed using the muon analysis
module of Mantid Arnold  _et al._ (2014).

To understand the magnetization reversal in Co2VO4, we carried out two sets of
first-principles electronic structure calculations: i) delocalized and ii)
localized. The delocalized calculations were performed using DFT in
conjunction with generalized gradient approximation (GGA) for exchange-
correlation functional in all-electron projected augmented plane wave (PAW)
formalism implemented in Vienna Simulation Package (VASP) including spin-orbit
coupling (SOC) Kresse and Furthmüller (1996); Kresse and Joubert (1999). We
used plane wave cut-off energy of 500 eV and $4\times 4\times 4$ k-mesh for
the Brillouin-zone integration. Structure optimization is performed by
randomly replacing 50% V_B with Co_B in the bulk CoV2O4 structure that has the
lowest total energy among all the considered configurations.

##  III Results and Discussions

Figure 1 shows the temperature ($T$) dependence of the zero field-cooled (ZFC)
and field-cooled (FC) magnetization ($M$) of Co2VO4 at applied magnetic field
$B$ = 0.03, 0.25, and 5 T. Three features are identified for $B=0.03$ T.
First, a sharp rise is observed at $T_{\rm{C}}=168(1)$ K (C-collinear)
corresponding to ferrimagnetic response and second, with an onset of a
decrease at $T_{\rm{NC}}=138(8)$ K (NC- non collinear), for both FC and ZFC.
The third feature occurs at the temperature range $T_{\rm{MR}}\sim 65-130$ K,
where the magnetization reverses, dependent on the cooling protocol (i.e., FC
or ZFC). Also, below 20 K there is an upturn in magnetization, suggestive of a
meta-magnetic behavior. Magnetization reversal does not occur for fields
higher than $\sim 0.25$ T and eventually, the ZFC and FC curves overlap for
fields higher than 5 T and the temperature dependence becomes reversible.

Figure 4: (a) The magnitude of $A$\- and $B$\- sub-lattice moments obtained
from Rietveld refinement for $B=0$ T. (b)Total moment (per f.u.) calculated by
adding moments FM moments. Temperature dependence of total moment emulates the
magnetization reversal as observed in magnetization measurement.

The magnetization at $T_{\rm{C}}$ for $B=0.1$ T is
$$0.2\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$/f.u.$ (not shown), which
is lower than that of CoV2O4 Huang  _et al._ (2012). A high spin configuration
$S=3/2\,($3\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$)$ for Co_A^2+ and
$S=1$ $($2\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$)$ for V_B^3+ have
been reported for CoV2O4 Ishibashi  _et al._ (2017). This indicates the
reduced magnetization in Co2VO4 is due to antiparallel Co_A and Co_B/V_B
(ferri-magnetic) assuming high spin configuration $S=3/2$, for Co2+ and
$S=1/2$ for V_B implying an oxidation state of $4+$ for V. As discussed below,
DFT calculations confirm the valence states of Co_B and V_B as well their
magnetic ordering in the localized solution.

The neutron powder diffraction patterns for 140 K and 25 K are presented in
Fig. 2 (a) and (b). Consistent with the aforementioned magnetic
susceptibility, Fig. 3 shows the emergence of the (111) and (220) magnetic
reflections at $T=168(1)$ K. At a lower temperature ($=138(2)$ K), a weak
(200) magnetic reflections is also observed (inset of Fig. 2 (a)). We
emphasize that the (200) is forbidden by $Fd\bar{3}m$. In the case of MnV2O4
observation of the (200) is linked to spin canting of the spinel-$B$-site and
a structural transition (Garlea  _et al._ , 2008). The temperature dependence
of the three magnetic Bragg reflections is shown in Fig. 3. These curves are
fitted to a power-law function. The critical temperature obtained for
reflections (111) and (220) is $T_{\rm{C}}$=167.8$\pm$0.4 K and for the (200)
at $T_{\rm{NC}}$ = 138.5$\pm$7.5 K. We also note that the critical exponents
associated with (111) and (200) are different, at 0.71$\pm$0.02 and
0.5$\pm$0.1, respectively.

Figure 5: (a) Magnitude of $A$\- and $B$\- sub-lattice moments obtained from Rietveld refinement for $B=2$ T. (Inset) Total moment (per f.u.) calculated by adding moments along the $c$-direction. Temperature dependence of total moment emulates magnetization for fields higher than 0.25 T. Table 1: Magnetic structure factors computed are presented as $|$F${}_{hkl}|^{2}$ (normalized) for the magnetic reflections ($hkl$) using Basis vectors (BVs) of the Irreducible representation(IR) belonging o $A$-site and $B$-site. Site | Irrep | BVs |  $|$F${}_{111}|^{2}$ |  $|$F${}_{200}|^{2}$ |  $|$F${}_{220}|^{2}$ |  $|$F${}_{331}|^{2}$
---|---|---|---|---|---|---
$A$-site | $\Gamma 10$ | $\psi_{A}^{1}$ | (0 0 1) |  |  |  |
8b |  | (FM) | (0 0 1) | 0.68 | 0 | 1.00 | 0.27
|  |  | (0 0 1) |  |  |  |
|  | $\psi_{B}^{1}$ | (0 0 1) |  |  |  |
|  | (FM) | (0 0 1) | 0.51 | 0 | 0 | 0.2
$B$-site | $\Gamma 10$ |  | (0 0 1) |  |  |  |
16c |  |  | (-1 -1 0) |  |  |  |
|  | $\psi_{B}^{2}$ | (1 1 0) |  |  |  |
|  | (AFM) | (-1 1 0) | 0.1 | 0.04 | 0.03 | 0.04
|  |  | (1 -1 0) |  |  |  |

To restrict the refinement of the neutron powder data, we exclude the
irreducible basis vectors, on both the $A$-and $B$-site, that are inconsistent
with the strongest observed peaks in the powder diffraction pattern. These
basis vectors that are (for $Fd\bar{3}m$ and propagation vector $k=(0,0,0)$))
founds using BasIREPS Rodriguez-Carvajal (1993) and SARAhWills (2000). We find
that the simplest model consistent with both magnetization and powder
diffraction data, consists of basis vectors from the IR $\Gamma 10$ with $A$\-
and $B$\- sub-lattice moments arranged anti-parallel. The magnetic
representation in this case corresponds to Shubnikov space group
$I4_{1}/am\textquoteright d\textquoteright$ (representation mGM4+) Perez-Mato
_et al._ (2015); Stokes  _et al._ . This picture is consistent with
observations of other spinel vanadates Garlea  _et al._ (2008); Ishibashi  _et
al._ (2017) as well as recommendations made for magnetic interactions in
Co2VO4 Menyuk  _et al._ (1960). Table 1 lists $|$F${}_{hkl}|^{2}$ for
prominent magnetic reflections calculated using these basis vectors. Note that
the intensity on (200) could be explained only by $\psi_{B}^{2}$ (AFM), while
(220) has no contributions from $\psi_{B}^{1}$ (FM). As seen in table 1,
ferromagnetic order on the $A$-site ($\psi_{A}^{1}$) gives rise to intensity
in both the (220) and (111) reflections while ferromagnetic order on the
$B$-site ($\psi_{B}^{1}$) only gives rise to intensity on (111). The
ferromagnetic ordered moment on the $A$-site can therefore be explicitly
distinguished from that of the $B$-site, and the relative magnitudes of the
ordered moments can be directly extracted from the refinement. While the net
ferrimagnetic moment can be directly deduced from neutron powder diffraction,
it cannot distinguish individual contributions from Co_B and V_B. We note that
combination of $\psi_{B}^{1}$ (c-axis) and $\psi_{B}^{2}$ ( ab-pane) leads to
the commonly observed 2-in-2-out configuration in pyrochlore structures Garlea
_et al._ (2008).

The refinements of neutron powder diffraction data at ($B=0$ T) at the
temperature range $5-250$ K yield the sub-lattice magnetic moments as shown in
Fig. 4 (a). Refined moments on $A$\- and $B$-sites are labeled as
$\mu_{\mathrm{A}(\mathrm{FM})}$, $\mu_{\mathrm{B}(\mathrm{FM})}$ and
$\mu_{\mathrm{B}(\mathrm{AFM})}$, respectively. The total moment
$\mu_{\rm{total}}=\mu_{\mathrm{A}(\mathrm{FM})}+\mu_{\mathrm{B}(\mathrm{FM})}$
is shown in Fig. 4 (b). Contributions from $\mu_{\mathrm{B}(\mathrm{AFM})}$ (
AFM see Tab. 1) cancel out and thus do not contribute to the total moment. As
shown, $\mu_{\rm{total}}$ as a function of temperature exhibits magnetic
behavior that resembles the magnetization shown in Fig. 1 (a). Indeed Fig. 4
(b) shows three identified temperatures in magnetization, $T_{\rm{C}}$,
$T_{\rm{NC}}$, and $T_{\rm{MR}}$. Similar analysis of the diffraction patterns
under magnetic field $B=2$ T also yield distinguishable
$\mu_{\mathrm{A}(\mathrm{FM})}$ and $\mu_{\mathrm{B}(\mathrm{FM})}$. However,
$\mu_{\rm{total}}$ at $B=2$ T does not show magnetization reversal (Fig. 5) as
$\mu_{\mathrm{A}(\mathrm{FM})}$ is always larger than
$\mu_{\mathrm{B}(\mathrm{FM})}$ for all temperatures, consistent with the
observation above $0.25$ T (Fig. 1). It is not possible to extract
$\mu_{\mathrm{B}(\mathrm{AFM})}$ reliably in these measurements, as the
defused signal from ethanol-methanol mixture overshadows the weak (200) peak.

Figure 6: Initial polarization (a) and relaxation rate (b) extracted from fits
to the ZF-$\mu^{+}$SR polarization spectra. For $T<T_{\rm{C}}$, fits are to
Eq. (2), with $\beta$ as a free parameter below $T_{\rm{MR}}$ (red points).
Above $T_{\rm{C}}$, fits are carried out to Eq. (1) (green points). Inset of
(a) $\beta$ vs $T$ where $\beta$ reaches 0.5 at 12 K.

To determine possible dynamic origin of the distinct critical behavior on the
$A$\- and $B$-sites, employ polarized muon spectroscopy. The ZF-$\mu^{+}$SR
measurements were performed at various points in the region 10$-250$ K. In the
paramagnetic region ($T>T_{\rm{C}}$), the ZF-$\mu^{+}$SR polarization spectra
shows temperature-independent relaxation with full polarization. This region
is parameterized with static Kubo–Toyabe times exponential decay (SKT-ED),
accounting for nuclear and electronic moments Kubo and Toyabe (1967).

$P_{z}(t)=P_{z_{1}}(0)\left(e^{-\delta^{2}t^{2}/2}\left(1-\delta^{2}t^{2}\right)\frac{2}{3}+\frac{1}{3}\right)e^{-\lambda
t}\\\ +P_{z_{2}}(0),$ (1)

where $\delta$ accounts for relaxation rate due to the coupling of nuclear
moments and muon and $\lambda$ for relaxation rate due to electron moments and
muon coupling. At $T_{\rm{C}}$, a sharp loss of the initial polarization is
observed. This is typical for a magnetically ordered system, where the missing
polarization is associated with strong electronic moments, giving rise to an
unresolved precession signal owing to the finite muon pulse width at the
source (ISIS). Below $T_{\rm{C}}$, a one-component stretched exponential decay
(Eqn. 2) function was sufficient to parameterize the ordered region, i.e.,

$P_{z}(t)=P_{z_{1}}(0)e^{-(\lambda t)^{\beta}}+P_{z_{2}}(0).$ (2)

In Eq. (2), $\lambda$ is the relaxation rate and $\beta$\- the stretching
exponent. When $\beta=1$, equation 2 turns into simple exponential decay. We
find that above $T_{\rm{MR}}$, the best fit to the data is obtained with
$\beta$ fixed at 1. However, below $T_{\rm{MR}}$, acceptable fits could only
be obtained fitting $\beta$ as a free parameter

Figure 6 shows the initial polarization and relaxation rate extracted from
ZF-$\mu^{+}$SR data as a function of temperature. The polarization parameter
($P_{s}(0)$) decreases to 2/3 of its initial value below $T_{\rm{C}}$.
Intriguingly, there is a further loss of polarization as the temperature is
reduced below $T_{\rm{C}}$, reaching a minimum at $T_{\rm{NC}}$ and only
regaining the 1/3 value of the initial polarization at $T_{\rm{MR}}$. The
broad dip between $T_{\rm{C}}$ and $T_{\rm{MR}}$ suggests that muons are
experiencing changes in the magnetic structure right through the region. This
is consistent with the corresponding peak in the relaxation rate, suggesting
spin fluctuation/reorientation process is driving an enhanced
$T_{1}$-relaxation of the muon signal through this temperature region.
Notably, there is no sharp peak in the relaxation rate at $T_{\rm{C}}$, as one
would expect for a typical magnetic phase transition. Fitting the low-
temperature data (T$<60$ K) with $\beta$ as a free parameter shows $\beta$
decreasing steadily, reaching a value $\approx 0.5$ at 12 K. We note that
$\beta=0.5$ is a signature of glassy transition Campbell  _et al._ (1994). The
glassy transition at low temperatures is typical of inverse spinels Nayak  _et
al._ (2015); Thota and Seehra (2013).

Initial polarization and relaxation data for LF-$\mu^{+}$SR measurements up to
0.3 T, carried out at 10 K, 125 K and 250 K, are shown in Figure 7. With
$T_{1}$-spin relaxation, $\lambda$, persisting at all temperatures to the
highest fields measured. This suggests that spin dynamics are present at all
temperatures, both in the paramagnetic regime above $T_{\rm{C}}$ and also in
the ordered state. We note that the strongest relaxation is measured at 125 K,
close to $T_{\rm{MR}}$, a region where electron delocalization is predicted by
DFT, and at this temperature the relaxation rate (Fig. 7b) appears to increase
with the applied field in contrast to results at measured at 10 K. A clear
recovery of the initial polarization is seen at the highest fields measured,
although full decoupling is not achieved, with only 50% of the polarization
being recovered. We suggest this is the result of both large internal fields
and spin dynamics below $T_{\rm{C}}$ in this material.

Figure 7: (a) Initial polarization (b) relaxation rate extracted from fits to
the LF-$\mu^{+}$SR polarization spectra with Eq. (2). At 125 K, complete
decoupling was realized with tiny field of 0.005 T. Due to instrument effects
$P_{s}(0)$ goes beyond 1 for higher fields.

To supplement the experimental results, we conduct delocalized and localized
electronic structure calculations. For the delocalized calculations, we employ
standard DFT with GGA functional. In the localized calculations, we add the
on-site electron correlation following the Dudarev et al.’s scheme Dudarev
_et al._ (1998) in which only the effective value of on-site correlation,
($U_{eff}=U-J=4.5$ eV for $3d$ electrons of Co and V Lee  _et al._ (2017)) is
meaningful. To be on equal footing, we also performed another calculation
using the same value of $U_{eff}$ and find the delocalized solution, which has
higher total energy than in the localized solution. As expected, the net
magnetic moments do not change significantly with the use of $U_{eff}$ in the
delocalized results.

As shown below, the localized calculations are akin to the low-temperature
ground state whereas the delocalized corresponds to the high-temperature
ferrimagnetic phase. Both localized and delocalized calculations show similar
magnetic moments for Co_A, while they are drastically different for Co_B and
V_B. These changes in the local magnetic moments are consistent with the low-
temperature neutron diffraction results (Table 2). The major contribution to
the net magnetic moment is from spin (Table 2), while the orbital
contributions are small and slight fluctuations for Co_B are found (not
shown). In the localized solution, we find canted spin-structures for Co_B and
V_B in qualitative agreement with the experimental results at 25 K (Table 2).
A notable prediction is the antiferromagnetic alignment of Co_A moments with
respect to Co_B with a small canting in the planar direction (Table 2). For
Co_A, the canting is not resolved experimentally. At the $B$-site, the V_B
moments follow the Co_B moments. These are remarkable changes in the
frustrated magnetic structures of the well-known spinel vanadates viz., FeV2O4
MacDougall  _et al._ (2012) and MnV2O4 Garlea  _et al._ (2008), in which V_B
moments are 2-in 2-out as observed experimentally. The calculated net magnetic
moment parallel to the $c$-direction is negative, in agreement with the low-
temperature data. The discrepancy between theory and experiment in the
magnitude of net magnetic moment is likely due to the choice of atomic sphere
sizes used in the calculations, i.e., Co_A moment
$\sim$2.6\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$$ is smaller than the
experiment $3.07\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$. We find $3d$
Co/V and O($2p$) hybridization, which we argue may affect the magnetic moments
on both Co and V. In calculations, the canting is small which results in a
large magnetic moment
($\sim$3.7\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$$) along the
$z$-direction, which qualitatively agrees with the experiment (see Table 2).
Another possibility for the slight discrepancy is random occupancy of Co and V
in the $B$-sites.

Interestingly, the magnetic moments obtained with the delocalized solutions
agree with the higher temperature data, indicating a more itinerant scenario
Bhandari  _et al._ (2021). The magnetic moments on the $B$-site are sensitive
to the choice of $U_{eff}$ (i.e., localized or delocalized), while the moments
on the $A$-site remain relatively unchanged. Inter-atomic charge transfer from
Co_B to V_B sites reduces the effective Co_B moments. This delicate balance
between Co_A, Co_B, and V_B magnetic moments in the localized and delocalized
limits presumably leads to the MR.

To demonstrate that the electronic configuration of Co is $3d^{7}$ (i.e.,
Co2+) and that of V_B is $3d^{1}$ (i.e., V4+), we examine the partial density
of states (PDOS) of each atom in the localized calculations. Figure 8 shows
very well split-off $e_{g}\downarrow$ and $t_{g}\uparrow$ $3d$-states of Co_A
around the Fermi level due to the tetrahedral crystal field of oxygen atoms.
In Co_B and V_B, the splittings of the three-fold $t_{2g}$-states are less
obvious. Because of the distorted octahedral crystal field effectKismarahardja
_et al._ (2011), these states split into an $a_{1g}$ and doubly degenerate
$e_{g}^{{}^{\prime}}$ resulting in $3d^{7}$, $S=3/2$ (occupying
$e_{g}^{{}^{\prime}}\uparrow$) for Co_B2+ as in Co_A2+ and $3d^{1}$, $S=3/2$
(occupying $a_{1g}$) for V_B4+ in the localized limit as inferred from the
current and the earlier experiments Menyuk  _et al._ (1960). An interesting
change is in vanadium oxidation state from V${}^{3+}(3d^{2})$, which is common
in the standard spinel compounds, to V${}^{4+}(3d^{1})$ with $S=1/2$
consistent with the magnetic moment Menyuk  _et al._ (1960). We note that
Co_B2+ and V_B2+ moments change significantly in the delocalized solutions
(Table 2). This is expected due to a narrow $t_{2g}$-splitting in both atoms
at $B$-sites.

Figure 8: Spin resolved partial density of states (PDOS) of Co_A, Co_B, and V_B. The positive and negative values indicate PDOS of majority and minority spin electrons, respectively. The spins of Co_A are aligned antiferromagnetically with respect to Co_B and V_B. Both Co at $A$\- and $B$ sites have $3d^{7}$ valence-electrons indicating the formation of Co2+ ion. The distorted octahedral crystal field splits $t_{2g}$ into an $a_{1g}$ and doubly degenerate $e_{g}$ states at site $B$. For Co_B, two electrons occupy the lower $e_{g}$ states forming $S=3/2$ state, while for V_B, one electron occupies $a_{1g}$ yielding V4+ oxidation state with $S=1/2$. Table 2: Canted magnetic moment components (spin + orbital) ($\mu_{x},\mu_{y},\mu_{z}$) of individual atom and total magnetic moments ($\mu_{\rm{total}}$) (per f.u.) in units of $\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$ in Co2VO4 obtained from Rietveld refinement from zero field data and theory (GGA + SOC and GGA + SOC + U calculations). For oxygens, only the calculated moments are given. Site | Atom | Experiment | Theory
---|---|---|---
|  | 140 K | 25 K | GGA+SOC | GGA+U+SOC
$A$-site |  Co_A | ($0,0,2.18(6)$) | ($0,0,3.07(6)$) | ($0,0,2.48$) | ($0.80,0,2.74$)
|  Co_B | NA | NA | ($0,0,-0.43$) | ($-0.81,|0.13|,-2.72$)
$B$-site |  V_B | NA | NA | ($0,0,-1.29$) | ($-0.34,|0.02|,-1.02$)
|  Co_B + V_B | ($0,0,-1.68(2)$) | ($|$0.88(8)$|$, $|$0.88(8)$|$, $-3.13(1)$) | ($0,0,-1.72$) | ($-1.15,|0.15|,-3.74$)
X-site | O | NA | NA | ($0,0,0.09$) | ($0,0,0.02$)
$\mu_{\rm{total}}\,($\text{\,}\mathrm{\SIUnitSymbolMicro}_{\rm{B}}$/f.u.)$ |  | ($0,0,0.50(8)$) | ($0,0,-0.06(7)$) | (0, 0, 1.14) | ($-0.32,0,-0.90$)

##  IV Conclusions

Our interest in this less studied inverse cobalt vanadium spinel springs from
the highly unconventional temperature-dependent magnetization of the compound,
where a spontaneous magnetization reversal takes place. We have performed
magnetization measurements and revealed at least three distinct anomalies.
Neutron powder diffraction analysis reveals two antiparallel magnetic
structures, each belonging to magnetic ions on $A$\- and $B$-sites. The
relative balance of these two structural components in essence determines the
magnetization. However, the order parameter of the magnetic order on one site
develops more moderately than that on the other site. While in itself unusual,
a tipping of the magnetic balance, caused by such site-specific magnetic
fluctuation, gives rise to a spontaneous flipping of the magnetization as the
temperature is lowered. The magnetization reversal determined at zero fields
is sensitive to an applied magnetic field, such that above $B\sim 0.25$ T
instead of a reversal a minimum in the magnetization is apparent at $T_{\rm
MR}$.

The root cause of this site-specificity is unveiled by DFT results. The DFT
calculations were performed assuming delocalized and localized spin carriers.
We argue that the magnetization reversal is a consequence of the itineracy of
Co and V electrons in the $B$-site. In particular, the delocalized
calculations, which we associate with the behavior of the high temperature of
the system (around $T_{\rm C}$), predict relatively small net magnetic moments
on the $B$-site compared to the $A$-site giving rise to the net ferrimagnetic
moment observed near $T_{\rm C}$. The DFT also shows that the Co moments on
the $A$-site are less sensitive to temperature change with electronic
configuration Co${}^{2+}(3d^{7})$ with $S=3/2$. For the $B$-site, the DFT
finds mixed oxidation states for Co and V that at low temperatures tend
towards their localized values, namely, Co2+ ($S=3/2)$ (as in the $A$-site)
and V4+ ($S=1/2$). This suggests that the MR is likely driven by delocalized
to localized cross-over of the $3d$-electrons of the Co/V atoms in the
$B$-site. We note that quantitative disagreements between theory and
experiment are expected owing to the complexities of this inverse spinel
Co2VO4 and the random distribution of Co and V in the $B$-site. Polarized muon
spectroscopy as a function of temperature is consistent with the neutron
diffraction and magnetization results but adds new insights that have been
predicted by our DFT calculations. The muon spectroscopy indicates highly
delocalized spin-carriers at high temperatures that gradually tend to
localization at lower temperatures.

##  V Acknowledgement

Part of this research at Ames Laboratory is supported by the U.S. Department
of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering under Contract No. DE-AC02-07CH11358. The electronic structure and
magnetism employed in this work are developed by D.P and his group in the
Critical Materials Institute, an Energy Innovation Hub led by the Ames
Laboratory and funded by the U. S. Department of Energy, Office of Energy
Efficiency and Renewable Energy, Advanced Manufacturing Office. Use of the
Spallation Neutron Source at the Oak Ridge National Laboratory is supported by
the U.S. Department of Energy, Office of Basic Energy Sciences, Scientific
Users Facilities Division. Experiments at the ISIS Pulsed Neutron and Muon
Source were supported by a beamtime allocation from the Science and Technology
Facilities Council. We thank the Helmholtz-Zentrum Berlin für Materialien und
Energie for the allocation of neutron beamtimes at BER II and bulk properties
measurements at CoreLab Quantum Materials.

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