Automated Detection of Cat Facial Landmarks]Automated Detection of Cat Facial Landmarks

George<EMAIL_ADDRESS>
Ilan<EMAIL_ADDRESS>
Anna<EMAIL_ADDRESS>
[]Information Systems Department, University of Haifa, Haifa, Israel

The field of animal affective computing is rapidly emerging, and analysis of facial expressions is a crucial aspect. One of the most significant challenges that researchers in the field currently face is the scarcity of high-quality, comprehensive datasets that allow the development of models for facial expressions analysis. One of the possible approaches is the utilisation of facial landmarks, which has been shown for humans and animals. In this paper we present a novel dataset of cat facial images annotated with bounding boxes and 48 facial landmarks grounded in cat facial anatomy. We also introduce a landmark detection convolution neural network-based model which uses a magnifying ensembe method. Our model shows excellent performance on cat faces and is generalizable to human facial landmark detection.

§ INTRODUCTION

There is a huge body of work addressing automated human facial analysis, which has a plethora of applications in affective computing, healthcare, biometry, human-computer interaction and many other fields [1, 2]. One of the cornerstones of most of the approaches is localization of keypoints, also known as fiducial points or facial landmarks [3]. Their location provides important information that can be used for face alignment, feature extraction, facial expression recognition, head pose estimation, eye gaze tracking and many more tasks [4, 5, 6]. Automated facial action unit recognition systems also often use facial landmarks [7, 8].

The type and number of landmarks varies depending on the specific application at hand: for coarse tasks such as face recognition or gaze direction assessment a few landmarks may suffice, however for facial expression analysis, depending on the species, up to eighty landmarks may be needed, leading to the need for automation of landmark detection [9, 10]. Despite its being seemingly simple task, this computer vision problem has proven to be extremely challenging due to inherent face variability as well as effects of pose, expression, illumination, etc. Accordingly, a huge body of research addresses the automation of facial landmark detection and localization.

Research concerned with automation of animal behavior analysis has so far lagged behind the human domain. However, this is beginning to change, partly due to introduction of deep-learning platforms such as the DeepLabCut [11], DeepPoseKit [12] and many more, which allow to automate animal motion tracking and recognition of keypoints located on animals’ body.  However, recently an increasing number of works go ‘deeper’ than tracking addressing recognition of animals’ affective states, including emotions, stress and pain. A comprehensive survey of state-of-the-art of these methods is provided in [13], focusing mainly on
facial expressions. Since they are produced by all mammals, they are one of the most important channels of communication in this context [14, 15]. Facial analysis of animals presents immense challenges due to the great variability of textures, shapes, breeds and morphological structure in the realm of animals. This leads to the need for addressing the problem of automated facial landmark detection in animals.

This brings about unique challenges not present in the human domain, such as more complicated data collection protocols and ground truth establishment in the absence of verbal communication with animals [13]. But most importantly, the variabilities across and within lead to the need of exploring feasible, economic, in terms of computational resources and time, approaches for solving the landmark localization problem in the animal domain.

In [16], Yang et al. used different deep learning approaches to classify pain in cats. Despite the high accuracy of the classification, their work lacks the explicability of the results obtained. The approach based on landmarks allows implementing action units into the procedure of pain or emotion recognition, which makes it more explainable, in contrast to the “black box" deep learning approach. In [17], landmarks were used to identify important facial areas for the same task of pain classification. The presented results clearly demonstrate the most significant aspects by which pain can be determined in cats, and, therefore, can provide additional insights in understanding cats behaviour and their well-being. While such explainability is not critically important for humans, where we can fully rely on our understanding of the expressions of human emotions and on the direct feedback, this is not so for animals.

Cats are of specific interest in the context of pain, as they are one of the most challenging species in terms of pain assessment and management due to a reduced physiological tolerance and adverse effects to common veterinary analgesics [18], a lack of strong consensus over key behavioural pain indicators [19] and human limitations in accurately interpreting feline facial expressions [20]. Three different manual pain assessment scales have been developed and validated in English for domestic cats:
the UNESP-Botucatu multidimensional composite pain scale [21], the Glasgow composite measure pain scale (CMPS) [22], and the Feline Grimace Scale (FGS) [23]. The latter was further used for a comparative study in which human’s assignment of FGS to cats during real time observations and then subsequent FGS scoring of the same cats from still images were compared. It was shown that there was no significant difference between the scoring methods [24], indicating that facial images can be a reliable medium from which to assess pain.

Finka and colleagues [25] were the first to apply 48 geometric landmarks to identify and quantify facial shape change associated with pain in cats. These landmarks were based on both the anatomy of cat facial musculature, and the range of facial expressions generated as a result of facial action units (CatFACS) [26].  The authors manually annotated the landmarks, and used statistical methods (PCA analysis) to establish a relationship between PC scores and a well-validated measure of pain in cats.
Feighelstein et al. [27] used the dataset and landmark annotations from [25] to automate pain recognition in cats using machine learning models based on geometric landmarks grouped into multivectors, reaching accuracy of above 72%, which was comparable to an alternative approach using deep learning. This indicates that the use of the 48 landmarks from [25] can be a reliable method for pain recognition from cat facial images. The main challenge in this method remains the automation of facial landmarks detection. In general, one of the most crucial gaps in animal affective computing today, as emphasized in [13], is the lack of publically available datasets. To the best of our knowledge, all datasets containing animal faces available today have very few facial landmarks, as discussed in the next section.

To address these gaps, this paper makes two main contributions. First, we introduce the Cat Facial Landmarks in the Wild (CatFLW) dataset containing images of cats with 48 facial landmarks, annotated using an AI-assisted annotation method and grounded with the cat facial anatomy. Secondly, we present an Ensemble Landmark Detector (ELD), a baseline model which shows excellent performance on cats. The model also performs well on human facial landmark WFLW dataset, compared to state-of-the-art models.

§ CATFLW DATASET

To promote the development of automated facial landmark detectors in cats, we have created the CatFLW dataset inspired by existing ones for humans and animals. The main motivation for its creation was the relatively small amount of similar datasets or a low number of facial landmarks in existing ones.

§.§ Related Datasets

Our dataset is based on the original dataset collected by Zhang et al. [28], that contains 10,000 images of cats annotated with 9 facial landmarks. It includes a wide variety of different cat individuals of different breeds, ages and sexes in different conditions, which can provide good generalization when training computer vision models, however some images from it depict several animals, have visual interference in front of animal faces or cropped (according to our estimates 10-15%). Figure <ref> shows the examples of such inapplicable images. It is also worth noting that a number of images contain inaccurate annotations with significant errors, which can also lead to incorrect operation of computer vision models trained on this data.

In [28] 9 landmarks are labeled for each image (two for the eyes, one for the nose and three for the each ear), which is sufficient for detecting and analyzing general information about animal faces (tilt of the head or direction of movement), but is not enough for analyzing complex movements of facial muscles.

Examples of not suitable images from [28]. The image is considered inapplicable when it is impossible to visually determine the location of the facial landmark with full confidence or uncertainty is created due to the presence of several animals.

Sun et al. [29] used the same dataset in their work, expanding the annotation to 15 facial landmarks.
It slightly better displays the features of the cats' faces structure, containing two more landmarks for each eye, one landmark for the bridge of the nose and one for the mouth in addition to the 9 landmarks presented in [28]. However,
to the best of our knowledge, only 1,706 out of the declared 10,000 images are publicly available.

Khan et al. [30] collected the AnimalWeb dataset, consisting of an impressive number of 21,900 images annotated with 9 landmarks. Despite the wide range of represented species, only approximately 450 images can be attributed to feline species. Moreover, the annotation system was chosen differently: here there are two landmarks for each eye, one for the nose and four for the mouth.

Dataset     Animal     Size     Facial Landmarks

Khan et al. [30]     Various     21900     9

Zhang et al. [28]     Cat     10000     9

Liu et al. [31]     Dog     8351     8

Cao et al. [32]     Various     5517     5

Mougeot et al. [33]     Dog     3148     3

Sun et al. [29]     Cat     1706     15

Yang et al. [34]     Sheep     600     8

Ours (CatFLW)     Cat         48

Comparison of animal facial landmarks datasets

Other relevant datasets, containing less than 9 facial landmarks, are mentioned in Table <ref>, including those in [31, 32, 33, 34]. For comparison, popular datasets for human facial landmark detection [35, 36] have several dozens of landmarks. There are also other datasets suitable for face detection and recognition of various animal species [37, 38, 39, 40, 41], but they have no facial landmark annotations.

Two conclusions follow from the above: firstly, the validity of the assumption about the lack of datasets with a comprehensive number of facial landmarks for animals and cats in particular, and secondly, these are differences in the annotations between datasets and in the choice of landmarks positions in accordance with the morphological structure of animal faces.

§.§ Guidelines and Annotation Process

The CatFLW dataset consists of images selected from the dataset in [28] using
the following inclusion criterion which optimize the training of landmark detection models: image should contain a single fully visible cat face, where the cat is in non-laboratory conditions (`in the wild'). Figure <ref> shows the examples of images which were used in the dataset.

Images with bounding boxes and 48 facial landmarks from CatFLW.

We should also consider the issue of partial occlusion of faces, where all the landmarks are not visible, but their location can be estimated from the logic of the structure of the face. Such images, however, were discarded due to considerations of working with convolution neural networks, which largely focus on the visual boundaries and textures of images, as a result of which, when these models are trained on images with incompletely visible faces, bias and incorrect behavior may occur during further predictions. As the practice of working with such models shows, subsequently, when predicting on images where the face is partially occluded, the model nevertheless “guesses" the positions of the landmarks based on the geometric relationship between all the landmarks as a whole embedded in it during training.

After selection and filtering of images, a bounding box and 48 facial landmarks were placed on each image using the Labelbox platform [42].

Bounding box. Each bounding box is defined by the two coordinates of the upper left and lower right corners in the coordinate system of the image. In the current dataset, it is not tied to facial landmarks (this is usually done by selecting the outermost landmarks and calculating the difference in their coordinates), but is visually placed so that the entire face of the animal fits into the bounding box as well as about 10% of the space around the face. This margin is made because when studying detection models on the initial version of the dataset, it was noticed that some of them tend to crop the faces and choose smaller bounding boxes. In the case when face detection is performed for the further localization of facial landmarks, the clipping of faces can lead to the disappearance from the image of important parts of the cat's face, such as the tips of the ears or the mouth. Such an annotation still makes it possible to construct bounding boxes by outermost landmarks if needed.

Landmarks. 48 facial landmarks introduced in Finka et al. [25] and used in Feighelstein et al. [27] were manually placed on each image (shown on the Figure <ref>). These landmarks were specifically chosen for their relationship with underlying musculature and anatomical features, and relevance to CatFACS [26]. The landmarks are grouped into four semantic groups in the sense of their physical location on the face: Lower Face (cheeks, whiskers and nose), Jaw (chin and mouth), Upper Face (eyes), and Ears. For each of them, the position guideline and, if possible, the associated facial muscles and action units are determined. Thus, the structure of facial landmarks is quite strictly tied to the anatomical features of the cat's face, which allows associating morphological characteristics with landmarks, their mutual position and movement. Detailed information about the landmarks' placement can be found in the appendices to the article [25].

Annotated cat face with the landmarks enumerated.

Annotation Process. The process of landmark annotation had several stages: at first, 10% images were annotated by an annotator who has an extensive experience labeling facial landmarks for various animals, then they were annotated by second expert with the same annotation instructions. The landmarks were then compared to verify the internal validity and reliability via the Inter Class Correlation Coefficient ICC2 [43], and reached a strong agreement between the annotators with a score of 0.998.

Finally, the remaining part of images were annotated by the first annotator using the 'human-in-the-loop' method described below. After the annotation, a review and correction of landmarks and bounding boxes were performed, all images were re-filtered and verified.

§.§ AI-assisted Annotation

Khan et al. [30] spent approximately 5,408 man-hours annotating all the images in the AnimalWeb dataset (each annotation was obtained by taking the median value of the annotations of five or more error-prone volunteers). Roughly, it is 3 minutes for the annotation of one image (9 landmarks) or 20 seconds for one landmark (taken per person). Our annotation process took $\sim$140 hours, which is about 4.16 minutes per image (48 landmarks) and only 5.2 seconds per landmark. Such performance is achieved due to a semi-supervised “human-in-the-loop" method that uses the predictions of a gradually-trained model as a basis for annotation.

Left: The distribution of annotation time per image for different batches. The predictions of the first model significantly reduced the annotation time, while the subsequent refinement by the second model reduced the variance, without changing the median value too much. Right: The percentage of images from the batch for which a predicted landmark was not shifted. The more accurate the model's predictions are, the less time is needed for annotation: correctly predicted landmarks do not need to be shifted.

To assess the impact of AI on the annotation process, 3 batches were created from the data. The first one consisted of two hundred images and was annotated without AI assistance. Next, we trained the Ensemble Landmark Detector (ELD) model on the dataset from [25] (about 600 images) combined with the first annotated batch ($\sim$800 images total). Details about the architecture of the model will be given in the next section. After that, we used this model (v1) to predict the landmarks on the second batch ($\sim$900 images). Based on the predictions, landmarks were annotated for the second batch. This approach made it possible to reduce the average time for annotation of one image between the first and second batches by 35%, since for annotation it became necessary only to adjust the position of the landmark, and not to place it from scratch. However, the predictions obtained turned out to be inaccurate, so the first and second batches were combined to train the next version of the model (v2) with the same architecture ($\sim$1700 images total). After training, the v2 model was used to predict the landmarks on the third batch, and then an annotation was performed based on these predictions. The results obtained for the time spent on annotation of one image are shown in Figure <ref>(left). Increasing the accuracy of predicting the position of a landmark reduces the time required to adjust it, sometimes to zero if the annotator considers the position to be completely correct. Figure <ref>(right) shows the distribution of the percentage of images in which a specific facial landmark was not shifted by the annotator: between the two versions of the model, the increase is approximately doubled.

§ ENSEMBLE LANDMARK DETECTOR

The problem of regression of facial landmarks is formulated as follows: having an image $\textbf{x} \in \mathbb{R}^{H \times W \times C}$, it is needed to create a function $f$ (in this case parameterized by a neural network) that maps the space $\mathbb{R}^{H \times W \times C}$ into the space of $K=48$ landmarks $L^K$: $f(\textbf{x}) = [\textbf{y}_1(\textbf{x}),\ldots,\textbf{y}_K(\textbf{x})] \in L^K$. Each landmark $\textbf{y}_i(\textbf{x})$ is defined by two coordinates in the $\textbf{x}$ coordinate system.

§.§ Architecture

Grishchenko et al. [44] point out that “using a single regression network for the entire face leads to degraded quality in regions that are perceptually more significant (e.g. lips, eyes)". Indeed, regression models, with correct training and compliance with the bias-variance tradeoff, are able to provide accurate results only with a relatively small dimension of the output vector. In the case of large dimensions, the model tends to “generalize" or average the output vector: this explains the fact that when obscuring a part of the face, hidden landmarks are still determined from the general geometry of the face and relative position with other landmarks, rather than from the image. Moreover, in the case of the human face, the landmarks are relatively “firmly connected" with each other (with the exception of the eyes and the lower part of the face). This leads to the fact that even with an inaccurate prediction of the regression model, the landmarks will be approximately in the right places, giving a relatively small error in general.

In [44], the authors distinguish three regions of interest on the human face: two eyes and lips, since they have the greatest mobility. By analogy, we will use the magnification of regions to refine landmark coordinates on cats' faces, but in a cascade way. For our model, we will use an ensemble architecture with five regions of interest, since in addition to the eyes and the whiskers area, cats are characterized by high ear mobility.

Face Detection. The first stage of landmark detection is localization and cropping of the cat's face from the input image. For this, the image is rescaled with filling to a resolution of 224x224 and is fed as an input to the face detector. As a detector, we used an EfficientNetV2-based [45] model with a custom head. Figure <ref> shows its structure.

EfficientNetV2-based model structure. The model takes a 224x224 RGB image and outputs the coordinates vector.

The model was trained on the CatFLW bounding boxes using the Adam optimizer, MSE loss, and reduction of the learning rate from 1e-4 to 1e-6 for 200 epochs. After detecting and cropping the image by the bounding box, the resulting image is again rescaled to 224x224 and passed for detection of regions.

Regions Detection.  In the next step, we used a model similar to the face detector, except that the output layer has size of 10. Thus, we detect the coordinates of 5 centers of regions of interest: eyes, nose (whiskers area) and two ears. We used taking average corresponding coordinates from CatFLW to get 5 landmarks from 48 as training data. As a result, the region detection model was trained with the same hyperparameters as the face detection model.

For augmentation, we used a slightly modified Negatively Correlated Augmentation (NCA) from <cit.>. Its idea is to build three distributions for the head tilt angles (roll, yaw and pitch) for the training set. Further, negatively correlated sampling is made for these distributions: the greater the angle of inclination, the more times the image will be sampled and augmented and vice versa. We will take as the basis for the method just 5 detected landmarks, and not 48 ground truth, since in the case of a different number of landmarks (other datasets or other animals), we would have to restructure our algorithm; in the case of five landmarks, there is no dependence on the annotation framework. Since it is not possible to determine the exact 3D angles of the tilt of the head by five landmarks, we used similar geometric relations to interpret the tilts.

As a roll angle, we used the angle of inclination of the line connecting the landmarks on the eyes. Taking the left eye as $\mathbf{L}_0$ and the right eye as $\mathbf{L}_1$, we have:

\begin{equation*}
\begin{aligned}
dx = L_{1_x} - L_{0_x}, \quad dy = L_{1_y} - L_{0_y}, \quad roll = \arctan\left(\frac{dy}{dx}\right). \\
\end{aligned}
\end{equation*}

Assuming the landmark on the nose as $\mathbf{L}_2$, on the left ear as $\mathbf{L}_3$ and on the right ear as $\mathbf{L}_4$, we can determine the yaw ratio:

\begin{equation*}
\begin{gathered}
l_1 = \lVert\mathbf{L}_3 - \mathbf{L}_0\rVert, \quad l_2 = \lVert\mathbf{L}_4 - \mathbf{L}_1\rVert, \\
yaw = \begin{cases}
0, & \text{if } l_1 = l_2; \\
-\arcsin\left(1 - \frac{l_1}{l_2}\right), & \text{if } l_1 < l_2; \\
\arcsin\left(1 - \frac{l_2}{l_1}\right), & \text{if } l_1 > l_2.
\end{cases} \\
\end{gathered}
\end{equation*}

And finally, for the pitch ratio, we have:

\begin{equation*}
\begin{gathered}
l_3 = \lVert\mathbf{L}_2 - \mathbf{L}_0\rVert, \quad l_4 = \lVert\mathbf{L}_2 - \mathbf{L}_1\rVert, \quad l_{12} = \frac{l_1 + l_2}{2}, \quad l_{34} = \frac{l_3 + l_4}{2}, \\
pitch = \frac{l_{34}}{l_{12}} - 1.
\end{gathered}
\end{equation*}

Further, three probability density function for the ratio distributions are constructed from the training set and the number of samples is calculated similarly to the process in <cit.>. With each sampling, changes in brightness, contrast, sharpness with random parameters, as well as random noise are applied to the image.

Ensemble Detection. After determining the centers of the regions of interest, the image is aligned by the eyes to reduce the spread on the roll tilt angles and five fixed-size regions are cropped out of it. For the eyes, the region size is set as a quarter of the image (56 pixels), for the nose and ears — half of the image (112 pixels). These sizes were taken empirically so that all associated landmarks were located within the regions in most cases. The regions are then resized to 224x224 to meet the input size of the EfficientNetV2 model.

Then the landmarks are divided into groups according to their corresponding regions. There are 8 landmarks for each eye, 5 for each ear, and 22 for the nose and whiskers region. Examples of cropped regions with corresponding landmarks are shown in Figure <ref>.

Five cropped regions of interest and corresponding landmarks.

All landmarks are then detected by an ensemble of five models, each of which has a structure similar to the detector, except that the size of the output layer is equal to the number of landmarks multiplied by two (the coordinates of the corresponding landmarks). After that, the coordinates of the landmarks are descaled several times to match the coordinate system of the original image and are combined into a final vector of 48 landmarks. Hyperparameters for all models were chosen the same, similar to the hyperparameters of the detector model. The final structure of the Ensemble Landmark Detector is shown in Figure <ref>.

Ensemble Landmark Detector. Provided ensemble architecture and region centers could be changed depending on the face morphology.

§.§ Experiments

We present an ablation study of various versions of our model on CatFLW in order to 1) spur a challenge of detecting landmarks on our dataset, and 2) demonstrate the effectiveness of the proposed features of the ELD model.

To evaluate the accuracy of the models, we will use the popular Normalized Mean Error (NME) metric, which is defined as

\begin{equation*}
NME = \frac{1}{M \cdot N} \sum_{i=1}^{N} \sum_{j=1}^{M} \frac{\left\Vert {x_i}^j - {x'_i}^j \right\Vert}{iod_i},
\end{equation*}

where $M$ is the number of landmarks in the image, $N$ is the number of images in the dataset, $iod_i$ is the inter-ocular distance (distance between the outer corners of the two eyes), ${x_i}^j$ and ${x'_i}^j$ — the coordinates of the predicted and the ground truth landmark respectively.

§.§.§ Ablation Study

Face Detection. Our goal in developing the CatFLW dataset and ELD model was to make an end-to-end system capable of working in the wild, that is, on unannotated images. Due to the fact that most of the datasets for detecting landmarks do not focus on detecting faces, but use ready-made bounding boxes at the data preprocessing stage, we do not deviate from this pipeline in this section later, however, we provide data for detecting landmarks in the case of using our manually labeled bounding boxes and bounding boxes determined by a detector trained on our dataset.

From the direct comparison given in Table <ref>, we can say that when using a facial detector, the accuracy drops significantly. This is explained by the fact that in the case of pre-cropped faces, the scale and approximate position of the landmarks in the image are preserved. In the case of a detector, there may be variations in the scale of the detected faces, as well as “cutting-off" meaningfully important facial parts potentially containing landmarks.

Face Detection     NME

No     2.98

Yes     6.11

Comparison of landmark detection using training data obtained by facial detector and with pre-cropped faces. In both cases EfficientNetV2B0 backbone was used.

Backbones. The EfficientNet models were chosen as a backbone for the ELD because of the balance between the number of parameters and the accuracy of performance. Since such backbones are rarely used for regression problems on their own, we could only refer to the accuracy on image classification datasets [46, 47, 48] and to the use of these models as backbones in other landmark detection models. For the experiment, we have selected several popular and most effective backbones and measured the accuracy of the prediction of the ELD model as a whole, where each of its components is a certain backbone. The results are shown in the Table <ref>.

Backbone     Param. (M)     NME

MobileNetV2     3.5     3.09

EfficientNetB0     5.3     2.97

EfficientNetV2B0     7.2      2.98

EfficientNetB1     7.9     2.98

EfficientNetV2B1     8.2      2.96

EfficientNetB2     9.2     2.91

EfficientNetV2B2     10.2      2.93

EfficientNetB3     12.3     2.94

DenseNet169     14.3     2.92

EfficientNetV2B3     14.5     2.91

EfficientNetV2S     21.6     2.83

ConvNeXtTiny     28.6     2.78

ResNet101     44.7     2.91

Evaluation of landmark detection on CatFLW using different backbones.

It can be seen that more complex backbones show better results, but a slight increase in accuracy is caused by a significant increase in the number of parameters. Considering that the ELD contains six similar models, we do not provide more results for complex backbones due to their inefficiency for detection.

§.§.§ WFLW Dataset

We additionally evaluated our method on the popular human face WFLW (Wider Facial Landmarks in-the-wild) dataset [10]. It contains 7,500 training images and 2,500 test images annotated with 98 landmarks.

For ensemble detection, we disable our face detection model, since the bounding boxes for this dataset are commonly preset by landmarks. We do not change the architecture of the model, besides dividing the human face into four regions of interest: eyes, nose and mouth. We separately detect 33 landmarks on the lower part of the face, as they are not suitable for our magnifying method. The same hyperparameters were used to train the models as for training similar models on CatFLW.

We compare ELD with several state-of-the-art methods, as shown in the Table <ref>.

Method     Backbone     NME

DeCaFa [49]     U-Net     4.62

HRNet [50]     HRNet-W18     4.6

Awing [51]     Hourglass     4.36

DAG [52]     HRNet-W18     4.21

LUVLi [53]     DU-Net     4.37

PIPNet [54]     ResNet-101     $4.31^*$

ADNet [55]     Hourglass     4.14

HIH [56]     Hourglass     4.08

DTLD [57]     ResNet-18     $4.08^*$

RePFormer [58]     ResNet-101     4.11

SPIGA [59]     Hourglass     4.06

STAR [60]     Hourglass     4.02

LDEQ [61]     DEQ     3.92

ELD     EfficientNetV2B1     $4.65$

ELD     EfficientNetV2B1     $4.05^*$

Evaluation of landmark detection on WFLW. * denotes that a subset of landmarks were used in the evaluation.

Some of the results for our model are given without the detection of the landmarks on the lower border of the face. Due to the impossibility of using the magnifying method, the detection accuracy on jaw landmarks is significantly worse than on the rest. Table <ref> shows the NME for landmarks taken on individual parts of the face.

The provided results demonstrate that our model handles localized groups of landmarks better, being significantly inferior in the case of non-localized ones. When detecting a partial set of landmarks, our model demonstrates results comparable to the state-of-the-art.

Left Eye     Right Eye     Nose     Mouth     Jaw

4.0     3.98     3.54     4.37     5.83

NME on different face parts using EfficientNetV2B1 as a backbone.

§ CONCLUSIONS AND FUTURE WORK

The field of animal affective computing is only just beginning to emerge. One of the most significant obstacles that researchers in this field currently face is the scarcity of high-quality, comprehensive datasets, as highlighted in [13]. The structure of annotations in the proposed dataset allows researchers to rely on the features of the anatomy of the feline face, which makes it possible to use it not only for the detection of facial landmarks, but also for a deeper analysis of the internal state of cats. It is our hope that the contributions of this paper will support the ongoing efforts to advance the field, and specifically automated facial analysis for a variety of species.

Due to the magnifying method working on semantically grouped landmarks, the ELD model is able to solve the landmark detection problem with high accuracy in a more general sense than only on cat faces, which was demonstrated on a popular human WFLW dataset. The limitation of our approach is the need for a close grouping of landmarks, which is not fulfilled, for example, for jaw landmarks on the same dataset. One of the possible solutions is to further divide the “spaced" landmarks into groups, which, however, may entail a loss of detection efficiency.

Future research includes extending the CatFLW dataset with more images and improving the ELD model performance. To do this, we plan to explore new backbones, use new approaches to images augmentation and possible parallelization of computations. We will also investigate how well the model generalizes to other species, such as dogs and monkeys.

We believe that the proposed approaches can serve as a guideline for similar solutions that can solve the problems of identification, detection of emotions and internal states of various animals in an effective and non-invasive way.

§ DATA AVAILABILITY

The dataset generated during and/or analysed during the current study is available from the corresponding author on reasonable request.

§ ACKNOWLEDGEMENTS

The research was supported by the Data Science Research
Center at the University of Haifa. We thank Ephantus Kanyugi for his contribution with data annotation and management. We thank Yaron Yossef and Nareed Farhat for their technical support.

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# Average number of zeros of characters of finite groups

Sesuai Yash Madanha  Department of Mathematics and Applied Mathematics,
University of Pretoria, Pretoria, 0002, South Africa<EMAIL_ADDRESS>
###### Abstract.

There has been some interest on how the average character degree affects the
structure of a finite group. We define, and denote by $\mathrm{anz}(G)$, the
average number of zeros of characters of a finite group $G$ as the number of
zeros in the character table of $G$ divided by the number of irreducible
characters of $G$. We show that if $\mathrm{anz}(G)<1$, then the group $G$ is
solvable and also that if $\mathrm{anz}(G)<\frac{1}{2}$, then $G$ is
supersolvable. We characterise abelian groups by showing that
$\mathrm{anz}(G)<\frac{1}{3}$ if and only if $G$ is abelian.

###### Key words and phrases:

zeros of characters, solvable groups, supersolvable groups, nilpotent groups,
abelian groups

###### 2010 Mathematics Subject Classification:

Primary 20C15, 20D10

Dedicated to the memory of Kay Magaard

##  1\. Introduction

Let $G$ be a finite group and ${\mathrm{Irr}}(G)$ be the set of complex
irreducible characters of $G$. Let $T(G)$ be the sum of degrees of complex
irreducible characters of $G$, that is,
$T(G)=\sum_{\chi\in{\mathrm{Irr}}(G)}\chi(1)$. Denote by $k(G)$ the number of
conjugacy classes of $G$. Then $k(G)=|{\mathrm{Irr}}(G)|$. Define the average
character degree of $G$ by

$\mathrm{acd}(G):=\dfrac{T(G)}{|{\mathrm{Irr}}(G)|}$.

Recently, a lot of authors have investigated the average character degree of
finite groups and how it influences the structure of the groups (see [13, 15,
14]). In fact, Magaard and Tong-Viet [13] proved that $G$ is solvable whenever
$\mathrm{acd}(G)<2$ and they conjectured that their result still holds if
$\mathrm{acd}(G)\leqslant 3$. This conjecture was settled by Isaacs, Loukaki
and Moretó [10, Theorem A] and they obtained some sufficient conditions for a
group to be supersolvable and also to be nilpotent. In the same article, the
authors of [10] conjectured that the best possible bound is when
$\mathrm{acd}(G)<\frac{16}{5}$ for $G$ to be solvable. Moretó and Nguyen
showed in [15, Theorem A], that indeed this was the best bound. We shall state
the results on average character degrees with the best bounds below.

######  Theorem 1.1.

[15, Theorem A] Let $G$ be a finite group. If $\mathrm{acd}(G)<\frac{16}{5}$,
then $G$ is solvable.

######  Theorem 1.2.

[10, Theorem B] Let $G$ be a finite group. If $\mathrm{acd}(G)<\frac{3}{2}$,
then $G$ is supersolvable.

######  Theorem 1.3.

[10, Theorem C] Let $G$ be a finite group. If $\mathrm{acd}(G)<\frac{4}{3}$,
then $G$ is nilpotent.

More work on average character degrees of finite groups is found in [8, 6, 7].
Another invariant that has been studied is the so-called average class size.
We shall refer the reader to [5, 10, 18] for more bibliographic information.

In this article, we investigate the corresponding problem for zeros of
characters of finite groups. We have to define a new invariant first. Recall
that if $\chi(g)=0$ for some $g\in G$ and $\chi\in{\mathrm{Irr}}(G)$, we say
$\chi$ vanishes on $g$. So $\chi$ vanishes on conjugacy classes. In the same
spirit, we define $\mathrm{nz}(G)$ to be the number of zeros in the character
table of $G$ and the average number of zeros of characters of $G$ by:

$\mathrm{anz}(G):=\dfrac{\mathrm{nz}(G)}{|{\mathrm{Irr}}(G)|}$.

Since linear characters do not vanish on any conjugacy class,
$\mathrm{anz}(G)=0$ for an abelian group $G$. A classical theorem of Burnside
[9, Theorem 3.15] shows that $\chi(g)=0$ for some $g\in G$ and a non-linear
irreducible character $\chi$, that is, $\chi$ vanishes on some conjugacy
class. This means $\mathrm{anz}(G)>0$ for non-abelian groups. We show that
zeros of characters influence the structure of a finite group.

We begin by proving a result analogous to [13, Theorem 1.1]:

######  Theorem A.

If $N$ is a minimal non-abelian normal subgroup of $G$, then there exists a
non-linear character $\chi\in{\mathrm{Irr}}(G)$ such that $\chi_{N}$ is
irreducible and $\chi$ vanishes on at least two conjugacy classes of $G$.

We need Theorem A to prove our second result below which corresponds to
Theorem 1.1:

######  Theorem B.

Let $G$ be a finite group. If $\mathrm{anz}(G)<1$, then $G$ is solvable.

Note that since $\mathrm{anz}(\mathrm{A}_{5})=1$, this bound is sharp.

We show that the converse of Theorem B does not necessarily hold. Let $Q$ be a
Sylow $3$-subgroup of $S={\mathrm{PSL}}_{3}(7)$ and suppose $G=N_{S}(Q)$. We
have that $\mathrm{anz}(G)=7/6>1$.

In Theorems 1.2 and 1.3, the bounds are optimal since
$\mathrm{acd}(\mathrm{A}_{4})=\frac{3}{2}$ and
$\mathrm{acd}(\mathrm{S}_{3})=\frac{4}{3}$. Since
$\mathrm{anz}(\mathrm{A}_{4})=\frac{1}{2}$ and
$\mathrm{anz}(\mathrm{S}_{3})=\frac{1}{3}$, the following corresponding
results hold:

######  Theorem C.

Let $G$ be a finite group. If $\mathrm{anz}(G)<\frac{1}{2}$, then $G$ is
supersolvable.

######  Theorem D.

Let $G$ be a finite group. If $\mathrm{anz}(G)<\frac{1}{3}$, then $G$ is
nilpotent.

Note that for an abelian group $G$, $\mathrm{acd}(G)=1$ and
$\mathrm{anz}(G)=0$. We show in Remark 4.1 that there is no number $c>1$ such
that $\mathrm{acd}(G)<c$ implies that $G$ is abelian. Contrary to
$\mathrm{acd}(G)$, we show that there exists a bound $c>0$, such that
$\mathrm{anz}(G)<c$ implies that $G$ is abelian. Hence we obtain a non-trivial
characterisation of abelian groups in terms of zeros of characters:

######  Theorem E.

Let $G$ be a finite group. Then $G$ is abelian if and only if
$\mathrm{anz}(G)<\frac{1}{3}$.

When $G$ is of odd order we obtain a bound that is better than that in Theorem
C. The following result is analogous to [10, Theorem D(a)].

######  Theorem F.

Let $G$ be a finite group of odd order. If $\mathrm{anz}(G)<1$, then $G$ is
supersolvable.

However, the bound in Theorem F might not be optimal. Indeed, in [10, Theorem
D(a)], the bound is best possible since there exists a group $G$ of odd order
which is not supersolvable such that $\mathrm{acd}(G)=\frac{27}{11}$ and also
$\mathrm{anz}(G)=\frac{16}{11}$ ($G$ is the unique non-abelian group of order
$75$). Hence we propose the following conjecture:

######  Conjecture 1.

Let $G$ be a finite group of odd order. If $\mathrm{anz}(G)<\frac{16}{11}$,
then $G$ is supersolvable.

The article is organized as follows. In Section 2, we list some preliminary
results. In Section 3, we prove Theorems A and B and in Section 4 we prove
Theorems C to F. We also show that there exist an infinite family of non-
abelian nilpotent groups $G$ such that $\mathrm{acd}(G)<\frac{4}{3}$ in this
last section.

##  2\. Preliminaries

######  Lemma 2.1.

Suppose that $N$ is a minimal normal non-abelian subgroup of a group $G$. Then
there exists an irreducible character $\theta$ of $N$ such that $\theta$ is
extendible to $G$ with $\theta(1)\geqslant 5$. In particular, if $N$ is simple
such that

  * (a)

$N$ is isomorphic to $\mathrm{A}_{5}$, then $\theta(1)=5$

  * (b)

$N$ is isomorphic to $\mathrm{A}_{n}$, $n\geqslant 7$, then $\theta(1)=n-1$

  * (c)

$N$ is isomorphic to a finite group of Lie type defined over a finite field
distinct from the Tits group ${}^{2}\mathrm{F}_{4}(2)^{\prime}$ and
${\mathrm{PSL}}_{2}(5)$, then $\theta$ is the Steinberg character.

###### Proof.

The first assertion is the statement of [13, Theorem 1.1] and the second
assertion follows from the proof of [13, Theorem 1.1]. ∎

######  Lemma 2.2.

[9, Corollary 6.17] Let $N$ be a normal subgroup of $G$ and let
$\chi\in{\mathrm{Irr}}(G)$ be such that $\chi_{N}=\theta\in{\mathrm{Irr}}(N)$.
Then the characters $\beta\chi$ for $\beta\in{\mathrm{Irr}}(G/N)$ are
irreducible, distinct for distinct $\beta$ and are all of the irreducible
constituents of $\theta^{G}$.

Recall that
${\mathrm{Irr}}(G{\mid}K)=\\{\chi\in{\mathrm{Irr}}(G):K\nsubseteq\ker\chi\\}$.
For $\chi\in{\mathrm{Irr}}(G)$ we write $n\upsilon(\chi)$ for the number of
conjugacy classes of $G$ on which $\chi$ vanishes. Then
$\mathrm{nz}(G{\mid}K):=\sum_{\chi\in{\mathrm{Irr}}(G\mid K)}n\upsilon(\chi)$.

######  Lemma 2.3.

Let $K$ be a normal subgroup of $G$ such that $K\leqslant G^{\prime}$. If
$\mathrm{anz}(G)<1$, then $\mathrm{anz}(G/K)\leqslant\mathrm{anz}(G)$.

###### Proof.

Let $a=\mathrm{anz}(G/K)$. Note that
$k(G)=|{\mathrm{Irr}}(G)|=|{\mathrm{Irr}}(G/K)|+|{\mathrm{Irr}}(G{\mid}K)|$.
If $|{\mathrm{Irr}}(G/K)|=c$, $|{\mathrm{Irr}}(G{\mid}K)|=d$,
$\mathrm{nz}(G/K)=m$ and $\mathrm{nz}(G{\mid}K)=n$, then $a=\frac{m}{c}$.
Since $K\leqslant G^{\prime}$, ${\mathrm{Irr}}(G{\mid}K)$ is a set of non-
linear irreducible characters of $G$ and also using Burnside’s Theorem,
$|{\mathrm{Irr}}(G{\mid}K)|=d\leqslant n=\mathrm{nz}(G{\mid}K)$. Since the
character table of $G/K$ is a sub-table of the character of $G$, we have that
$\mathrm{nz}(G/K)\leqslant\mathrm{nz}(G)$. Hence
$1>\mathrm{anz}(G)=\frac{m+n}{c+d}\geqslant\frac{m+d}{c+d}\geqslant\frac{m}{c}=a$
as required. ∎

We show that Theorem B holds for perfect groups.

######  Lemma 2.4.

Let $G$ be a finite group such that $G=G^{\prime}$. Then
$\mathrm{anz}(G)\geqslant 1$.

###### Proof.

Since $G$ has one linear character, it is sufficient to show that there exists
a non-linear irreducible character of $G$ that vanishes on at least two
conjugacy classes. Suppose the contrary, that is, every non-linear irreducible
character of $G$ vanishes on exactly one conjugacy class. Then by [2,
Proposition 2.7], $G$ is a Frobenius group with a complement of order $2$ and
an abelian odd-order kernel, that is, $G$ is solvable, contradicting the
hypothesis that $G$ is perfect. Hence the result follows. ∎

Let $G$ be a finite group and $\chi\in{\mathrm{Irr}}(G)$. Recall that
$\upsilon(\chi):=\\{x\in G\mid\chi(x)=0\\}$.

######  Lemma 2.5.

Let $G$ be a non-abelian finite group and $\chi\in{\mathrm{Irr}}(G)$ be non-
linear. Suppose that $N$ is a normal subgroup of $G$ such that
$G^{\prime}\leqslant N<G$. If $\chi_{N}$ is not irreducible, then the
following two statements hold:

  * (a)

There exists a normal subgroup $K$ of $G$ such that $N\leqslant K<G$ and
$G{\setminus}K\subseteq\upsilon(\chi)$.

  * (b)

If $(G{\setminus}G^{\prime})\cap\upsilon(\chi)$ consists of $n$ conjugacy
classes of $G$, then

$|G{:}G^{\prime}|-|K{:}G^{\prime}|\leqslant n$.

###### Proof.

For (a), note that since $G/N$ is abelian, it follows that $\chi$ is a
relative $M$-character with respect to $N$ by [9, Theorem 6.22]. This means
that there exists $K$ with $N\leqslant K\leqslant G$ and
$\psi\in{\mathrm{Irr}}(K)$ such that $\chi=\psi^{G}$ and
$\psi_{N}\in{\mathrm{Irr}}(N)$. Hence $G{\setminus}K\subseteq\upsilon(\chi)$,
Since $\chi_{N}$ is not irreducible we have that $K<G$ and the result follows.

For (b), let $g_{1},g_{2},\dots,g_{m}$ be a complete set of representatives of
the cosets of $G^{\prime}$ in $G$, with $g_{1},g_{2},\dots,g_{k}$ a complete
set of representatives of the cosets of $G^{\prime}$ in $K$ so that
$m=|G:G^{\prime}|$ and $k=|K:G^{\prime}|$. Since $\chi$ vanishes on
$G{\setminus}K$, we have that $\chi$ vanishes on
$g_{k+1},g_{k+2},\dots,g_{m}$. But these elements are not conjugate and so
$m-k\leq n$ as required. ∎

In the following results we will consider groups with an irreducible character
that vanishes on one conjugacy class. We denote by $k_{G}(N)$, the number of
conjugacy classes of $G$ in $N$ where $N$ is a subset of $G$. We first prove
an easy lemma:

######  Lemma 2.6.

Let $N$ be a normal subgroup of $G$. If $k_{G}(G{\setminus}N)=1$, then
$|G{:}N|=2$ and $G$ is a Frobenius group with an abelian kernel $N$ of odd
order.

###### Proof.

Since $k_{G}(G{\setminus}N)=1$, all the elements in $G{\setminus}N$ are
conjugate. It follows that if $x\in G{\setminus}N$, then the size of the
conjugacy class containing $x$ , say $c$, is $|G|-|N|$. Since $N$ is a proper
subgroup of $G$, $c\geq|G|-|G|/2=|G|/2$. But the size of any conjugacy class
of a non-trivial group is a proper divisor of the group, so we deduce that
$|N|=|G|/2$. Then $C_{G}(x)=\langle x\rangle$ is of order $2$, so $G=\langle
x\rangle N$ is a Frobenius group with a complement $\langle x\rangle$ of order
$2$ and hence the Frobenius kernel $N$ is abelian of odd order. ∎

We also need this result:

######  Proposition 2.7.

Let $G$ be an almost simple group. If $\chi\in{\mathrm{Irr}}(G)$ vanishes on
exactly one conjugacy class, then one of the following holds:

  * (a)

$G={\mathrm{PSL}}_{2}(5)$, $\chi(1)=3$ or $\chi(1)=4$;

  * (b)

$G\in\\{\mathrm{A}_{6}{:}2_{2},~{}\mathrm{A}_{6}{:}2_{3}\\}$, $\chi(1)=9$ for
all such $\chi\in{\mathrm{Irr}}(G)$;

  * (c)

$G={\mathrm{PSL}}_{2}(7)$, $\chi(1)=3$;

  * (d)

$G={\mathrm{PSL}}_{2}(8){:}3$, $\chi(1)=7$;

  * (e)

$G={\mathrm{PGL}}_{2}(q)$, $\chi(1)=q$, where $q\geq 5$;

  * (f)

$G=$ ${}^{2}\rm{B_{2}}(8){:}3$, $\chi(1)=14$.

###### Proof.

It was shown in [11, Theorem 5.2] and the proof of [12, Theorem 1.2] that all
irreducible characters of $G$ that vanishes on exactly one conjugacy class are
primitive, the result follows. Hence the list in [12, Theorem 1.2] is a
complete one. The result then follows. ∎

######  Lemma 2.8.

A finite group $G$ has exactly one non-linear irreducible character if and
only if $G$ is isomorphic to one of the following:

  * (a)

$G$ is an extra-special $2$-group.

  * (b)

$G$ is a Frobenius group with an elementary abelian kernel of order $p^{n}$
and a cyclic complement of order $p^{n}-1$, where $p$ is prime and $n$ a
positive integer.

Moreover, $\mathrm{anz}(G)=\frac{m-1}{m+1}$ for some integer $m\geqslant 2$.

###### Proof.

The statements (a) and (b) follows from [19, Theorem]. Let $G$ be an extra-
special $2$-group of order $2^{2k+1}$ with $|G/G^{\prime}|=|G/Z(G)|=2^{2k}$.
Then $G$ has $2^{2k}$ linear characters and so $2^{2k}+1$ irreducible
characters. Note that the non-linear irreducible character $\chi$ of $G$ is
fully ramified with respect to $Z(G)$. In particular, $\chi$ vanishes on
$G{\setminus}Z(G)$. Hence $\chi$ vanishes on at least $2^{2k}-1$ conjugacy
classes. Since $G$ has the identity and a non-trivial central element which
are non-vanishing elements, $\chi$ vanishes on exactly $2^{2k}-1$ conjugacy
classes. Then $\mathrm{anz}(G)=\frac{2^{2k}-1}{2^{2k}+1}$ and the result
follows.

Suppose $G$ is a Frobenius group with an elementary abelian kernel
$G^{\prime}$ of order $p^{n}$ and a cyclic complement of order $p^{n}-1$. Note
that $|G/G^{\prime}|=p^{n}-1$ and so $G$ has $p^{n}-1$ linear characters.
Hence $G$ has $p^{n}$ irreducible characters. Since the non-linear irreducible
character $\chi$ of $G$ is induced from irreducible character of $G^{\prime}$,
$\chi$ vanishes on $G{\setminus}G^{\prime}$. It follows that $\chi$ vanishes
on at least $p^{n}-2$ conjugacy classes. Since $G^{\prime}$ has one non-
trivial conjugacy class of $G$, $\chi$ does not vanish on $G^{\prime}$.
Otherwise, $\chi$ vanishes on $G{\setminus}\\{1\\}$, a contradiction. Since
$G$ has $p^{n}$ conjugacy classes, we have that $\chi$ vanishes on exactly
$p^{n}-2$ conjugacy classes and so $\mathrm{anz}(G)=\frac{p^{n}-2}{p^{n}}$ as
required. ∎

We shall need the following two results to prove Theorem F.

######  Lemma 2.9.

[16] A finite group $G$ has exactly two non-linear irreducible characters if
and only if $G$ is isomorphic to one of the following:

  * (a)

$G$ is an extra-special $3$-group.

  * (b)

$G$ is a $2$-group of order $2^{2k+2}$, $|G^{\prime}|=2$, $|Z(G)|=4$, and $G$
has two non-linear irreducible characters with equal degree $2^{k}$.

  * (c)

$G$ is a Frobenius group with an elementary abelian kernel of order $9$ and
Frobenius complement $Q_{8}$.

  * (d)

$G$ is a Frobenius group with an elementary abelian kernel of order $p^{k}$
and a cyclic Frobenius complement of order $(p^{k}-1)/2$, where $p$ is odd
prime.

  * (e)

$G/Z(G)$ is a Frobenius group with an elementary abelian kernel of order
$p^{k}$ and a cyclic Frobenius complement of order $p^{k}-1$, where $p$ is
prime and $|Z(G)|=2$.

######  Lemma 2.10.

[17, Theorem 2.6] Let $G$ be a finite group of odd order. Then $G$ has an
irreducible character that vanishes on exactly two conjugacy classes if and
only if $G$ is one of the following groups:

  * (a)

$G$ is a Frobenius group with a complement of order $3$.

  * (b)

There are normal subgroups $M$ and $N$ of $G$ such that: $M$ is a Frobenius
group with the kernel $N$; $G/N$ is a Frobenius group of order $p(p-1)/2$ with
the kernel $M/N$ and a cyclic complement of order $(p-1)/2$ for some odd prime
$p$. In this case, $\chi_{M}$ is irreducible.

######  Lemma 2.11.

Let $\ell,m,n,b$ be positive integers with $\ell,n\geqslant 2$ and $b\geqslant
2\ell+1$. Suppose that $b=mn$. Then

$b-m\geqslant\ell+1$.

###### Proof.

Suppose $b$ is odd. If $b$ is prime, then $b-1\geqslant 2\ell+1-1>\ell+1$. If
$b$ is not prime, then the greatest $m$ is when $n=3$ and $b-m\geqslant
b-\frac{b}{3}\geqslant\frac{2}{3}b>\frac{b}{2}=\ell$. If $b$ is even, then
$b\geqslant 2\ell+2$ and the greatest $m$ is when $n=2$. Hence $b-m\geqslant
2\ell+2-(\ell+1)=\ell+1$ as required. ∎

##  3\. Non-solvable and solvable groups

###### Proof of Theorem A.

We begin the proof by showing that we may assume that $N$ is simple. For if
$N=T_{1}\times T_{2}\cdots\times T_{k}$, where $T_{i}\cong T$, $T$ is a non-
abelian simple group, for $i=1,2,\dots,k$ and $k\geqslant 2$, then by Lemma
2.1, there exists $\theta_{i}\in{\mathrm{Irr}}(T_{i})$ such that
$\chi_{N}=\theta_{1}\times\theta_{2}\times\cdots\times\theta_{k}$, where
$\chi\in{\mathrm{Irr}}(G)$. Since $\theta_{1}(1)>1$, there exists $x\in T_{1}$
such that $\theta_{1}(x)=0$ by Burnside’s Theorem. But $\chi$ vanishes on
$(x,1,\dots,1)$ and $(x,x,1,\dots,1)$. Since these two elements are not
conjugate in $G$, $\chi$ vanishes on at least two conjugacy classes of $G$.

We may assume that $N$ is simple. Let $C=C_{G}(N)$. We claim that $C=1$.
Otherwise, $N\times C$ is a normal subgroup of $G$. There exists a non-linear
$\theta\in{\mathrm{Irr}}(N)$ such that $\theta_{N}=\chi$ by Lemma 2.1. There
exists $x\in N$ such that $\theta(x)=0$. Then $\chi(xc)=0$ for any $c\in C$.
Let $c\in C{\setminus}\\{1\\}$. Since $x$ and $xc$ are not conjugate in $G$,
$\chi$ vanishes on at least two conjugacy classes. Hence $C=1$ and we have
that $G$ is almost simple.

By Lemma 2.1, there exists $\chi$ such that $\chi_{N}$ is irreducible and
$\chi(1)\geqslant 5$. If $\chi$ vanishes on two conjugacy classes, then the
result follows. We may assume that $\chi$ vanishes on one conjugacy class. By
Proposition 2.7 and considering the choice of $\chi$ from Lemma 2.1, we are
left with the following cases:
$G\in\\{{\mathrm{PGL}}_{2}(q),\mathrm{A}_{6}{:}2_{2},\mathrm{A}_{6}{:}2_{3}\\}$(note
that ${\mathrm{PSL}}_{2}(9)\cong\mathrm{A}_{6}$). Note that
$N={\mathrm{PSL}}_{2}(q)$. We will choose another irreducible character of $N$
that extends to $G$. Obviously that alternative character vanishes on at least
two conjugacy classes. For $G=\mathrm{S}_{5}$, let
$\theta\in{\mathrm{Irr}}(\mathrm{A}_{5})$ be such that $\theta(1)=4$. Then
$\theta$ is extendible to ${\mathrm{Aut}}(\mathrm{A}_{5})=\mathrm{S}_{5}$.
Using the Atlas [3], we have that $\chi_{G}$ vanishes on two elements of
distinct orders. Hence the result follows.

For $G\in\\{\mathrm{A}_{6}{:}2_{2},\mathrm{A}_{6}{:}2_{3}\\}$, using the Atlas
[3], we can choose an irreducible character $\theta$ of $N$ of degree $10$
that extends to ${\mathrm{Aut}}(\mathrm{A}_{6})$. Lastly, $\chi$ vanishes on
more than two conjugacy classes of $G$. Hence the result follows.

Suppose $G={\mathrm{PGL}}_{2}(q)$, where $q\geqslant 7$. It is well known that
${\mathrm{PSL}}_{2}(q)$ has irreducible characters of degree $q-1$ and $q+1$.
By [20, Theorem A], an irreducible character $\theta$ of degree $q+1$ is
extendible to ${\mathrm{Aut}}({\mathrm{PSL}}_{2}(q))$ except when
$N={\mathrm{PSL}}_{2}(3^{f})$ and $G={\mathrm{PGL}}_{2}(3^{f})$, with $f$ an
odd positive integer (case (iii) of [20, Theorem A]). If
$N={\mathrm{PSL}}_{2}(3^{f})$ and $G={\mathrm{PGL}}_{2}(3^{f})$ with $f$ an
odd positive integer, then we choose an irreducible character $\theta$ of
${\mathrm{PSL}}_{2}(3^{f})$ of degree $q-1$ which is extendible to
${\mathrm{Aut}}({\mathrm{PSL}}_{2}(3^{f}))$. Thus the result follows. ∎

###### Proof of Theorem B.

We shall prove our result by induction on $|G|$. We may assume that $G$ is
non-solvable. If $G=G^{\prime}$, then the result follows by Lemma 2.4, so
$G\neq G^{\prime}$.

Let $G^{\infty}$ be the solvable residual of $G$. Note that $G^{\infty}$ is
perfect. If $N<G^{\infty}\leqslant G^{\prime}$ is a minimal normal subgroup of
$G$, then $G^{\infty}/N$ is perfect and so $G/N$ is non-solvable. But
$\mathrm{anz}(G/N)<1$ by Lemma 2.3 and hence $G/N$ is solvable by induction, a
contradiction.

We may assume that $N=G^{\infty}$ is a non-abelian minimal normal subgroup of
$G$. By Theorem A, there exists $\chi\in{\mathrm{Irr}}(G)$ such that
$\chi_{N}$ is irreducible and $\chi$ vanishes on two conjugacy classes.
Suppose the two conjugacy classes are represented by elements $g_{1}$ and
$g_{2}$ of $G$. Since $\chi_{G^{\prime}}$ is irreducible, we have that the
character $\beta\chi$ is irreducible for every linear $\beta$ of $G$ by Lemma
2.2. The $\beta\chi$’s are distinct for distinct characters $\beta$. We show
that every character of the form $\beta\chi$ also vanishes on $g_{1}$ and
$g_{2}$. Then $\beta\chi(g_{i})=\beta(g_{i})\chi({g_{i}})=\beta(g_{i})\cdot
0=0$, where $i\in\\{1,2\\}$.

Hence for every linear character $\beta$ of $G$, there is a corresponding non-
linear irreducible character of $G$ of the form $\beta\chi$ that vanishes on
two conjugacy classes of $G$. Let $a$ be the number of linear characters of
$G$ and $b$ be the number of non-linear irreducible characters not of the form
$\beta\chi$ (note that these irreducible characters may vanish on more than
one conjugacy class). Then $|{\mathrm{Irr}}(G)|=2a+b$ and
$\mathrm{nz}(G)=2a+b+c$, where $c$ is a non-negative integer. Therefore
$\mathrm{anz}(G)=\frac{2a+b+c}{2a+b}\geqslant\frac{2a+b}{2a+b}\geqslant\frac{2a}{2a}=1$,
concluding our argument. ∎

##  4\. Supersolvable, nilpotent and abelian groups

###### Proof of Theorem C.

Suppose $G$ is non-abelian, that is, $G^{\prime}>1$. We shall use induction on
$|G|$ to show that $G$ is supersolvable. By Theorem B, $G$ is solvable since
$\mathrm{anz}(G)<\frac{1}{2}<1$. Let $N\leqslant G^{\prime}$ be a minimal
normal subgroup of $G$. By Lemma 2.3,
$\mathrm{anz}(G/N)\leqslant\mathrm{anz}(G)<\frac{1}{2}$. Using the inductive
hypothesis, $G/N$ is supersolvable. If $N$ is cyclic or $N\leqslant\Phi(G)$,
the Frattini subgroup, then $G$ is supersolvable and the result follows.

Suppose that $\chi\in{\mathrm{Irr}}(G)$ be non-linear such that
$\chi_{G^{\prime}}$ is irreducible. By Lemma 2.2, for every linear character
$\beta_{i}\in{\mathrm{Irr}}(G/G^{\prime})$, there exists a non-linear
$\beta_{i}\chi\in{\mathrm{Irr}}(G)$, that is, the number of linear characters
of $G$ is less than or equal to the number of non-linear irreducible
characters of $G$. Every non-linear irreducible character of $G$ vanishes on
at least one conjugacy class by [9, Theorem 3.15]. Let
$|{\mathrm{Irr}}(G)|=a+b$, where $a$ is the number of the non-linear
characters and $b$ is the number of linear characters and
$\mathrm{nz}(G)=a+c$, where $c$ is non-negative integer. Then
$\mathrm{anz}(G)=\frac{a+c}{a+b}\geqslant\frac{a+c}{2a}\geqslant\frac{1}{2}$,
since $a\geqslant b$. This contradicts our hypothesis.

Hence every non-linear irreducible character $\chi$ of $G$ is such that
$\chi_{G^{\prime}}$ is not irreducible. Note that $G$ is an $M$-group by [9,
Theorems 6.22 and 6.23 ] since $G/N$ is supersolvable and $N$ is abelian. By
Lemma 2.5, there exists a normal subgroup $K$ of $G$ such that
$G^{\prime}\leqslant K<G$ and $G{\setminus}K\subseteq\upsilon(\chi)$. If
$\upsilon(\chi)$ is a conjugacy class, then $G$ is a Frobenius group with an
abelian kernel and a complement of order two by Lemma 2.6. Thus by [2,
Proposition 2.7], every irreducible character of $G$ vanishes on at most one
conjugacy class. Note that $G$ has two linear characters. Since
$\mathrm{anz}(G)<\frac{1}{2}$, $G$ can only have one non-linear character. It
follows that $G$ has three conjugacy classes. This implies that
$G\cong\mathrm{S}_{3}$. Now $\mathrm{S}_{3}$ is supersolvable group and we are
done.

Therefore every non-linear irreducible character of $G$ vanishes on at least
$\ell$ conjugacy classes, where $\ell\geqslant 2$. Suppose $G$ has an
irreducible character $\chi$ that vanishes on exactly $\ell$ conjugacy
classes. Let $b$ be the number of linear characters of $G$ and $m$ be the
number of non-linear irreducible characters of $G$. If $b\leqslant 2\ell-1$,
then $\mathrm{anz}(G)\geqslant\frac{m\ell}{b+m}\geqslant\frac{1}{2}$,
contradicting our hypothesis. We may assume that $b=2\ell$. Then $G$ has only
one non-linear irreducible character $\chi$. By Lemma 2.8, $G$ is a Frobenius
group with an elementary abelian kernel of order $p^{n}$ and cyclic complement
of order $p^{n}-1$ for some prime $p$. Also note that
$|{\mathrm{Irr}}(G)|=2\ell+1$. By Lemma 2.8,
$\mathrm{anz}(G)=\frac{2\ell-1}{2\ell+1}\geqslant\frac{1}{2}$ since
$\ell\geqslant 2$, a contradiction.

Suppose that $b\geqslant 2\ell+1$. Then by Lemma 2.5, there exists a normal
subgroup $K$ of $G$ such that $G^{\prime}\leqslant K<G$ and
$G{\setminus}K\subseteq\upsilon(\varphi)$ for every non-linear irreducible
character $\varphi$ of $G$. We also have that
$|G{:}G^{\prime}|-|K{:}G^{\prime}|=b-|K{:}G^{\prime}|\geqslant\ell+1$ by Lemma
2.11, again contradicting our hypothesis that $G$ has an irreducible character
that vanishes on exactly $\ell$ conjugacy classes. This concludes our proof. ∎

###### Proof of Theorem D.

Suppose $\mathrm{anz}(G)<\frac{1}{3}$. By Theorem C, $G$ is supersolvable
group. In particular, $G$ is an $M$-group. Suppose that
$\chi\in{\mathrm{Irr}}(G)$ be non-linear such that $\chi_{G^{\prime}}$ is
irreducible. Then by the argument in the second paragraph of the proof of
Theorem C, $\mathrm{anz}(G)\geqslant\frac{1}{2}>\frac{1}{3}$, a contradiction.
Hence every non-linear irreducible character $\chi$ of $G$ is such that
$\chi_{G^{\prime}}$ is not irreducible. There exists a normal subgroup $K$ of
$G$ such that $G^{\prime}\leqslant K<G$ and
$G{\setminus}K\subseteq\upsilon(\chi)$ by Lemma 2.5. If $\upsilon(\chi)$ is a
conjugacy class, then $G$ is a Frobenius group with an abelian kernel and a
complement of order two using Lemma 2.6. By [2, Proposition 2.7], we have that
every irreducible character of $G$ vanishes on at most one conjugacy class.
Since $G$ has two linear characters, $\mathrm{anz}(G)\geqslant\frac{1}{3}$,
contradicting our hypothesis.

Hence every non-linear irreducible character of $G$ vanishes on at least
$\ell$ conjugacy classes, where $\ell\geqslant 2$. Suppose $G$ has an
irreducible character $\chi$ that vanishes on exactly $\ell$ conjugacy
classes. Let $b$ be the number of linear characters of $G$. If $b\leqslant
3\ell-1$, then $\mathrm{anz}(G)\geqslant\frac{1}{3}$, a contradiction. We may
assume that $b=3\ell$. Then $G$ has only one non-linear irreducible character
$\chi$. By Lemma 2.8, $G$ is a Frobenius group with an elementary abelian
kernel of order $p^{n}$ and a cyclic complement of order $p^{n}-1$ for some
prime $p$. By Lemma 2.8, $\mathrm{anz}(G)=\frac{2\ell-1}{2\ell+1}>\frac{1}{3}$
since $\ell\geqslant 2$, a contradiction.

Suppose that $b\geqslant 3\ell+1$. Then by Lemma 2.5, there exists a normal
subgroup $K$ of $G$ such that $G^{\prime}\leqslant K<G$ and
$G{\setminus}K\subseteq\upsilon(\varphi)$ for every non-linear irreducible
character $\varphi$ of $G$. We also have that
$|G{:}G^{\prime}|-|K{:}G^{\prime}|=b-|K{:}G^{\prime}|\geqslant\ell+1$ by Lemma
2.11, contradicting our hypothesis that $G$ has an irreducible character that
vanishes on exactly $\ell$ conjugacy classes. This concludes our proof. ∎

######  Remark 4.1.

Let $\mathcal{L}=\\{G\mid G$ is an extra-special $2$-group of order $2^{2k+1}$
for some positive integer $k\\}$ and $G\in\mathcal{L}$. Since
$|G/G^{\prime}|=2^{2k}$ and $G$ has only one non-linear irreducible character,
we have $\mathrm{acd}(G)=\frac{2^{2k}+2^{k}}{2^{2k}+1}$. Note that
$\mathcal{L}$ is an infinite family of nilpotent groups $G$ such that
$\mathrm{acd}(G)<\frac{4}{3}$. Hence $\mathrm{acd}(G)\rightarrow 1$ and
$G\rightarrow\infty$. In other words there does not exist $c>1$ such that
$\mathrm{acd}(G)<c$ implies that $G$ is abelian.

Our last result shows that there does not exist a non-abelian nilpotent group
$G$ such that $\mathrm{anz}(G)<\frac{1}{3}$. We shall restate Theorem E below.

######  Theorem 4.2.

Let $G$ be a finite group. Then $G$ is abelian if and only if
$\mathrm{anz}(G)<\frac{1}{3}$.

###### Proof.

If $G$ is abelian, then $\mathrm{anz}(G)=0<\frac{1}{3}$. Suppose that
$\mathrm{anz}(G)<\frac{1}{3}$. Then $G$ is nilpotent group by Theorem D. Using
induction on $|G|$, we have that $G/N$ is abelian for some minimal normal
subgroup $N$ of $G$, that is, $N=G^{\prime}$. If $\chi_{G^{\prime}}$ is
irreducible, then $\chi$ is linear. Hence every non-linear irreducible
character $\chi$ is such that $\chi_{G^{\prime}}$ is not irreducible.

By Lemma 2.5, there exists a normal subgroup $K$ of $G$ such that
$G^{\prime}\leqslant K<G$ and $G{\setminus}K\subseteq\upsilon(\chi)$. If
$\upsilon(\chi)$ is a conjugacy class, then $G$ is a Frobenius group with an
abelian kernel and a complement of order two by Lemma 2.6, a contradiction
since $G$ is nilpotent.

Hence every non-linear irreducible character of $G$ vanishes on at least
$\ell$ conjugacy classes, where $\ell\geqslant 2$. Suppose $G$ has an
irreducible character $\chi$ that vanishes on $\ell$ conjugacy classes. Let
$b$ be the number of linear characters of $G$. If $b\leqslant 3\ell-1$, then
$\mathrm{anz}(G)\geqslant\frac{1}{3}$, a contradiction. We may assume that
$b=3\ell$. Then $G$ has only one non-linear irreducible character $\chi$. By
Lemma 2.8, $G$ is an extra-special group $2$-group. Using the second part of
Lemma 2.8, we have that $\mathrm{anz}(G)=\frac{2\ell-1}{2\ell+1}>\frac{1}{3}$
since $\ell\geqslant 2$, a contradiction.

Suppose that $b\geqslant 3\ell+1$. Then by Lemma 2.5, there exists a normal
subgroup $K$ of $G$ such that $G^{\prime}\leqslant K<G$ and $G\setminus
K\subseteq\upsilon(\varphi)$ for every non-linear irreducible character
$\varphi$ of $G$. We have that
$|G{:}G^{\prime}|-|K{:}G^{\prime}|=b-|K{:}G^{\prime}|\geqslant\ell+1$ by Lemma
2.11, contradicting our hypothesis that $G$ has an irreducible character that
vanishes on exactly $\ell$ conjugacy classes. This concludes our proof. ∎

###### Proof of Theorem F.

Suppose that $G$ is non-abelian. Note that $G$ is solvable. Also note that
every non-linear irreducible character of $G$ vanishes on at least $\ell$
conjugacy classes, $\ell\geqslant 2$. Suppose that $G$ has an irreducible
character $\chi$ that vanishes on two conjugacy classes. By Lemma 2.10, either
$G$ is a Frobenius group with a complement of order $3$ or there are normal
subgroups $M$ and $N$ of $G$ such that: $M$ is a Frobenius group with the
kernel $N$; $G/N$ is a Frobenius group of order $p(p-1)/2$ with the kernel
$M/N$ and a cyclic complement of order $(p-1)/2$ for some odd prime $p$. If
$G$ is a Frobenius group with a complement of order $3$, then $G$ can only
have at most two non-linear irreducible characters. Otherwise
$\mathrm{anz}(G)\geqslant\frac{8}{7}$, a contradiction (note that a group of
odd order has an even number of non-linear irreducible characters). By Lemma
2.9, $G$ is the group in the case (d) of Lemma 2.9 with $(p^{k}-1)/2=3$, that
is, $p^{k}=7$ and $|G|=21$. Hence $G$ is supersolvable. We may assume that $G$
is the group in the second case above. Then $M=G^{\prime}$, $\chi_{M}$ is
irreducible and by Lemma 2.2, for every linear character $\alpha$ of $G$,
there exists a corresponding $\alpha\chi\in{\mathrm{Irr}}(G)$ and hence
$\mathrm{anz}(G)\geqslant 1$, contradicting our hypothesis.

Hence every non-linear irreducible character of $G$ vanishes on at least
$\ell$ conjugacy classes, $\ell\geqslant 3$. Suppose that $G$ has irreducible
character vanishing on exactly $\ell$ conjugacy classes. If there exists
$\chi$ such that $\chi_{G^{\prime}}$ is irreducible, then for every linear
character $\alpha$ of $G$, there exists a corresponding
$\alpha\chi\in{\mathrm{Irr}}(G)$ and hence $\mathrm{anz}(G)\geqslant 1$,
contradicting our hypothesis. We may assume that for every irreducible
character $\chi$ of $G$, $\chi_{G^{\prime}}$ is reducible. By Lemma 2.5, there
exists a normal subgroup $K$ of $G$ such that $G^{\prime}\leqslant K<G$ and
$G{\setminus}K\subseteq\upsilon(\varphi)$ for every non-linear irreducible
character $\varphi$ of $G$. If $b\leqslant 2\ell-1$, then $G$ has at most two
non-linear irreducible characters. Otherwise,
$\mathrm{anz}(G)\geqslant\frac{3\ell}{2\ell+2}>1$, a contradiction. Hence $G$
has exactly two non-linear irreducible characters. By Lemma 2.9, $G$ is a
Frobenius group with an elementary abelian kernel $G^{\prime}$ of order
$p^{k}$ and a cyclic Frobenius complement of order $(p^{k}-1)/2$, where $p$ is
odd prime. Note that every non-linear irreducible character of $G$ vanishes on
$G{\setminus}G^{\prime}$. Thus $b=(p^{k}-1)/2$ and since $|G{:}G^{\prime}|=b$,
we have that $b-1\leqslant\ell$ using Lemma 2.5. Hence
$\mathrm{anz}(G)\geqslant\frac{2(b-1)}{b+2}>1$, a contradiction to our
hypothesis.

If $b\geqslant 2\ell+1$, we have that
$|G{:}G^{\prime}|-|K{:}G^{\prime}|=b-|K{:}G^{\prime}|\geqslant\ell+1$ by Lemma
2.11, a contradiction to the hypothesis that $G$ has an irreducible character
that vanishes on exactly $\ell$ conjugacy classes. ∎

## Acknowledgements

The author would like to thank the reviewer for providing simpler arguments in
the proofs of several lemmas and Theorem A.

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