# Investigation of ensemble methods for the detection of deepfake face
manipulations

Giatsoglou, Nikolaos
CERTH
<EMAIL_ADDRESS>Papadopoulos, Symeon
CERTH
<EMAIL_ADDRESS>Kompatsiaris, Ioannis
CERTH
<EMAIL_ADDRESS>
###### Abstract

The recent wave of AI research has enabled a new brand of synthetic media,
called deepfakes. Deepfakes have impressive photorealism, which has generated
exciting new use cases but also raised serious threats to our increasingly
digital world. To mitigate these threats, researchers have tried to come up
with new methods for deepfake detection that are more effective than
traditional forensics and heavily rely on deep AI technology. In this paper,
following up on encouraging prior work for deepfake detection with attribution
and ensemble techniques, we explore and compare multiple designs for ensemble
detectors. The goal is to achieve robustness and good generalization ability
by leveraging ensembles of models that specialize in different manipulation
categories. Our results corroborate that ensembles can achieve higher accuracy
than individual models when properly tuned, while the generalization ability
relies on access to a large number of training data for a diverse set of known
manipulations.

##  1 Introduction

Figure 1: Examples of face forgeries created with different generation and
manipulation methods. It is evident that different methods can create
deepfakes of varying quality. The examples are from the FaceForensics++ [1],
the ForgeryNet [2], and the Generated-Faces-in-the-Wild [3] datasets, except
for the stable diffusion images that were generated directly from the website
of Stability.AI (https://dreamstudio.ai).

New technologies bring new challenges. This is especially true for deepfakes,
a type of synthetic media with increased photorealism, which has been recently
made possible with advancements in machine learning (ML) and artificial
intelligence (AI) [4]. On one side, deepfakes promise to revolutionize the
media industry, including films and games, and to offer outlets of creativity
to independent creators and Internet users. On the other side, they pose
serious threats, for example, by easily generating and spreading political
propaganda, revenge porn, and other types of harmful content. Since many
deepfake cyberattacks rely on manipulating fake identities, research to date
has been primarily focusing on human face forgeries [5], examples of which are
shown in Fig. 1.

After acknowledging the serious threat of deepfakes, the research community
has shown increased interest on appropriate detection methods. While media
forensics based on conventional statistical and ML methods have been proposed,
a consensus has emerged that deep neural networks are more effective at
detecting the subtle artifacts of deepfakes [1]. In particular, deep learning
models can automatically learn highly discriminative features for deepfake
detection without relying on laborious handcrafted feature engineering.

Deep learning techniques, however, also have limitations. First, while they
achieve high detection accuracy on known manipulations, their accuracy drops
considerably on unseen manipulations [6] and is sensitive to processing
operations such as compression and resizing [1]. This is the generalization
problem of deepfake detection. Second, since forgeries are refined and become
undetectable to the human eye, detection models need to rely on increasingly
subtle artifacts and their decisions are hard to justify and explain. This is
the explainability problem of deepfake detection. Third, deepfake generation
methods continually improve and new methods come up, so that forensics turn
into an arms-race kind of problem. Finally, since deepfakes are ultimately
assorted pixels like real images and videos, it is conceivable that they will
be impossible to detect in the near future. Currently, there are efforts to
embed provenance information in media to securely track all possible
manipulations but the adoption of these technologies requires compliant
capturing devices and is at a preliminary stage111https://c2pa.org/. In the
meantime, detection algorithms are worthwhile.

From the ML perspective, deepfake detection is typically treated as a
classification problem, either at the video- or the image-/frame-level,
supplemented by related tasks such as temporal and spatial segmentation. The
deepfake class however is not clearly defined. Focusing on facial
manipulations, a deepfake can be understood as a face that deviates from a
representative collection of real human faces. Due to the photorealism of
deepfakes, high level artifacts are becoming difficult if not impossible to
detect; hence detection algorithms try to detect the subtle artifacts of the
generating algorithms. Since different generation algorithms produce different
artifacts, we believe that deepfake detection is best framed as an attribution
problem.

Attribution techniques have been used in the literature of generative
adversarial network (GAN) images [7], where different GAN architectures are
identified with specific fingerprints, and are also applicable to face
forgeries. An attractive approach to attribution is through an ensemble of AI
models that specialize in different forgeries, which can readily adapt to new
manipulations by incorporating new models to the ensemble. In addition,
ensembles are popular in the deepfake literature because they can pool the
performance of the base models and potentially generalize to unseen
manipulations. Indeed, we hypothesize that a sufficiently large ensemble of
manipulation methods can generalize better by detecting similarities of the
unseen manipulations with the known manipulations. There are many ways,
however, to build an ensemble and it is not clear which design is optimal. Due
to this, the aim of our work is to experimentally investigate different
designs of ensemble architectures for the detection of deepfakes.

##  2 Related Work

This section aims to give a brief overview of the deepfake detection
literature in order to better situate our work within it. Section 2.1
describes the basic forensic tasks for deepfakes. Section 2.2 describes the
main technical approaches for detection with an emphasis on deep neural
networks. Finally, Section 2.3 zooms in on the ensemble techniques and their
potential for deepfake detection.

###  2.1 Overview of deepfake forensic tasks

While traditional media forensics address operations of adding, editing, and
removing objects from media, deepfakes cover a variety of manipulations of
increased photorealism that are powered by AI technology. These include
synthesis of completely artificial images based on a reference collection, a
text input, or sketch, as well as style transfers and facial manipulations
[4]. For the detection of these forgeries, the literature in divided in two
branches: i) detection of GAN-generated images, and ii) detection of face
manipulations in videos [4]. Focusing on face manipulations, they can be
further distinguished in multiple subtypes such as face-swaps, i.e., replacing
a face with another one, face editing, i.e., changing the characteristics of a
face without replacing its identity, reenactment, i.e., changing the
expression of a face in alignment with another face, while taxonomies are
continually refined. Another approach, proposed by [2], is to split the face
forgeries into two broad categories, identity-remained and identity-replaced,
depending on if the identity in the tampered video is maintained. This
distinction is useful because identity-remained forgeries are more challenging
to detect due to the lack of blending artifacts.

The base forensic task for face manipulations is detection, i.e., recognizing
which images or video contain fake faces, which can be formulated as a binary
classification problem. [2] has extended this task to ternary classification
which tries to characterize the type of forgery as identity-remained or
identity-swapped, in an effort to provide more information for forgeries.
Following this line, more fine-grained tasks are possible that distinguish the
exact subtype of forgery or the underlying generation algorithm. Identifying
the generation algorithm is called source attribution and is popular in the
literature of GAN-generated images. This task tries to fingerprint the
artifacts of GAN models and/or the noise patterns of real cameras left in
authentic images [7]. Attribution however is less common in the literature of
face manipulation: to the authors’ knowledge, [8] is the only work where
attribution of face-swap models is the main focus. This is partly explained
because existing datasets frequently omit attribution labels to mimic a more
realistic setting but the trend may be reversed with newer datasets like
ForgeryNet [2]. In addition, [9] has produced preliminary evidence that more
fine-grained discrimination of manipulations can lead to better detection and
generalization to unseen manipulations.

Beyond classification problems, deepfake forensics also include segmentation
tasks either in the temporal domain, i.e., finding which frames of a video are
forged, or the spatial domain, i.e., finding which areas of a frame are
forged. Since faces need to be extracted from images, face recognition is also
relevant and used in the preprocessing step of the detection pipeline.
Although this preprocessing step is frequently taken for granted, false
positives can have a negative impact on the pipeline, as explored in [10].
With newer deepfake datasets, algorithms are also expected to discern and keep
track of multiple identities. Finally, explainability of detection, i.e.,
justifying the decision of detection algorithms to humans, is a widely
acknowledged problem but hard to evaluate and with no standardized tasks.
Currently, the literature uses heatmaps, produced by standard explainable AI
(XAI) techniques, that can reveal potentially tampered areas [11]. These
heatmaps are evaluated qualitatively [12] and are not straightforward to
interpret, considering the photorealism of deepfakes and the subtlety of the
detected artifacts.

Our approach: Our work focuses on detection but from an attribution
perspective. Inspired by [9], we believe that such as approach will become
important for generalization as more data for known manipulations become
available with newer datasets, e.g., [2]. We further believe that attribution
can be useful for explainabilty, directing human users to suspected types of
forgery.

Figure 2: Representative approaches for deepfake detection with handcrafted
and learned features. Notice that ensemble and multimodal techniques can be
used with either type of features.

###  2.2 Overview of deepfake detection methods

Initial approaches to deepfake detection used handcrafted features, such as
scale-invariant feature transform (SIFT) vectors, image spectra, media stream
descriptions, distorted facial landmarks, and biological signals [13]. While
these methods are easier to explain to humans, they have become less popular
than neural models that automatically learn features from pixels due to their
ability for optimal feature extraction, e.g., see [14]. In addition, deep
neural networks are considered more effective than their shallow counterparts
[15], especially after the key experimental work of [1]. Deep learning models
are capable of learning both low- and high-level features of media, due to the
pooling operations of subsequent layers, but are prone to overfitting: while
they display > 90% accuracy on the test set of the datasets used for training,
their accuracy is significantly decreased on unseen datasets. A significant
amount of work has thus focused on finding architectures that can generalize
better.

From a technical perspective, deepfake detectors operate at the image-/frame-
or the video-level. Frame-level detectors detect intra-frame inconsistencies
and aggregate the results to the whole video. They are typically based on
different flavors of convolutional neural networks (CNNs) [15] and, more
recently, on visual transformers (ViTs) that split an image into multiple
patches, e.g., [16]. While this approach cannot detect temporal and
audiovisual inconsistencies, frame-level detector are lightweight and
practical. On the other hand, video-level detectors operate on sequences of
frames and, possibly, audio. They typically extract features for consecutive
frames which are then passed in a temporal-aware model such as a recurrent
neural network (RNN), a long short-term memory (LSTM), or a transformer [15].
Other approaches include 3D CNNs, capsule networks, optical flow models, and
fusion techniques to incorporate audio information [15]. Video-level detection
has more potential to detect inconsistencies but it is significantly more
intensive in terms of computational resources.

Our approach: Our work focuses on frame-level detection due to its usefulness
for general image-level detection and its use to incorporate in ensembles.
Lightweight methods are also worthwhile for application in resource contrained
environments, e.g., edge devices.

###  2.3 Overview of ensemble techniques for deepfake detection

Ensembles of CNNs have become popular for deepfake detection, after they were
found in the top solutions of the DeepFake Detection Challenge (DFDC)
competition 222https://www.kaggle.com/competitions/deepfake-detection-
challenge/leaderboard. Ensemble learning is an ML technique which combines
multiple models so that their ensemble becomes more accurate and robust than
each individual model [17]. An ensemble architecture comprises the base
learners and a combination layer, which can be either another learner or an
aggregation function which does not require training. By combining multiple
simpler models, ensembles aspire to create more accurate and robust models
that generalize better. To achieve this, base models have to be reasonably
accurate and diverse.

[18, 19] proposed ensembles of CNNs with different architectures for
diversity. In particular, [18] used 7 different architectures 333The
architectures used were XceptionNet, ResNet101, InceptionResNetV2, MobileNet,
InceptionV3, DenseNet121, and DenseNet169. and combined their output with
another CNN meta learner, while [19] experimented with 26 isolated CCNs and
combined the ResNet models, which were found the least reliable, in order to
boost their performance. In contrast, [20] used a single CNN architecture,
Efficient-B4, and diversified it with attention and different training
procedures (supervised and unsupervised mode with triple loss). These works
achieved high accuracy on the datasets used for training but were not tested
on out-of-distribution datasets.

[21, 21] were inspired by the solutions of the DFDC challenge and experimented
with the top solutions. For example, [21] reused the architecture of the WM
team, which uses an ensemble of 3 models based on XceptionNet and
EfficientNet, and incorporated attention maps for explainability and
augmentation of the training dataset. [22] mixed the top solutions of DFDC and
experimented with different mixing strategies, in order to achieve higher
accuracy on the DFDC dataset. Other related works include [23], which combined
existing models that detect physiological signals, and [24], which ensembled
models that were trained on different facial landmarks. These approaches added
inductive bias in learning, which helps with their intepretability but also
have the limitation of hand-crafted features.

Finally, [25] comprehensively studied the impact of different fusion
techniques on performance, which is frequently unexplored. They used an
ensemble of 6 models444The architectures used were ResNet50, XceptionNet,
EfficientNet-B4 with/out attention and with supervised/unsupervised training.
and tested various parametric and non-parametric fusion methods. They found
that parametric techniques work much better than non-parametric technique but
their decisions are not easy to interpret and not readily extensible.

Our approach: From the above, we see that ensemble models are popular in the
literature as a way to boost the performance of already successful models but
they have not been studied consistently due to the large degrees of freedom in
the ensemble design. In this work, we want to compare how different
configurations of ensembles work and their impact on the generalization of
deepfake detection.

Figure 3: The architecture of our deepfake detector. The preprocessor samples
frames from an input video file with a constant rate and extracts faces from
them. A clustering approach that identifies faces belonging to the same
identity is used in order to avoid false results in the face extraction, like
in [10]. The ensemble detector employs a collection of $N$ AI models to
analyze the faces and a combiner to reach a robust score. Finally, the score
aggregator aggregates the scores of each frame and face to arrive at a video-
level score.

##  3 Methodology

Our deepfake detector operates at the frame level and its architecture is
shown in Fig. 3. The preprocessor is responsible for sampling video frames at
a constant rate, recognizing and cropping the present human faces, and
normalizing them to a specific size. After the faces are extracted, they are
clustered to unique identities and outliers are discarded, according to the
approach of [10]. This step is taken in order to avoid false positives from
the recognition module. Subsequently, the faces pass through the frame
detector, which tries to detect the presence of forgeries, and potentially
attribute the manipulation method. The resulting scores describe the
confidence of the faces being fake and are eventually aggregated to derive a
score for the whole video.

In the following, we focus on the design and performance of the frame
detector, on which the whole pipeline depends. In particular, we first define
the problem of deepfake detection formally from an ML persepctive, and then
describe the ensemble design that we investigate.

###  3.1 Problem setup

We denote a face by the triplet $(x,y,z)$ where $x$ is the face image, $y$ a
multiclass attribution label, and $z$ a binary detection label. In more
detail, the image $x$ is an array of dimensions $(C,H,W)$ where $C$ is the
number of color channels, and $H$, $W$ are the image’s height and width.
Considering a selection of $K$ possible manipulations, the attribution label
takes $K+1$ values where value $0$ represents a real face and the remaining
values the $K$ manipulations. Finally, the detection label is binary where
values $0$ and $1$ represent a real and a fake face respectively. If we know
the attribution label, it is easy to derive the detection label through the
following function:

$z=g(y)=\begin{cases}0&\text{if }y=0\\\ 1&\text{if }y>0\end{cases}.$ (1)

Based on the above, we can define the following tasks:

Detection task: We seek a function $\hat{p}(z|x)$ that approximates the
posterior distribution $p(z|x)$, so that:

$\hat{z}=\arg\max~{}\hat{p}(z|x)$ (2)

Attribution task: We seek a function $\hat{p}(y|x)$ that approximates the
posterior distribution $p(y|x)$, so that:

$\hat{y}=\arg\max~{}\hat{p}(y|x)$ (3)

With a dataset offering suitable labels, it is straightforward to train single
models for the detection or the attribution tasks, e.g., using the binary and
the multiclass cross-entropy loss. Notice that the attribution models can be
easily converted to the detection task by employing the function $g$ as
$\hat{z}=g(\hat{y})$.

###  3.2 Binary detection ensemble

For the binary detection ensemble, we group $N$ detection models, denoted by
$\hat{p}_{i}(z|x)$, whose training has been diversified. We then soft-combine
the models’ outputs so that

$\hat{p}(z|x)=\frac{\sum_{i=1}^{N}\hat{p}_{i}(z|x)}{N}.$ (4)

The final result is given by (2). This ensemble cannot be used for attribution
because it does not distinguish the different manipulations.

###  3.3 Multiclass attribution ensemble

For the multiclass attribution ensemble, we group $N$ attribution models,
denoted by $\hat{p}_{i}(y|x)$, whose training again has been diversified. We
then soft-combine the models’ outputs so that

$\hat{p}(y|x)=\frac{\sum_{i=1}^{N}\hat{p}_{i}(y|x)}{N}.$ (5)

The final result is given by (3) and can be used for detection by converting
to binary via the function $g$.

###  3.4 One-manipulation-vs-real ensemble

For the one-manipulation-vs-real ensemble, each model specializes in one of
the $K$ manipulation types and is trained as a binary classifier that
discriminate between the $i$-th manipulation and the real class. Letting
$s_{i}$ be the score of the $i$-th manipulation and $t$ a threshold, the final
decision is given through max pooling as

$\hat{y}=\begin{cases}\arg\max s_{i}&\text{if }\max\\{s_{i}\\}>t\\\ 0&\text{if
}\max\\{s_{i}\\}<t\end{cases},$ (6)

which can be converted to binary via the function $g$. This ensemble is very
convenient because it can be easily extended with new models that specialize
in different manipulations.

###  3.5 One-manipulation-vs-rest ensemble

For the one-manipulation-vs-rest ensemble, each model specializes again in one
of the $K$ manipulations but discriminates against all the remaining
manipulations. Again, considering the threshold $t$ to discern the real faces,
the final decision is taken via (6) The one-vs-rest ensemble is similar to the
one-vs-real ensemble but delineates the boundaries of each manipulation class
more clearly. The downside is that it requires the knowledge of the other
manipulation classes during training, hence it is not as extensible.

##  4 Results

In this section, we present the results of our experiments. Section 4.1 states
our experimental setup, including the used models, hyperparameters, and
training procedure. The following sections then describe our results on the
intra-dataset attribution task (Section 4.2), the intra-dataset detection task
(Section 4.3), and the cross-dataset detection task (Section 4.4).

###  4.1 Experimental setup

We trained all models with the FaceForensics++ dataset which contains ground
truth values about the manipulations. We used the official split of
FaceForensics++555https://github.com/ondyari/FaceForensics/tree/master/dataset/splits
and preprocessed the videos as described in Section 3 to create a dataset of
face images. In particular, we sampled the videos at $1$ fps and extracted the
faces with the MTCNN face detection module from the facenet-pytorch
library666https://pypi.org/project/facenet-pytorch/, which is popular in the
literature. The faces are subsequently resized to $(224,224)$ which is
compatible with the model used for detection, and augmented during training
for robustness similarly to the winning solution of the DFDC
challenge777https://github.com/selimsef/dfdc_deepfake_challenge#augmentations.

For the ensemble, we used the EfficientNet-B0 CNN model [26] with pretrained
weights on ImageNet for the base classifiers. The EfficientNet-B0 architecture
was selected because it is performant and sufficiently lightweight in
resources to construct big ensembles. The models were then trained with the
PyTorch framework, using binary and multiclass cross-entropy loss, weight
decay $5*10^{-}4$, and the Adam optimizer with learning rate 0.001. The models
were trained on the raw quality videos of the FaceForensics++ dataset for 40
epochs with an early stopping criterion of no improvement in validation
accuracy after 5 consecutive epochs. To address the class imbalance, we
oversampled the minority classes during training and used the balanced
accuracy metric to evaluate the models.

In more details, for evaluation, we tested our models in both the intra-
dataset scenario, i.e., on the test set of the FaceForensics++ dataset, and
the cross-dataset scenario, i.e., on the Celeb-DF, DFDC preview, DFDC, and
OpenForensics datasets. We evaluated both the attribution and detection task
in the intra-dataset scenario, and only the detection task in the cross-
dataset scenario, since the classes of the other datasets do not match. The
balanced accuracy metric is also different in the two tasks: for detection, it
is defined as:

$BA_{det}=0.5~{}p(z=0|x=0)+0.5~{}p(z=1|x=1)$ (7)

while for attribution, it is defined as:

$BA_{att}=0.5~{}p(y=0|x=0)+0.5~{}\frac{\sum_{i=1}^{K}p(y=i|x=i)}{K}.$ (8)

The intuition behind (7) and (8) is as follows. Since we do not know the prior
distribution of the real and fake faces, we give equal weights to the real and
fake class, and we evenly distribute the weight of the fake class to the
manipulation categories in the attribution task. In this manner, we avoid bias
towards the fake classes and evaluate our models on their performance on all
classes.

Figure 4: Balanced attribution accuracy for the raw and high quality version
of the FaceForensics++ test dataset with different thresholds of the one-vs-
real and one-vs-rest ensemble.

###  4.2 Intra-dataset attribution task

We first evaluate our models on the intra-dataset attribution task. Before
deriving our results, we investigated appropriate thresholds $t$ for the one-
vs-real and one-vs-rest ensembles. In particular, we evaluated the balanced
accuracy for the attribution task on the raw and high quality version of
FaceForensics++ for different values of $t$ spread evenly between $0.05$ and
$0.95$ with step $0.05$. The results are shown in Fig. 4. We see that for the
raw version of FaceForensics++, the accuracy increases monotonically with
higher thresholds up to the value 0.9. This implies that the ensembles are
both confident and accurate in detecting manipulations so that higher
thresholds avoid false positive detections for real images. In addition, the
one-vs-rest ensemble outperforms the one-vs-real ensemble for all thresholds.
The behavior is different however for the high quality version of
FaceForensics++ where the accuracy of both ensembles is lower and a better
trade-off is achieved in the mid-range thresholds. We also see that the
performance of the two ensembles is much closer and the best ensemble depends
on the threshold value. For better generalization and simplicity, we selected
the natural mid-range threshold $0.5$.

With these thresholds, the results of the intra-dataset attribution task are
shown in Table 1, which compares single attribution models with ensembles of 6
models, as is the number of classes in FaceForensics++. For the raw quality
dataset, we see that the accuracy of all models is in the $>90\%$ range. In
addition, the accuracy of the single models has small variation and their
ensemble is better that each individual one. Finally, the one-vs-real ensemble
is better than the one-vs-rest ensemble, as also seen in Fig. 4.

In the high quality dataset however, the behavior changes appreciably. First,
there is a large variation in the accuracies of the single models, which
highlights the importance of hyperparameter optimization for generalization.
Indeed, considering that the single models differed only on the random
parameters of the training procedure, the best single model managed to extract
better features for generalization due to pure luck. Second, the accuracy of
their ensemble is close to but not better than the accuracy of the best single
model. This indicates that ensembles is a practical way to enhance non-optimal
individual models but to ensure best ensemble performance, parametric
combination is needed. Third, the accuracies of the one-vs-real and one-vs-
rest ensembles are higher and that the one-vs-rest ensemble outperforms the
one-vs-real ensemble, as also seen in Fig. 4. This implies that the better
discriminative ability of the one-vs-rest ensemble is helpful for
generalization.

Finally, we see that the results on the low quality dataset are low, in some
cases lower than pure chance. This shows that our model cannot distinguish
manipulation artefacts when the images are heavily compressed and that a more
sophisticated approach is needed for their training. Nevertheless, in the next
section we evaluate if the detection models are still able to better detect
general manipulation artefacts from authentic faces.

| Single models | Ensembles
---|---|---
| Multiclass | Multiclass | One vs real | One vs rest
FF++raw | 93.35 - 96.25 | 96.45 | 94.43 | 91.88
FF++high | 76.29 - 84.78 | 83.51 | 83.95 | 85.74
FF++low | 43.89 - 53.69 | 52.28 | 46.00 | 46.76

Table 1:  Balanced accuracy for the attribution task evaluated on the test
split of the FaceForensics++ datasets.

###  4.3 Intra-dataset detection task

Table 2 shows the intra-dataset detection performance of our models. In this
case, we include the multiclass attribution models that are converted to
binary labels to check if the increased discrimination helps the detection
task. The table includes both single multiclass models and their ensemble. In
the raw quality version of the dataset, we see again that all models achieve
accuracy $>90\%$. The single multiclass models can potentially achieve higher
accuracy than their binary counterparts although their range of accuracies is
wider. The ensembles of both types of models achieve higher performance than
individual model, higher also than the one-vs-real and one-vs-rest ensembles.
We note again that the one-vs-real ensemble performs better in the raw quality
dataset.

As was the case in Section 4.2, the behavior is markedly changed in the high
quality dataset. Again, we note a significant variation in the accuracies of
the single models, both for the binary and the multiclass cases, and that
their ensembles do not achieve better performance than the individual models.
Most importantly, we see that the multiclass models achieve significantly
higher accuracy than the binary detectors, confirming the observations of [9].
Even more impressingly, the one-vs-real and one-vs-rest ensembles achieve the
highest accuracies, suggesting the ability of specialized models to better
detect forgeries than aggregate ones.

Finally, for the low quality dataset, we again note lower accuracies,
suggesting the weakness of our models to detect discriminative features under
heavy compression, however, the results are improved compared with the
attribution task. This implies that despite the poor attribution ability, the
models are still able to distinguish some general forgeries from authentic
faces. In this case, the one-vs-real achieves the best accuracy along with a
single attribution model.

Dataset | Single models | Ensembles
---|---|---
Binary | Multiclass | Binary | Multiclass | One vs real | One vs rest
FF++raw | 94.67 - 95.81 | 94.32 - 96.61 | 96.27 | 96.77 | 95.26 | 93.15
FF++high | 70.71 - 80.81 | 77.02 - 86.68 | 75.50 | 83.94 | 85.71 | 87.41
FF++low | 59.60 - 63.16 | 60.11 - 65.07 | 62.58 | 62.68 | 65.36 | 63.27

Table 2:  Balanced accuracy for the detection task evaluated on the test split
of the FaceForensics++ datasets.

###  4.4 Cross-dataset detection task

We conclude with our results on the cross-dataset detection task, shown in
Table 3. In general, we note that the accuracies are low for all datasets,
suggesting the poor generalization ability of our models. Again, we see that
the ensemble versions of the single models do not guarantee better performance
than the individual models. Indeed, this has been a constant characteristic in
all datasets except for the raw version of FaceForensics++ where our models
had the best performance. This could be a prerequisite condition for the
superiority of the averaging ensemble, otherwise a parametric approach is
required. Additionally, the one-vs-real ensemble achieves almost always the
best performance among all ensembles but, interestingly, it is comparable or
even lower than the best performance of individual models.

Despite the above, we must highlight that the perfomances are too close to
random chance to extract safe conclusions. The poor generalizibility of the
ensemble can be attributed to the age of the FaceForensics++ dataset. In
particular, except for CelebDF, all the unseen datasets were published after
FaceForensics++ and contained deepfakes of higher quality and diversity.
Additionally, the manipulations of FaceForensics++ are too few to achieve good
generalization. We believe that using a more recent and diverse dataset like
ForgeryNet, the generalization ability of the ensembles can be significantly
improved.

| Single models | Ensembles
---|---|---
| Binary | Multiclass | Binary | Multiclass | One vs real | One vs rest
CelebDF | 59.48 - 68.84 | 62.08 - 66.02 | 65.97 | 65.57 | 65.46 | 54.13
DFDC preview | 55.08 - 63.29 | 54.17 - 60.39 | 59.82 | 57.28 | 64.49 | 54.15
DFDC | 53.04 - 56.35 | 51.87 - 56.00 | 54.61 | 53.18 | 55.82 | 54.36
OpenForensics | 45.50 - 57.15 | 46.22 - 55.65 | 53.32 | 51.14 | 54.89 | 49.17

Table 3:  Balanced accuracy for the detection task on unseen datasets.

##  5 Conclusion

Deepfakes are a product of the recent AI revolution, which we currently do not
know how to handle. Fearing the potential damage to society and the
individual, the research community has sought to find adequate solutions for
the detection of deepfake media but no definitive solution has yet emerged. In
this paper, following on encouraging and under-explored leads in the
literature, we have investigated the potential of ensemble models to
successfully detect face forgeries through attribution, aspiring to generalize
on unseen manipulations. Our results have shown that, when properly tuned,
ensembles can indeed achieve superior performance than individual models but a
small number of manipulations is not sufficient for good generalization. In
the future, we plan to enhance our solution with greater diversity of
manipulations, specifically, the ForgeryNet dataset which we believe will
unlock more opportunities for generalization.

## Acknowledgment

This research was supported by the EU H2020 projects TruBlo (Grant Agreement
957228) and AI4Media (Grant Agreement 951911).

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the following, $T_{n}=\sum_{j=1}^{n}\tau_{n}$, and we use the notation
$d^{n}\tau=\prod_{k=1}^{n}\intop_{0}^{\infty}d\tau_{k}$)

$\displaystyle({\rm
H}_{1t}^{(n-1)})_{m\tilde{t}_{m^{\prime},n-1}}=(-1)^{n-1}\intop_{0}^{\infty}d^{n-1}\tau\sum_{\alpha_{1},\dots\alpha_{n}=2}^{4}$
(290) (291) $\displaystyle
I_{n}^{\alpha_{1}\dots\alpha_{n}}(\\{T(\\{\tau\\})\\})e^{(E_{*}-E_{m}^{ZM})T_{n-1}}$
$\displaystyle~{}~{}\times\braket{m}{\mathbf{g}_{\alpha_{1}}(T_{n-1})\dots\mathbf{g_{\alpha_{n}}}(0)}{m^{\prime}};$
(292)
$\displaystyle(G^{(ij)})_{\tilde{t}_{mi},\tilde{t}_{m^{\prime}j}}=(-1)^{(i+j)}\intop_{0}^{\infty}d^{i+j-1}\tau\sum_{\alpha_{1},\dots\alpha_{i+j}=2}^{4}$
(293) (294)
$\displaystyle{\color[rgb]{1,0,0}\tau_{j}}I_{i+j}^{\alpha_{1}\dots\alpha_{i+j}}(\\{T(\\{\tau\\})\\})e^{(E_{*}-E_{m}^{ZM})T_{i+j-1}}$
(295) (296)
$\displaystyle\times\braket{m}{:\mathbf{g}_{\alpha_{1}}(T_{i+j-1})\dots\mathbf{g}_{\alpha_{i+j}}(0)}{m^{\prime}}.$
(297)

We give the explicit forms of the objects $I_{n}^{k_{1},\dots k_{n}}$ below:

$\displaystyle I_{3}^{ijk}(\\{T\\})$ $\displaystyle=$ $\displaystyle
V_{ijk}(T_{2},T_{1},0);$ (298) $\displaystyle I_{4}^{ijkl}(\\{T\\})$
$\displaystyle=$ $\displaystyle V_{ijkl}(T_{3},T_{2},T_{1},0)$ (302)
$\displaystyle~{}~{}~{}~{}-V_{ij}(T_{3},T_{2})V_{kl}(T_{1},0);$ $\displaystyle
I_{5}^{ijklm}(\\{T\\})$ $\displaystyle=$ $\displaystyle
V_{ijklm}(T_{4},\dots,T_{1},0);$ (308) $\displaystyle\hskip-36.135pt-
V_{ijk}(T_{4},T_{3},T_{2})V_{lm}(T_{1},0)$ $\displaystyle\hskip-36.135pt-
V_{ij}(T_{4},T_{3})V_{klm}(T_{2},T_{1},0)$ $\displaystyle
I_{6}^{ijklmn}(\\{T\\})$ $\displaystyle=$ $\displaystyle
V_{ijklmn}(T_{5},\dots,T_{1},0)$ (318) $\displaystyle\hskip-36.135pt-
V_{ijkl}(T_{5},T_{4},T_{3},T_{2})V_{mn}(T_{1},0)$
$\displaystyle\hskip-36.135pt-
V_{ijk}(T_{5},T_{4},T_{3})V_{lmn}(T_{2},T_{1},0)$
$\displaystyle\hskip-36.135pt-
V_{ij}(T_{5},T_{4})V_{klmn}(T_{3},T_{2},T_{1},0)$
$\displaystyle\hskip-36.135pt+V_{ij}(T_{5},T_{4})V_{kl}(T_{3},T_{2})V_{mn}(T_{1},0);$
$\displaystyle I_{7}^{ijklmno}(\\{T\\})$ $\displaystyle=$ $\displaystyle
V_{ijklmno}(T_{6},\dots,T_{1},0)$ (334) $\displaystyle\hskip-36.135pt-
V_{ijklm}(T_{6},T_{5},T_{4},T_{3},T_{2})V_{no}(T_{1},0)$
$\displaystyle\hskip-36.135pt-
V_{ijkl}(T_{6},T_{5},T_{4},T_{3})V_{mno}(T_{2},T_{1},0)$
$\displaystyle\hskip-36.135pt-
V_{ijk}(T_{6},T_{5},T_{4})V_{lmno}(T_{3},T_{2},T_{1},0)$
$\displaystyle\hskip-36.135pt-
V_{ij}(T_{6},T_{5})V_{klmno}(T_{4},T_{3},T_{2},T_{1},0)$
$\displaystyle\hskip-36.135pt+V_{ijk}(T_{6},T_{5},T_{4})V_{lm}(T_{3},T_{2})V_{no}(T_{1},0)$
$\displaystyle\hskip-36.135pt+V_{ij}(T_{6},T_{5})V_{klm}(T_{4},T_{3},T_{2})V_{no}(T_{1},0)$
$\displaystyle\hskip-36.135pt+V_{ij}(T_{6},T_{5})V_{kl}(T_{4},T_{3})V_{mno}(T_{2},T_{1},0).$

Similarly to perturbation theory, the integrands in eqs. (292)-(297) simplify
considerably by introducing explicit time ordering and exploiting the
resulting permutation symmetry. However, additional care is needed in
symmetrizing the intrgration domain. Since the interaction consists of
multiple types of vertices (quadratic, cubic and quartic terms), it turned out
convenient to keep the $\tau=0$ operator distinguished, while the remaining
operators are assumed equivalent. In particular, we can write

$\displaystyle({\rm
H}_{1t}^{(3)})_{m\tilde{t}_{m^{\prime},3}}=-\intop_{0}^{\infty}d^{3}T$ (335)
(336)
$\displaystyle\left(\sum_{2\leq\alpha_{1}\leq\alpha_{2}\leq\alpha_{3}\leq\alpha_{4}}^{4}\frac{3!}{n_{2}!n_{3}!n_{4}!}V^{\mathrm{comp}}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}(\\{T\\})\right.$
(337) (338)
$\displaystyle~{}~{}\times\braket{m}{\hat{\mathcal{T}}\mathbf{g}_{\alpha_{1}}(T_{3})\dots\mathbf{g_{\alpha_{n}}}(0)}{m^{\prime}}\Biggr{)}$
(339) (340)
$\displaystyle+\frac{1}{2}\Theta(T_{1}-\min(T_{2},T_{3}))\Biggr{(}\sum_{\alpha_{1},\alpha_{2}=2}^{4}$
(341) (342) $\displaystyle
V_{\alpha_{1}\alpha_{1}}(T_{3},T_{2})V_{\alpha_{2}\alpha_{2}}(T_{1},0)$ (343)
(344)
$\displaystyle\braket{m}{\hat{\mathcal{T}}\mathbf{g}_{\alpha_{1}}(T_{3})\mathbf{g}_{\alpha_{1}}(T_{2})\mathbf{g_{\alpha_{2}}}(T_{1})\mathbf{g_{\alpha_{2}}}(0)}{m^{\prime}}\Biggr{)}$
(345)

where $\hat{\mathcal{T}}$ denotes time ordering so that larger time arguments
are arranged to the left. The variables $n_{2}$, $n_{3}$, $n_{4}$ refer to the
number of quadratic, cubic and quartic vertices excluding the one with zero
argument, while $\Theta(T)$ is the Heaviside unit step function. The
compressed correlator $V^{comp}_{\alpha_{1},\dots\alpha_{n}}$ is obtainted
from $V_{\alpha_{1},\dots\alpha_{n}}$ in the following way. First we remove
any disconnected Feynman diagrams appearing in
$V_{\alpha_{1},\dots\alpha_{n}}$. Second, we distribute diagrams which can be
transformed into each other by a permutation of the first $n-1$ vertices into
equivalence classes. We choose a representative diagram from each class so
that $\alpha_{1}\leq\alpha_{2}\leq\dots\leq\alpha_{n-1}$, and multiply it with
the number of diagrams in the class. Thus extra symmetry factors corresponding
to this partial symmetrization are obtained by an explicit counting of
equivalent diagrams.

The subset of Feynman diagrams containing only quartic vertices, for the
lowest few orders are pictured in Figures 19 and 20. While restricting the
evaluation to topologically different diagrams greatly reduces the complexity
of evaluations, one has to keep track of the time arguments in the zero mode
correlator, which have to be arranged “manually” into the correct time order
in their numerical evaluation.

From the viewpoint of Feynman diagrams, the $n$-point functions contain
disconnected pieces. (see e.g., Fig. 20 (a)). The explicit disconnected terms
of $I_{n}^{k_{1}\dots k_{n}}$ only partially cancel these terms, as the time
ordering of the latter is partially fixed. Nonetheless, these terms combine in
a nice way. Cancellation occurs whenever the disconnected correlator factors
can be arranged into at least two subsets, so that the narrowest time interval
encompassing all time arguments inside a single subset is disjoint from the
corresponding intervals of the other subsets (Figure 21). In other words,
there is a cutoff in the separation of disconnected pieces. Together with the
exponential decay of the propagator, this feature ensures that the dominant
contribution to the $\tau$-integral comes from the the region with all $\tau$
variables being small. When the $\tau$ integrands are symmetrized under $\tau$
ordering, the regions where the disconnected terms give a contribution needs
to be symmetrized explicitly.

The above method is suitable for calculating the ground state and zero-
momentum one-particle states of the interacting theory.

Things become more complicated if one is to calculate general excited states.
One could in principle include nontrivial oscillator states. When states
having non-trivial oscillator content are included among $\mathcal{H}_{1}$
states, one obtains Feynman diagrams with external lines, which adds to the
complexity of calculating the associated matrix elements.

###  C.2 Choice of $E_{*}$

All ‘conventional’ tail states depend on the energy parameter $E_{*}$.
Technically $E_{*}$ appears as the (a priori unknown) exact energy of the
eigenvalue that we seek. In Fig. 1, the emphasis was on the numerical
convergence in $N_{ZM}$ and so we took $E_{*}$ from reference data originally
obtained with the method described in [1]. In general it is tempting to
perform a self-consistent iteration to fix this energy parameter. However, we
argue that such an iteration is actually not favorable in general. The stage
of the derivation in which $E_{*}$ is identified with the exact energy
eigenvalue involves the full, non-truncated Hamiltonian. There is no reason
why an iteration in a restricted Hilbert space should be optimal in obtaining
the eigenvalue. A better point of view may be to consider $E_{*}$ as a
variational parameter, and look for the optimal set of tails by minimizing the
ground state energy with respect to $E_{*}$.

###  C.3 Feynman diagrams

Practically, we generate the Feynman diagrams as follows. In our setup the
relevant diagrams have no external legs, so we will restrict our attention to
these “vacuum” diagrams.

  1. 1.

For a given number of vertices $n$, generate all distinct ordered sets of
vertex ranks (no. of legs) from which at least one diagram can be assembled.
For the $\phi^{4}$ model, $2\leq\text{rank}\leq 4$ for each vertex and the sum
of all ranks has to be even. For example, there are six relevant ordered sets
for $n=3$. Henceforth we introduce the notation $(r_{1},r_{2},\dots,r_{n})$
for the set of diagrams having $n$ vertices with ranks $r_{1}$,…, $r_{n}$.

  2. 2.

To each collection of vertices, e.g. $\left(2,2,4,4\right)$, construct a list
of all possible Feynman diagrams by connecting the vertices in all possible
ways, such that no line starts and ends on the same vertex. Disconnected
diagrams are also kept. All possibilities are easily generated in Mathematica
as long as the number of vertices is not too large.

  3. 3.

Distribute the appropriate symmetry factors to the graphs. Each graph is
multiplied by

$S(\textit{graph})=\left(\frac{\prod_{i\in\textit{vertices}}\mathcal{R}_{i}!}{\prod_{i,j\in\textit{vertices};\>j>i}\mathcal{P}_{ij}!}\right)$
(346)

where $\mathcal{R}_{i}$ is the number of legs of vertex $i$, and
$\mathcal{P}_{ij}$ is the number of propagators between vertices $i$ and $j$.

  4. 4.

If $V^{\mathrm{comp}}$ is calculated, we select the connected Feynman diagrams
appearing in $V_{\alpha_{1},\dots\alpha_{n}}$. We distribute diagrams which
can be transformed into each other by a permutation of the first $n-1$
vertices into equivalence classes. We choose a representative diagram from
each class so that $\alpha_{1}\leq\alpha_{2}\leq\dots\leq\alpha_{n-1}$, and
multiply it with the number of diagrams in the class.

  5. 5.

Finally, the integrand corresponding to a given set of graphs is generated in
real (Euclidean) space. The formula is then exported to $C++$ syntax using
Mathematica’s CForm function. When the number of diagrams approaches the order
of $10^{4}$, the corresponding source code is in the order of megabytes, and
it becomes convenient to divide the integrand into a number of separate $C++$
functions and compile a static library. This greatly reduces compilation time.

(a) Partial cancellation for the lowest order disconnected diagrams. Here
$T_{3}\geq T_{2}\geq T_{1}\geq 0$. The top diagram is cancelled by the
explicit subtraction in $I_{4}^{ijkl}(\\{\tau\\})$, but the other two are not.

(b) An example of higher orders (arising in $I_{7}$): disconnected diagrams
with disjoint argument configurations (like the one depicted) cancel. Here
time increases from left to right (in particular, $T_{2}<T_{1}$.) The
correlator $\langle V(T_{6})V(T_{5})\rangle$ is completely separated in its
arguments from the other correlators which results in the cancellation by
$V(T_{6},T_{5})V(T_{4},T_{3},T_{2},T_{1},0)$.

Figure 21: Partial cancellation of disconnected terms

###  C.4 Feynman diagram data

####  C.4.1 First Krylov order (NLO): $2$ and $3$ vertices

At this order, the lowest order contributing diagrams have vertex content
$(2,2)$, $(3,3)$ and $(4,4)$, respectively. There are six relevant ordered
sets involving three vertices. A single possible Feynman diagram corresponds
to each set. They are depicted in Figure 5.

####  C.4.2 Second Krylov order (2NLO): $4$ and $5$ vertices

For $4$ vertices, there are $9$ different ordered rank sets. The corresponding
numbers of diagrams are shown in Table 2. For $5$ vertices, there are $12$
different ordered rank sets. The respective numbers of diagrams are depicted
in Table 2.

4-Vertex set | # diagrams | 5-Vertex set | # diagrams
---|---|---|---
(2,2,2,2) | 6 | (2,2,2,2,2) | 22
(2,2,2,4) | 3 | (2,2,2,2,4) | 21
(2,2,3,3) | 6 | (2,2,2,3,3) | 29
(2,2,4,4) | 6 | (2,2,2,4,4) | 29
(2,3,3,4) | 6 | (2,2,3,3,4) | 37
(2,4,4,4) | 6 | (2,2,4,4,4) | 46
(3,3,3,3) | 10 | (2,3,3,3,3) | 48
(3,3,4,4) | 10 | (2,3,3,4,4) | 58
(4,4,4,4) | 15 | (2,4,4,4,4) | 84
|  | (3,3,3,3,4) | 72
|  | (3,3,4,4,4) | 101
|  | (4,4,4,4,4) | 158
Table 2: Number of diagrams with $4$ and $5$ vertices before symmetrization

####  C.4.3 Third Krylov order (3NLO): $6$ and $7$ vertices

For $6$ vertices, there are $16$ different ordered rank sets. The
corresponding numbers of diagrams are shown on Table 3. For $7$ vertices,
there are $20$ different ordered rank sets. The corresponding numbers of
diagrams are shown on Table 4.

6-Vertex set | # diagrams | # connected | # compressed
---|---|---|---
(2,2,2,2,2,2) | 130 | 60 | 1
(2,2,2,2,2,4) | 130 | 90 | 2
(2,2,2,2,3,3) | 190 | 150 | 10
(2,2,2,2,4,4) | 209 | 163 | 12
(2,2,2,3,3,4) | 262 | 237 | 29
(2,2,2,4,4,4) | 346 | 309 | 37
(2,2,3,3,3,3) | 352 | 306 | 33
(2,2,3,3,4,4) | 449 | 425 | 123
(2,2,4,4,4,4) | 691 | 636 | 68
(2,3,3,3,3,4) | 574 | 532 | 32
(2,3,3,4,4,4) | 856 | 828 | 233
(2,4,4,4,4,4) | 1430 | 1360 | 74
(3,3,3,3,3,3) | 760 | 640 | 10
(3,3,3,3,4,4) | 1093 | 1023 | 60
(3,3,4,4,4,4) | 1819 | 1746 | 175
(4,4,4,4,4,4) | 3355 | 3150 | 42
Table 3: Number of diagrams with $6$ vertices before and after symmetrization. The ”compressed” column indicates the number of inqeuivalent connected diagram classes 7-Vertex set | # diagrams | # connected | # compressed
---|---|---|---
(2,2,2,2,2,2,2) | 822 | 360 | 1
(2,2,2,2,2,2,4) | 1005 | 630 | 3
(2,2,2,2,2,3,3) | 1402 | 1050 | 15
(2,2,2,2,2,4,4) | 1662 | 1250 | 19
(2,2,2,2,3,3,4) | 2159 | 1856 | 61
(2,2,2,2,4,4,4) | 3093 | 2622 | 88
(2,2,2,3,3,3,3) | 2878 | 2484 | 94
(2,2,2,3,3,4,4) | 3975 | 3669 | 382
(2,2,2,4,4,4,4) | 6453 | 5862 | 216
(2,2,3,3,3,3,4) | 5140 | 4746 | 139
(2,2,3,3,4,4,4) | 8188 | 7842 | 1129
(2,2,4,4,4,4,4) | 14613 | 13790 | 372
(2,3,3,3,3,3,3) | 6720 | 5940 | 70
(2,3,3,3,3,4,4) | 10466 | 9946 | 503
(2,3,3,4,4,4,4) | 18497 | 17960 | 1659
(2,4,4,4,4,4,4) | 35865 | 34350 | 361
(3,3,3,3,3,3,4) | 13440 | 12300 | 35
(3,3,3,3,4,4,4) | 23453 | 22483 | 608
(3,3,4,4,4,4,4) | 45103 | 43810 | 1103
(4,4,4,4,4,4,4) | 93708 | 90075 | 195
Table 4: Number of diagrams with $6$ vertices before and after symmetrization.
The ”compressed” column indicates the number of inqeuivalent connected diagram
classes

##  Appendix D Matrix elements for universal tails

Using the notations from Appendix C, the matrix elements of the Hamiltonian
and the Gram matrix between the universal tail states can be written as
($T_{n}=\sum_{j=1}^{n}\tau_{n}$, and we use the notation
$d^{n}\tau=\prod_{k=1}^{n}\intop_{0}^{\infty}d\tau_{k}$))

$\displaystyle\braket{\widetilde{t}_{i_{1},\cdots,i_{m}}}{\widetilde{\mathbf{V}}_{j}}{\widetilde{t}_{k_{1},\cdots,k_{n}}}$
$\displaystyle=$ $\displaystyle(-1)^{m+n}\intop d^{m+n}\tau$ (349)
$\displaystyle\hskip-72.26999pt\times
I_{m+n+1}^{i_{1},\cdots,i_{m},j,k_{1},\cdots,k_{n}}(\\{T(\\{\tau\\})\\})$
$\displaystyle\braket{\widetilde{t}_{i_{1},\cdots,i_{m}}}{\widetilde{\mathbf{H}}_{0,osc}^{(m)}}{\widetilde{t}_{k_{1},\cdots,k_{n}}}$
$\displaystyle=$
$\displaystyle-\braket{\widetilde{t}_{i_{1},\cdots,i_{m}}}{\widetilde{\mathbf{V}}_{k_{1}}}{\widetilde{t}_{k_{2},\cdots,k_{n}}}$
(351)
$\displaystyle\braket{\widetilde{t}_{i_{1},\cdots,i_{m}}}{\widetilde{t}_{k_{1},\cdots,k_{n}}}$
$\displaystyle=$ $\displaystyle(-1)^{m+n}\intop d^{m+n-1}\tau$ (355)
$\displaystyle\hskip-72.26999pt\times\tau_{n}I_{m+n}^{i_{1},\cdots,i_{m},k_{1},\cdots,k_{n}}(\\{T(\\{\tau\\})\\})$

In the above, the quantities $I_{n}^{k_{1},\dots k_{n}}$ were defined in eqs.
(298).

Note that the time arguments are in explicit time order and so the
disconnected pieces do not completely cancel the ones implicit in the
$n$-point functions $V_{ijk\dots}$. Since our $K=3$ basis spans $40$ states in
the oscillator subspace, in principle we would need to calculate a total of
$4100$ matrix elements corresponding to the operators
$\widetilde{\mathbf{V}}_{2}$ $\widetilde{\mathbf{V}}_{3}$,
$\widetilde{\mathbf{V}}_{4}$, $\widetilde{\mathbf{H}}_{0,osc}^{(m)}$ and the
Gram matrix $\widetilde{\mathbf{G}}$ (their representations are symmetric
matrices). However, due to their structure, it is sufficient to compute a
total of $1312$ matrix elements, $574$ of which are $12$ dimensional
integrals.

##  Appendix E Euclidean finite volume propagator

In this appendix we compute the finite volume propagators. For the massless
case, the finite volume two-point function is defined as

$G_{0}(t,x)=\left\langle\left|\left[\tilde{\varphi}_{+}(t,x),\tilde{\varphi}_{-}(0,0)-\right]\right|\right\rangle=\frac{1}{2L}\sum_{n\neq
0}\frac{e^{ik_{n}x-|k_{n}|t}}{|k_{n}|},$ (356)

which can be summed up analytically to yield

$G_{0}(t,x)=-\frac{1}{4\pi}\log\left[1+e^{\frac{-4\pi}{L}t}-2e^{\frac{-2\pi}{L}t}\cos\left(\frac{2\pi}{L}x\right)\right].$
(357)

In the massive case, the finite volume propagator $G_{m}(t,x)$ for a free
boson of mass $m$ is calculated in the following way. It follows from the
massive mode expansion that $G_{m}(t,x)$ can be written as

$G_{m}(t,x)=\frac{1}{2L}\sum_{n\neq
0}\frac{e^{ik_{n}x-\omega_{n}t}}{\omega_{n}}.$ (358)

In the following it will be convenient to explicitly subtract the zero mode
contribution

$G_{m}(t,x)=\frac{1}{2L}\sum_{n\in\mathbb{Z}}\frac{e^{ik_{n}x-\omega_{n}t}}{\omega_{n}}-\frac{1}{2Lm}e^{-mt},$
(359)

which allows us to write $G_{m}(t,x)$ in terms of a contour integral

$\displaystyle G_{m}(t,x)=-\frac{e^{-mt}}{2mL}$
$\displaystyle+\frac{1}{8\pi}\intop_{C}\frac{e^{ipL}}{e^{ipL}-1}(e^{ipx}+e^{-ipx})\frac{e^{-t\sqrt{p^{2}+m^{2}}}}{\sqrt{p^{2}+m^{2}}}dp,$
(360)

where we have also exploited the $x\rightarrow-x$ symmetry of the sum. The
contour $C$ consists of many disconnected pieces, encircling all poles on the
real axis in the positive direction. This contour is first deformed into two
horizontal sections. One section goes slightly below the real axis from
$-\infty-i\epsilon$ towards $+\infty-i\epsilon$ ($\epsilon>0$), while the
other goes above the real axis from $+\infty+\epsilon$ to $-\infty+\epsilon$.
From the contour, we subtract (and then add explicitly) the large volume
approximation of the sum,

$G_{m,\infty}=\frac{1}{4\pi}\intop_{-\infty}^{\infty}dp\frac{\cos(px)e^{-t\sqrt{p^{2}+m^{2}}}}{\sqrt{p^{2}+m^{2}}},$
(361)

after which the contours can be transformed into two large semicircles,
avoiding the vertical square root branch cuts above and below the real axis.

In the following we will rely on the identity

$\displaystyle
K_{0}(\sqrt{\rho^{2}+\tau^{2}})=\intop_{0}^{\infty}e^{-\sqrt{\rho^{2}+\tau^{2}}\cosh\theta}d\theta$
$\displaystyle=\frac{1}{2}\intop_{-\infty}^{\infty}e^{-r\cosh
u\cosh\theta}d\theta$ (362) (363)
$\displaystyle=\frac{1}{2}\intop_{-\infty}^{\infty}e^{-r\cosh^{2}u\cosh\theta-r\cosh
u\sinh u\sinh\theta}d\theta.$ (364)

where $K_{0}(x)$ is the modified Bessel function of the second kind.

Let us first study the large-volume asymptotic $G_{m,\infty}(t,x)$. Changing
the integration variable to $p=m\sinh\theta$, we obtain

$G_{m,\infty}=\frac{1}{4\pi}\intop_{-\infty}^{\infty}\cos(mx\sinh\theta)e^{-mt\cosh\theta}.$
(365)

Comparing this form to the integral eq. (364), we obtain

$G_{m,\infty}=\frac{1}{2\pi}K_{0}\left(m\sqrt{t^{2}+x^{2}}\right).$ (366)

The contribution of the contours tightened around the branch cuts can be
written in the form

$G_{m,\Delta}=\frac{1}{4\pi}\intop_{0}^{\infty}\frac{4\cos(mt\sinh\theta)\cosh(mx\cosh\theta)}{e^{mL\cosh\theta}-1}.$
(367)

Expanding the denominator in the small parameter $e^{-mL\cosh\theta}$, we
obtain an infinite series of terms of the form (364). In this way we get
($0<x<L$)

$\displaystyle G_{m}(t,x)$ $\displaystyle=$ $\displaystyle
G_{m,\infty}+G_{m,\Delta}$ (368) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}K_{0}\left(m\sqrt{x^{2}+t^{2}}\right)$ (376)
$\displaystyle+\frac{1}{2\pi}K_{0}\left(m\sqrt{(L-x)^{2}+t^{2}}\right)+\sum_{n=1}^{\infty}\delta
G_{n},$ $\displaystyle\delta
G_{n}=\frac{1}{2\pi}K_{0}\left(m\sqrt{(nL+x)^{2}+t^{2}}\right)$
$\displaystyle+\frac{1}{2\pi}K_{0}\left(m\sqrt{((n+1)L-x)^{2}+t^{2}}\right).$

This form is manifestly periodic in $x$ with period $L$.

##  Appendix F Evaluating the integrals

To obtain the numerical matrix elements, the integration was performed using
the globally adaptive Cuhre algorithm of the Cuba package. This is a very
competitive algorithm for multidimensional integrals of moderate dimensions,
with the error generally reducing as $N_{\mathrm{eval}}^{-1}$ instead of
$N_{\mathrm{eval}}^{-1/2}$ common in Monte Carlo methods.

The integration used around $50$ million function evaluations for each matrix
element. This results in a relative precision of order $10^{-3}$ for matrix
elements of the third order tail states.

For the actual computations it was advantageous to use polar coordinates in
the time subspace for $4>2$ dimensional integrals. The parameterization that
we used is as follows:

$\displaystyle t_{1}$ $\displaystyle=r\cos\varphi_{1}$ $\displaystyle t_{2}$
$\displaystyle=t_{1}\tan\varphi_{1}\cos\varphi_{2}$ $\displaystyle\vdots$
$\displaystyle t_{D/2-1}$
$\displaystyle=t_{D/2-2}\tan\varphi_{D/2-2}\cos_{D/2-1}$ $\displaystyle
t_{D/2}$ $\displaystyle=t_{D/2-1}\tan\varphi_{D/2-1}$

and where we introduce $r=R_{0}\rho$, with a rescale parameter $R_{0}=40$.

For the numerical integration, we have to transform the integral into the unit
hypercube. The angles are naturally bounded as

$\left\\{\varphi_{1},\dots,\varphi_{D/2}\right\\}\quad\in\left[0,\frac{\pi}{2}\right],$
(377)

while the space coordinates are bounded as

$\left\\{x_{1},\dots,x_{D/2}\right\\}\quad\in\left[0,L\right].$ (378)

We also restrict $\rho$ to the interval $[0,1]$, so $R_{0}$ amounts to a
temporal cutoff. It is chosen so as the results are independent of it. For the
integration, these variables are rescaled to their barred counterparts
$\bar{\varphi}_{i},\bar{x_{i}}$, so that they are bounded in the
$\left[0,1\right]$ interval. Thus the final Jacobi determinant takes the form

$J=R_{0}^{D/2-1}\rho^{D/2-1}L^{D/2}\left(\frac{\pi}{2}\right)^{D/2-1}\prod_{k=1}^{D/2-2}\left(\sin\varphi_{k}\right)^{\frac{D}{2}-1-k},$
(379)

(note that (379) is expressed using the non-rescaled angles, but the
integration is understood with respect to the barred ones, hence the extra
numerical prefactor.)

For the modified Bessel functions, the C++ code uses the implementation of the
alglib package. A technical point is that $\mathrm{K}_{0}\left(x\right)$
behaves as $-\ln x+\mathrm{const}$ in the $x\rightarrow 0$ limit. Therefore,
it was necessary to define a regularized function with a momentum cutoff
$\delta_{\text{cut}}^{-1}$

$\displaystyle\tilde{\mathrm{K}}_{0}\left(x\right)$
$\displaystyle=\begin{cases}\mathrm{K}_{0}\left(x\right),&x>\delta_{\text{cut}}\\\
\mathrm{K}_{0}\left(\delta_{\text{cut}}\right),&x<=\delta_{\text{cut}}\end{cases}$

and $\tilde{\mathrm{K}}_{0}$ is used instead of $\mathrm{K}_{0}$ in the
numerical computations.

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