# SML: Semantic Meta-learning for Few-shot Semantic Segmentation

Ayyappa Kumar Pambala1, Titir Dutta1, Soma Biswas1

###### Abstract

The significant amount of training data required for training Convolutional
Neural Networks has become a bottleneck for applications like semantic
segmentation. Few-shot semantic segmentation algorithms address this problem,
with an aim to achieve good performance in the low-data regime, with few
annotated training images. Recently, approaches based on class-prototypes
computed from available training data have achieved immense success for this
task. In this work, we propose a novel meta-learning framework, Semantic Meta-
Learning (SML) which incorporates class level semantic descriptions in the
generated prototypes for this problem. In addition, we propose to use the well
established technique, ridge regression, to not only bring in the class-level
semantic information, but also to effectively utilise the information
available from multiple images present in the training data for prototype
computation. This has a simple closed-form solution, and thus can be
implemented easily and efficiently. Extensive experiments on the benchmark
PASCAL-5i dataset under different experimental settings show the effectiveness
of the proposed framework.

##  1 Introduction

Image segmentation is one of the fundamental problems in the field of computer
vision. Traditional supervised segmentation methods (Long, Shelhamer, and
Darrell 2015; Badrinarayanan, Kendall, and Cipolla 2017; Lin et al. 2017) give
impressive results when large amounts of annotated data are available.
However, this requirement of labeled training data for segmentation task is
quite difficult, since annotating each and every pixel for huge amount of
image data is highly expensive and cumbersome. On the other hand, humans can
identify any novel concept very easily even with very few examples of the
same. Few-shot semantic segmentation (Wang et al. 2019; Zhang et al. 2019;
Siam and Oreshkin 2019) tries to address this problem, by working in the very
low data regime, utilizing few annotated images from each class.

Meta-learning or learning-to-learn approaches have achieved very good
performance for the problem of few shot learning (Vinyals et al. 2016; Snell,
Swersky, and Zemel 2017), and also for the segmentation application (Dong and
Xing 2018; Rakelly et al. 2018a). Training meta-learning algorithms
constitutes two stages of learning: (1) base-learner, which learns to predict
an individual task at the episode level and (2) meta-learner, which learns to
generalize by learning across a large number of training tasks/episodes.
Significant amount of research has been done along these lines, but recently,
the approaches based on computing class representatives or class prototypes
have been very successful (Wang et al. 2019; Tian et al. 2020).

In this work, we propose a novel meta-learning framework, termed as Semantic
Meta-Learning or SML, to address the few-shot semantic segmentation task by
utilizing class-specific semantic information. The proposed SML approach is
also based on computing the prototypes corresponding to each class in the
training data. But, in contrast to other prototype-based meta-learning
approaches in literature (Wang et al. 2019; Dong and Xing 2018), SML does not
compute the class prototypes as the average representation of all the visual
feature embeddings. Instead, they are learnt by incorporating the semantic
knowledge of the particular class (obtained automatically from the class
names) into the visual information obtained from the images by the base-
learner. In addition, we propose to utilise the visual feature embeddings
obtained from multiple training images of the same class individually while
computing the class prototypes, instead of averaging them. Both these steps
effectively bring the visual embeddings of the same class images closer to one
another, and also maintain semantically meaningful intra-class distances
between the class prototypes, even when few training images per class are
available. Inspired by the seminal work in (Bertinetto et al. 2019), these
objectives are achieved through learning a linear function between the visual
feature embeddings and the semantic information or attributes using standard
ridge regression method. Thus, in the proposed SML framework, this linear
function has a closed-form solution to compute the prototypes, which makes the
computation very efficient. Extensive experiments on the benchmark PASCAL-5i
dataset with different experimental settings show that the proposed SML
framework is effective for few-shot semantic segmentation task. SML also
performs favorably with respect to the state-of-the-art, even for weaker
annotations. Thus, the contributions of this work are as follows:

  1. 1.

We propose a novel end-to-end meta-learing framework, SML, which can
effectively integrate the attribute information and the visual feature
embeddings for the meta-learner to generalize to new classes during testing.

  2. 2.

SML also utilises multiple images in the training data more effectively for
computing better class-prototypes.

  3. 3.

Extensive evaluation on the PASCAL-5i shows that the proposed SML framework
performs better or comparable to the state-of-the-art.

The rest of the paper is organized as follows. The relevant literature is
described in Section 2. The problem definition and the proposed approach are
described in details in Section 3 and Section 4 respectively. The experimental
details are described in Sections 5 and the paper concludes with a brief
summary in Section 6.

Figure 1: Illustration of the proposed Semantic Meta-Learning (SML) framework
for few-shot semantic segmentation. This illustration is for a single episode
for 1-way 2-shot segmentation task. The base-learner learns the class-
prototypes by integrating the visual information from the image (using
average-masked features ${\phi}$) and the attribute information $\mathbf{a}$,
which is accomplished using a linear function $h_{W}$. These class-prototypes
are used to obtain the final prediction over the query set.

##  2 Related work

Few-shot learning is a very active area of research in the field of computer
vision, and several approaches have been proposed. In this section, we give
pointers to the relevant meta-learning based approaches in literature which
addresses few-shot classification and semantic segmentation.

#### Few-shot learning:

Meta-learning based few-shot learning approaches can be broadly divided into
(a) metric learning based approaches (Vinyals et al. 2016; Snell, Swersky, and
Zemel 2017), (b) optimization based approaches (Finn, Abbeel, and Levine 2017;
Ravi and Larochelle 2017; Bertinetto et al. 2019), etc. The goal of these
approaches is to leverage the support set images to learn to classify the
images in the query set. Matching network (Vinyals et al. 2016) learns using
soft weighted nearest neighbour scores obtained from support images.
Prototypical network (Snell, Swersky, and Zemel 2017) evaluates support class
means as the class-prototypes and the classification is performed on the query
examples using Euclidean distance measure. Relation network (Sung et al. 2018)
learns to find relations between small number of images at each episode, which
contains both support and query images. Recently, in differentiable solver
(Bertinetto et al. 2019), the network is learned using ridge regression.
Though the proposed method is inspired from (Bertinetto et al. 2019), we
propose to additionally fuse the semantic information of the classes for the
goal of few-shot semantic segmentation.

#### Few-shot segmentation:

Some of the early approaches in the literature for few-shot segmentation,
follow the strategy in (Vinyals et al. 2016; Snell, Swersky, and Zemel 2017)
to learn network parameters from the support images and perform pixel wise
classification task on the query images. Co-FCN (Rakelly et al. 2018a) learns
to segment the query images by fusing feature information of support images. A
metric learning based prototype learning is used in (Dong and Xing 2018) for
few-shot segmentation. Weight imprinting mechanism of new classes using
adaptive masked proxies is used in (Siam and Oreshkin 2019). CANet (Zhang et
al. 2019) uses iterative optimization solution and MetaSegNet (Tian et al.
2020) uses a differentiable optimization solver for the segmentation task.
Recently, prototypical networks (Snell, Swersky, and Zemel 2017) for few-shot
classification is adapted in (Dong and Xing 2018; Wang et al. 2019) to perform
the segmentation task. PANet (Wang et al. 2019) uses masked average pooling to
obtain the prototypes, and also proposes a reverse alignment regularizer to
learn the prototypes using a swapped-setting of support and query images.
PPNet (Liu et al. 2020) uses multiple class prototypes with unlabelled data
for the segmentation task.

Our method is inspired from (Wang et al. 2019) to obtain better class
prototypes by incorporating the semantic class-descriptions (using the class
names), into the base-learners to learn the segmentation task. In essence, the
base-learner exploits both visual information and semantic/attribute knowledge
to obtain better prototypes and also to efficiently cluster the features of
images of the same class. In the next section, we explain the proposed
approach.

##  3 Problem Definition

Here, we discuss in details the proposed SML framework for the task of few-
shot semantic segmentation. First, we describe the notations used and the
problem statement. Let the annotated training data be denoted as
$\mathcal{D}_{base}=\\{\mathbf{I},\mathbf{M}\\}$, where, $\mathbf{M}$
represents the ground truth mask corresponding to the input image
$\mathbf{I}$. Let $\mathcal{C}_{base}$ be the set of base classes for which
the annotated data is available. During testing, the goal is to segment images
from a set of novel classes $\mathcal{C}_{novel}$, given only few images from
each class. Let the number of images available per class during testing be
denoted by $K$, and $K$ is usually less than or equal to $5$. The testing
classes do not overlap with the training ones, i.e.
$\mathcal{C}_{base}\cap\mathcal{C}_{novel}=\phi$.

To address this task, a meta-learning based approach is adopted, where we
attempt to imitate the testing scenario as closely as possible during
training. The entire training process is split into sub-tasks or episodes,
referred to as episode-based training (Vinyals et al. 2016; Snell, Swersky,
and Zemel 2017). Each task or episode $\mathcal{E}=(\mathcal{S},\mathcal{Q})$
consists of a support set $\mathcal{S}$ and a query set $\mathcal{Q}$. To
emulate the testing scenario, each episode is constructed to be a $C$-way,
$K$-shot segmentation task, i.e., given a support set of $K$ images and their
ground truth masks from each of $C$ classes, the goal is to segment the images
in the query set containing objects from one of these $C$ classes. Thus,
$\mathcal{S}$ contains randomly selected $C$ classes from the set
$\mathcal{C}_{base}$ with $K$ images from each class, i.e.,
$\mathcal{S}=\bigcup\limits_{c=1}^{C}\\{\mathbf{I}^{s}_{i,c},\mathbf{M}^{s}_{i,c}\\}_{i=1}^{K}$.
Similarly, the query set
$\mathcal{Q}=\\{\mathbf{I}_{j}^{q},\mathbf{M}^{q}_{j}\\}_{j=1}^{N_{q}}$
contains $N_{q}$ number of query images from the classes present in its
support set. The set of images in the support and query sets in every episode
is strictly non-overlapping. The network is trained by sampling several
episodes in a meta-learning fashion.

In this work, segmentation is performed by computing semantically meaningful
prototypes for each class. Given the class names present in the support (or
query) set, any existing pre-trained language model, such as FastText (Joulin
et al. 2016) or Word2Vec (Mikolov et al. 2013) can be utilised to obtain the
semantic or attribute vectors automatically corresponding to those object
classes. Let $\mathbf{a}_{c}\in\mathbb{R}^{d_{a}}$ denote the attribute vector
corresponding to the $c^{th}$ class. All image pixels in the support and query
set which belong to the $c^{th}$ class will have the same semantic
representation. Note that, this information can be easily obtained given just
the class names in $\mathcal{D}_{base}$, and does not require any additional
information. Attribute information is widely used in applications like ZSL
(Akata et al. 2016), but it has been relatively less explored in the context
of few-shot learning.

### Main Idea of SML framework

An illustration of the SML framework for few-shot segmentation is shown in
Figure 1. The proposed framework is learnt in a meta-learning fashion
consisting of several episodes, and Figure 1 illustrates the base learner for
$1$-way, $2$-shot scenario. It has two main modules: 1) feature-extractor
module and 2) attribute-injector module. For each episode, given the support
images, their visual features are extracted using the feature extractor
network. Their masks are used to obtain the foreground and background regions,
which are in turn used to compute the average foreground ($\phi_{cow}$) and
background ($\phi_{bg}$) feature embeddings for each support image by average
mask pooling. The class names in the support set is used by the attribute
injector module to incorporate the semantic information into the base learner.
Specifically, this is achieved by using a linear function ($h_{W}$) to
effectively combine the information from the attribute vectors and the visual
features, towards the goal to obtain better class prototypes ($h_{W}(.)$).
These final prototypes of the class ($h_{W}({\bf a}_{cow})$) and the
background ($h_{W}({\bf a}_{bg})$) are used to perform segmentation on the
images in the query set. The loss computed between the query prediction and
its ground truth mask is used to update the base-learner, which constitute the
feature extractor and the linear function $h_{W}$. The meta-learner is learnt
over several training episodes to generalize to segment novel classes with few
examples per class. We will discuss these modules in details in the next
section.

The main difference between the proposed SML framework and other prototype-
based few-shot segmentation approaches (Wang et al. 2019; Dong and Xing 2018)
is the use of attribute information. We feel that semantic information is
extremely beneficial, specially in the low-data regime. Instead of relying
solely on the visual information from the images, SML effectively combines it
with the semantic information for obtaining improved class prototypes. In
addition, the features from multiple support images are not averaged, rather
they individually contribute to the prototype computation. This helps to bring
the features from the same class closer to one another, thus effectively
utilising all the additional support images. The closed form solution of the
linear function makes the learning very efficient. Extensive experiments and
analysis have been presented in Section 5, which justifies the effectiveness
of this framework. Now, we will discuss the design of loss functions and meta-
learning based training methodology followed in our work.

##  4 Proposed SML Framework

In this section, we will describe the different components of the proposed SML
framework. The proposed meta-learning framework learns over a collection of
episodes, and thus gets trained with data of all classes
($\mathcal{C}_{base}$) in the training set. On the other hand, the _base
learner_ learns over a single episode with data from randomly selected
$C$-classes. The base learner consists of a feature extractor module and an
attribute injector module as described below.

#### 1\. Feature-extractor module:

Given an input image $\mathbf{I}$ from support or query, the feature
extraction module $f_{\theta}$ extracts its 3-d feature representation
$f_{\theta}(\mathbf{I})\in\mathbb{R}^{h\times w\times d}$. Here,
$f_{\theta}(\mathbf{I})$ is the upsampled feature map and has the same height
($h$) and width ($w$) as the original image and and $d$ denotes the number of
channels of the final convolutional layer. The set of learnable parameters in
the feature extractor module is denoted as $\theta$. This 3-d visual feature
representation of an image can be considered as the $d$-dimensional embedding
of each of the image pixels, which either belong to the class of interest or
the background.

As per the standard meta-learning set-up (Vilalta and Drissi 2002), in each
episode, the base learner first processes the data in the support set
$\mathcal{S}$, and then computes the losses on the query set $\mathcal{Q}$.
Given the support set
$\mathcal{S}=\bigcup\limits_{c=1}^{C}\\{\mathbf{I}^{s}_{i,c},\mathbf{M}^{s}_{i,c}\\}_{i=1}^{K}$,
for every image, its 3-d feature representation
$f_{\theta}({\mathbf{I}_{i,c}^{s}})$ is computed using the feature extractor
module. This obviously contains the feature representation of both the object
(or, the foreground) and the background in the image. The foreground and
background features can be separated using the mask $\mathbf{M}_{i,c}^{s}$
provided with the input image. As in several other works (Siam and Oreshkin
2019; Wang et al. 2019), we also use average mask-pooling operation to obtain
the foreground (or, background) embedding as follows:

${\phi}_{i,c}^{s}=\frac{1}{|\mathcal{X}|}\sum_{\mathcal{X}}{f_{\theta}(\mathbf{I}_{i,c}^{s})\odot\mathbf{1}_{(\mathbf{M}_{i,c}^{s}==c)}}\text{,
for foreground}$
${\phi}_{i,bg}^{s}=\frac{1}{|\mathcal{Y}|}\sum_{\mathcal{Y}}{f_{\theta}(\mathbf{I}_{i,c}^{s})\odot\mathbf{1}_{(\mathbf{M}_{i,c}^{s}\neq
c)}}\text{, for background}$ (1)

Here, $\odot$ represents the point-wise multiplication along the dimension of
$d$. $\mathcal{X}$ and $\mathcal{Y}$ denote the respective sets of spatial
locations in the image $\mathbf{I}_{i,c}^{s}$ for which the corresponding
indicator functions are activated. Thus, $\phi_{i,c}^{s}$ and
$\phi_{i,bg}^{s}\in\mathbb{R}^{d}$ can be considered as the mask-pooling of
foreground and background features separately in the visual embedding space
evaluated as the mean over their spatial spread in the image. We normalize the
features, $\phi_{i,c}^{s}=\frac{\phi_{i,c}^{s}}{||\phi_{i,c}^{s}||_{2}}$ and
$\phi_{i,bg}=\frac{\phi_{i,bg}^{s}}{||\phi_{i,bg}^{s}||_{2}}$ in our
experiments.

#### 2\. Attribute-injector module:

In some of the recent successful approaches (Wang et al. 2019), the image
embeddings are solely utilised to obtain the class-representatives or
prototypes. We propose to augment this information with the semantic knowledge
for better generalization to unseen classes with few examples during testing.
Here, the semantic information is given by the attribute vectors,
$\mathbf{a}_{c}\in\mathbb{R}^{d_{a}}$, which are the class-name embeddings for
the $c^{th}$ class. These attributes can be automatically extracted from a
pre-trained FastText (Joulin et al. 2016) or Word2Vec (Mikolov et al. 2013)
language model. Here, $d_{a}$ denotes the attribute dimension. Given these
semantic attributes of the classes in the training data, this module
incorporates this information to compute better class prototypes for the
segmentation task. In this work, we use a linear function for the attribute
injector as given below:

$\displaystyle h_{W}(\mathbf{a}_{c})$ $\displaystyle=\mathbf{W}\mathbf{a_{c}}$
(2)

where, $\mathbf{W}\in\mathbb{R}^{d\times d_{a}}$ is the weight matrix
containing all the trainable parameters. Similarly, we also obtain the latent
space representation of the background of the images as
$h_{W}(\mathbf{a}_{bg})=\mathbf{W}\mathbf{a}_{bg}$; where $\mathbf{a}_{bg}$ is
the FastText or Word2Vec embedding of the word _background_ , and thus is the
same for all images.

In order to effectively incorporate the semantic information with the
embeddings obtained from the images using equation (1) in the latent space, we
use the standard ridge regression (Bertinetto et al. 2019; Verma and Rai
2017). Thus, the loss function for learning $\mathbf{W}$ can be expressed as

$\displaystyle\mathcal{L}_{W}$
$\displaystyle=||\Phi-\mathbf{W}\mathbf{A}||_{2}^{2}+\lambda||\mathbf{W}||_{2}^{2}$
(3)

where,
$\Phi=\big{[}\\{\phi_{i,c}^{s}|\phi_{i,bg}\\}_{i=1}^{K}\big{]}_{c=1}^{C}\in\mathbb{R}^{d\times
2|\mathcal{S}|}$ and
$\mathbf{A}=\big{[}\\{\mathbf{a}_{i,c}^{s}|\mathbf{a}_{i,bg}^{s}\\}_{i=1}^{K}\big{]}_{c=1}^{C}\in\mathbb{R}^{d_{a}\times
2|\mathcal{S}|}$. $\lambda$ is a learnable hyper-parameter which is set
experimentally to balance the L2-regularizer on the parameters $\mathbf{W}$.
The optimum set of parameters of the attribute injector module $\mathbf{W}$ is
obtained by minimizing this loss function which has a closed-form solution
given by

$\mathbf{W}={\Phi}\mathbf{A}^{T}(\mathbf{A}\mathbf{A}^{T}+\lambda\mathbf{I}_{d_{a}})^{-1}$
(4)

where, $\mathbf{I}_{d_{a}}$ is an identity matrix of dimension $d_{a}$.

### Learning the base-learner parameters

The goal is to leverage the query set images and its corresponding ground
truth masks to learn the parameters of the feature extractor module and the
linear function parameters $\mathbf{W}$, such that during testing, semantic
segmentation can be performed using only few images per class. In each
episode, given the support images with the corresponding masks from
$C$-classes, $\Phi$ and $\mathbf{A}$ can be computed, from which the linear
function parameters $\mathbf{W}$ are learnt as discussed before. Note that,
for each class $c$, the images corresponding to this class will have different
feature representations ($\phi_{i,c}^{s}$) because of the intra-class
variability, but they will all have the same semantic representation
corresponding to the class name $\mathbf{a}_{c}$ for both support and query.
This unique semantic representation of the $c^{th}$ class is considered its
class prototype $h_{W}(\mathbf{a}_{c})=\mathbf{W}\mathbf{a}_{c}$ in this work.

Now, given an image from the query set $\mathbf{I}_{j}^{q}$, we first compute
its visual representation by passing it through the feature extractor module
$f_{\theta}({\mathbf{I}}_{j}^{q})$ and the feature corresponding to each pixel
is normalized. Next, the normalized feature representation
$f_{\theta}(\mathbf{I_{j}^{q}})$ is convolved with the normalized class-
prototypes for all the $C$-classes selected in that episode as follows:

$\displaystyle\mathbf{S}_{j;c}^{q}$
$\displaystyle=f_{\theta}(\mathbf{I}_{j}^{q})*h_{W}(\mathbf{a}_{c}),\text{ for
}c=\\{1,...,C\\}\cup\\{bg\\}$ (5)

Here, $*$ stands for convolution operation and
$h_{W}(\mathbf{a}_{c})=\mathbf{W}\mathbf{a}_{c}$ is the class-prototype of the
$c^{th}$ class. We have used the same notations for the normalized
representations for simplicity. Thus,
$\mathbf{S}_{j;c}^{q}\in\mathbb{R}^{h\times w}$ denotes the 2-d mask of the
query image with the class-similarities, which is finally used for
classification. Here, in addition to the foreground classes, similarity with
the background class prototype _bg_ is also computed. To segment the query
image, we perform pixel-wise classification of the query feature
$\mathbf{S}_{j;c}^{q}$. For this, we compute the logit-score for the pixel at
location $(m,n)$ as the probability of the evaluated similarity feature
$\mathbf{S}_{j;c}^{q}$ to belong to class $c$ as

$\displaystyle
p(\mathbf{S}_{j;c}^{q}(m,n))=\frac{exp(\alpha\mathbf{S}_{j;c}^{q}(m,n)+\beta)}{\sum\limits_{c\in{\\{1,..,C\\}\cup\\{bg\\}}}{exp(\alpha\mathbf{S}_{j;c}^{q}(m,n)+\beta)}}$
(6)

where $\alpha,\beta$ are the scaling and bias parameters, respectively. We
learn the feature extractor module of the base learner by minimizing the
cross-entropy loss function

$\displaystyle\mathcal{L}_{CE}$
$\displaystyle={\sum\limits_{(m,n)\in\mathbf{I}_{j}^{q}}{-\log
p(\mathbf{S}_{j;c}^{q}(m,n))}}$ (7)

Inspired by (Wang et al. 2019), we also interchange the samples in support
$\mathcal{S}$ and query $\mathcal{Q}$ set to enhance the segmentation
performance. Thus, we learn another set of parameters $\bar{\mathbf{W}}$ on
the newly-constructed $\mathcal{S}$ and $\mathcal{Q}$ using (4). Following
similar steps as shown above, we compute the reverse alignment loss
$\mathcal{L}_{R}$ as in (Wang et al. 2019) which is given by,

$\mathcal{L}_{R}={\sum\limits_{(m,n)\in\mathbf{I}^{s}}{-\log
p({\mathbf{S}}_{c}^{s}(m,n)})}$ (8)

Note that subscripts are removed for the similarity predictions and for the
images to avoid clutter. The objective used to learn the feature extractor is
given by

$\min_{\theta}\sum\limits_{(\mathcal{S},\mathcal{Q})}\big{(}\mathcal{L}_{CE}+\mathcal{L}_{R}\big{)}$
(9)

This completes the learning of the base-learner for a single episode. To
summarize, in each episode, first, the attribute injector module parameters
$\mathbf{W}$ are learnt to generate class prototypes using the previously
learnt base-learner, which are used to further fine-tune the feature extractor
parameters $\theta$ by computing the pixel wise cross-entropy loss. Using
several such training episodes, the meta-learner learns to generalize and
segment images containing novel objects during testing.

### Prediction

Once the meta-learner is trained using all the data over a number of training
episodes, it can be used to perform segmentation on novel images in an
episode, $\mathcal{E}=(\mathcal{S},\mathcal{Q})$ which contains images from
$C_{novel}$. As in training, $\mathbf{W}$ computed using the visual image
features of the support set and the attributes of the classes present in the
testing set. Using this, the class prototypes are obtained. Finally, the
prediction on the features extracted from the query image
$\mathbf{I}_{j}^{q}\in\mathcal{Q}$ is performed as,

$\hat{\mathbf{S}}_{j}^{q}(m,n)=\underset{c\in\\{1,..,C\\}\cup\\{bg\\}}{\mathrm{arg\text{
}max}}~{}~{}\alpha\mathbf{S}_{j,c}^{q}(m,n)+\beta.$ (10)

##  5 Experiments

In this section, we describe the experiments performed to evaluate the
effectiveness of the proposed SML framework.

Method | Backbone | 1-shot | 5-shot
---|---|---|---
split-0 | split-1 | split-2 | split-3 | Mean | split-0 | split-1 | split-2 | split-3 | Mean
OSLM (Shaban et al. 2017) | VGG16 | 33.6 | 55.3 | 40.9 | 33.5 | 40.8 | 35.9 | 58.1 | 42.7 | 39.1 | 43.9
co-FCN (Rakelly et al. 2018a) | VGG16 | 36.7 | 50.6 | 44.9 | 32.4 | 41.1 | 37.5 | 50.0 | 44.1 | 33.9 | 33.9
SG-One (Zhang et al. 2020) | VGG16 | 40.2 | 58.4 | 48.4 | 38.4 | 46.3 | 41.9 | 58.6 | 48.6 | 39.4 | 47.1
AMP (Siam and Oreshkin 2019) | VGG16 | - | - | - | - | 43.4 | - | - | - | - | 46.9
PANet (Wang et al. 2019) | VGG16 | 42.3 | 58.0 | 51.1 | 41.2 | 48.1 | 51.8 | 64.6 | 59.8 | 46.5 | 55.7
SML (Ours) | VGG16 | 43.0 | 59.0 | 51.3 | 41.4 | 48.7 | 52.3 | 64.9 | 61.0 | 50.4 | 57.1
PANet (Wang et al. 2019) | RN50 | 44.0 | 57.5 | 50.8 | 44.0 | 49.1 | 55.3 | 67.2 | 61.2 | 53.2 | 59.2
PPNet (Liu et al. 2020) | RN50 | 47.8 | 58.7 | 53.8 | 45.6 | 51.5 | 58.3 | 67.8 | 64.8 | 56.7 | 61.9
CANet (Zhang et al. 2019) | RN50 | 52.5 | 65.9 | 51.3 | 51.9 | 55.4 | 55.5 | 67.8 | 51.9 | 53.2 | 57.1
SML (Ours) | RN50 | 47.4 | 59.7 | 53.8 | 44.4 | 51.3 | 56.0 | 67.8 | 62.1 | 54.0 | 60.0
Table 1: Performance evaluation of the proposed SML and comparison with the other state-of-the-art methods for both 1-way 1-shot and 1-way 5-shot experimental protocols on PASCAL-5i data. The results are reported in terms of mean-IoU. Method | Backbone | 1-shot | 5-shot
---|---|---|---
FG-BG (Rakelly et al. 2018b) | VGG16 | 55.0 | -
Fine-Tuning (Rakelly et al. 2018b) | VGG16 | 55.1 | 55.6
OSLSM (Shaban et al. 2017) | VGG16 | 61.3 | 61.5
co-FCN (Rakelly et al. 2018a) | VGG16 | 60.1 | 60.2
PL (Dong and Xing 2018) | VGG16 | 61.2 | 62.3
A-MCG (Hu et al. 2019) | VGG16 | 61.2 | 62.2
SG-One (Zhang et al. 2020) | VGG16 | 63.9 | 65.9
AMP (Siam and Oreshkin 2019) | VGG16 | 62.2 | 63.8
PANet (Wang et al. 2019) | VGG16 | 66.5 | 70.7
SML (Ours) | VGG16 | 66.8 | 71.0
CANet (Zhang et al. 2019) | RN50 | 66.2 | 69.6
SML (Ours) | RN50 | 67.1 | 72.2
Table 2: Evaluation (binary mean-IoU) of segmentation results of proposed SML
and comparison with other state-of-the-art for 1-way 1-shot protocol on
PASCAL-5i data.

#### Dataset Used:

We use PASCAL-5i (Shaban et al. 2017) dataset to evaluate the model
performance. PASCAL-5i (Shaban et al. 2017) is derived from PASCAL VOC 2012
(Everingham et al. 2010) and SBD (Hariharan et al. 2011). This dataset
contains $20$ classes. All the $20$-classes are divided into $4$ splits, with
$5$-categories per split. As is the standard practice, the proposed SML model
is trained on $3$ splits (Shaban et al. 2017) and evaluated on the $4^{th}$
split.

#### Word-embeddings:

We use two semantic encodings in our work: (1) Word2vec (Mikolov et al. 2013)
is trained on Google News dataset (Wang, Ye, and Gupta 2018) which contains
$3$-million words; (2) FastText (Joulin et al. 2016) is trained on Common-
Crawl dataset (Mikolov et al. 2018). We use these pre-trained models for
extracting the class-name embeddings in our work.

#### Evaluation metric:

Two widely-used metrices, Mean-IoU (Shaban et al. 2017; Zhang et al. 2020) and
Binay-IoU (Rakelly et al. 2018a; Dong and Xing 2018; Hu et al. 2019) are used
to report the segmentation performance of the proposed SML. Mean-IoU
calculates mean of Intersection-over-Union for all the foreground classes.
Binary-IoU calculates the Intersection-over-Union by treating all the
foreground classes as one class and background class as one class.

#### Implementation Details:

We implement our proposed approch using PyTorch (Paszke et al. 2017). We use
TITAN-X $2040$ $12$GB GPU to run our experiments. We use VGG-$16$ (Simonyan
and Zisserman 2015), and ResNet-$50$ (He et al. 2016) models, pre-trained on
ImageNet (Russakovsky et al. 2015) as the feature extractor module for SML. In
case of VGG-16, the image features are extracted from the output of $5^{th}$
convolution block. For ResNet-$50$, the image features are extracted from the
$4^{th}$ convolution block. In our implementation, we modify ResNet-50 as
follows: first two residual blocks use convolution with stride $1$, and last
two blocks are designed with dialated convolutions with $2$, $4$, to increase
the receptive field. The number of episodes used to train the proposed SML is
$30k$. The SGD-solver with weight decay of $5\times 10^{-4}$ and momentum of
$0.9$ is used for the optimization. We apply initial learning rate of $1\times
10^{-3}$ and $1.75\times 10^{-3}$ for VGG-$16$ and ResNet-$50$ respectively.
The learning rate is reduced by a factor of $0.1$ after every $10k$
iterations. The learnable parameter $\lambda$ in equation (4) is initialized
to 100. We empirically set $\alpha=10$ and $\beta=1$ in equation (6). We have
used pre-trained Word2Vec model to extract the semantic embedding from class-
names in our results and only used FastText for additional analysis. The
proposed SML framework is evaluated over $1k$ testing episodes. To mitigate
the sensitivity of the model to random initialization, we repeat the
experiments $5$ times and report the mean-IoU.

(a)

(b)

(c)

(d)

Figure 2: 1-way 1-shot segmentation results using proposed SML - (a) Support
set image; (b) GT: query image with ground-truth mask; (c) and (d) are
segmentation results using VGG-16 and ResNet-50 as the feature extractors.
Best viewed in color and when zoomed.
Figure 3: 1-way 5-shot segmentation results using proposed SML. First five columns shows the support set images with ground-truth masks. The remaining columns show the query image with ground-truth mask and segmentation results using VGG-16 and ResNet-50 as the feature extractors respectively. Best viewed in color and when zoomed. Method | split-0 | split-1 | split-2 | split-3 | Mean
---|---|---|---|---|---
PANet (VGG-16) | - | - | - | - | 45.1
SML (VGG-16) | 40.5 | 54.8 | 47.2 | 39.5 | 45.5
SML (ResNet-50) | 43.2 | 55.6 | 49.5 | 44.4 | 48.1
Table 3: Mean-IoU performance of proposed SML for 2-way 1-shot segmentation
task.

### Comparison with state-of-the-art methods

We compare the proposed SML with several recent state-of-the-art methods -
CANet (Zhang et al. 2019), PANet (Wang et al. 2019), PPNet (Liu et al. 2020),
AMP (Siam and Oreshkin 2019), SG-One (Zhang et al. 2020) etc. Here we discuss
the results for different testing protocols.

#### 1-way 1-shot and 1-way 5-shot:

We present the results for 1-way 1-shot and 1-way 5-shot semantic segmentation
in Table 1 for both VGG-$16$ and ResNet-$50$ backbones as the feature
extractor. The mean-IoU performances reported in the table for all the other
approaches are taken directly from (Wang et al. 2019; Liu et al. 2020). We
observe that for VGG-16 backbone, the proposed SML outperforms all the other
approaches for both the protocols. The performance improvement compared to the
other approaches is significantly more for 5-shot protocol. This implies that
SML is able to effectively use all the images in the support set to improve
the segmentation performance. For ResNet-$50$ backbone, SML performs at par
with the recent PPNet (Liu et al. 2020). Though CANet (Zhang et al. 2019)
performs better, it is evaluated with multi-scale input unlike the others, as
also noted in (Liu et al. 2020).

We also compare the SML performance with the state-of-the-art methods using
binary-IoU as the evaluation criteria. These results are reported in Table 2.
The results for all the other methods are taken directly from (Wang et al.
2019; Siam and Oreshkin 2019). We observe that SML outperforms all other
approaches for both the backbones. Specifically, for ResNet-$50$, it gives an
improvement of $0.9\%$ and $2.6\%$ for 1-shot and 5-shot respectively.

We present the segmentation results for few images using SML in Figure 2 and
Figure 3 for 1-way 1-shot and 1-way 5-shot experimental protocols
respectively. We observe that SML provides good segmentation results even with
significant background clutter. In general, ResNet-$50$ gives better results
than VGG-$16$ backbone, which is also validated by the quantitative results.

#### 2-way 1-shot:

We further investigate the performance of SML for $2$-way $1$-shot protocol.
From Table 3, we observe that SML performs favorably compared to PANet (Wang
et al. 2019) for the same VGG-$16$ backbone.

Method | 1-shot | 5-shot
---|---|---
D | BB | S | D | BB | S
PANet (VGG-16) | 48.1 | 45.1 | 44.8 | 55.7 | 52.8 | 54.6
SML (VGG-16) | 48.7 | 45.5 | 45.0 | 57.1 | 53.7 | 55.8
SML (ResNet-50) | 51.3 | 47.9 | 49.2 | 60.0 | 55.1 | 57.6
Table 4: Mean-IoU of proposed SML with different image-annotations. D: Dense,
BB: Bounding box, S: Scribble.

As expected, the results are significantly better with ResNet-$50$ backbone.

Additional Analysis:  Here we further analyze the proposed SML framework to
better understand it.
1\. Evaluation with weaker annotations:  We experiment with weaker annotations
of the images which are cheaper and easy to obtain (Lin et al. 2016), instead
of the pixel-by-pixel dense annotations, which is extremely expensive and
time-consuming. We observe from Table 4 that even for weaker annotations the
proposed SML performs better than the PANet architecture (Wang et al. 2019).
2\. Evaluation with different semantic embedding:  All experiments in this
paper are reported with Word2Vec as the embedding-framework. We also performed
experiment with FastText (Joulin et al. 2016). Those results, using both
embeddings for $1$-way $1$-shot and $1$-way $5$-shot protocols, are summarized
in Table 5. We observe that though both perform well, Word2Vec embeddings are
slightly better compared to FastText for our application.

Semantic Knowledge | 1-shot | 5-shot
---|---|---
Word2Vec | 51.3 | 60.0
FastText | 50.9 | 59.5
Table 5: Mean-IoU for 1-way 1-shot and 5-shot segmentation tasks with
different semantic-embeddings.

##  6 Conclusion

In this paper we have proposed a novel Semantic Meta-Learning (SML) framework
which utilises the semantic information of object classes for the task of few-
shot semantic segmentation. Towards that goal, we introduced a novel
attribute-injector module in a traditional meta-learning setting. We performed
extensive experiments with different protocols and observed that the proposed
framework performs similar or better as compared to the state-of-the-art.

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# Non-equatorial scalar rings supported by magnetized Schwarzschild-Melvin
black holes

Shahar Hod  The Ruppin Academic Center, Emeq Hefer 40250, Israel  The Hadassah
Institute, Jerusalem 91010, Israel

###### Abstract

It has recently been demonstrated that magnetized black holes in composed
Einstein-Maxwell-scalar-Gauss-Bonnet field theories with a non-minimal
negative coupling of the scalar field to the Gauss-Bonnet curvature invariant
may support spatially regular scalar hairy configurations. In particular, it
has been revealed that, for Schwarzschild-Melvin black-hole spacetimes, the
onset of the near-horizon spontaneous scalarization phenomenon is marked by
the numerically computed dimensionless critical relation
$(BM)_{\text{crit}}\simeq 0.971$, where $\\{M,B\\}$ are respectively the mass
and the magnetic field of the spacetime. In the present paper we prove, using
analytical techniques, that the boundary between bald Schwarzschild-Melvin
black-hole spacetimes and hairy (scalarized) black-hole solutions of the
composed Einstein-Maxwell-scalar-Gauss-Bonnet theory is characterized by the
exact dimensionless relation
$(BM)_{\text{crit}}=\sqrt{{{\sqrt{6}-2}\over{2\sqrt{6}}}+\sqrt{{{\sqrt{6}-1}\over{2}}}}$
for the critical magnetic strength. Intriguingly, we prove that the critical
dimensionless magnetic parameter $(BM)_{\text{crit}}$ corresponds to
magnetized black holes that support a pair of linearized non-minimally coupled
thin scalar rings that are characterized by the non-equatorial polar angular
relation $(\sin^{2}\theta)_{\text{scalar-
ring}}={{690-72\sqrt{6}+4\sqrt{3258\sqrt{6}-7158}}\over{789}}<1$. It is also
proved that the classically allowed angular region for the negative-coupling
near-horizon spontaneous scalarization phenomenon of magnetized Schwarzschild-
Melvin spacetimes is restricted to the black-hole poles,
$\sin^{2}\theta_{\text{scalar}}\to 0$, in the asymptotic large-strength
magnetic regime $BM\gg 1$.

##  I Introduction

The celebrated no-hair conjecture in black-hole physics NHC ; JDB  has
asserted that static scalar field configurations cannot be supported in black-
hole spacetimes that contain spatially regular absorbing horizons. Early
mathematical investigations of the Einstein-matter field equations Bek1 ; Sot2
; Her1 ; Sot3 ; BekMay ; Hod1  have revealed, in accord with the spirit of
this influential conjecture, that black holes cannot support scalar fields
which are minimally coupled to the Ricci scalar of the curved spacetime.
Similar conclusions have been obtained for scalar fields with a non-trivial
(non-minimal) coupling to the Ricci curvature scalar BekMay ; Hod1 .

However, recent mathematical studies Sot5 ; Sot1 ; GB1 ; GB2 ; ChunHer ; SotN
; Hodsg1 ; Hodsg2 ; Hodca ; Done ; Herrecnum ; BeCo ; Brih ; Hernn  of the
coupled Einstein-matter field equations have revealed the physically
intriguing fact that black holes with spatially regular horizons can support
hairy matter configurations which are made of scalar fields with a direct non-
minimal coupling to the Gauss-Bonnet invariant ${\cal G}$ of the curved black-
hole spacetime Notechar ; Hersc1 ; Hersc2 ; Hodsc1 ; Hodsc2 ; Moh ; Hodnrx .

The critical boundary between bald black holes and hairy (scalarized) black-
hole spacetimes in Einstein-Gauss-Bonnet field theories is marked by the
presence of ‘cloudy’ configurations, linearized non-minimally coupled scalar
fields which are supported by the familiar black-hole solutions of the
Einstein field equations. In particular, depending on the matter content of
the theory and the assumed symmetry of the spacetime, cloudy non-minimally
coupled scalar field configurations have been studied in Schwarzschild,
Reissner-Nordström, Kerr, and Kerr-Newman black-hole spacetimes Sot5 ; Sot1 ;
GB1 ; GB2 ; ChunHer ; SotN ; Hodsg1 ; Hodsg2 ; Hodca ; Done ; Herrecnum ; BeCo
; Brih ; Hernn .

The spontaneous scalarization phenomenon of black holes in Einstein-Gauss-
Bonnet field theories is closely related to the fact that the Klein-Gordon
wave equation of the non-minimally coupled scalar field $\phi$ contains an
effective spatially-dependent mass term. This mass term, which has the compact
linearized form $-\eta\phi{\cal G}$ [see Eq. (10) below] Noteetaa , may become
negative in the vicinity of the black-hole horizon, implying that the
effective potential of the composed black-hole-scalar-field system may behave
as an attractive potential well that binds the scalar field to the near-
horizon region of the central black hole.

Intriguingly, it has recently been demonstrated in the physically important
work Hernw  that, in Einstein-Gauss-Bonnet field theories, the phenomenon of
black-hole spontaneous scalarization can be triggered by the presence of
magnetic fields. In particular, it has been demonstrated numerically in Hernw
that, for Schwarzschild-Melvin black-hole spacetimes, the spontaneous
scalarization phenomenon is magnetically-induced in the sense that only black
holes in the dimensionless strong magnetic regime

$BM>(BM)_{\text{crit}}\simeq 0.971\ $ (1)

can support scalar field configurations with a non-minimal negative coupling
to the Gauss-Bonnet invariant of the curved magnetized spacetime (Here $M$ and
$B$ are respectively the mass and the magnetic field of the black-hole
spacetime).

The main goal of the present compact paper is to explore, using analytical
techniques, the onset of the spontaneous scalarization phenomenon in
magnetized Schwarzschild-Melvin black-hole spacetimes of the composed
Einstein-Maxwell-scalar-Gauss-Bonnet field theory with negative values of the
non-minimal coupling parameter $\eta$.

In particular, below we shall derive a remarkably compact analytical
expression for the critical magnetic strength $(BM)_{\text{crit}}$ which marks
the boundary between scalarless Schwarzschild-Melvin black holes and
scalarized (hairy) magnetized black-hole configurations. In addition, we shall
reveal the physically intriguing fact that the black-hole solutions of the
composed Einstein-Maxwell-scalar-Gauss-Bonnet field theory can support pairs
of infinitesimally thin static scalar rings which are located above and below
the equator of the central supporting black hole.

##  II Description of the system

We shall study, using analytical techniques, the onset of the negative-
coupling spontaneous scalarization phenomenon of magnetized Schwarzschild-
Melvin black holes in the composed Einstein-Maxwell-scalar-Gauss-Bonnet field
theory whose action is given by the integral expression Hernw

$S={{1}\over{16\pi}}\int d^{4}x\sqrt{-g}\Big{[}R-F_{\mu\nu}F^{\mu\nu}-{1\over
2}\nabla_{\alpha}\phi\nabla^{\alpha}\phi+f(\phi){\cal G}\Big{]}\ ,$ (2)

where $F_{\mu\nu}$ is the Maxwell electromagnetic tensor of the spacetime. The
term $f(\phi){\cal G}$, which is essential for the existence of spontaneously
scalarized black-hole solutions, represents the direct (non-minimal) coupling
between the massless scalar field $\phi$ and the Gauss-Bonnet invariant

${\cal G}\equiv
R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\ $ (3)

of the black-hole spacetime.

The magnetized Schwarzschild-Melvin black-hole solution of the composed field
theory (2) is characterized by the curved line element Hernw ; Melv ; HF ;
Notecord ; Noteunits

$ds^{2}=\Lambda^{2}[-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\theta^{2}]+\Lambda^{-2}r^{2}\sin^{2}\theta
d\varphi^{2}\ ,$ (4)

where

$\Lambda=\Lambda(r,\theta;B)=1+{1\over 4}B^{2}r^{2}\sin^{2}\theta\ .$ (5)

The metric function

$f(r)=1-{{2M}\over{r}}\ ,$ (6)

whose simple root

$r_{\text{H}}=2M\ $ (7)

determines the radius of the black-hole horizon, is expressed in terms of the
mass $M$ of the spacetime.

From Eqs. (3), (4), (5), and (6) one obtains the (rather cumbersome) near-
horizon dimensionless functional expression Hernw ; NoteMev

$\displaystyle M^{4}{\cal G}(r\to
r_{\text{H}}^{+},\theta;M,B)=\Big{[}4\big{(}1+B^{2}M^{2}\sin^{2}\theta\big{)}^{8}\Big{]}^{-1}\times\Big{\\{}3\Big{[}B^{8}M^{8}\sin^{8}\theta-2B^{4}M^{4}\sin^{4}\theta+\big{[}1-B^{4}M^{4}\sin^{2}(2\theta)\big{]}^{2}\Big{]}+$
$\displaystyle
24\cos^{2}\theta\big{(}B^{8}M^{8}\sin^{6}\theta-B^{6}M^{6}\sin^{4}\theta+B^{2}M^{2}\big{)}+16B^{4}M^{4}\cos^{4}\theta\big{(}1-6B^{2}M^{2}\sin^{2}\theta\big{)}\Big{\\}}\
$ (8)

for the magnetically-dependent Gauss-Bonnet invariant of the Schwarzschild-
Melvin black hole.

The action (2) of the composed Einstein-Maxwell-scalar-Gauss-Bonnet field
theory yields the characteristic Klein-Gordon differential equation Hernw

$\nabla^{\nu}\nabla_{\nu}\phi=\mu^{2}_{\text{eff}}\phi\ $ (9)

for the scalar field, where the presence of the magnetically-dependent
effective mass term in (9),

$\mu^{2}_{\text{eff}}(\theta;M,B)=-\eta\cdot{\cal G}(\theta;M,B)\ ,$ (10)

is a direct outcome of the non-minimal coupling between the scalar field of
the theory and the Gauss-Bonnet invariant of the spacetime. The physical
parameter $\eta$ in (10) plays the role of an expansion coefficient in the
weak-field functional behavior Hernw

$f(\phi)={1\over 2}\eta\phi^{2}\ $ (11)

of the scalar coupling function. This physical parameter Noteett  determines
the strength of the direct non-minimal coupling between the massless scalar
field and the Gauss-Bonnet invariant (II) of the magnetized Schwarzschild-
Melvin black hole.

##  III Onset of negative-coupling near-horizon spontaneous scalarization in
magnetized Schwarzschild-Melvin black-hole spacetimes

In the present section we shall study the onset of the negative-coupling
spontaneous scalarization phenomenon in the composed Einstein-Maxwell-scalar-
Gauss-Bonnet field theory (2). In particular, we shall reveal the interesting
fact that, at the onset of the magnetically-induced spontaneous scalarization
phenomenon, Schwarzschild-Melvin black-hole spacetimes can support non-
equatorial thin rings which are made of massless scalar fields with a negative
coupling to the Gauss-Bonnet invariant of the magnetized spacetime.

The presence of an effective negative mass term (a binding potential well) in
the Klein-Gordon wave equation (9) of the non-minimally coupled scalar field
provides a necessary condition for the existence of spatially regular
supported field configurations (bound-state scalar clouds) in the black-hole
spacetime GB1 ; Done ; ChunHer ; Hodca ; Hernn . Intriguingly, from Eq. (II)
one learns that, depending on the values of the dimensionless magnetic
parameter $BM$ of the spacetime and the polar angle $\theta$, the effective
mass term (10) in the Klein-Gordon differential equation (9) may become
negative in the vicinity of the horizon of the Schwarzschild-Melvin black
hole.

In particular, the onset of the near-horizon spontaneous scalarization
phenomenon in the magnetized black-hole spacetime (4) is marked by the
critical functional relation Hodca ; Hernn ; Hodjp

$\text{min}\\{\mu^{2}_{\text{eff}}(\theta;M,B)\\}\to 0^{-}\ .$ (12)

For negative values of the non-minimal coupling parameter $\eta$ of the
composed Einstein-Maxwell-scalar-Gauss-Bonnet field theory (2), the
characteristic relation (12) yields the critical functional relation

$\text{min}\\{G(\theta;M,B)\\}\to 0^{-}\ $ (13)

at the onset of the spontaneous scalarization phenomenon.

Interestingly, and most importantly for our analysis, as we shall now show
explicitly, the set of coupled equations [see Eq. (13)]

$G(\theta;M,B)=0\ $ (14)

with

${{dG(\theta;M,B)}\over{d\theta}}=0\ ,$ (15)

which determine the onset of the near-horizon spontaneous scalarization
phenomenon of the magnetized black-hole spacetime (4), can be solved
analytically.

To see this, it is convenient to define the dimensionless variables

$\beta\equiv(BM)^{2}\ \ \ \ ;\ \ \ \ x\equiv\sin^{2}\theta\ ,$ (16)

in terms of which the near-horizon Gauss-Bonnet invariant (II) can be written
in the dimensionless form Notesim

$\displaystyle M^{4}{\cal G}(r\to r_{\text{H}}^{+},x;\beta)=\big{[}4(1+\beta
x)^{8}\big{]}^{-1}\times\Big{\\{}3\Big{[}\beta^{4}x^{4}-2\beta^{2}x^{2}+\big{[}1-4\beta^{2}x(1-x)\big{]}^{2}\Big{]}+$
$\displaystyle
24(1-x)\big{(}\beta^{4}x^{3}-\beta^{3}x^{2}+\beta\big{)}+16\beta^{2}(1-x)^{2}\big{(}1-6\beta
x\big{)}\Big{\\}}.\ $ (17)

Substituting (III) into Eqs. (14) and (15), one obtains the coupled equations

$(16\beta^{2}+24\beta+3)-8\beta(12\beta^{2}+7\beta+3)x+2\beta^{2}(24\beta^{2}+84\beta+17)x^{2}-72\beta^{3}(\beta+1)x^{3}+27\beta^{4}x^{4}=0\
$ (18)

and Notedu

$-2(12\beta^{2}+7\beta+3)+\beta(24\beta^{2}+84\beta+17)x-54\beta^{2}(\beta+1)x^{2}+27\beta^{3}x^{3}=0\
$ (19)

for the critical dimensionless variables
$\\{x_{\text{crit}},\beta_{\text{crit}}\\}$.

Interestingly, the coupled polynomial equations (18) and (19), which determine
the onset of the near-horizon spontaneous scalarization phenomenon of the
magnetized Schwarzschild-Melvin black hole (4), can be solved analytically to
yield the remarkably compact critical solution

$\beta_{\text{crit}}={1\over
2}-{{1}\over{\sqrt{6}}}+\sqrt{{{\sqrt{6}-1}\over{2}}}\ $ (20)

with

$x_{\text{crit}}={{690-72\sqrt{6}+4\sqrt{3258\sqrt{6}-7158}}\over{789}}\ .$
(21)

From the analytically derived expression (21) one deduces that the onset of
the spontaneous scalarization phenomenon at the critical magnetic strength
(20) is characterized by the presence of a pair of infinitesimally thin non-
equatorial scalar rings which are supported by the magnetized black hole at
the unique polar angles $\theta^{-}_{\text{ring}}=63.177^{\circ}$ and
$\theta^{+}_{\text{ring}}=116.823^{\circ}$ [see Eqs. (16) and (21)] Notering .

##  IV The classically allowed polar angular region for the spontaneous
scalarization phenomenon of magnetized Schwarzschild-Melvin black holes

In the present section we shall reveal the physically interesting fact that
the negative-coupling near-horizon spontaneous scalarization phenomenon of
Schwarzschild-Melvin black holes is characterized by two critical values of
the dimensionless magnetic strength parameter $BM$. These critical magnetic
strengths mark the boundaries between three qualitatively different spatial
behaviors of the composed magnetized-black-hole-nonminimally-coupled-scalar-
field configurations.

We shall now discuss the three qualitatively different regimes of the magnetic
parameter $BM$:

(1) Case I: In the dimensionless weak magnetic regime

$\beta<\beta^{-}_{\text{crit}}\equiv{1\over
2}-{{1}\over{\sqrt{6}}}+\sqrt{{{\sqrt{6}-1}\over{2}}}\ ,$ (22)

the near-horizon Gauss-Bonnet invariant (II) of the Schwarzschild-Melvin black
hole is positive definite in the entire polar angular range
$\theta\in[0,\pi]$. Thus, Schwarzschild-Melvin black holes in the
dimensionless magnetic regime (22) cannot support negatively coupled massless
scalar fields in their near-horizon region.

(2) Case II: In the intermediate-strength magnetic regime

${1\over
2}-{{1}\over{\sqrt{6}}}+\sqrt{{{\sqrt{6}-1}\over{2}}}\equiv\beta^{-}_{\text{crit}}\leq\beta\leq\beta^{+}_{\text{crit}}\equiv
1\ ,$ (23)

the near-horizon Gauss-Bonnet invariant (II) of the Schwarzschild-Melvin black
hole becomes negative in the polar angular range

$\displaystyle{1\over
3}\Bigg{[}2+{{2-2\sqrt{{{2}\over{3}}}}\over{\beta}}-{{\sqrt{12-{{12-4\sqrt{6}}\over{\beta}}-{{8\sqrt{6}-11}\over{\beta^{2}}}}}\over{\sqrt{3}}}\Bigg{]}\leq\sin^{2}\theta\leq{1\over
3}\Bigg{[}2+{{2-2\sqrt{{{2}\over{3}}}}\over{\beta}}+{{\sqrt{12-{{12-4\sqrt{6}}\over{\beta}}-{{8\sqrt{6}-11}\over{\beta^{2}}}}}\over{\sqrt{3}}}\Bigg{]}\
.$ (24)

The polar range (24) defines, in the dimensionless magnetic regime (23), the
classically allowed angular region for the spontaneous scalarization
phenomenon of negatively-coupled scalar fields in the near-horizon region of
the magnetized Schwarzschild-Melvin black hole.

Interestingly, from the analytically derived relation (24) one learns that the
classically allowed angular width for the near-horizon spontaneous
scalarization phenomenon is infinitesimally thin in the near-critical magnetic
regime $\beta/\beta^{-}_{\text{crit}}\to 1^{+}$. In particular, defining the
near-critical relation

$\beta=\beta^{-}_{\text{crit}}\cdot(1+\epsilon)\ \ \ ,\ \ \ 0\leq\epsilon\ll
1\ ,$ (25)

one finds from (24) the non-trivial (non-linear) critical functional behavior

$\Delta(\sin^{2}\theta)={{\sqrt{{32\alpha\over
3}\Big{(}3\sqrt{2}-2\sqrt{3}+6\alpha\Big{)}}}\over{2+3\sqrt{2}\alpha-\alpha^{2}}}\cdot\sqrt{\epsilon}\
\ \ ,\ \ \ \alpha\equiv\sqrt{\sqrt{6}-1}\ $ (26)

for the infinitesimally thin classically allowed angular interval.

The classically allowed angular region (24) for the near-horizon spontaneous
scalarization phenomenon of the magnetized Schwarzschild-Melvin black holes in
the intermediate magnetic regime (23) is a monotonically increasing function
of the magnetic strength parameter $\beta$. In particular, it is characterized
by the limiting property

${{15-4\sqrt{6}}\over{9}}\leq\sin^{2}\theta\leq 1\ \ \ \ \text{for}\ \ \ \
\beta=\beta^{+}_{\text{crit}}\ $ (27)

for the critical magnetic strength $\beta=\beta^{+}_{\text{crit}}$.

It is interesting to point out that the black-hole configuration with the
critical magnetic strength $\beta^{+}_{\text{crit}}=1$ is unique in the sense
that the classically allowed polar angular region for the negative-coupling
near-horizon spontaneous scalarization phenomenon extends in this case all the
way to the equator of the magnetized black hole.

(3) Case III: In the dimensionless strong magnetic regime

$1\equiv\beta^{+}_{\text{crit}}<\beta\ ,$ (28)

the near-horizon Gauss-Bonnet invariant (II) of the Schwarzschild-Melvin black
holes becomes negative in the polar angular range

$\displaystyle{1\over
3}\Bigg{[}2+{{2-2\sqrt{{{2}\over{3}}}}\over{\beta}}-{{\sqrt{12-{{12-4\sqrt{6}}\over{\beta}}-{{8\sqrt{6}-11}\over{\beta^{2}}}}}\over{\sqrt{3}}}\Bigg{]}\leq\sin^{2}\theta\leq{1\over
3}\Bigg{[}2+{{2+2\sqrt{{{2}\over{3}}}}\over{\beta}}-{{\sqrt{12-{{12+4\sqrt{6}}\over{\beta}}+{{8\sqrt{6}+11}\over{\beta^{2}}}}}\over{\sqrt{3}}}\Bigg{]}\
.$ (29)

The polar range (29) defines the classically allowed angular region for the
negative-coupling near-horizon spontaneous scalarization phenomenon of
strongly magnetized [see Eq. (28)] Schwarzschild-Melvin black holes.

Interestingly, one finds that, in the strong magnetic regime (28), the two
polar boundaries of the classically allowed region (29) are monotonically
decreasing functions of the dimensionless magnetic parameter $\beta$. In
particular, one finds that the classically allowed angular interval (29)
gradually shrinks to the infinitesimally thin polar region

${{1-\sqrt{{{2}\over{3}}}}\over{\beta}}\leq\sin^{2}\theta\leq{{1+\sqrt{{{2}\over{3}}}}\over{\beta}}\
\ \ \ \ \text{for}\ \ \ \ \ \beta\to\infty\ $ (30)

in the asymptotically large magnetic regime. One therefore concludes that the
classically allowed polar region for the negative-coupling near-horizon
spontaneous scalarization phenomenon of the magnetized Schwarzschild-Melvin
spacetime is restricted to the vicinity of the black-hole poles in the
asymptotic large-strength magnetic regime $\beta\gg 1$.

##  V Summary and discussion

It has recently been shown Hernw  that magnetized black holes may support non-
minimally coupled scalar hairy configurations. In particular, the recently
published important work Hernw  has revealed the physically interesting fact
that, in composed Einstein-Maxwell-scalar-Gauss-Bonnet field theories with
negative values of the non-minimal coupling parameter $\eta$, there exists a
critical magnetic strength [see Eq. (1)],

$(BM)_{\text{crit}}\simeq 0.971\ ,$ (31)

above which Schwarzschild-Melvin black holes can support near-horizon massless
scalar field configurations with a non-minimal direct coupling to the
magnetically-dependent Gauss-Bonnet curvature invariant (II).

In the present paper we have studied, using analytical techniques, the onset
of the negative-coupling near-horizon spontaneous scalarization phenomenon in
magnetized Schwarzschild-Melvin black-hole spacetimes. The main results
derived in this paper and their physical implications are as follows:

(1) We have proved that the critical black-hole magnetic strength
$({BM})_{\text{crit}}$, which marks the boundary between bald Schwarzschild-
Melvin black holes and hairy (scalarized) black holes in the Einstein-Maxwell-
scalar-Gauss-Bonnet field theory (2) with a negative non-minimal Gauss-Bonnet-
scalar-field coupling, is given by the compact dimensionless analytical
expression [see Eqs. (16) and (20)]

$(BM)_{\text{crit}}=\sqrt{{1\over
2}-{{1}\over{\sqrt{6}}}+\sqrt{{{\sqrt{6}-1}\over{2}}}}\ .$ (32)

It is worth emphasizing the fact that the analytically derived critical black-
hole magnetic strength (32) agrees remarkably well with the corresponding
numerical value (31) of the critical magnetic field as originally presented in
the physically interesting work Hernw .

(2) We have revealed the fact that non-equatorial thin scalar rings can be
supported in magnetized black-hole spacetimes. In particular, one finds that,
in the dimensionless critical limit

$BM\to[(BM)_{\text{crit}}]^{+}\ ,$ (33)

the effective mass term (10) of the composed Schwarzschild-Melvin-black-hole-
nonminimally-coupled-massless-scalar-field system becomes negative in a pair
of narrow non-equatorial polar rings which are characterized by the
analytically derived angular relation [see Eqs. (16), (21), (25) and (26)]

$\sin^{2}\theta_{\text{ring}}={{18+8(3\sqrt{2}-2\sqrt{3}){\alpha}}\over{\big{(}2+3\sqrt{2}\alpha-\alpha^{2}\big{)}^{2}}}\
\ \ ,\ \ \ \alpha\equiv\sqrt{\sqrt{6}-1}\ .$ (34)

This physically interesting property of the Schwarzschild-Melvin curved
spacetime implies that magnetized black holes can support, in the critical
limit (33), cloudy configurations of the non-minimally coupled massless scalar
fields in the form of two infinitesimally thin non-equatorial scalar rings
which are characterized by the polar angles [see Eq. (34)]

$\theta^{-}_{\text{ring}}=63.177^{\circ}\ \ \ ;\ \ \
\theta^{+}_{\text{ring}}=116.823^{\circ}\ .$ (35)

(3) It has been revealed that the Schwarzschild-Melvin black hole with the
critical magnetic strength $BM=1$ is unique in the sense that the near-horizon
angular region for which the Gauss-Bonnet invariant becomes non-positive (thus
allowing for the onset of the negative-coupling spontaneous scalarization
phenomenon) extends in this case all the way to the equator of the magnetized
black hole [see Eq. (24)].

(4) We have revealed the fact that, for strong magnetic fields in the $BM\gg
1$ regime, the classically allowed angular region for the negative-coupling
spontaneous scalarization phenomenon of magnetized Schwarzschild-Melvin
spacetimes gradually shrinks as the value of the dimensionless magnetic
strength $BM$ increases. In particular, we have proved that the near-horizon
spontaneous scalarization phenomenon is restricted to the narrow near-polar
angular interval [see Eqs. (16) and (30)] Notesymm

${{\sqrt{1-\sqrt{{{2}\over{3}}}}}\over{BM}}\leq\theta_{\text{scalar}}\leq{\sqrt{{1+\sqrt{{{2}\over{3}}}}}\over{BM}}\
\ \ \ \ \text{for}\ \ \ \ \ BM\gg 1\ $ (36)

in the asymptotic large-strength magnetic regime $BM\gg 1$.

ACKNOWLEDGMENTS

This research is supported by the Carmel Science Foundation. I would like to
thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for helpful
discussions.

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