# CAM-Based Methods Can See through Walls

Magamed Taimeskhanov<EMAIL_ADDRESS>
Université Côte d’Azur
Laboratoire J.A. Dieudonnée, CNRS
Nice, France Ronan Sicre<EMAIL_ADDRESS>
Centrale Méditerranée
Aix Marseille Univ., CNRS, LIS
Marseille, France Damien Garreau<EMAIL_ADDRESS>
Julius-Maximilians University
Institute of Computer Science
Würzburg, Germany Corresponding author.

###### Abstract

CAM-based methods are widely-used post-hoc interpretability method that
produce a saliency map to explain the decision of an image classification
model. The saliency map highlights the important areas of the image relevant
to the prediction. In this paper, we show that most of these methods can
incorrectly attribute an important score to parts of the image that the model
cannot see. We show that this phenomenon occurs both theoretically and
experimentally. On the theory side, we analyze the behavior of GradCAM on a
simple masked CNN model at initialization. Experimentally, we train a VGG-like
model constrained to not use the lower part of the image and nevertheless
observe positive scores in the unseen part of the image. This behavior is
evaluated quantitatively on two new datasets. We believe that this is
problematic, potentially leading to mis-interpretation of the model’s
behavior.

Keywords: Interpretability, Computer Vision, Convolutional Neural Networks,
Class Activation Maps.

##  1 Introduction

The recent advances of machine learning pervade all applications, including
the most critical. However, deep learning models intrinsically possess many
parameters, have complicated architectures, and rely on many non-linear
operations, preventing the users to get a good grasp of the rationale behind
particular decisions. These models are often called “black boxes” for these
reasons (Benítez et al., 1997). In this respect, there is a growing need for
interpretability of the models that are used, which gave birth to the field of
eXplainable AI (XAI). When the model to explain is already trained, our main
topic of interest, this is often called post-hoc interpretability (Lipton,
2018; Zhang et al., 2021; Linardatos et al., 2021).

In the specific case of image classification, the explanations provided to the
user often take the form of a saliency map superimposed to the original image,
for instance simply looking at the gradient with respect to the input of the
network (Simonyan et al., 2013). The message is simple: the areas highlighted
by the saliency maps are used by the network for the prediction. When the
first layers of the network are convolutional layers (Fukushima, 1980), one
can take advantage of this and look at the activations of the filters
corresponding to the class prediction that we are trying to explain. Indeed,
these first layers act like a bank of filters on the input image, and the
degree to which they are activated gives us information on the behavior of the
network. Thus the first layers possess a certain degree of interpretability,
even though it can be challenging to aggregate the information coming from
different filters. In any case, the next layers generally consist in a fully-
connected neural network, thus suffering from the same caveats as other
models. In addition, this second part of the network is equally important for
the prediction, but is not taken into account in the explanations we provide
if we simply look at activation values.

To solve this problem, a natural idea is to weight each activation map
depending on how the second part of the network uses it. In the case of a
single additional layer, this is called _class activation maps_ (CAM, Zhou et
al., 2016). The methodology was quickly generalized by Selvaraju et al.
(2017), using the _average gradient_ values of the subsequent layers instead,
giving rise to GradCAM, arguably one of the most popular posthoc
interpretability method for CNNs. Many extensions are proposed in the
following years, we list them in Appendix A and refer to Zhang et al. (2023)
for a recent survey. Without being too technical, for all these methods, the
explanations provided consist in a weighted average of the activation maps.

A close inspection of each of these methods reveals that the coefficient
associated to each individual map is global, in the sense that the same
coefficient is applied to the whole map. The main message of this paper is
that this can be problematic, since different parts of the activation map may
be used differently by the subsequent layers. Worse, some parts may even be
unused by the subsequent network and still highlighted in the final
explanation (see Figure 1). Thus we believe that, while giving apparently
more-than-satisfying results in practice, CAM-based methods should be used
with caution, keeping in mind that some parts of the image may be highlighted
whereas they are not even seen by the network.

heightadjust=all, valign=c []

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Figure 1: Example of GradCAM failure on a VGG-like model trained on the
ImageNet dataset (masked $[\mathbf{VGG}]$, see Figure 4). _Left:_ original
image; _Middle:_ GradCAM explanation before up-sampling; _Right:_ original
image with GradCAM explanation overlayed as a heatmap. The network does not
have access to the red part of the image, but GradCAM does highlight some
pixels in this area.

###  1.1 Related work

This paper is inspired by a line of recent works concerned with the
reliability of saliency maps claiming that solely relying on the visual
explanation provided by a saliency map can be misleading (Kindermans et al.,
2019; Ghorbani et al., 2019). Ghorbani et al. (2019) introduce a method for
altering the input data with imperceptible perturbations which do not change
the predicted label, yet generating different saliency maps. On the other
hand, Kindermans et al. (2019) show that numerous saliency methods generate
incorrect scores for the input features when the model prediction is invariant
to translation of the input data by a constant. It is important to note that
neither of these studies specifically challenges the reliability of CAM-based
methods.

This perspective on saliency maps is supported by the work of Adebayo et al.
(2018), which introduces a randomization-based sanity check indicating that
some existing saliency methods are independent of both the model and the data.
We note that GradCAM passes the sanity checks proposed by Adebayo et al.
(2018). Draelos and Carin (2021), proposing HiResCAM, are less positive
regarding GradCAM pointing out, as we do, that the use of a global coefficient
can produce positive explanations where there should not be. Compared to our
work, they provide few theoretical explanations and perform experiments on
model which are not using parts of the input image.

Taking another angle, Heo et al. (2019) directly attacks the reliability of
GradCAM saliency maps by adversarial model manipulation, _i.e._ , fine-tuning
a model with the purpose of making GradCAM saliency maps unreliable. This is
achieved by using a specific loss function tailored to this effect. Our
approach is different, as we simply force a strong form of sparsity in the
model’s parameters, not targeting a specific interpretability method.

###  1.2 Organization of the paper

We start by looking at GradCAM in Section 2. For a given simple CNN
architecture described in Section 2.1, we derive closed-form expressions for
its explanations in Section 2.2. Leveraging these expressions, we prove in
Section 2.3 that GradCAM explanations are positive at initialization, even
though a large part of the weights are set to zero.

In Section 3, we demonstrate experimentally that this phenomenon remains true
after training. To this extent, we proceed in two steps. First, we train to a
reasonable accuracy a VGG-like model on ImageNet (Deng et al., 2009) which
does not see the lower part of input images, described in Section 3.1. Then,
we create two datasets (Section 3.2) consisting in superposition of images of
the same class. We show experimentally in Section 3.3 that CAM-based methods
applied to this model wrongly highlights a large portion of the lower part of
the images, misleading the user by showing that the lower part is used for the
prediction whereas, by construction it is not. The code for all experiments
and the datasets are available online.111https://github.com/MagamedT/cam-can-
see-through-walls We conclude in Section 4.

Conv $\mathcal{C}$ $\mathbf{A}\in$
---
$\mathbb{R}^{V\times h\times w}$
ReLU $\mathbf{B}\in$
---
$\mathbb{R}^{V\times h\times w}$
max pool $\mathcal{M}$ $\mathbf{C}\in$
---
$\mathbb{R}^{V\times h^{\prime}\times w^{\prime}}$
flatten $\mathbf{C}^{\prime}\in$
---
$\mathbb{R}^{Vh^{\prime}w^{\prime}}$
$\operatorname{\mathcal{F}}$$y_{c}\in\mathbb{R}$ $\bm{\xi}\in$
---
$[0,1]^{H\times W}$
Input$g\colon\mathbb{R}^{H\times W}\to\mathbb{R}^{V\times h\times
w}$$f\colon\mathbb{R}^{V\times h\times w}\to\mathbb{R}$

Figure 2: The model used for the derivation of feature importance scores,
$\operatorname{[\mathbf{CNN}]}$. The number of filters in the convolutional
layer $\mathcal{C}$ is $V\in\mathbb{N}^{\star}$. The size of the max pooling
filters $k^{\prime}\in\mathbb{N}^{\star}$ is implicitly defined such that
$(h^{\prime},w^{\prime})=\frac{1}{k^{\prime}}(h,w)$ in $\mathbb{N}^{\star}$.
The fully-connected neural network
$\operatorname{\mathcal{F}}\left(\cdot\right)$ takes $\mathbf{C}^{\prime}$ as
input and processes it through $L$ layers with ReLU activation functions to
produce a raw score $y_{c}$, without converting this score into a
“probability.”

##  2 Mathematical description

The model used for the theoretical analysis done in Section 2.3 is described
in Section 2.1, the derivation of GradCAM coefficients in Section 2.2.

###  2.1 A simple CNN

Let us describe mathematically the model we consider, denoted by
$\operatorname{[\mathbf{CNN}]}$ and depicted in Figure 2. On a high-level,
$\operatorname{[\mathbf{CNN}]}$ is a $(L+1)$-layers network, consisting in a
single convolution / max pooling layer, followed by a $L$-layers fully-
connected neural network with ReLU activations. Thus the case $L=1$
corresponds to a single convolutional / max pooling layer followed by a linear
transformation.

More precisely, we consider a grayscale image $\bm{\xi}\in[0,1]^{H\times W}$
as input. For instance, if we consider the MNIST dataset (LeCun et al., 1998),
$(H,W)=(28,28)$. We note that our analysis can be easily extended to RBG
images. The convolutional layer $\operatorname{\mathcal{C}}\left(\cdot\right)$
consists of $V$ filters $\mathbf{F}=(\mathbf{F}_{1},\ldots,\mathbf{F}_{V})$,
represented as a collection of $V$ matrices of shape $k\times k$.

Formally, the output of the convolution step,
$\mathbf{A}\vcentcolon=\operatorname{\mathcal{C}}\left(\bm{\xi}\right)\in\mathbb{R}^{V\times
h\times w}$, is given by:

$\forall
v\in[V],\,\forall(i,j)\in[h]\times[w],\qquad\mathbf{A}_{i,j}^{(v)}=\sum_{p,q=1}^{k}\bm{\xi}_{i+p-1,j+q-1}\mathbf{F}_{p,q}^{(v)}\,,$
(1)

where $(h,w)=(H-k+1,W-k+1)$.

In practice, the filter weights are initialized randomly, typically i.i.d.
uniform or Gaussian with proper scaling. There are two main trends on how to
scale the variance, either Glorot (Glorot and Bengio, 2010) (also called
Xavier), or He (He et al., 2015). The later with uniform distribution is
default for the CNN layer used in PyTorch. However, we assume from now on
i.i.d. Gaussian $\operatorname{\mathcal{N}}\left(0,\tau^{2}\right)$
initialization in our analysis for mathematical convenience.

After the convolution step, we apply a ReLU non-linearity, denoted by
$\operatorname{\sigma}\vcentcolon=\max(0,\cdot)$. We define the _rectified
activation maps_
$\mathbf{B}\vcentcolon=\operatorname{\sigma}\left(\mathbf{A}\right)\in\mathbb{R}^{V\times
h\times w}$, where $\operatorname{\sigma}$ is applied coordinate-wise. Next,
we consider a down-sampling layer, here a ($k^{\prime}\times k^{\prime}$) max
pooling $\operatorname{\mathcal{M}}\left(\cdot\right)$. One can see that the
output of the max pooling,
$\mathbf{C}\vcentcolon=\operatorname{\mathcal{M}}\left(\mathbf{B}\right)\in\mathbb{R}^{V\times
h^{\prime}\times w^{\prime}}$, is given by:

$\forall
v\in[V],\,\forall(i^{\prime},j^{\prime})\in[h^{\prime}]\times[w^{\prime}],\,\mathbf{C}_{i^{\prime},j^{\prime}}^{(v)}=\max\left(\mathbf{B}_{k^{\prime}(i^{\prime}-1)+1:k^{\prime}i^{\prime},k^{\prime}(j^{\prime}-1)+1:k^{\prime}j^{\prime}}^{(v)}\right)\,,$
(2)

where $(h^{\prime},w^{\prime})=\frac{1}{k^{\prime}}(h,w)$. Note that we assume
$k^{\prime}$ to divide $h$ and $w$ for simplicity.

Finally, let us describe recursively the fully-connected part of
$\operatorname{[\mathbf{CNN}]}$, denoted by
$\operatorname{\mathcal{F}}\left(\cdot\right)$:

$\begin{aligned}
\operatorname{\mathcal{F}}\colon&\mathbb{R}^{Vh^{\prime}w^{\prime}}\to\mathbb{R}\\\
&\mathbf{C}^{\prime}\xmapsto{}h_{L}\left(\mathbf{C}^{\prime}\right)\end{aligned}\quad\text{with}\quad\begin{cases}h_{0}\left(\mathbf{x}\right)=\mathbf{x},\\\
h_{\ell}\left(\mathbf{x}\right)=\mathbf{W}^{(\ell)}\operatorname{\sigma}\left(h_{\ell-1}\left(\mathbf{x}\right)\right)&\text{for
}\ell\in[L],\end{cases}$ (3)

where $\mathbf{W}^{(\ell)}\in\mathbb{R}^{d_{\ell}\times d_{\ell-1}}$ is a
weight matrix connecting layer $(\ell-1)$ and $\ell$ with $\ell\in[L]$ and
$d_{\ell}$ the size of layer $\ell$. Note that we set
$d_{0}=Vh^{\prime}w^{\prime}$, and $d_{L}=1$, since we see the output of our
model as the un-normalized logit associated to a given class of a prediction
problem. We also, denote by
$\mathbf{a}^{(\ell)}\vcentcolon=h_{\ell}\left(\mathbf{C}^{\prime}\right)$ the
non-rectified activation of layer $\ell$ and
$\mathbf{r}^{(\ell)}\vcentcolon=\operatorname{\sigma}\left(\mathbf{a}^{(\ell)}\right)$
its rectified counterpart.

#### Summary.

The model we consider can be described concisely as
$\operatorname{\mathcal{F}}\left(\operatorname{\mathcal{M}}\left(\operatorname{\sigma}\left(\operatorname{\mathcal{C}}\left(\cdot\right)\right)\right)\right)$.
As explained in introduction, given the nature of CAM-based explanations, it
is convenient to split $\operatorname{[\mathbf{CNN}]}$ in two functions $f$
and $g$ for the computations of the next Section 2.2. More precisely, we write

$\begin{aligned} \operatorname{[\mathbf{CNN}]}\colon\,&[0,1]^{H\times
W}\to\mathbb{R}\\\ &\bm{\xi}\xmapsto{}f\circ
g(\bm{\xi})\end{aligned}\quad\text{with}\quad\begin{cases}g(\bm{\xi})\vcentcolon=\operatorname{\sigma}\left(\operatorname{\mathcal{C}}\left(\bm{\xi}\right)\right)\\\
f(\mathbf{C})\vcentcolon=\operatorname{\mathcal{F}}\left(\operatorname{\mathcal{M}}\left(\mathbf{C}\right)\right)\,.\end{cases}$
(4)

Recall that we refer to Figure 2 for an illustration.

###  2.2 Closed-form expression

The original idea of CAM (Zhou et al., 2016) was limited to computing the
saliency map as a linear combination of the feature maps in the last
convolutional layer when $L=1$. Later, GradCAM (Selvaraju et al., 2017)
removed the architecture constraints by computing the average gradient of each
feature map with respect to $\mathbf{B}$. In our notation, we have:

######  Definition 1 (GradCAM)

For an input $\bm{\xi}$ and model $\operatorname{[\mathbf{CNN}]}$, the GradCAM
feature scores are given by

$\displaystyle[\mathbf{GC}]\vcentcolon=\operatorname{\sigma}\left(\sum_{v=1}^{V}\alpha_{v}\mathbf{B}^{(v)}\right)\in\mathbb{R}_{+}^{h\times
w}\,,$

where each
$\alpha_{v}\vcentcolon=\operatorname{GAP}\left(\nabla_{\mathbf{B}^{(v)}}f(\mathbf{B})\right)\in\mathbb{R}$.
Here, GAP denotes the _global average pooling_ , that is, the average of all
values, and $\operatorname{\sigma}$ the ReLU as before.

Definition 1 is of course to be taken coordinate-wise. We note that, in
practice, $[\mathbf{GC}]$ is up-sampled and normalized to produce a saliency
map with the _same shape_ as the input image. To be more precise, what we
define as $[\mathbf{GC}]$ is the middle panel of Figure 1, whereas the final
user will nearly always visualize the right panel. The most important thing to
notice in Definition 1 is that $\alpha$ is a global coefficient.

We now show why this can be an issue. Looking at Definition 1, whenever the
underlying model is not too complicated, one can actually hope to derive a
closed-form expression for the feature importance scores of $[\mathbf{GC}]$ as
a function of the model’s parameters. This is achieved by:

######  Proposition 2 ($\alpha$ coefficients for GradCAM, $V=1$)

Recall that the $\mathbf{a}$ vectors denote the non-rectified activation and
$\mathbf{W}$ the weights of the linear part of
$\operatorname{[\mathbf{CNN}]}$. Then, for input $\bm{\xi}$, the
$[\mathbf{GC}]$ coefficient $\alpha$ is given by

$\displaystyle\alpha=\frac{1}{hw}\sum_{\begin{subarray}{c}i,j=1\end{subarray}}^{h^{\prime},w^{\prime}}\sum_{i_{1},\ldots,i_{L-1}=1}^{d_{1},\ldots,d_{L-1}}\mathds{1}_{\mathbf{a}_{i_{1}}^{(1)},\ldots,\mathbf{a}_{i_{L-1}}^{(L-1)}>0}\prod_{p=1}^{L}(\mathbf{W}_{i_{p},i_{p-1}}^{(p)})^{\top}\,,$

where we set $i_{0}\vcentcolon=(i,j)$ and $i_{L}=1$.

From Proposition 2, we immediately deduce a closed-form expression for GradCAM
explanations. We note that Proposition 2 can be readily extended to an
arbitrary number of filters $V>1$, in which case the $\mathbf{a}$ and
$\mathbf{W}$ should be interpreted as corresponding to the relevant $v\in[V]$.

The proof of Proposition 2 can be found in Appendix C. In Appendix A, we
describe mathematically several other CAM-based methods in the setting of
$\operatorname{[\mathbf{CNN}]}$: XGradCAM (Fu et al., 2020), GradCAM++
(Chattopadhay et al., 2018), HiResCAM (Draelos and Carin, 2021), ScoreCAM
(Wang et al., 2020) and AblationCAM (Desai and Ramaswamy, 2020). A close
inspection of these definitions reveals that they also use global weighting
coefficients applied to the corresponding activation maps, with the notable
exception of HiResCAM.

Figure 3:  Illustration of Theorem 3 on an MNIST (LeCun et al., 1998) digit
(_left panel_). We set to zero the lower part of $\mathbf{W}$ for
$[\mathbf{CNN}]$, initialize the filter values and remaining weights to i.i.d.
$\operatorname{\mathcal{N}}\left(0,1\right)$, and run GradCAM to get a
saliency map (_right panel_). Even though our network does not see the red
part of the image, GradCAM does highlight some pixels in this area, as
predicted by Theorem 3.

###  2.3 Theoretical analysis

Leveraging the results of Section 2.2, we are able to describe precisely the
behavior of GradCAM at initialization for $\operatorname{[\mathbf{CNN}]}$,
specifically when the classifier part of our model comprises a single layer
($L=1$). This analysis is justified by existing works (Lee et al., 2019; Du et
al., 2019; Allen-Zhu et al., 2019b, a; Zou et al., 2020) showing that neural
networks often stay “near initialization” during training. As promised, we
conduct this analysis when the network does not have access to the lower part
of the image. Our main result is:

######  Theorem 3 (Expected GradCAM scores, $L=1$, masked $[\mathbf{CNN}]$)

Let $\bm{\xi}\in[0,1]^{H\times W}$ be an input image. Let
$\mathbf{m}\vcentcolon=\bm{\xi}_{i:i+k-1,j:j+k-1}$ be the patch of $\bm{\xi}$
corresponding to index $(i,j)\in[h]\times[w]$. Assume that $h^{\prime}$ is
even, and $\mathbf{W}_{:,-\frac{h^{\prime}}{2}:,:}=0$. Assume that the filter
values and the non-zero weights are initialized i.i.d.
$\operatorname{\mathcal{N}}\left(0,\tau^{2}\right)$. Then, if the number of
filters $V$ is greater than $20$, we have the following expected lower bound
on the GradCAM explanation for pixel $(i,j)$:

$\operatorname{\mathbb{E}}\left[[\mathbf{GC}]_{i,j}\right]=\operatorname{\mathbb{E}}\left[\operatorname{\sigma}\left(\sum_{v=1}^{V}\alpha_{v}\mathbf{B}_{i,j}^{(v)}\right)\right]\geq\frac{V-20}{\sqrt{V}}\sqrt{\frac{h^{\prime}w^{\prime}}{16\pi}}\frac{\tau^{2}}{hw}\left\lVert\mathbf{m}\right\rVert_{2}\,,$
(5)

where the expectation in the previous inequality is taken with respect to
initialization of the filters and the remaining weights of the linear layer.

Setting $\mathbf{W}_{:,-\frac{h^{\prime}}{2}:,:}$ to $0$ disables the weights
within $\mathbf{W}$ that are linked to the lower half part of the activation
map $\mathbf{C}_{:,-\frac{h^{\prime}}{2}:,:}$, effectively preventing
$\operatorname{[\mathbf{CNN}]}$ from accessing the lower half of $\mathbf{C}$.
As a consequence, $\operatorname{[\mathbf{CNN}]}$ does not see the lower half
of $\xi$, up to side effects. The main consequence of Theorem 3 is that, when
the number of filters associated to the class to explain is large enough,
$[\mathbf{GC}]_{i,j}$ is positive in expectation if some pixels are activated
in the receptive field associated to $(i,j)$. Thus GradCAM highlights all
parts of the image where there is some “activity,” even though this
information is not used by the network in the end. We illustrate Theorem 3 in
Figure 3. The main limitation of this analysis is its focus on the behavior at
initialization: we investigate in the following whether this behavior also
happens after training. Another limitation is the restriction to a single
linear layer, but we note that taking $L=1$ in the fully connected part of
$\operatorname{[\mathbf{CNN}]}$ is a dominant approach since ResNet (He et
al., 2016).

The proof of Theorem 3 can be found in Appendix D. The key ingredient of the
proof is obtaining a probabilistic control of
$\sum_{q}\alpha_{q}\mathbf{B}^{(q)}$. We note that a similar analysis is
possible for other expressions of $\alpha$, thus other CAM-based methods.

##  3 Experiments

We know ask the following question: are the consequences of Theorem 3 true
after training, and for a more realistic model? To this extent, we train a
CNN-based model which by construction cannot access some specified part of the
input which we call the _dead zone_ (see Figure 4, details in Section 3.1).
Clearly, since the dead zone does not influence the output, it should not
contain positive model explanations. To test whether this is true, we create
two datasets (Section 3.2). Each item of the first one is composed of two
images from ImageNet with the same label in both the seen and the unseen part
of the image. The second dataset is built using generative models on the same
categories with two objects in each image located in the seen and unseen part
as well. We then check whether CAM-based methods wrongly highlight areas in
the dead zone in Section 3.3.

$[0,1]^{3\times 224\times 224}$ $\bm{\xi}$ $\times 2\rightarrow\times
3$$\times 2\rightarrow\times 3$white$\mathbf{W}_{1}$ conv $+$ ReLU
---
$3\times 3$
max-pool
---
$2\times 2$
$\mathbb{R}^{4096\times(256\times 14\times 14)}$$\mathbf{W}$
$\mathbb{R}^{4096}$$\mathbb{R}^{1000}$ $\mathbf{y}$

Figure 4: Our trained masked VGG16-based model on ImageNet with $87.0\%$ top-5
accuracy. The down weights $\mathbf{W}_{:,\,:,\,-9:,\,:}$ are set to $0$ and
not updated during training. Only the up weights $\mathbf{W}_{:,\,:,\,:5,\,:}$
and the other parameters undergo training. This setting implies that every red
part in the channels does not impact the prediction scores, meaning that they
are not used. Symbol $\times 2\rightarrow\times 3$ means the model first uses
the green block twice, with each time having $2$ consecutive convolutions.
Then, it uses the green block three times, with each time having $3$
consecutive convolutions. There is no max pooling after the last convolution.

###  3.1 Model

Figure 5: Plots of the training loss and the top-5 validation accuracy of our
unmasked and masked VGG16-like models on Imagenet-1k (2012) (Deng et al.,
2009). The unmasked $[\mathbf{VGG}]$ (baseline) and our masked
$[\mathbf{VGG}]$ yield $87.0\%$, resp. $90.4\%$ top-5 accuracy and $66.5\%$
resp. $71.5\%$ top-1 accuracy on the validation set.

#### Model definition.

The CNN used in our experiments is a modification of a classical VGG16
architecture (Simonyan and Zisserman, 2015) which we call $[\mathbf{VGG}]$.
Whereas the original VGG16 model is composed of $5$ convolutional blocks
including either $2$ or $3$ convolutional layers with ReLU and max pooling,
followed by $3$ dense layers. In $[\mathbf{VGG}]$ we remove the last max
pooling (in the fifth convolutional block) and we further apply a mask on
selected neurons of the first dense layer so the layer can not see the lower
part of the activation maps, see Figure 4 for more details.

#### Masking.

We forbid the network from seeing the dead zone in a very simple way: in the
first dense layer $\mathbf{W}$, which has size $4096\times(256\times 14\times
14)$, we permanently set to $0$ a band of height $9$ corresponding to the
lower weights. Formally, this means setting $\mathbf{W}_{:,\,:,\,-9:,\,:}=0$,
which is denoted in red above $\mathbf{W}$ in Figure 4. Effectively, we are
building a wall that stops all information flowing from the last convolutional
layer to the remainder of the network. Since the weights $\mathbf{W}$ are
directly connected to the final activation map
$\mathbf{B}\in\mathbb{R}_{+}^{256\times 14\times 14}$, this masking
effectively zeroes out the lower sections in each channel denoted by
$\mathbf{B}_{:,\,-9:,\,:}$. We can trace back the zeroed activations in
$\mathbf{B}$ to the preceding activation map $\mathbf{C}$, pinpointing the
exact patches in $\mathbf{C}$ that correspond (after convolution) to the
features observed in the zeroed activation of $\mathbf{B}$. Because of the
side effects in the computation of convolutions, this area of $\mathbf{C}$ is
slightly smaller: some pixel activation will still play a role in the model’s
prediction. Repeating this process until we reach the original image yields a
dead zone of height $54$ pixels, highlighted in red above $\bm{\xi}$ in Figure
4, which covers $24\%$ of the image area. As we mentioned earlier, the other
main difference with VGG16 is the removing of the final max pooling layer.
This leads to a larger activation layer, allowing us to set weights to zero
without hindering too much the network’s ability, see Figure 5. We note that
$[\mathbf{VGG}]$ bears a strong resemblance to
$\operatorname{[\mathbf{CNN}]}$. The main difference is that the convolutional
layer of $\operatorname{[\mathbf{CNN}]}$ is replaced by several convolutional
blocks in $[\mathbf{VGG}]$, see Figure 4.

#### Training.

We train $[\mathbf{VGG}]$ on Imagenet-1k (Deng et al., 2009) using classical
data augmentation recipe, _i.e._ , random flip and random crop. As
optimization algorithm, we use stochastic gradient descent with momentum,
weight decay, and a learning rate scheduler. To observe the slight accuracy
drop induced by masking a significant part of images, we train a baseline
model without masking. We report the train loss and the validation accuracy
across training in Figure 5.

#### Comparison to SOTA.

We also compare the validation top-1 and top-5 accuracy of the VGG16 model
found in the PyTorch repository. Our $[\mathbf{VGG}]$ without max pooling and
no masking offers the same performance: $71.5\%$ top-1 and $90.4\%$ top-5
accuracy on the validation set. As we mention in Figure 5, our model
$[\mathbf{VGG}]$ with masking has lower performance, which is expected as a
fourth of the input image, $\bm{\xi}_{:,\,171:224,\,:}$, is unseen by the
model. We obtain $66.5\%$, resp. $71.5\%$, top-1 and $87.0\%$, resp. $90.4\%$,
top-5 accuracy on the validation set for our masked $[\mathbf{VGG}]$, resp.
unmasked $[\mathbf{VGG}]$. Nevertheless, we see that $[\mathbf{VGG}]$ is a
realistic network able to predict ImageNet classes with reasonable accuracy.
We believe that the drop in accuracy is only minor because ImageNet images are
centered, and there is enough information in the upper part of the image to
achieve near-perfect prediction.

###  3.2 Proposed datasets

STACK-MIX |  |  |  |
---|---|---|---|---
STACK-GEN |  |  |  |
Figure 6: Sampled images from both of our datasets, _i.e._ , STACK-MIX and
STACK-GEN.

#### Objective.

To assess how much CAM-based saliency maps emphasize irrelevant areas of an
image, we introduce two new datasets in which we control the positions of the
image elements using two techniques: cutmix (Yun et al., 2019) and generative
model. More precisely, we produce two datasets, called STACK-MIX and STACK-
GEN. Where each image contains two objects, one in the bottom part of the
image which is the dead zone for $[\mathbf{VGG}]$, and the second subject at
the top of the image. Therefore, the subject at the center of the image will
be mainly responsible for the top-1 predicted score by our masked
$[\mathbf{VGG}]$.

#### STACK-MIX.

We first generate labels for our datasets by ramdomly sampling $100$ classes
from the $398$ first labels of Imagenet, which corresponds to animals.

The first dataset, called STACK-MIX, consists of $100$ images featuring one
image from each of the $100$ classes. Each example $\bm{\xi}$ is created by
mixing, in a cutmix (Yun et al., 2019) fashion, two images
$(\bm{\xi}_{1},\bm{\xi}_{2})$ with same label and sampled randomly in the
validation set of Imagenet as follows:

$\bm{\xi}\vcentcolon=\begin{pmatrix}(\bm{\xi}_{1})_{:,\,:170,\,:}\\\
(\bm{\xi}_{2})_{:,\,171:224,\,:}\end{pmatrix}\in[0,1]^{3\times 224\times
224}\,,$ (6)

meaning that we create a composite image $\bm{\xi}$ by superposing an upper
vertical slice, taken from the top region of $\bm{\xi}_{1}$ with size $3\times
170\times 224$, with a lower vertical slice, taken from the bottom region of
$\bm{\xi}_{2}$ with size $3\times 54\times 224$. Finally, the quality of the
generated images is verified through manual inspection. This dataset lacks
realism due to the distinct separation between the two subjects. We address
this issue with the help of generative models.

#### STACK-GEN.

The second dataset, called STACK-GEN, consists of $100$ images featuring one
image from each of the same $100$ classes. It was generated using ChatGPT +
DALL·E 3 (Brown et al., 2020; Ramesh et al., 2021) by sampling prompts of the
following form: “A photo of {animal name} stacked on top of {same animal
name}.” The word “stacked” determines the positions of the subjects in the
generated image, which proceeds as follows: first, ChatGPT refines the
original prompt to enhance its suitability for DALL·E 3, then the image is
generated. We then preprocess the generated images by selectively editing them
to minimize the background and centering the focus on the two animals. This
editing involves cropping the images to a 1:1 ratio, ensuring one animal is
predominantly within the dead zone as defined by our $[\mathbf{VGG}]$, while
the other is positioned in the upper part of the new image. Figure 6 shows
examples of the created images. Recall that both datasets are available on the
github repository of the project.

Input image |  |  |  |
---|---|---|---|---
GradCAM |  |  |  |
GradCAM++ |  |  |  |
XGradCAM |  |  |  |
ScoreCAM |  |  |  |
AblationCAM |  |  |  |
EigenCAM |  |  |  |
HiResCAM |  |  |  |
Figure 7: Saliency maps given by the considered CAM-based methods for
$[\mathbf{VGG}]$. With the notable exception of HiResCAM, all method highlight
parts of images from STACK-GEN and STACK-MIX which are unseen by the network.
The lower part in red is unseen by the model.

###  3.3 Results

Table 1: Table showing the activation in the unseen part of the image measured by $\mu(\cdot)\times 100$ for several CAM-based methods on both datasets. methods |  STACK-MIX $\downarrow$ |  STACK-GEN $\downarrow$
---|---|---
GradCAM (Selvaraju et al., 2017) |  $22.7\pm 13.4$ | $21.6\pm 11.6$
GradCAM++ (Chattopadhay et al., 2018) |  $28.8\pm 8.1$ | $28.5\pm 7.9$
XGradCAM (Fu et al., 2020) |  $23.8\pm 9.0$ | $22.8\pm 9.0$
ScoreCAM (Wang et al., 2020) |  $19.9\pm 10.3$ | $18.5\pm 10.6$
AblationCAM (Desai and Ramaswamy, 2020) |  $21.0\pm 9.9$ | $20.8\pm 9.6$
EigenCAM (Bany Muhammad and Yeasin, 2021) |  $51.7\pm 19.7$ | $55.8\pm 21.6$
HiResCAM (Draelos and Carin, 2021) |  $0.0\pm 0.0$ | $0.0\pm 0.0$

For our $[\mathbf{VGG}]$, we generate saliency maps from various CAM-based
methods on our two datasets, STACK-MIX and STACK-GEN, using the predicted
category for each example. Then, we measure how much of the CAM-based saliency
maps emphasize the unseen part, _i.e._ , the dead zone. We use the metric
$\mu(\cdot)$ defined for a upscaled saliency map
$\mathbf{\Lambda}\in\mathbb{R}^{224\times 224}_{+}$ as follows:

$\mu(\mathbf{\Lambda})\vcentcolon=\frac{\left\lVert\mathbf{\Lambda}_{171:224,\,:}\right\rVert_{2}}{\left\lVert\mathbf{\Lambda}\right\rVert_{2}}\,,$
(7)

where $\left\lVert\cdot\right\rVert_{2}$ is the $\ell^{2}$-norm and the lower
part of the image $\bm{\xi}_{:,\,171:224,\,:}$ is unseen by our
$[\mathbf{VGG}]$. We note that for a saliency map $\mathbf{\Lambda}$, the
lower $\mu(\mathbf{\Lambda})$, the better.

The results can be found in Table 1 and Figure 7. We observe that every CAM-
based methods, except HiResCAM, highlights unseen parts of an image to some
extent. Moreover, the observation are consistent over both datasets.

##  4 Conclusion

In this paper, we looked into several CAM-based methods, with a particular
focus on GradCAM. We showed that they can highlight parts of the input image
that are provably not used by the network. This was also showed theoretically,
looking at the behavior of GradCAM for a simple, masked CNN at initialization:
the saliency map is positive in expectation, even in areas which are unseen by
the network. Experimentally, this phenomenon appears to remain true, even on a
realistic network trained to a good accuracy on ImageNet.

As future work, we would like to extend the theory to a ResNet-like
architecture and other CAM-based methods, such as LayerCAM (Jiang et al.,
2021). We also would like to multiply the number of images in our two new
datasets, with the hope that this framework can become a standard check for
saliency maps explanations.

## Acknowledgements

This work was funded in part by the French Agence Nationale de la Recherche
(grant number ANR-19-CE23-0009-01 and ANR-21-CE23-0005-01). Most of this work
was realized while DG was employed at Université Côte d’Azur. We thank Jenny
Benois-Pineau for her valuable insights.

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##  Appendix A Other definitions

The work of Fu et al. (2020) proposed an alternative to $[\mathbf{GC}]$ by re-
scaling the gradient in the weighting coefficient $\alpha$. The main objective
of this alternative is to obtain a method that satisfies two axioms
(sensitivity and conservation properties) defined by Fu et al. (2020), which,
according to their findings, enhances visualization performance w.r.t.
specific metrics.

######  Definition 4 (XGradCAM)

In our notation, for an input $\bm{\xi}$ and model
$\operatorname{[\mathbf{CNN}]}$, the XGradCAM feature scores are given by

$\displaystyle[\mathbf{XC}]\vcentcolon=\operatorname{\sigma}\left(\sum_{v=1}^{V}\alpha_{v}^{(2)}\mathbf{B}^{(v)}\right)\in\mathbb{R}_{+}^{h\times
w}\,,$

where each
$\alpha_{v}^{(2)}\vcentcolon=\operatorname{GAP}\left(\frac{\mathbf{B}^{(v)}}{\left\lVert\mathbf{B}^{(v)}\right\rVert_{1}}\odot\nabla_{\mathbf{B}^{(v)}}f(\mathbf{B})\right)$
is the _global average pooling_ of the gradient of $f$ at $\mathbf{B}^{(v)}$
rescaled by the normalized activation
$\frac{\mathbf{B}^{(v)}}{\left\lVert\mathbf{B}^{(v)}\right\rVert_{1}}$.

GradCAM++ (Chattopadhay et al., 2018) then modified GradCAM by computing
rectified gradient together with second and third order derivative
information. This alternative appears to produce better saliency maps in cases
where the input image contains multiple subjects.

######  Definition 5 (GradCAM++)

In our notation, for an input $\bm{\xi}$ and model
$\operatorname{[\mathbf{CNN}]}$, the GradCAM++ feature scores are given by

$\displaystyle[\mathbf{C+}]\vcentcolon=\operatorname{\sigma}\left(\sum_{v=1}^{V}\alpha_{v}^{(3)}\mathbf{B}^{(v)}\right)\in\mathbb{R}_{+}^{h\times
w}\,,$

where each
$\alpha_{v}^{(3)}\vcentcolon=\operatorname{GAP}\left(\frac{\partial_{\mathbf{B}^{(v)}}^{2}f(\mathbf{B})}{2\partial_{\mathbf{B}^{(v)}}^{2}f(\mathbf{B})+\left\lVert\mathbf{B}^{(v)}\right\rVert_{1}\partial_{\mathbf{B}^{(v)}}^{3}f(\mathbf{B})}\odot\operatorname{\sigma}\left(\nabla_{\mathbf{B}^{(v)}}f(\mathbf{B})\right)\right)$
is the _global average pooling_ of the gradient of $f$ at $\mathbf{B}^{(v)}$
rescaled by second and third order derivatives defined as

$\partial_{\mathbf{B}^{(v)}}^{n}f(\mathbf{B})\vcentcolon=\left(\partial_{\mathbf{B}^{(v)}_{i,j}}^{n}f(\mathbf{B})\right)_{i,j\in[h]\times[w]}\,.$

More recently, HiResCAM (Draelos and Carin, 2021) proposed to replace the
averaging of the gradient over the map by the _element-wise_ multiplication
between gradient and activation. In our notation:

######  Definition 6 (HiResCAM)

For an input $\bm{\xi}$ and model $\operatorname{[\mathbf{CNN}]}$, the
HiResCAM feature scores are given by

$\displaystyle[\mathbf{HC}]\vcentcolon=\operatorname{\sigma}\left(\sum_{v=1}^{V}\alpha^{(4)}\odot\mathbf{B}^{(v)}\right)\in\mathbb{R}_{+}^{h\times
w}\,,$

where
$\alpha^{(4)}\vcentcolon=\nabla_{\mathbf{B}^{(v)}}f(\mathbf{B})\in\mathbb{R}^{h\times
w}$ and $\odot$ is the element-wise matrix product.

The work of Wang et al. (2020) got rid of the dependency on gradients of the
weighting coefficients $\alpha$.

######  Definition 7 (ScoreCAM)

In our notation, for an input $\bm{\xi}$ and model
$\operatorname{[\mathbf{CNN}]}$, the ScoreCAM feature scores are given by

$\displaystyle[\mathbf{SC}]\vcentcolon=\operatorname{\sigma}\left(\sum_{v=1}^{V}\alpha_{v}^{(5)}\mathbf{B}^{(v)}\right)\in\mathbb{R}_{+}^{h\times
w}\,.$

In the previous display, $\alpha^{(5)}=\mathrm{softmax}(\beta^{(5)})$, where
$\beta_{v}^{(5)}\vcentcolon=\operatorname{[\mathbf{CNN}]}\left(\bm{\xi}\odot
H^{(v)}\right)-\operatorname{[\mathbf{CNN}]}\left(\bm{\xi}\right)$,
$H^{(v)}\vcentcolon=s\left(\mathrm{Up}\left(\mathbf{B}^{(v)}\right)\right)$
the upsampled and normalized activation map $\mathbf{B}^{(v)}$, $s(\cdot)$ a
normalization function that maps matrix values into $[0,1]$, and
$\mathrm{Up}(\cdot)$ an upsampling function which resize a matrix to the size
of $\bm{\xi}$.

Finally, AblationCAM (Desai and Ramaswamy, 2020) removes each activation to
observe the impact on the prediction. Ablations resulting in larger drops
receive higher weights. AblationCAM is similar to ScoreCAM: a sort of masking
is performed to observe changes in prediction and no gradient is required.
However, both methods require lots of forward passes through the network.

######  Definition 8 (AblationCAM)

In our notation, for an input $\bm{\xi}$ and model
$\operatorname{[\mathbf{CNN}]}$, the AblationCAM feature scores are given by

$\displaystyle[\mathbf{AC}]\vcentcolon=\operatorname{\sigma}\left(\sum_{v=1}^{V}\alpha_{v}^{(6)}\mathbf{B}^{(v)}\right)\in\mathbb{R}_{+}^{h\times
w}\,,$

where each
$\alpha_{v}^{(6)}\vcentcolon=\frac{\mathbf{y}-\mathbf{y}_{v}}{\mathbf{y}}$,
$\mathbf{y}\vcentcolon=\operatorname{[\mathbf{CNN}]}\left(\bm{\xi}\right)$ the
predicted score by our model and $\mathbf{y}_{v}$ is the output of
$\operatorname{[\mathbf{CNN}]}$ when the activation map $\mathbf{B}^{(v)}$ is
zero out.

It should be clear that the method we use to compute the weighting
coefficients $\alpha$ of $[\mathbf{GC}]$, in Proposition 2, yields the
coefficients $\alpha^{(4)}$ for $[\mathbf{HC}]$ and $\alpha^{(2)}$ for
$[\mathbf{XC}]$, since in proving Proposition 2 we have computed
$\nabla_{\mathbf{B}^{(v)}}f(\mathbf{B})$, which is the challenging part.

##  Appendix B Technical results

As a consequence of Eq. (1), $\mathbf{A}$ (seens as a vector of size $h\times
w$) is a centered Gaussian vector, with covariance matrix given by

$\forall(i,j),(i^{\prime},j^{\prime})\in([h]\times[w])^{2},\,\,\left(\Sigma^{A}\right)_{(i,j),(i^{\prime},j^{\prime})}=\operatorname{Tr}\left(\xi_{i:i+k,j:j+k}^{\top}\xi_{i^{\prime}:i^{\prime}+k,j^{\prime}:j^{\prime}+k}\right)\,.$
(8)

Because of this simple remark, we can describe precisely the distribution of
$\mathbf{A}$ (and, further, $\mathbf{B}$), thanks to the following lemmas. We
denote by $\phi$ the density of the standard Gaussian distribution and $\Phi$
its cumulative distribution function.

######  Lemma 9 (Expectation of rectified Gaussian)

Let $X\sim\operatorname{\mathcal{N}}\left(\mu,\tau^{2}\right)$, we get the
following expectation for the rectified Gaussian
$X^{+}\vcentcolon=\operatorname{\sigma}\left(X\right)$:

$\displaystyle\operatorname{\mathbb{E}}\left[X^{+}\right]=\mu\Phi\left(\frac{\mu}{\tau}\right)+\tau\phi\left(\frac{-\mu}{\tau}\right)\,.$

Proof  See Beauchamp (2018).

######  Lemma 10 (Law of convolution)

Let $\mathbf{m}\in\mathbb{R}^{k\times k}$ and
$\mathbf{F}\sim\operatorname{\mathcal{N}}\left(0,\tau^{2}\mathrm{Id}_{k}\right)$,
then the convolution $\mathbf{F}\star\mathbf{m}$ has distribution
$\operatorname{\mathcal{N}}\left(0,(\tau\left\lVert\mathbf{m}\right\rVert_{2})^{2}\right)$.

Proof  First let us remind that:

$\displaystyle\mathbf{F}\star\mathbf{m}$
$\displaystyle=\sum_{i,j=1}^{k}\mathbf{F}_{i,j}\mathbf{m}_{i,j}\,.$

This lemma is derived from rapid calculations that proceed as follows,
utilizing the fact that the elements $\left(\mathbf{F}_{i,j}\right)_{i,j}$ are
independent and identically distributed (i.i.d.):

$\displaystyle\operatorname{\mathbb{E}}\left[\mathbf{F}\star\mathbf{m}\right]$
$\displaystyle=0\,,$
$\displaystyle\operatorname{Var}\left(\mathbf{F}\star\mathbf{m}\right)$
$\displaystyle=\tau^{2}\sum_{i,j=1}^{k}\mathbf{m}_{i,j}^{2}=\tau^{2}\left\lVert\mathbf{m}\right\rVert_{2}^{2}\,.$

Straightforward computations yields:

######  Lemma 11 (Moments of squared rectified Gaussian)

Let $X\sim\operatorname{\mathcal{N}}\left(0,\tau^{2}\right)$ of density
$f(\cdot)$, we get the following two moments for the squared rectified
Gaussian $(X^{+})^{2}$:

$\displaystyle\operatorname{\mathbb{E}}\left[(X^{+})^{2}\right]=\frac{\tau^{2}}{2}\quad\text{and
}\operatorname{Var}\left((X^{+})^{2}\right)=\frac{5}{4}\tau^{4}\,.$

##  Appendix C Proof of Proposition 2

#### Notation.

Before starting the proof, let us recall some notation. In Section 2.1, we
defined $\mathbf{a}^{(u)}\vcentcolon=h_{u}\left(\mathbf{C}^{\prime}\right)$
the non-rectified activation of layer $u$,
$\mathbf{r}^{(u)}\vcentcolon=\operatorname{\sigma}\left(\mathbf{a}^{(u)}\right)$
its rectified counterpart. We also defined $\mathbf{C}$ the max pooled
rectified activation $\mathcal{M}(\mathbf{B})$. In this proof, we write
$f_{u}$ the function that maps the input of the $u$-th layer
$\mathbf{r}^{(u-1)}$ to its output $\mathbf{r}^{(u)}$ as follows
$f_{u}(\mathbf{r}^{(u-1)})\vcentcolon=\operatorname{\sigma}\left(\mathbf{W}^{(u)}\mathbf{r}^{(u-1)}\right)$
with $u\in[L]$.

Now let us turn to the computation of
$\nabla_{\mathbf{B}}f(\mathbf{B})\in\mathbb{R}^{(h\times w)\times 1}$ with
$\mathbf{B}\in\mathbb{R}^{h\times w}$. By the chain rule, we get
$\nabla_{\mathbf{B}}f(\mathbf{B})=\leavevmode\nobreak\
\nabla_{\mathbf{B}}\left(\operatorname{\mathcal{F}}\circ\mathcal{M}\right)(\mathbf{B})=\nabla_{\mathbf{B}}\mathcal{M}\cdot\nabla_{\mathcal{M}(\mathbf{B})}\operatorname{\mathcal{F}}$
where:

$\displaystyle\nabla_{\mathbf{C}}\operatorname{\mathcal{F}}$
$\displaystyle=\nabla_{\mathbf{C}}f_{1}\cdot\nabla_{f_{1}(\mathbf{C})}\left(\mathbf{W}^{(L)}f_{L-1}\circ\cdots\circ
f_{2}\right)$
$\displaystyle=\left(\nabla_{\mathbf{C}}f_{1}\right)\cdot\left(\nabla_{\mathbf{r}^{(1)}}f_{2}\right)\cdots\left(\nabla_{\mathbf{r}^{(L-2)}}f_{L-1}\right)\cdot\left(\mathbf{W}^{(L)}\right)\,.$

We compute $\nabla_{\mathbf{r}^{(i)}}f_{i+1}\in\mathbb{R}^{d_{i}\times
d_{i+1}}$ as follows:

$\displaystyle\nabla_{\mathbf{r}^{(i)}}f_{i+1}=\nabla_{\mathbf{r}^{(i)}}\operatorname{\sigma}\left(\mathbf{W}^{(i+1)}\mathbf{r}^{(i)}\right)=\left(\mathbf{W}^{(i+1)}\right)^{\top}\nabla_{\mathbf{a}^{(i+1)}}\operatorname{\sigma}\left(\mathbf{a}^{(i+1)}\right)\,.$

Now let us remark that, for all $x\in\mathbb{R}^{d}$,
$\partial_{i}\operatorname{\sigma}\left(x\right)_{j}=\partial_{i}\operatorname{\sigma}\left(x_{j}\right)=\mathds{1}_{x_{j}>0}\mathds{1}_{i=j}$,
since $\operatorname{\sigma}$ is applied element-wise. Thus

$\displaystyle\nabla_{\mathbf{a}^{(i+1)}}\operatorname{\sigma}\left(\mathbf{a}^{(i+1)}\right)=\begin{pmatrix}\mathds{1}_{\mathbf{a}^{(i+1)}_{1}>0}&0&\cdots&0\\\
0&\mathds{1}_{\mathbf{a}^{(i+1)}_{2}>0}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\mathds{1}_{\mathbf{a}^{(i+1)}_{d_{i+1}}>0}\end{pmatrix}\,.$

Finally,

$\displaystyle\nabla_{\mathbf{r}^{(i)}}f_{i+1}$
$\displaystyle=\begin{pmatrix}\mathbf{W}^{(i+1)^{\top}}_{1,\,:}&\cdots&\mathbf{W}^{(i+1)^{\top}}_{d_{i+1},\,:}\end{pmatrix}\nabla_{\mathbf{a}^{(i+1)}}\operatorname{\sigma}\left(\mathbf{a}^{(i+1)}\right)$
$\displaystyle=\begin{pmatrix}\mathds{1}_{\mathbf{a}^{(i+1)}_{1}>0}\mathbf{W}^{(i+1)^{\top}}_{1,\,:}&\cdots&\mathds{1}_{\mathbf{a}^{(i+1)}_{d_{i+1}}>0}\mathbf{W}^{(i+1)^{\top}}_{d_{i+1},\,:}\end{pmatrix}\,.$

Now to compute $\nabla_{\mathbf{C}}\operatorname{\mathcal{F}}$ let us start
from the end with $\nabla_{\mathbf{r}^{(L-2)}}f_{L-1}\mathbf{W}^{(L)^{\top}}$:

$\displaystyle\nabla_{\mathbf{r}^{(L-2)}}f_{L-1}\mathbf{W}^{(L)^{\top}}$
$\displaystyle=\begin{pmatrix}\mathds{1}_{\mathbf{a}^{(L-1)}_{1}>0}\mathbf{W}^{(L-1)^{\top}}_{1,\,:}&\cdots&\mathds{1}_{\mathbf{a}^{(L-1)}_{d_{L-1}}>0}\mathbf{W}^{(L-1)^{\top}}_{d_{L-1},\,:}\end{pmatrix}\mathbf{W}^{(L)^{\top}}$
$\displaystyle=\begin{pmatrix}\sum_{i_{L-1}=1}^{d_{L-1}}\mathds{1}_{\mathbf{a}^{(L-1)}_{i_{L-1}}>0}\mathbf{W}^{(L-1)^{\top}}_{i_{L-1},\,:}\mathbf{W}^{(L)}_{1,i_{L-1}}\end{pmatrix}\,.$

Similarly,

$\displaystyle\nabla_{\mathbf{r}^{(L-3)}}f_{L-2}$
$\displaystyle=\begin{pmatrix}\mathds{1}_{\mathbf{a}^{(L-2)}_{1}>0}\mathbf{W}^{(L-2)^{\top}}_{1,\,:}&\cdots&\mathds{1}_{\mathbf{a}^{(L-2)}_{d_{L-2}}>0}\mathbf{W}^{(L-2)^{\top}}_{d_{L-2},\,:}\end{pmatrix}\,,$

so the next computation gives

$\displaystyle\nabla_{\mathbf{r}^{(L-3)}}f_{L-2}\nabla_{\mathbf{r}^{(L-2)}}f_{L-1}\mathbf{W}^{(L)^{\top}}$
$\displaystyle=\nabla_{\mathbf{r}^{(L-3)}}f_{L-2}\begin{pmatrix}\sum_{i_{L-1}=1}^{d_{L-1}}\mathds{1}_{\mathbf{a}^{(L-1)}_{i_{L-1}}>0}\mathbf{W}^{(L-1)^{\top}}_{i_{L-1},\,:}\mathbf{W}^{(L)}_{1,i_{L-1}}\end{pmatrix}$
$\displaystyle=\nabla_{\mathbf{r}^{(L-3)}}f_{L-2}\begin{pmatrix}\sum_{i_{L-1}=1}^{d_{L-1}}\mathds{1}_{\mathbf{a}^{(L-1)}_{i_{L-1}}>0}\mathbf{W}^{(L-1)^{\top}}_{i_{L-1},\,1}\mathbf{W}^{(L)}_{1,i_{L-1}}\\\
\vdots\\\
\sum_{i_{L-1}=1}^{d_{L-1}}\mathds{1}_{\mathbf{a}^{(L-1)}_{i_{L-1}}>0}\mathbf{W}^{(L-1)^{\top}}_{i_{L-1},\,L-2}\mathbf{W}^{(L)}_{1,i_{L-1}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\sum_{i_{L-2},i_{L-1}=1}^{d_{L-2},d_{L-1}}\mathds{1}_{\mathbf{a}^{(L-2)}_{i_{L-2}},\,\mathbf{a}^{(L-1)}_{i_{L-1}}>0}\mathbf{W}^{(L-2)^{\top}}_{i_{L-2},\,:}\mathbf{W}^{(L-1)^{\top}}_{i_{L-1},\,i_{L-2}}\mathbf{W}^{(L)}_{1,i_{L-1}}\end{pmatrix}\,.$

Now we can conclude by straightforward induction:

$\displaystyle\nabla_{\mathbf{C}}\operatorname{\mathcal{F}}$
$\displaystyle=\begin{pmatrix}\sum_{i_{1},\ldots,i_{L-1}=1}^{d_{1},\ldots,d_{L-1}}\mathds{1}_{\mathbf{a}^{(1)}_{i_{1}},\ldots,\mathbf{a}^{(L-1)}_{i_{L-1}}>0}(\mathbf{W}_{i_{1},:}^{(1)})^{\top}(\mathbf{W}_{i_{2},i_{1}}^{(2)})^{\top}\ldots(\mathbf{W}_{i_{L-1},i_{L-2}}^{(L-1)})^{\top}\mathbf{W}_{1,i_{L-1}}^{(L)}\end{pmatrix}\,.$

Now to finish we need to compute $\nabla_{\mathbf{B}}\mathcal{M}(\mathbf{B})$,
to do so, we suppose that the convolutional layer $\mathcal{C}$ returns an
reshaped rectified activation map $\mathbf{B}$ such that we get the following
gradient:

$\displaystyle\begin{pmatrix}\mathbf{D}_{1}&0&0\\\ 0&\mathbf{D}_{2}&0\\\
\vdots&&\vdots\\\ 0&\cdots&\mathbf{D}_{h^{\prime}\times
w^{\prime}}\end{pmatrix}\in\mathbb{R}^{((k^{\prime})^{2}\times
h^{\prime}\times w^{\prime})\times(h^{\prime}\times
w^{\prime})}=\mathbb{R}^{(h\times w)\times(h^{\prime}\times w^{\prime})}$

with $\mathbf{D}_{i}\in\mathbb{R}^{(k^{\prime})^{2}\times 1}$ being the $i$-th
block of indicator functions defined as follows:

$\displaystyle\mathbf{D}_{i}\vcentcolon=\begin{pmatrix}\mathds{1}_{\max{\mathbf{m}_{i}}=\mathbf{m}_{i,(1,1)}}\\\
\mathds{1}_{\max{\mathbf{m}_{i}}=\mathbf{m}_{i,(1,2)}}\\\ \vdots\\\
\mathds{1}_{\max{\mathbf{m}_{i}}=\mathbf{m}_{i,(2,1)}}\\\ \vdots\\\
\mathds{1}_{\max{\mathbf{m}_{i}}=\mathbf{m}_{i,(k^{\prime},k^{\prime})}}\end{pmatrix}\,,$

where $\mathbf{m}_{i,(p,q)}\vcentcolon=\left(\mathbf{m}_{i}\right)_{p,q}$ and
$\mathbf{m}_{i}\vcentcolon=\mathbf{B}_{k^{\prime}(i^{\prime}-1)+1:k^{\prime}i^{\prime},k^{\prime}(j^{\prime}-1)+1:k^{\prime}j^{\prime}}\in\mathbb{R}_{+}^{k^{\prime}\times
k^{\prime}},\,(i^{\prime},j^{\prime})\vcentcolon=((i-1)/k^{\prime}+1,\,(i-1\mod
k^{\prime})+1)$ is the $i$-th patch of $\mathbf{B}$ (the patch are ordered
from left to right, starting from the top of $\mathbf{B}$). Also note that
$\cdot/\cdot$ is the integer division.

Finally, we can compute $\nabla_{\mathbf{B}}f(\mathbf{B})$ as follows:

$\displaystyle\nabla_{\mathbf{B}}f(\mathbf{B})$
$\displaystyle=\nabla_{\mathbf{B}}\mathcal{M}\cdot\nabla_{\mathcal{M}(\mathbf{B})}\operatorname{\mathcal{F}}$
$\displaystyle=\begin{pmatrix}\mathbf{D}_{1}\rho_{1}\\\ \vdots\\\
\mathbf{D}_{h^{\prime}\times w^{\prime}}\rho_{h^{\prime}\times
w^{\prime}}\end{pmatrix}$ $\displaystyle\in\mathbb{R}^{(h\times w)\times 1}$

where $\rho_{b}\in\mathbb{R}$, $b\in[h^{\prime}\times w^{\prime}]$ is defined
as follows:

$\displaystyle\rho_{b}\vcentcolon=\sum_{i_{1},\ldots,i_{L-1}=1}^{d_{1},\ldots,d_{L-1}}\mathds{1}_{\mathbf{a}^{(1)}_{i_{1}},\ldots,\mathbf{a}^{(L-1)}_{i_{L-1}}>0}(\mathbf{W}_{i_{1},b}^{(1)})^{\top}(\mathbf{W}_{i_{2},i_{1}}^{(2)})^{\top}\ldots(\mathbf{W}_{i_{L-1},i_{L-2}}^{(L-1)})^{\top}\mathbf{W}_{1,i_{L-1}}^{(L)}\,.$

To compute $\alpha$, we simply take the average of the components of the
previous display, $\nabla_{\mathbf{B}}f(\mathbf{B})$, which yields

$\displaystyle\alpha$
$\displaystyle=\frac{1}{hw}\sum_{k=1}^{(k^{\prime})^{2}}\sum_{j=1}^{h^{\prime}\times
w^{\prime}}\mathbf{D}_{j,k}\rho_{j}$
$\displaystyle=\frac{1}{hw}\sum_{j=1}^{h^{\prime}\times
w^{\prime}}\left(\sum_{k=1}^{(k^{\prime})^{2}}\mathbf{D}_{j,k}\right)\rho_{j}$
$\displaystyle=\frac{1}{hw}\sum_{j=1}^{h^{\prime}\times
w^{\prime}}\rho_{j}\,,$

since $\sum_{k=1}^{(k^{\prime})^{2}}\mathbf{D}_{j,k}=1$ with fixed $j$ (it is
also the case in practice when using automatic differentiation) and
$\mathbf{D}_{j,k}\vcentcolon=\left(\mathbf{D}_{j}\right)_{k}$.

##  Appendix D Proof of Theorem 3

Before jumping into the proof, let us remark that the left-hand side of Eq.
(5) is non-negative, thus there is nothing to prove if
$\left\lVert\mathbf{m}\right\rVert_{2}=0$. Thus we will assume that
$\left\lVert\mathbf{m}\right\rVert_{2}>0$ from now on.

#### Separating the randomness.

There are two sources of randomness: the weights of the filters
$\mathbf{F}=(\mathbf{F}^{(1)},\ldots,\mathbf{F}^{(V)})$ and the coefficients
of the linear layer $\mathbf{W}$. We start this proof by dissociating these
two sources of randomness. More precisely, following Definition 1, and using
the law of total expectation, we compute the expected GradCAM heatmap at
coordinates $(i,j)$ as

$\operatorname{\mathbb{E}}\left[\operatorname{\sigma}\left(\sum_{q=1}^{V}\alpha_{q}\mathbf{B}_{i,j}^{(q)}\right)\right]=\operatorname{\mathbb{E}}\left[\mathbb{E}\left[\operatorname{\sigma}\left(\sum_{q=1}^{V}\alpha_{q}\mathbf{B}_{i,j}^{(q)}\right)\middle|\mathbf{F}\right]\right]\,.$
(9)

Using the computed GradCAM coefficient in Proposition 2 with $L=1$, we get
$\alpha_{q}=\frac{1}{hw}\sum_{p=1}^{h^{\prime}w^{\prime}}\mathbf{W}_{p}^{(q)}$.
Since we assume that the weights of the linear layer are i.i.d.
$\operatorname{\mathcal{N}}\left(0,\tau^{2}\right)$, for the upper part, and
$0$ for the lower part, $\alpha_{q}$ is a centered Gaussian with variance
$(\frac{\tau}{hw})^{2}\frac{h^{\prime}w^{\prime}}{2}$. Now, we recall that
$\mathbf{B}$ does not depend on $\mathbf{W}$. Therefore, conditionally to
$\mathbf{F}^{(q)}$, $\alpha_{q}\mathbf{B}_{i,j}^{(q)}$ is a centered Gaussian
with variance

$\left(\mathbf{B}_{i,j}^{(q)}\right)^{2}\left(\frac{\tau}{hw}\right)^{2}\frac{h^{\prime}w^{\prime}}{2}\,.$

We deduce that

$\sum_{q=1}^{V}\alpha_{q}\mathbf{B}_{i,j}^{(q)}|\mathbf{F}\sim\operatorname{\mathcal{N}}\left(0,\sum_{q=1}^{V}\left(\mathbf{B}_{i,j}^{(q)}\right)^{2}\left(\frac{\tau}{hw}\right)^{2}\frac{h^{\prime}w^{\prime}}{2}\right)\,.$
(10)

#### Computing expectations.

Since, conditionally to $\mathbf{F}$, $\sum_{q}\alpha_{q}\mathbf{B}^{(q)}$ is
a a centered Gaussian with variance given by Eq. (10), we can use Lemma 9 to
compute

$\displaystyle\mathbb{E}\left[\operatorname{\sigma}\left(\sum_{q=1}^{V}\alpha_{q}\mathbf{B}_{i,j}^{(q)}\right)\middle|\mathbf{F}\right]$
$\displaystyle=\sqrt{\left(\frac{\tau}{hw}\right)^{2}\frac{h^{\prime}w^{\prime}}{2}\sum_{q=1}^{V}\left(\mathbf{B}_{i,j}^{(q)}\right)^{2}}\phi(0)$
$\displaystyle=\frac{\tau}{2hw\sqrt{\pi}}\sqrt{h^{\prime}w^{\prime}\sum_{q=1}^{V}\left(\mathbf{B}_{i,j}^{(q)}\right)^{2}}\,.$

Coming back to Eq. (9), we deduce that

$\operatorname{\mathbb{E}}\left[\operatorname{\sigma}\left(\sum_{q=1}^{V}\alpha_{q}\mathbf{B}_{i,j}^{(q)}\right)\right]=\frac{\tau}{2hw\sqrt{\pi}}\sqrt{h^{\prime}w^{\prime}}\operatorname{\mathbb{E}}\left[\sqrt{\sum_{q=1}^{V}\left(\mathbf{B}_{i,j}^{(q)}\right)^{2}}\right]\,.$
(11)

#### Lower bound.

Up to the best of our knowledge, there is no closed-form expression for Eq.
(11), and we now proceed to find a lower bound to this expression. Let
$Y\vcentcolon=\sum_{q=1}^{V}\left(\mathbf{B}_{i,j}^{(q)}\right)^{2}$. By Lemma
10 and 11,
$\operatorname{\mathbb{E}}\left[Y\right]=V\frac{(\tau\left\lVert\mathbf{m}\right\rVert_{2})^{2}}{2}$
and
$\operatorname{Var}\left(Y\right)=\frac{5}{4}V(\tau\left\lVert\mathbf{m}\right\rVert_{2})^{4}$
with $\mathbf{m}\vcentcolon=\bm{\xi}_{i:i+k-1,j:j+k-1}$ a patch at index
$(i,j)$ in the image $\bm{\xi}$. By Chebyshev’s inequality, for any $t>0$,

$\operatorname{\mathbb{P}}\left(Y\geq\operatorname{\mathbb{E}}\left[Y\right]-t\right)\geq
1-\frac{\operatorname{Var}\left(Y\right)}{t^{2}}\,.$ (12)

Let us set $t=\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}>0$ in the
previous display, in this way

$\frac{\operatorname{Var}\left(Y\right)}{t^{2}}=\frac{5}{4}V\tau^{4}\left\lVert
m\right\rVert^{4}\cdot\left(\frac{V^{2}\tau^{4}\left\lVert
m\right\rVert^{4}}{2^{2}\cdot 2^{2}}\right)^{-1}=\frac{20}{V}\,.$

Since $\left\lVert\mathbf{m}\right\rVert_{2}>0$,

$\operatorname{\mathbb{P}}\left(Y\geq\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}\right)\geq
1-\frac{20}{V}\,.$

We now conclude the proof writing

$\displaystyle\operatorname{\mathbb{E}}\left[\operatorname{\sigma}\left(\sum_{q=1}^{V}\alpha_{q}\mathbf{B}_{i,j}^{(q)}\right)\right]$
$\displaystyle=\frac{\tau}{2hw\sqrt{\pi}}\sqrt{h^{\prime}w^{\prime}}\left(\mathbb{E}\left[\sqrt{Y}\middle|Y\geq\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}\right]\operatorname{\mathbb{P}}\left(Y\geq\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}\right)\right.$
$\displaystyle\qquad\left.+\mathbb{E}\left[\sqrt{Y}\middle|Y<\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}\right]\operatorname{\mathbb{P}}\left(Y<\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}\right)\right)$
$\displaystyle\geq\frac{\tau}{2hw\sqrt{\pi}}\sqrt{h^{\prime}w^{\prime}}\mathbb{E}\left[\sqrt{Y}\middle|Y\geq\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}\right]\operatorname{\mathbb{P}}\left(Y\geq\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}\right)\qquad\text{(since
$Y>0$)}$
$\displaystyle\geq\frac{\tau}{2hw\sqrt{\pi}}\sqrt{h^{\prime}w^{\prime}}\operatorname{\mathbb{E}}\left[\sqrt{\frac{\operatorname{\mathbb{E}}\left[Y\right]}{2}}\right]\left(1-\frac{20}{V}\right)$
$\displaystyle=\frac{\sqrt{Vh^{\prime}w^{\prime}}}{4\sqrt{\pi}}\frac{\tau^{2}}{hw}\left(1-\frac{20}{V}\right)\left\lVert
m\right\rVert_{2}\,.$

##  Appendix E Additional Experiments

In this section, we present two additional sets of experiments, one on STACK-
MIX (Figure 8), the other on STACK-GEN (Figure 9).

Input image |  |  |  |
---|---|---|---|---
GradCAM |  |  |  |
GradCAM++ |  |  |  |
XGradCAM |  |  |  |
ScoreCAM |  |  |  |
AblationCAM |  |  |  |
EigenCAM |  |  |  |
HiResCAM |  |  |  |
Figure 8: Saliency maps given by the considered CAM-based methods for $[\mathbf{VGG}]$. With the notable exception of HiresCAM, they all highlight parts of images from STACK-GEN which are unseen by the network (this is denoted by the red, rectangular shape in the lower part of the image). Input image |  |  |  |
---|---|---|---|---
GradCAM |  |  |  |
GradCAM++ |  |  |  |
XGradCAM |  |  |  |
ScoreCAM |  |  |  |
AblationCAM |  |  |  |
EigenCAM |  |  |  |
HiResCAM |  |  |  |
Figure 9: Saliency maps given by the considered CAM-based methods for
$[\mathbf{VGG}]$. With the notable exception of HiresCAM, they all highlight
parts of images from STACK-MIX which are unseen by the network (this is
denoted by the red, rectangular shape in the lower part of the image).


# Dynamical mass generation of spin-2 fields in de Sitter space for an $O(N)$
symmetric model at large $N$

Nahomi Kan<EMAIL_ADDRESS>National Institute of Technology, Gifu College,
Motosu-shi, Gifu 501-0495, Japan  Kiyoshi Shiraishi
<EMAIL_ADDRESS>Graduate School of Sciences and Technology for
Innovation, Yamaguchi University, Yamaguchi-shi, Yamaguchi 753–8512, Japan

###### Abstract

We consider the strong-coupling phase in a model of $O(N)$ spin-2 field theory
in de Sitter spacetime and the effective mass of spin-2 fields therein. In the
strong-coupling phase, the Higuchi bound limits the mass parameter in the
theory. The analysis using the large $N$ approximation finds the critical
value of the mass parameter with numerical calculation.

###### pacs:

03.70.+k, 04.25.-g, 04.62.+v, 11.10.-z, 11.10.Kk 11.90.+t, 12.90.+b,

##  I Introduction

The massive spin-2 field theory has been studied for quite some time FP .
However, in recent years, the theory of massive graviton Hinterbichler ;
deRham  as a spin-2 particle has attracted attention as a kind of modified
gravitation theories. As a key to solving the cosmological constant problem,
it is also considered significant to investigate the relationship between the
mass of the graviton and a small cosmological constant. Besides, bigravity
theories SMvS  and multigravity theories are also considered, and in special
cases, it is also speculated that spin-2 particles (not yet discovered) other
than the graviton partly plays the role of dark matter CCL .

On the other hand, the existence of the Higuchi bound Higuchi1 ; Higuchi2  is
known for the mass of a spin-2 particle in maximally symmetric spacetime. It
has been shown that a negative norm state appears in the spin-2 field theory
below the critical mass-squared, $\frac{D-2}{D(D-1)}R$ (where $R$ is the
scalar curvature of the de Sitter space), and the theory becomes unstable
(except for the massless case). This is a distinctive feature of the spin-2
field theory that is not seen in other spin fields. Therefore, in some sense,
it is speculated that the study of spin-2 theory is a very important key point
related to both the dark energy problem and the dark matter problem.

Now, for example, the masses of the (spin 1/2, 1) particles in the standard
model is obtained by the Higgs mechanism. On the other hand, how is the mass
of spin-2 particles determined? In this paper, as a very bold assumption, we
consider a model in which the dynamical mass is determined from the
interaction of $N$ spin-2 fields $h^{a}_{\mu\nu}$ $(a=1,\dots,N)$. Consider
$N$ spin-2 fields $h_{\mu\nu}^{a}$ with $O(N)$ invariant interaction:

$\mathcal{L}_{int}=-\frac{\lambda}{8N}(h^{a\,\mu\nu}h^{a}_{\mu\nu}-h^{a\mu}_{\mu}h^{a\nu}_{\nu})^{2}\,,$
(1)

where $\lambda$ is the coupling constant. Then, the vacuum expectation value

$\langle\sigma\rangle\equiv\langle
m^{2}+\frac{\lambda}{2N}(h^{a\,\mu\nu}h^{a}_{\mu\nu}-h^{a\mu}_{\mu}h^{a\nu}_{\nu})\rangle\,,$
(2)

(where $m$ is the mass parameter) is the mass squared of the spin-2 fields in
the strong-coupling vacuum. For the spin-zero field model, the large $N$
approximation as a self-consistent approach has been devised since half a
century ago CJP ; KK ; AKS ; SAT , which makes it possible to study the
strong-coupling phase relatively easily, and we also uses this method in this
paper.111The large $N$ scalar field theory in de Sitter space has been studied
in Refs. Serreau ; GS ; LR , and recently, that in three dimensional AdS space
has been investigated in Ref. BH . Since the mass in the Lagrangian and the
effective mass in the strong-coupling vacuum are generally different, we are
most interested in how the parameter restriction by the Higuchi bound in de
Sitter spacetime works.

Various no-go theories have been found for massless spin-2 field interactions
BDGH . However, earlier discussions have already reported that the theory of
spin-2 with perfectly sound interactions requires an infinite number of
massive fields anyway DPS ; BHR . The relationship between the string theory,
which involves infinite particle states, and the massive spin-2 theory has
been investigated LMMS . Therefore, although the model treated in the present
paper does not have a number of theoretic properties such as UV completeness,
we have begun our study with the expectation that the theory will have better
properties in the emerging vacuum in nonperturvative regime.

In our model, defined in the next section, we proceed with the study assuming
that the $N$ spin-2 fields are independent of the graviton. However, the
mixing with the graviton is conceivable as a field of the same spin-2, so in
the future, we will manage to construct an effective model related to
cosmology and elementary particle theory, after taking the mixing properly. Of
course, the existence of large-$N$ spin-2 fields with a degenerate mass is
problematic in the literal sense. We expect that even in this simple model we
can see some features of the model with a finite number of fields. We would
like to leave such research as a subject for future papers.

The structure of the present paper is as follows. In Sec. II, we introduce the
spin-2 model considered in this paper. In Sec. III, we consider the case of
four dimensional de Sitter spacetime. The critical value of the renormalized
Lagrangian mass, avoiding the Higuchi bound, is examined. Finally, we discuss
the conclusion and future prospects in the last section. We have Appendix A
for the case of three dimensional de Sitter spacetime.

##  II $O(N)$ symmetric spin-2 model in de Sitter spacetime

Let us consider the following model. The action of $N$ massive symmetric
tensor field $h^{a}_{\mu\nu}$ on a $D$-dimensional maximally symmetric
spacetime with the constant scalar curvature $R\equiv g^{\mu\nu}R_{\mu\nu}$,
where the Ricci tensor $R_{\mu\nu}$ equals to $\frac{R}{D}g_{\mu\nu}$, is
written by AD1 ; AD2 ; BKP ; BGKP ; BGP

$\displaystyle S_{FP}$ $\displaystyle=$ $\displaystyle\int
d^{D}x\sqrt{-g}\Bigl{[}-\frac{1}{2}\nabla_{\mu}h^{a}_{\nu\rho}\nabla^{\mu}h^{a\,\nu\rho}+\nabla_{\nu}h^{a}_{\mu\rho}\nabla^{\mu}h^{a\,\nu\rho}-\nabla_{\mu}h^{a}\nabla_{\rho}h^{a\,\mu\rho}+\frac{1}{2}\nabla_{\mu}h^{a}\nabla^{\mu}h^{a}$
(3)
$\displaystyle+\frac{R}{D}\Bigl{(}h^{a\,\mu\nu}h^{a}_{\mu\nu}-\frac{1}{2}h^{a}h^{a}\Bigr{)}-\frac{1}{2}m_{0}^{2}(h^{a\,\mu\nu}h^{a}_{\mu\nu}-h^{a}h^{a})\Bigr{]}\,,$

where $h\equiv h^{\rho}_{\rho}=g^{\mu\nu}h_{\mu\nu}$, $\nabla_{\mu}$
represents the covariant derivative in terms of the metric $g_{\mu\nu}$, and
$m_{0}$ is the mass parameter (the bare Lagrangian mass). In addition, we
assume the following $O(N)$ invariant interaction:

$S_{int}=\int
d^{D}x\sqrt{-g}\,\Bigl{[}-\frac{\lambda_{0}}{8N}(h^{a\,\mu\nu}h^{a}_{\mu\nu}-h^{a}h^{a})^{2}\Bigr{]}\,,$
(4)

where $\lambda_{0}$ is the (bare) quartic self-interaction coupling constant.
In this paper, we will not consider the possibility of $O(N)$ symmetry
breaking, although it might be interesting.

To introduce the auxiliary field $\sigma$, we add the following action, which
is nondynamical itself, for $\sigma$:

$S_{aux}=\int d^{D}x\sqrt{-g}\,\left\\{\frac{N}{2\lambda_{0}}\left[\sigma-
m_{0}^{2}-\frac{\lambda_{0}}{2N}(h^{a\,\mu\nu}h^{a}_{\mu\nu}-h^{a}h^{a})\right]^{2}\right\\}\,.$
(5)

Note that the functional integral over $\sigma$ is a trivial Gaussian
integral.

It can be seen that the vacuum expectation value of $\sigma$ becomes the
square of the mass of the spin-2 fields. Performing a Gaussian integral over
the spin-2 fields $h^{a}_{\mu\nu}$ (and the corresponding ghost fields with
the gauge fixing) can then be performed in the partition function with the
total action $S=S_{FP}+S_{int}+S_{aux}$ and we obtain the one-loop effective
action up to an additive constant:

$S_{eff}=\int
d^{D}x\sqrt{-g}\,\left[\frac{N}{2\lambda_{0}}\sigma^{2}-\frac{Nm_{0}^{2}}{\lambda_{0}}\sigma-
NL_{0}(\sigma)\right]\equiv\int
d^{D}x\sqrt{-g}\,\Bigl{[}-NV(\sigma)\Bigr{]}\,,$ (6)

where DDLS ; DLS ; CD

$L_{0}(\sigma)\equiv\mathcal{V}_{D}^{-1}\left\\{\frac{1}{2}{\rm
Tr\,}\ln\left[\Delta(1,1)-2\frac{R}{D}+\sigma\right]-\frac{1}{2}{\rm
Tr\,}\ln\left[\Delta(\mbox{$\frac{1}{2},\frac{1}{2}$})-2\frac{R}{D}+\sigma\right]\right\\}\,,$
(7)

with $\mathcal{V}_{D}$ is the volume of the spacetime,
$\int\sqrt{-g}\,d^{D}x$. Here, the differential operators
$\Delta(\mbox{$\frac{1}{2},\frac{1}{2}$})$ and $\Delta(1,1)$ are defined as

$\Delta(\mbox{$\frac{1}{2},\frac{1}{2}$})\xi_{\mu}\equiv-\Box\xi_{\mu}+R_{\mu\nu}\xi^{\nu}\,,\quad\Delta(1,1)\phi_{\mu\nu}\equiv-\Box\phi_{\mu\nu}+R_{\mu\tau}\phi^{\tau}_{\nu}+R_{\nu\tau}\phi^{\tau}_{\mu}-2R_{\mu\rho\nu\tau}\phi^{\rho\tau}\,,$
(8)

respectively, where $R_{\mu\rho\nu\tau}$ denotes the Riemann tensor.

The equation of motion $\frac{\delta S}{\delta\sigma}=0$, which determines the
extremum of the effective potential, $\frac{dV}{d\sigma}=0$, leads to

$\frac{1}{\lambda_{0}}\sigma-\frac{m_{0}^{2}}{\lambda_{0}}-\frac{dL_{0}(\sigma)}{d\sigma}=0\,,$
(9)

which gives the relation of the leading order in the expansion in $1/N$, thus
it is independent of $N$. The expected value of $\sigma$ can be obtained by
solving this equation.

We will compute the one-loop contribution of the spin-2 fields by with the
spectrum of the Laplacian on a $D$-sphere $S^{D}$, as the Euclidean version of
$D$-dimensional de Sitter spacetime. Hereafter, we set the constant curvature
of the space as

$R_{\mu\rho\nu\tau}=\frac{\Lambda}{D-1}(g_{\mu\nu}g_{\rho\tau}-g_{\mu\tau}g_{\rho\nu})\,,\quad
R_{\mu\nu}=g^{\rho\tau}R_{\mu\rho\nu\tau}=\Lambda
g_{\mu\nu}\,,\quad\mbox{and}\quad R=D\Lambda\,,$ (10)

where $\Lambda$ is a positive cosmological constant, while the volume is given
by

$\mathcal{V}_{D}=\frac{2\pi^{\frac{D+1}{2}}}{\Gamma\left(\frac{D+1}{2}\right)}\left(\frac{\Lambda}{D-1}\right)^{-D/2}\,.$
(11)

Then, the first derivative of $L_{0}$ is formally given by DDLS ; DLS ; CD ;
RO1 ; RO2 ; Higuchi ; Allen ; MM

$\displaystyle\frac{dL_{0}(\sigma)}{d\sigma}$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{D+1}{2}\right)}{4\pi^{\frac{D+1}{2}}}\left(\frac{\Lambda}{D-1}\right)^{D/2}\times$
(12)
$\displaystyle\left\\{\sum_{\ell=0}^{\infty}d_{2}(\ell)\left[\frac{\Lambda}{D-1}(\ell+2)(\ell+D+1)+\sigma\right]^{-1}\right.$
$\displaystyle\left.-\sum_{\ell=0}^{\infty}d_{1}(\ell)\left[\frac{\Lambda}{D-1}\ell(\ell+D+1)+\sigma\right]^{-1}\right\\}\,,$

where the degeneracies are

$d_{2}(\ell)\equiv\frac{(D+1)(D-2)}{2}(\ell+1)(\ell+D+2)(2\ell+D+3)\frac{(\ell+D-1)!}{(D-1)!(\ell+3)!}\,,$
(13)

and

$d_{1}(\ell)\equiv(\ell+1)(\ell+D)(2\ell+D+1)\frac{(\ell+D-2)!}{(D-2)!(\ell+2)!}\,.$
(14)

Note that $\frac{dL_{0}}{d\sigma}$ contains divergences, which must be
addressed by parameter renormalization. We will show concrete methods and
numerical results for four dimensional spacetime in the next section, and the
results for three dimensional spacetime in Appendix A.

##  III four dimensions ($D=4$)

###  III.1 Calculation of one-loop contribution

For four dimensions, the expression (12) becomes222In the limit of
$\Lambda\rightarrow 0$, we find that
$\frac{dL_{0}(\sigma)}{d\sigma}=\frac{1}{8\pi^{2}}\int\frac{k^{3}dk}{k^{2}+\sigma}=\frac{g}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}+\sigma}$
with $g=2$. According to DDLS , the absence of the discontinuity between the
massive case and massless case at the one-loop level has been noted (thus,
$g=2$ instead of $g=5$).

$\frac{dL_{0}(\sigma)}{d\sigma}=\frac{\Lambda}{16\pi^{2}}\left\\{\sum_{\ell=0}^{\infty}d_{2}(\ell)\left[(\ell+2)(\ell+5)+\frac{3\sigma}{\Lambda}\right]^{-1}-\sum_{\ell=0}^{\infty}d_{1}(\ell)\left[\ell(\ell+5)+\frac{3\sigma}{\Lambda}\right]^{-1}\right\\}\,,$
(15)

where

$d_{2}(\ell)=\frac{5}{6}(\ell+1)(\ell+6)(2\ell+7)\quad\mbox{and}\quad
d_{1}(\ell)=\frac{1}{2}(\ell+1)(\ell+4)(2\ell+5)\,.$ (16)

Using the integration formula

$\frac{1}{z^{2}-\beta^{2}}=\int_{0}^{\infty}e^{-zt}\frac{\sinh\beta
t}{\beta}dt\,,$ (17)

$\frac{dL_{0}}{d\sigma}$ is expressed as

$\displaystyle\frac{dL_{0}(\sigma)}{d\sigma}$ $\displaystyle=$
$\displaystyle\frac{\Lambda}{16\pi^{2}}\left\\{\frac{5}{16}\int_{\epsilon}^{\infty}\frac{e^{-\frac{7}{2}t}(2-7e^{t}+7e^{2t})}{\left(\sinh\frac{t}{2}\right)^{4}}\frac{\sinh\beta_{2}(\sigma)t}{\beta_{2}(\sigma)}dt\right.$
(18)
$\displaystyle\left.-\frac{1}{16}\int_{\epsilon}^{\infty}\frac{e^{-\frac{5}{2}t}(1-5e^{t}+10e^{2t})}{\left(\sinh\frac{t}{2}\right)^{4}}\frac{\sinh\beta_{1}(\sigma)t}{\beta_{1}(\sigma)}dt\right\\}\,,$

where

$\beta_{2}(\sigma)\equiv\sqrt{\frac{9}{4}-\frac{3\sigma}{\Lambda}}\quad\mbox{and}\quad\beta_{1}(\sigma)\equiv\sqrt{\frac{25}{4}-\frac{3\sigma}{\Lambda}}\,.$
(19)

Here, we introduced a UV regulator $\epsilon$ in order to avoid divergences at
the lower limit in the integrals.

Now, noting that

$\frac{\sinh\beta t}{\beta}=t+\frac{\beta^{2}}{6}t^{3}+O(t^{5})\,,$ (20)

we renormalize the divergent integral by defining the renormalized parameters
as follows:333We adopt a kind of “minimal subtraction” of divergences in the
integral in this paper.

$\displaystyle\frac{m^{2}}{\lambda}$ $\displaystyle=$
$\displaystyle\frac{m_{0}^{2}}{\lambda_{0}}+\frac{\Lambda}{16\pi^{2}}\left\\{\frac{5}{16}\int_{\epsilon}^{\infty}\frac{e^{-\frac{7}{2}t}(2-7e^{t}+7e^{2t})}{\left(\sinh\frac{t}{2}\right)^{4}}\left(t+\frac{3}{8}t^{3}\right)dt\right.$
(21)
$\displaystyle\left.-\frac{1}{16}\int_{\epsilon}^{\infty}\frac{e^{-\frac{5}{2}t}(1-5e^{t}+10e^{2t})}{\left(\sinh\frac{t}{2}\right)^{4}}\left(t+\frac{25}{24}t^{3}\right)dt\right\\}\,,$

and

$\displaystyle\frac{1}{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda_{0}}+\frac{1}{32\pi^{2}}\left\\{\frac{5}{16}\int_{\epsilon}^{\infty}\frac{e^{-\frac{7}{2}t}(2-7e^{t}+7e^{2t})}{\left(\sinh\frac{t}{2}\right)^{4}}t^{3}dt\right.$
(22)
$\displaystyle\left.-\frac{1}{16}\int_{\epsilon}^{\infty}\frac{e^{-\frac{5}{2}t}(1-5e^{t}+10e^{2t})}{\left(\sinh\frac{t}{2}\right)^{4}}t^{3}dt\right\\}\,.$

Then, the equation of motion for $\sigma$ (the gap equation) is rewritten as

$\frac{1}{\lambda}\sigma-\frac{m^{2}}{\lambda}-\frac{dL(\sigma)}{d\sigma}=0\,,$
(23)

where

$\frac{dL(\sigma)}{d\sigma}\equiv\frac{dL_{0}(\sigma)}{d\sigma}-\left(\frac{m^{2}}{\lambda}-\frac{m_{0}^{2}}{\lambda_{0}}\right)+\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)\sigma\,.$
(24)

After the renormalization, $\frac{dL(\sigma)}{d\sigma}$ is calculable by the
integral by setting $\epsilon\rightarrow 0$.

Figure 1: $\frac{1}{\Lambda}\frac{dL}{d\sigma}$ for the four dimensional case
as the function of $\frac{\sigma}{\Lambda}$.

We show $\frac{1}{\Lambda}\frac{dL}{d\sigma}$ as the function of
$\frac{\sigma}{\Lambda}$ in Fig. 1, which is obtained by numerical integration
in the case of four dimensions. We should notice that $\frac{dL}{d\sigma}$ is
calculable without regard to the Higuchi bound for $\sigma>0$. It can be seen
that $L$ is convex downward as a function of $\sigma$, since
$\frac{1}{\Lambda}\frac{dL}{d\sigma}$ is monotonously increasing as shown in
Fig. 1. The minimum of $L$ is located at $\sigma/\Lambda\approx 1.5$.

###  III.2 Negative coupling constant $\lambda$

For the $O(N)$ scalar theory, it is reported that there appears the $O(N)$
symmetric stable ground state when the renormalized coupling $\lambda$ is
negative KK ; AKS ; SAT . Therefore, we first examine the case with
$\lambda<0$ in our model.444Most recently, a preprint Weller  has appeared.
The preprint includes a recent discussion on the negative coupling constant in
scalar field theory and some important references.

Figure 2: The relation between $\frac{m^{2}}{\Lambda}$ and
$\frac{\sigma}{\Lambda}$ for some specific values of the coupling
$\lambda(<0)$. The curves correspond to $|\lambda|=0,30,60,90,120$, whose
leftmost points line up from top to bottom.

Since the equation of motion can be read as, for $\lambda<0$,

$\frac{m^{2}}{\Lambda}=\frac{\sigma}{\Lambda}+|\lambda|\frac{1}{\Lambda}\frac{dL}{d\sigma}\qquad(\lambda<0)\,,$
(25)

we can exhibit the relation between $\frac{m^{2}}{\Lambda}$ and the vacuum
expectation value of $\frac{\sigma}{\Lambda}$ for some specific values of the
coupling $\lambda$ in Fig. 2 (where we use the simple $\sigma$ instead of
$\langle\sigma\rangle$ for the vacuum expectation value). We show the region
where $\sigma/\Lambda>2/3$, above the Higuchi bound Higuchi1 ; Higuchi2 . The
lower limit of $m^{2}/\Lambda$ as a function of $|\lambda|$, which yields
$\sigma/\Lambda>2/3$, is shown in Fig. 3.

Figure 3: The lower limit of $m^{2}/\Lambda$ for $\sigma/\Lambda>2/3$, as a
function of $|\lambda|$.

###  III.3 Positive coupling constant $\lambda$

For $\lambda>0$, the equation of motion for $\sigma$ reads

$\frac{m^{2}}{\Lambda}=\frac{\sigma}{\Lambda}-\lambda\frac{1}{\Lambda}\frac{dL}{d\sigma}\qquad(\lambda>0)\,.$
(26)

Figure 4: The relation between $\frac{m^{2}}{\Lambda}$ and
$\frac{\sigma}{\Lambda}$ for some specific values of the coupling
$\lambda(>0)$. The curves correspond to
$\lambda=0,30,60,90,120,150,180,210,240,270,300$, whose leftmost points line
up from left to right.

We show the relation between $\frac{m^{2}}{\Lambda}$ and the vacuum
expectation value of $\frac{\sigma}{\Lambda}$ for some specific values of the
coupling $\lambda$ in Fig. 4. We find the region where $\sigma/\Lambda>2/3$,
above the Higuchi bound Higuchi1 ; Higuchi2 . The increasing lines in the
figure correspond to the extremum of the action, which is continuously
connected to that in the limit of $\lambda\rightarrow 0$. The lower limit of
$m^{2}/\Lambda$ as a function of $\lambda$, which yields $\sigma/\Lambda>2/3$,
is shown in Fig. 5.

Figure 5: The lower limit of $m^{2}/\Lambda$ for $\sigma/\Lambda>2/3$, as a
function of $\lambda$.

In the present analysis, we assume that $O(N)$ symmetry is unbroken. However,
especially in the case of $m^{2}<0$, which realizes for $\lambda>\approx 289$,
the unbroken $O(N)$ symmetry is problematic as in the case with the classical
Higgs potential. More detailed analysis, such as Ref. SAT  for scalar theory,
is needed to clarify the symmetry breaking in our present model. However,
since our model deals with fields with spacetime indices, similar analyses of
Ref. SAT  must be reconstructed, and further research is left as a future
topic.

##  IV Conclusion and outlook

In this paper, in the $O(N)$ spin-2 model, the effective mass in a strong-
coupling vacuum is studied by the large $N$ approximation, and the critical
values of the mass parameter, which exceeds the Higuchi bound, are numerically
estimated. If we take a large enough $m^{2}$, we can find a strong-coupling
vacuum with an effective mass that exceeds the Higuchi bound.

Furthermore, for sufficiently large coupling constant, it is possible that the
effective mass is created when the mass parameter is zero (if $|\lambda|>31$
for $\lambda<0$ and if $\lambda>289$ for $\lambda>0$, in the case with $D=4$).
There is also the well-known problem of discontinuity Hinterbichler ; deRham ;
DV  at $m=0$,555Another discontinuity is also found at
$m^{2}=\frac{D-2}{D-1}\Lambda$ Hinterbichler ; deRham ; DLS . so this point
may require additional study in the future, though the absence of the
discontinuity at one-loop level is reported in Ref. DDLS .

The bound has been obtained in our toy model, but we hope it makes some sense
in the choice of models in realistic models (including an infinite number of
fields). Also, in such a case, the quantum effects of other matter fields
should inevitably come into play. After incorporating them, we would like to
consider the running of coupling constants and possible phase transitions. The
stability of the vacuum is the most important topic that should be
investigated further.

In this paper, we have also presented expressions in general $D$ dimensions,
but it may be considered that renormalization or evaluation of parameters in
higher dimensions should be done more carefully than in four dimensions, so we
would like to leave it as a future task. However, like the calculation of
quantum effects in higher dimensional theory (e.g., CW ), it may be possible
to perform numerical calculations in a similar model in odd-dimensional
spacetime. We would like to consider such a issue in the future. We assumed
that the $O(N)$ symmetry is not broken in this paper (for simplicity as in the
initial stage of the study), but we would like to consider the case where
$O(N)$ is broken in the future. We wish to examine the model when the
background spacetime is also the Nariai spacetime Nariai , a higher-
dimensional Kaluza–Klein spacetime, etc. Paying attention to whether the
anisotropy can be avoided or favored, we want to study their consequences. In
the future, we would like to research bold hypotheses such as the self-
consistent de Sitter universe (e.g., Refs. LR ; WA ; AW ) in the strong-
coupling phase.

##  Appendix A THREE DIMENSIONS ($D=3$)

###  A.1 Calculation of one-loop contribution

As in the four dimensional case, we find the following. The formal expression
of the one-loop calculation reads, for $D=3$,

$\frac{dL_{0}}{d\sigma}=\frac{\sqrt{\Lambda}}{4\sqrt{2}\pi^{2}}\left\\{\sum_{\ell=0}^{\infty}d_{2}(\ell)\left[(\ell+2)(\ell+4)+\frac{2\sigma}{\Lambda}\right]^{-1}-\sum_{\ell=0}^{\infty}d_{1}(\ell)\left[\ell(\ell+4)+\frac{2\sigma}{\Lambda}\right]^{-1}\right\\}\,,$
(27)

where

$d_{2}(\ell)=2(\ell+1)(\ell+5)\quad\mbox{and}\quad
d_{1}(\ell)=2(\ell+1)(\ell+3)\,.$ (28)

As the treatment in the previous section, we can find its integration form:

$\displaystyle\frac{dL_{0}(\sigma)}{d\sigma}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\Lambda}}{4\sqrt{2}\pi^{2}}\left\\{\frac{1}{4}\int_{\epsilon}^{\infty}\frac{e^{-\frac{5}{2}t}(-3+5e^{t})}{\left(\sinh\frac{t}{2}\right)^{3}}\frac{\sinh\beta_{2}(\sigma)t}{\beta_{2}(\sigma)}dt\right.$
(29)
$\displaystyle\left.-\frac{1}{4}\int_{\epsilon}^{\infty}\frac{e^{-\frac{3}{2}t}(-1+3e^{t})}{\left(\sinh\frac{t}{2}\right)^{3}}\frac{\sinh\beta_{1}(\sigma)t}{\beta_{1}(\sigma)}dt\right\\}\,,$

where

$\beta_{2}(\sigma)\equiv\sqrt{1-\frac{2\sigma}{\Lambda}}\quad\mbox{and}\quad\beta_{1}(\sigma)\equiv\sqrt{4-\frac{2\sigma}{\Lambda}}\,.$
(30)

The renormalization should be done as

$\frac{m^{2}}{\lambda}=\frac{m_{0}^{2}}{\lambda_{0}}+\frac{\sqrt{\Lambda}}{16\sqrt{2}\pi^{2}}\left\\{\int_{\epsilon}^{\infty}\frac{e^{-\frac{5}{2}t}(-3+5e^{t})}{\left(\sinh\frac{t}{2}\right)^{3}}tdt-\int_{\epsilon}^{\infty}\frac{e^{-\frac{3}{2}t}(-1+3e^{t})}{\left(\sinh\frac{t}{2}\right)^{3}}tdt\right\\}\,,$
(31)

and the coupling constant $\lambda_{0}=\lambda$ does not undergo
renormalization correction. Then, we set

$\frac{dL(\sigma)}{d\sigma}\equiv\frac{dL_{0}(\sigma)}{d\sigma}-\left(\frac{m^{2}}{\lambda}-\frac{m_{0}^{2}}{\lambda_{0}}\right)\,.$
(32)

Figure 6: $\frac{1}{\sqrt{\Lambda}}\frac{dL}{d\sigma}$ in the three
dimensional case as the function of $\frac{\sigma}{\Lambda}$.

Qualitatively, it is the same as in the case of four dimensions. Note,
however, that the dimensions of the coupling constants are different. We show
$\frac{1}{\sqrt{\Lambda}}\frac{dL}{d\sigma}$ as the function of
$\frac{\sigma}{\Lambda}$ in Fig. 6 in the case of three dimensions. It can be
seen that $L$ is convex downward as a function of $\sigma$ as in the four
dimensional case. The minimum of $L$ is located at
$\sigma/\sqrt{\Lambda}\approx 3$.

###  A.2 Negative coupling constant $\lambda$

The equation of motion can be read as, for $\lambda<0$,

$\frac{m^{2}}{\Lambda}=\frac{\sigma}{\Lambda}+\frac{|\lambda|}{\sqrt{\Lambda}}\frac{1}{\sqrt{\Lambda}}\frac{dL}{d\sigma}\,,$
(33)

where $\frac{|\lambda|}{\sqrt{\Lambda}}$ is dimensionless. We show the
relation between $\frac{m^{2}}{\Lambda}$ and $\frac{\sigma}{\Lambda}$ for some
specific values of $\frac{|\lambda|}{\sqrt{\Lambda}}$ in Fig. 7. We show the
region where $\sigma/\Lambda>1/2$, which is above the Higuchi bound in three
dimensions.

Figure 7: the relation between $\frac{m^{2}}{\Lambda}$ and
$\frac{\sigma}{\Lambda}$ for some specific values of the coupling
$\lambda(<0)$. The lines correspond to
$|\lambda|/\sqrt{\Lambda}=0,30,60,90,120$, whose leftmost points line up from
top to bottom.

The lower limit of the mass parameter, which yields $\sigma/\Lambda>1/2$, is
shown in Fig. 8.

Figure 8: The lower limit of $m^{2}/\Lambda$ for $\sigma/\Lambda>1/2$, as a
function of $|\lambda|/\sqrt{\Lambda}$.

###  A.3 Positive coupling constant $\lambda$

The equation of motion can be read as, for $\lambda>0$,

$\frac{m^{2}}{\Lambda}=\frac{\sigma}{\Lambda}-\frac{\lambda}{\sqrt{\Lambda}}\frac{1}{\sqrt{\Lambda}}\frac{dL}{d\sigma}\,.$
(34)

We show the relation between $\frac{m^{2}}{\Lambda}$ and
$\frac{\sigma}{\Lambda}$ for some specific values of
$\frac{\lambda}{\sqrt{\Lambda}}$ in Fig. 9. We show the region where
$\sigma/\Lambda>1/2$, which is above the Higuchi bound in three dimensions.

Figure 9: the relation between $\frac{m^{2}}{\Lambda}$ and
$\frac{\sigma}{\Lambda}$ for some specific values of the coupling
$\lambda(>0)$. The lines correspond to
$\lambda/\sqrt{\Lambda}=0,30,60,90,120,150,180,210,240,270,300,330,360$, whose
leftmost points line up from left to right.

The lower limit of the mass parameter, which yields $\sigma/\Lambda>1/2$, is
shown in Fig. 10.

Figure 10: The lower limit of $m^{2}/\Lambda$ for $\sigma/\Lambda>1/2$, as a
function of $\lambda/\sqrt{\Lambda}$.

###### Acknowledgements.

The authors thank the anonymous referees for pointing out an error in the
early manuscript and the importance of the negative self-coupling constant.

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