# Consistent Video-to-Video Transfer Using Synthetic Dataset

Jiaxin Cheng, Tianjun Xiao & Tong He
Amazon Web Services Shanghai AI Lab
<EMAIL_ADDRESS>

###### Abstract

We introduce a novel and efficient approach for text-based video-to-video
editing that eliminates the need for resource-intensive per-video-per-model
finetuning. At the core of our approach is a synthetic paired video dataset
tailored for video-to-video transfer tasks. Inspired by Instruct Pix2Pix’s
image transfer via editing instruction, we adapt this paradigm to the video
domain. Extending the Prompt-to-Prompt to videos, we efficiently generate
paired samples, each with an input video and its edited counterpart. Alongside
this, we introduce the Long Video Sampling Correction during sampling,
ensuring consistent long videos across batches. Our method surpasses current
methods like Tune-A-Video, heralding substantial progress in text-based video-
to-video editing and suggesting exciting avenues for further exploration and
deployment. https://github.com/amazon-science/instruct-video-to-
video/tree/main

Figure 1: InsV2V has versatile editing capabilities encompassing background,
object, and stylistic changing. Our method adopts a one-model-all-video
strategy, achieving comparable performance while necessitating only inference.
InsV2V eliminates the need to specify prompts for both original and target
videos, simplifying the process by requiring only an edit prompt, thereby
enhancing intuitiveness in video editing.

##  1 Introduction

Text-based video editing Wu et al. (2022); Zhao et al. (2023); Wang et al.
(2023a); Qi et al. (2023); Liu et al. (2023) has recently garnered significant
interest as a versatile tool for multimedia content manipulation. However,
existing approaches present several limitations that undermine their practical
utility. Firstly, traditional methods typically require per-video-per-model
finetuning, which imposes a considerable computational burden. Furthermore,
current methods require users to describe both the original and the target
video Wu et al. (2022); Zhao et al. (2023); Wang et al. (2023a); Qi et al.
(2023); Liu et al. (2023). This requirement is counterintuitive, as users
generally only want to specify what edits they desire, rather than providing a
comprehensive description of the original content. Moreover, these methods are
constrained to individual video clips; if a video is too long to fit into
model, these approaches fail to ensure transfer consistency across different
clips.

To overcome these limitations, we introduce a novel method with several
distinctive features. Firstly, our approach offers a universal one-model-all-
video transfer, freeing the process from per-video-per-model finetuning.
Moreover, our model simplifies user interaction by only necessitating an
intuitive editing prompt, rather than detailed descriptions of both the
original and target videos, to carry out desired alterations. Secondly, we
develop a synthetic dataset precisely crafted for video-to-video transfer
tasks. Through rigorous pairing of text and video components, we establish an
ideal training foundation for our models. Lastly, we introduce a sampling
method specifically tailored for generating longer videos. By using the
transferred results from preceding batches as a reference, we achieve
consistent transfers across extended video sequences.

We introduce Instruct Video-to-Video (InsV2V), a diffusion-based model that
enables video editing using only an editing instruction, eliminating the need
for per-video-per-model tuning. This capability is inspired by Instruct
Pix2Pix Brooks et al. (2023), which similarly allows for arbitrary image
editing through textual instructions. A significant challenge in training such
a model is the scarcity of naturally occurring paired video samples that can
reflect an editing instruction. Such video pairs are virtually nonexistent in
the wild, motivating us to create a synthetic dataset for training.

Our synthetic video generation pipeline builds upon a large language model
(LLM) and the Prompt-to-Prompt Hertz et al. (2022) method that is initially
designed for image editing tasks (Figure 2). We use an example-driven in-
context learning approach to guide the LLM to produce these paired video
descriptions. Additionally, we adapt the Prompt-to-Prompt (PTP) method to the
video domain by substituting the image diffusion model with a video
counterpart Ho et al. (2022c). This modification enables the generation of
paired samples that consist of an input video and its edited version,
precisely reflecting the relationships delineated by the editing prompts.

In addressing the limitations of long video editing in conventional video
editing methods, we introduce Long Video Sampling Correction (LVSC). This
technique mitigates challenges arising from fixed frame limitations and
ensures seamless transitions between separately processed batches of a lengthy
video. LVSC employs the final frames of the previous batch as a reference to
guide the generation of subsequent batches, thereby maintaining visual
consistency across the entire video. We also tackle issues related to global
or holistic camera motion by introducing a motion compensation feature that
uses optical flow. Our empirical evaluations confirm the effectiveness of LVSC
and motion compensation in enhancing video quality and consistency.

##  2 Related Work

Diffusion Models The advent of the diffusion model Sohl-Dickstein et al.
(2015); Ho et al. (2020) has spurred significant advancements in the field of
image generation. Over the course of just a few years, we have observed the
diffusion model making groundbreaking progress in a variety of fields. This
includes areas such as super-resolution Saharia et al. (2022c), colorization
Saharia et al. (2022a), novel view synthesis Watson et al. (2022), style
transfer Zhang et al. (2023), and 3D generation Poole et al. (2022); Tang et
al. (2023); Cheng et al. (2023b). These breakthroughs have been achieved
through various means. Some are attributable to the enhancements in network
structures such as Latent Diffusion Models (also known as Stable Diffusion)
Rombach et al. (2022), GLIDE Nichol et al. (2022), DALLE2 Ramesh et al.
(2022), SDXL Podell et al. (2023), and Imagen Saharia et al. (2022b). Others
are a result of improvements made in the training paradigm Nichol & Dhariwal
(2021); Song & Ermon (2019); Dhariwal & Nichol (2021); Song et al. (2020b; a).
Furthermore, the ability to incorporate various conditions during image
generation has played a crucial role. These conditions include elements such
as layout Cheng et al. (2023a); Rombach et al. (2022), segmentation Avrahami
et al. (2023; 2022); Balaji et al. (2022); Yang et al. (2023a), or even the
use of an image as reference Mou et al. (2023); Ruiz et al. (2023); Gal et al.
(2022).

Diffusion-based Text-Guided Image Editing Image editing is a process where we
don’t desire completely unconstrained generation but modifying an image under
certain guidance (i.e.a reference image) during its generation. Various
methods have been proposed to address this task. Simple zero-shot image-to-
image translation methods, such as SDEdit Meng et al. (2021) performed through
diffusion and denoising on reference image. Techniques that incorporate a
degree of optimization, such as Imagic Kawar et al. (2023), which utilizes the
concept of textual inversion Gal et al. (2022), and Null-text Inversion Mokady
et al. (2023), which leverages the Prompt-to-Prompt strategy Hertz et al.
(2022) to control the behavior of cross-attention in the diffusion model for
editing, have also been explored. These methods can impede the speed of
editing due to the necessity for per-image-per-optimization. Models like
Instruct Pix2Pix Brooks et al. (2023) have been employed to achieve image
editing by training on synthetic data. This approach adeptly balances editing
capabilities and fidelity to the reference image.

Diffusion-based Text-Guided Video Editing The success of the diffusion model
in image generation has been extended to video generation as well Ho et al.
(2022b); Harvey et al. (2022); Blattmann et al. (2023); Mei & Patel (2023); Ho
et al. (2022a), and similarly, text-guided video editing has sparked interest
within the community. Techniques akin to those in image editing have found
applications in video editing. For instance, Dreamix Molad et al. (2023) uses
diffusion and denoising for video editing, reminiscent of the approach in
SDEdit Meng et al. (2021). Strategies altering the behavior of the cross-
attention layer to achieve editing, like the Prompt-to-Prompt Hertz et al.
(2022), have been adopted by methods such as Vid2Vid-Zero Wang et al. (2023a),
FateZero Qi et al. (2023), and Video-p2p Liu et al. (2023). Recent
developments Wang et al. (2023b); Esser et al. (2023); Zhao et al. (2023)
leverage certain condition modalities extracted from the original video, like
depth maps or edge sketches, to condition the video generation. The related,
albeit non-diffusion-based, method Text2live Bar-Tal et al. (2022) also
provides valuable perspectives on video editing.

Figure 2: The pipeline for generating a synthetic dataset using a large
language model, whose outputs include the prompt triplet consisting of input,
edit, and edited prompts, as well as a corresponding pair of videos.
Visualization of generated videos can be found in Appendix B

##  3 Synthetic Paired Video Dataset

A crucial element for training a model capable of arbitrary video-to-video
transfer, as opposed to a per-video-per-model approach, lies in the
availability of ample paired training data. Each pair consists of an input
video and its corresponding edited version, providing the model with a diverse
range of examples essential for generalized performance. However, the scarcity
of naturally occurring video pairs with such correspondence poses a
significant challenge to the training process.

To address this, we introduce the concept of a trade-off between initial
training costs, including dataset creation, and long-term efficiency. We
advocate for the use of synthetic data, which, while incurring an upfront
cost, accurately maintains the required correspondence and fulfills the
conditions for effective video-to-video transfer learning. The merit of this
synthetic data generation approach is underscored by its potential to offset
the initial investment through substantial time savings and efficiency in the
subsequent inference stages. This approach contrasts with ‘per-vid-per-model’
methods that necessitate repetitive fine-tuning for each new video, making our
strategy both cost-effective and practical in diverse real-world applications.

The potential of synthetic data generation, well-documented in the realm of
text-guided editing Brooks et al. (2023), is thereby extended to video-to-
video transfer. This method allows us to construct the optimal conditions for
the model to learn, offering a practical solution to the inherent obstacles
associated with acquiring matching real-world video pairs.

###  3.1 Dataset Creation:

In order to generate the synthetic dataset, we leverage the approach of
Prompt-to-Prompt (PTP) Hertz et al. (2022), a proven method for producing
paired samples in the field of image editing. The PTP employs both self-
attention and cross-attention replacements to generate semantically aligned
edited images. In self-attention, the post-softmax probability matrix of the
input prompt replaces that of the edited prompt. The cross-attention
replacement specifically swaps the text embedding of the edited prompt with
that of the input prompt.

In the context of video-to-video transfer, we adapt PTP by substituting its
underlying image diffusion models with a video diffusion model. In addition,
we extend the self-attention replacement to temporal attention layers, a
critical modification for maintaining structural coherence between input and
edited videos. Figure 2 shows the overall pipeline for data generation. To
guide the synthetic data generation, we employ a set of paired text prompts,
comprising an input prompt, an edited prompt, and an edit prompt. The input
prompt corresponds to the synthetic original video, while the edited prompt
and the edit prompt represent the desired synthetic edited video and the
specific changes to be applied on the original video respectively.

###  3.2 Prompt Sources

Our synthetic dataset is constructed using paired prompts from two
differentiated sources, each serving a specific purpose in the training
process. The first source, LAION-IPTP, employs a fine-tuned GPT-3 model to
generate prompts based on 700 manually labeled captions from the LAION-
Aesthetics dataset Brooks et al. (2023). This yielded a set of 450,000 prompt
pairs, of which 304,168 were utilized for synthetic video creation. While the
GPT-3-based prompts offer a substantial volume of data, they originate from
image captions and thus have limitations in their applicability to video
generation. This led us to incorporate a second source, WebVid-MPT, which
leverages video-specific captions from the WebVid 10M dataset Bain et al.
(2021). Using MPT-30B Team (2023) in a zero-shot manner, we devised a set of
guidelines (see Appendix A for details) to generate the needed paired prompts,
adding an additional 100,000 samples to our dataset. Crucially, the WebVid-MPT
source yielded a threefold increase in the success rate of generating usable
samples compared to the LAION-IPTP source after sample filtering, reinforcing
the need for video-specific captions in the training process. The LAION-IPTP
prompt source demonstrated a success rate of 5.49% (33,421 successful
generations from 608,336 attempts). The WebVid-MPT prompt source showed a
higher success rate of 17.49% (34,989 successful generations from 200,000
attempts).

###  3.3 Implementation Details and Sample Selection Criteria

Implementation Details: We use public available text-to-video
model111https://modelscope.cn/models/damo/text-to-video-synthesis/summary for
generating synthetic videos. Each video has 16 frames, processed over 30 DDIM
Song et al. (2020a) steps by the diffusion model. The self-attention and
cross-attention replacements in the Prompt-to-Prompt model terminate at a
random step within the ranges of 0.3 to 0.45 and 0.6 to 0.85 respectively (out
of 30 steps). The classifier-free guidance scale is a random integer value
between 5 and 12.

Sample Selection Criteria: To ensure the quality of our synthetic dataset, we
employ a CLIP-based filtering. For each prompt, generation is attempted with
two random seeds, and three distinct clip scores are computed: CLIP Text
Score: Evaluates the cosine similarity between each frame and the text. CLIP
Frame Score: Measures the similarity between original and edited frames. CLIP
Direction Score: Quantifies the similarity between the transition of original-
frame-to-edited-frame and original-text-to-edited-text. These scores are
obtained for each of the 16 frames and averaged. A sample is preserved if it
meets the following conditions: CLIP Text Score $>$ 0.2 for both original and
edited videos, CLIP Direction Score $>$ 0.2, and CLIP Frame Score $>$ 0.5.
Samples that fail to meet these criteria are discarded.

Figure 3: (a) The architecture of InsV2V. For handling long videos processed
in multiple batches, our approach leverages the proposed LVSC to utilize the
final frames from the preceding batch as reference frames for the subsequent
batch. (b) The inflated convolutionals and attention layer, as well as
temporal attention layer can handle 5D video tensors by dynamically reshaping
them. (c) During each denoising iteration, the LVSC adjusts the predicted
noise $\tilde{\varepsilon}_{\theta}(z_{t})$ based on reference frames
$z_{t}^{ref}$ prior to executing the DDIM denoising.

##  4 Model Architecture

###  4.1 Preliminaries

Diffusion models learn to predict content in an image by iteratively denoising
an entirely random Gaussian noise. In the training process, the input image is
first corrupted by the Gaussian noise, termed diffusion. The model’s task is
to restore a diffused noisy image to its original form. This process can be
considered as the optimization of the variational lower bound on the
distribution $p(x)$ of the image $x$ with a $T$ step Markovian inverse
process. Various conditions $c$, such as text, images, and layouts, can be
incorporated into diffusion models during the learning process. The model
$\varepsilon_{\theta}$ we aim to train is conditioned on text and its training
loss can be expressed as

$L=\mathbb{E}_{x,\varepsilon\sim\mathcal{N}(0,1),t}\|\varepsilon-\varepsilon_{\theta}(x_{t},t,c)\|^{2}$
(1)

Where $x_{t}$ is the result of diffusing the input image $x$ at timestep
$t\in[1,T]$ using random noise $\varepsilon$. In practice, we employ the
Latent Diffusion Model (LDM) Rombach et al. (2022) as our backbone model and
condition it on input videos by concatenating the videos to the latent space,
as illustrated in Figure 3 (a). Instead of generating images in the RGB pixel
space, LDM employs a trained Vector Quantized Variational AutoEncoder (VQVAE)
Esser et al. (2021) to convert images to visual codes in latent space. This
allows the model to achieve better results with the same training resources.
Specifically, the image $x$ is first transformed into a latent code
$z=\mathrm{VQEnc}(x)$, and the model learns to predict $p(z)$ from random
noise. In the context of the video editing, two distinct conditions exist: the
editing instructions and the reference video, represented by $c_{T}$ and
$c_{V}$ respectively. The model is designed to predict $p(z)$ by optimizing
the following loss function:

$L=\mathbb{E}_{\mathrm{VQEnc}(x),\varepsilon\sim\mathcal{N}(0,1),t}\|\varepsilon-\varepsilon_{\theta}(z_{t},t,c_{V},c_{T})\|^{2}$
(2)

###  4.2 Inflate Image-to-Image Model To Video-to-Video Model

Given the substantial similarities between image-to-image transfer and video-
to-video transfer, our model utilizes a foundational pre-trained 2D image-to-
image transfer diffusion model. Using foundational model simplifies training
but falls short in generating consistent videos, causing noticeable jitter
when sampling frames individually. Thus, we transform this image-focused model
into a video-compatible one for consistent frame production. We adopt model
inflation as recommended in previous studies Wu et al. (2022); Guo et al.
(2023). This method modifies the single image diffusion model to produce
videos. The model now accepts a 5D tensor input $x\in\mathbb{R}^{b\times
c\times f\times h\times w}$. Given its architecture was designed for a 4D
input, we adjust the convolutional and attention layers in the model (Figure 3
(b)). Our inflation process involves: (1) Adapting convolutional and attention
layers to process a 5D tensor by reshaping it temporarily to 4D. Once
processed, it’s reverted to 5D. (2) Introducing temporal attention layers for
frame consistency. When these layers handle a 5D tensor, they reshape it to a
3D format, enabling pixel information exchange between frames via attention.

###  4.3 Sampling

During sampling, we employ an extrapolation technique named Classifier-Free
Guidance (CFG) Ho & Salimans (2022) to augment the generation quality. Given
that we have two conditions, namely the conditional video and the editing
prompt, we utilize the CFG method for these two conditions as proposed in
Brooks et al. (2023). Specifically, for each denoising timestep, three
predictions are made under different conditions: the unconditional inference
$\varepsilon_{\theta}(z_{t},\varnothing,\varnothing)$, where both the
conditions are an empty string and an all-zero video; the video-conditioned
prediction $\varepsilon_{\theta}(z_{t},c_{V},\varnothing)$; and the video and
prompt-conditioned prediction $\varepsilon_{\theta}(z_{t},c_{V},c_{T})$. Here,
we omit timestep $t$ for symbolic convenience. The final prediction is an
extrapolation between these three predictions with video and text classifier-
free guidance scale $s_{V}\geq 1$ and $s_{T}\geq 1$.

$\displaystyle\tilde{\varepsilon_{\theta}}(z_{t},c_{V},c_{T})$
$\displaystyle=\varepsilon_{\theta}(z_{t},\varnothing,\varnothing)$ (3)
$\displaystyle+s_{V}\cdot(\varepsilon_{\theta}(z_{t},c_{V},\varnothing)-\varepsilon_{\theta}(z_{t},\varnothing,\varnothing))$
$\displaystyle+s_{T}\cdot(\varepsilon_{\theta}(z_{t},c_{V},c_{T})-\varepsilon_{\theta}(z_{t},c_{V},\varnothing))$

###  4.4 Long Video Sampling Correction for Editing Long Videos

In video editing, models often face limitations in processing extended video
lengths in one go. Altering the number of input frames can compromise the
model’s efficacy, as frame count is usually preset during training. To manage
lengthy videos, we split them into smaller batches for independent sampling.
While intra-batch frame consistency is preserved, inter-batch consistency
isn’t guaranteed, potentially resulting in visible discontinuities at batch
transition points.

To address this issue, we propose Long Video Sampling Correction (LVSC):
during sampling, the results from the final $N$ frames of the previous video
batch can be used as a reference to guide the generation of the next batch
(Figure 3 (c)). This technique helps to maintain visual consistency across
different batches. Specifically, let
$z^{ref}=z^{prev}_{0}[:,-N:]\in\mathbb{R}^{1,N,c,h,w}$ denote the last N
frames from the transfer result of the previous batch. Here, to avoid
confusion with the batch size, we set the batch size to 1. The ensuing batch
is the concatenation of noisy reference frames and subsequent frames
$[z^{ref}_{t},z_{t}]$. On the model’s prediction
$\varepsilon_{\theta}(z_{t}):=\varepsilon_{\theta}(z_{t},t,c_{V},c_{T})$, we
implement a score correction and the final prediction is the summation between
raw prediction $\varepsilon_{\theta}(z_{t})$ and correction term
$\varepsilon_{t}^{ref}-\varepsilon_{\theta}(z_{t}^{ref})$, where
$\varepsilon_{t}^{ref}$ is the closed-form inferred noise on reference frames.
For notation simplicity, we use $\varepsilon(z_{t}^{ref})$ and
$\varepsilon(z_{t})$ to denote the model’s predictions on reference and
subsequent frames, though they are processed together by the model instead of
separate inputs.

$\displaystyle\varepsilon_{t}^{ref}$
$\displaystyle=\frac{z^{ref}_{t}-\sqrt{\bar{\alpha_{t}}}z^{ref}}{\sqrt{1-\bar{\alpha_{t}}}}\in\mathbb{R}^{1,N,c,h,w}$
(4) $\displaystyle\tilde{\varepsilon_{\theta}}(z_{t})$
$\displaystyle=\varepsilon_{\theta}(z_{t})+\frac{1}{N}\sum_{i=1}^{N}(\varepsilon_{t}^{ref}[:,i]-\varepsilon_{\theta}(z_{t}^{ref})[:,i])$
(5)

We apply averaging on the correction term when there are multiple reference
frames as shown in Equations 4 and 5, where $\bar{\alpha_{t}}$ is the
diffusion coefficient for timestep $t$
(i.e.$z_{t}=\mathcal{N}(\sqrt{\bar{\alpha_{t}}}z_{0},(1-\bar{\alpha_{t}})I)$).
In our empirical observations, we find that when the video has global or
holistic camera motion, the score correction may struggle to produce
consistent transfer results. To address this issue, we additionally introduce
a motion compensation that leverages optical flow to establish correspondences
between each pair of reference frames and the remaining frames in the batch.
We then warp the score correction in accordance with this optical flow with
details presented in Appendix D.

##  5 Experiments

###  5.1 Experimental Setup

Dataset For our evaluation, we used the Text-Guided Video Editing (TGVE)
competition dataset222https://sites.google.com/view/loveucvpr23/track4. The
TGVE dataset contains 76 videos that come from three different sources:
Videov, Youtube, and DAVIS Perazzi et al. (2016). Every video in the dataset
comes with one original prompt that describes the video and four prompts that
suggest different edits for each video. Three editing prompts pertain to
modifications in style, background, or object within the video. Additionally,
a multiple editing prompt is provided that may incorporate aspects of all
three types of edits simultaneously.

Metrics for Evaluation Given that our focus is on text-based video editing, we
look at three critical aspects. First, we assess whether the edited video
accurately reflects the editing instructions. Second, we determine whether the
edited video successfully preserves the overall structure of the original
video. Finally, we consider the aesthetics of the edited video, ensuring it is
free of imperfections such as jittering. Our evaluation is based on user study
and automated scoring metrics. In the user study, we follow TGVE competition
to ask users three key questions. The Text Alignment question: Which video
better aligns with the provided caption? The Structure question: Which video
better retains the structure of the input video? The Quality question:
Aesthetically, which video is superior? These questions aim to evaluate the
quality of video editing, focusing on the video’s alignment with editing
instructions, its preservation of the original structure, and its aesthetic
integrity. For objective metrics, we incorporate PickScore Kirstain et al.
(2023) that computes the average image-text alignment over all video frames
and CLIP Frame (Frame Consistency) Radford et al. (2021), which measures the
average cosine similarity among CLIP image embeddings across all video frames.
We prefer PickScore over the CLIP text-image score since it’s tailored to more
closely align with human perception of image quality, which is also noticed by
Podell et al. (2023).

###  5.2 Baseline Methods

We benchmark InsV2V against leading text-driven video editing techniques:
Tune-A-Video Wu et al. (2022), Vid2Vid-Zero Wang et al. (2023a), Video-P2P Liu
et al. (2023), and ControlVideo Zhao et al. (2023). Tune-A-Video has been
treated as a de facto baseline in this domain. Vid2Vid-Zero and Video-P2P
adopt the cross-attention from Prompt-to-Prompt (PTP)Hertz et al. (2022),
while ControlVideo leverages ControlNetZhang & Agrawala (2023). We test all
methods for 32 frames, but PTP-based ones, due to their computational demand,
are limited to 8 frames. Baselines are processed in a single batch to avoid
inter-batch inconsistencies and use latent inversion Mokady et al. (2023) for
structure preservation, which causes double inference time. Conversely, our
method retains the video’s structure more efficiently.

We also extend the comparison to include recent tuning-free video editing
methods such as TokenFlow Geyer et al. (2023), Render-A-VideoYang et al.
(2023b), and Pix2VideoCeylan et al. (2023). These methods, by eliminating the
necessity for individual video model tuning, present a comparable benchmark to
our approach. To ensure frame-to-frame consistency, these methods either adopt
cross-frame attention similar to Tune-A-VideoWu et al. (2022), as seen in
Ceylan et al. (2023); Yang et al. (2023b), or establish pixel-level
correspondences between features and a reference key frame as in Geyer et al.
(2023). This approach is effective in maintaining quality when there are minor
scene changes. However, in scenarios with significant differences between the
key and reference frames, these methods may experience considerable
degradation in video quality. This limitation is more clearly illustrated in
Figure 12 in the Appendix.

###  5.3 Model Details

Our model is adapted from a single image editing Stable Diffusion Brooks et
al. (2023) and we insert temporal attention modules after each spatial
attention layers as suggested by Guo et al. (2023). Our training procedure
makes use of the Adam optimizer with a learning rate set at $5\times 10^{-5}$.
The model is trained with a batch size of 512 over a span of 2,000 iterations.
This training process takes approximately 30 hours to complete on four NVIDIA
A10G GPUs.

During sampling, we experiment with varying hyperparameters for video
classifier-free guidance (VCFG) within the choice of [1.2, 1.5, 1.8], text
classifier-free guidance to 10 and video resolutions of 256 and 384. A
detailed visual comparison using these differing hyperparameters can be found
in the supplementary material (Appendix E). The hyperparameters combination
that achieves the highest PickScore is selected as the final sampling result.
Each video is processed in three distinct batches using LVSC with a fixed
frame count of 16 within a batch, including reference frames from preceeding
batch, and resulting in a total frame count of 32.

###  5.4 Long Video Score Correction and Motion Compensation

Table 1: Comparison of motion-aware MSE and CLIP frame similarity between the last frame of the preceding batch and the first frame of the subsequent batch on TGVE dataset. LVSC | MC | MAMSE (%) $\downarrow$ | CLIPFrame $\uparrow$
---|---|---|---
✗ | ✗ | 2.02 | 0.9072
✓ | ✗ | 1.44 | 0.9093
✓ | ✓ | 1.37 | 0.9095

To assess performance improvement, we employ CLIP Frame Similarity and Motion-
Aware Mean Squared Error (MAMSE) as evaluation metrics. Unlike traditional
MSE, MAMSE accounts for frame-to-frame motion by utilizing optical flow to
warp images, thereby ensuring loss computation in corresponding regions.
Incorporating Long Video Score Correction (LVSC) and Motion Compensation (MC)
has led to enhanced performance as reflected in Table 1. Further qualitative
comparison, detailing the benefits of LVSC and MC, are provided in Appendices
C and D.

###  5.5 User Study

Table 2: The first two columns display automated metrics concerning CLIP frame
consistency and PickScore. The final four columns pertain to a user study
conducted under the TGVE protocol, where users were asked to select their
preferred video when comparing the method against the TAV.

Method | CLIPFrame | PickScore | Text Alignment | Structure | Quality | Average
---|---|---|---|---|---|---
TAV* | 0.924 | 20.36 | - | - | - | -
CAMP* | 0.899 | 20.71 | 0.689 | 0.486 | 0.599 | 0.591
T2I_HERO* | 0.923 | 20.22 | 0.531 | 0.601 | 0.564 | 0.565
Vid2Vid-Zero | 0.926 | 20.35 | 0.400 | 0.357 | 0.560 | 0.439
Video-P2P | 0.935 | 20.08 | 0.355 | 0.534 | 0.536 | 0.475
ControlVideo | 0.930 | 20.06 | 0.328 | 0.557 | 0.560 | 0.482
TokenFlow | 0.940 | 20.49 | 0.287 | 0.563 | 0.624 | 0.491
Pix2Video | 0.916 | 20.12 | 0.468 | 0.529 | 0.538 | 0.511
Render-A-Video | 0.909 | 19.58 | 0.326 | 0.551 | 0.525 | 0.467
InsV2V (Ours) | 0.911 | 20.76 | 0.690 | 0.717 | 0.689 | 0.699

*: Scores from TGVE leaderboard.

Figure 4: The abbreviations on x-axis indicate user preferences across four
types of video editing within TGVE: Style, Background, Object, and Multiple.
Each title specifies the evaluation metrics used for the corresponding
figures. A ”+” symbol signifies that the user vote meets multiple criteri.
Additional qualitative results are presented in Appendix F

We conducted two separate user studies. The first followed the TGVE protocol,
where we used Tune-A-Video as the baseline and compared the outcomes of our
method with this well-known approach. However, we realized that this method of
always asking users to choose a better option might not fully represent the
reality, where both videos could either perform well or poorly. Thus, in the
second user study, we compare our method with seven publicly available
baselines. Instead of asking users to choose a better video, we asked them to
vote for the quality of text alignment, structural preservation, and aesthetic
quality for each transferred video. As evidenced by Table 2 and Figure 4, our
approach excelled in all metrics except CLIP Frame similarity. We contend that
CLIP Frame similarity is not an entirely apt metric because it only captures
semantic similarity between individual frames, which may not be constant
throughout a well-edited video due to changing scenes.

In our evaluation protocol, we additionally introduce a multi-metric
assessment, capturing cases where videos satisfy multiple evaluation criteria
concurrently. This composite measure addresses a key shortcoming of single
metrics, which may inadequately reflect overall editing performance. For
example, a high structure score might indicate that the edited video perfectly
preserves the original, but such preservation could come at the expense of
alignment with the editing instruction.

The results presented in Figure 4 further corroborate the advantages of our
approach. While baseline methods demonstrate respectable performance when text
alignment is not a prioritized criterion, they fall short when this element is
incorporated into the assessment. In contrast, our method not only excels in
aligning the edited video with the textual instructions but also maintains the
structural integrity of the video, resulting in a high-quality output. This
dual success underlines the robustness of our method in meeting the nuanced
requirements of video editing tasks.

##  6 Conclusion

This research tackles the problem of text-based video editing. We have
proposed a synthetic video generation pipeline, capable of producing paired
data required for video editing. To adapt to the requirements of video
editing, we have employed the model inflation technique to transform a single
image diffusion model into a video diffusion model. As an innovation, we’ve
introduced the Long Video Sampling Correction to ensure the generation of
consistent long videos. Our approach is not only efficient but also highly
effective. The user study conducted to evaluate the performance of our model
yielded very high scores, substantiating the robustness and usability of our
method in real-world applications.

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## Supplementary of Coherent Video-to-Video Transfer Using Synthetic Datasets

##  Appendix A Instruction For Zero-shot MPT Data Generation:

The instruction for MPT comprises three parts. The first is the task
description which delineates the objectives for the bot.

You are a bot to generate synthetic text data for generating video clip. You
will creatively generate paired prompt triplet for synthetic data generation
for video clip. You will be given an input prompt and you should return the
edit prompt and output prompt. The output prompt reflects the sentence after
applying edit prompt on the input prompt. Ensure that the prompt are proper
for generating video clip (i.e. prompt should describe a scene).Successful
editing do changing the main subject, modifying the context or setting or
altering the artistic style.Here are some examples of editing that are likely
to success (do not limit to the verb used in the follow. You must be creative
and the editing should be diverse):Edit of Landscape:Edit: Convert the
cityscape to a seascape.Edit: Turn the desert scene into a lush
forest.Replacement of Characters:Edit: Replace the cowboys with
astronauts.Edit: Turn the group of children into a group of elderly
people.Edit of Time:Edit: Switch the night scene to a day scene.Edit:
Transform the contemporary setting into a medieval setting.Addition of
Significant Elements:Edit: Add a full moon to the clear sky.Edit: Include a
rainbow in the cloudy scene.Edit of Weather or Season:Edit: Make the sunny day
into a snowfall.Edit: Transform the summer scene into autumn.Edit the Action
or Activity:Edit: Change the soccer game to a ballet performance.Edit: Replace
the cooking scene with a gardening scene.Edit of Artistic Style:Edit: Make it
look like a watercolor painting.Edit: It is now in the style of Van Gogh. (do
not only use Van Gogh)

The subsequent phase of the instruction involves presenting MPT with five
randomly selected examples from the LAION-IPTP dataset. These samples have
been previously successful in generating paired prompts that meet the CLIP
filter criteria. The depiction below illustrates this process:

Here are some success examples (please be creative and not limited to
examples)Input: Graham Wands - George Square, Glasgow, watercolourEdit: Turn
the watercolour into a pencil sketchOutput: Graham Wands - George Square,
Glasgow, pencil sketchInput: Pierre de Clausade, (French, 1910-1976), Winter
at the LakeEdit: make it a sunsetOutput: Pierre de Clausade, (French,
1910-1976), Sunset at the LakeInput: Rex Beanland, Charing Cross, watercolour,
9 12Edit: make it an oil paintingOutput: Rex Beanland, Charing Cross, oil
painting, 9 12Input: Mark Van Crombrugge, Old Milk Bottle and Grapes, oil, 31
x 59.Edit: Make the bottle transparent.Output: Mark Van Crombrugge,
Transparent Old Milk Bottle, oil, 31 x 59.Input: """Large Original Oil
painting on canvas. Beautiful portrait of a woman 24x24"""""""Edit: make the
woman a catOutput: """Large Original Oil painting on canvas. Beautiful
portrait of a cat 24x24""""

Finally, MPT is provided with video captions from the WebVid dataset that are
designated for processing. This marks the initiation of the generation process
for novel, paired prompts.

Generate triplet for following inputs:Merida, mexico - may 23, 2017: tourists
are walking on a roadside near catholic church in the street of mexico at
sunny summer day.Fun clown - 3d animationHappy family using laptop on bed at
home11th march 2017. nakhon pathom, thailand. devotees goes into a trance at
the wai khru ceremony at wat bang phra temple. what bang phra is famous for
its magically charged tattoos and amulets.Decorate with pineapple sweet cake
roll.Beautiful lake aerial viewFrankfurt, germany-circa 2013:traffic with
skyscrapers in background at night along the river main in frankfurt, time
lapseYoung positive couple laughing in the backyard in front of the large
house under falling snow. bearded man and attractive woman in warm clothes
have winter fun. the guy showing thumb upBroadcast twinkling squared diamonds,
multi color, abstract, loopable, 4kWheat harvesting. combine harvester gathers
the wheat crop on the field.

##  Appendix B Visualization of Synthetic Video Dataset

In Section 3, we introduce a pipeline designed to produce synthetic paired
videos. Further visual demonstrations of these results can be found in Figures
5 and 6. The generated paired videos maintain a similar structure, and the
editing can showcase the edit prompt.

Figure 5: Visualization of synthetic samples produced by the data generation
pipeline in Section 3. Each sample contains two generated videos: the upper
one represents the input prompt, while the lower one depicts the output
prompt. The actual videos comprise 16 frames while 6 subsampled frames are
displayed here for brevity.

Figure 6: Visualization of synthetic samples produced by the data generation
pipeline in Section 3. Each sample contains two generated videos: the upper
one represents the input prompt, while the lower one depicts the output
prompt. The actual videos comprise 16 frames while 6 subsampled frames are
displayed here for brevity.

##  Appendix C Long Video Sampling Detail Illustration and Examples

In Section 4.4, we addressed the challenge of ensuring consistency across
different batches when sampling long videos, particularly when each batch is
processed separately. This section introduces an illustrative explanation of
the Long Video Sampling Correction (LVSC) method in Figure 7. In LVSC, there
are $N$ overlapping reference frames between two consecutive batches, but this
method transcends a basic sliding window technique. Instead of allowing the
editing model unrestricted freedom in transferring edits, the reference frames
guide the appearance of subsequent frames during the editing process.

Figure 7: Schematic of Long Video Sampling Correction (LVSC) mechanism. The
diagram illustrates the interaction between reference latents from a preceding
batch and subsequent latents during the denoising process. The closed-form
inferred noise (Equation 4) is computed for the $N$ reference latents (shown
in blue), which then guides the correction of the actual denoised subsequent
latents (shown in orange).

Additionally, we present a visual comparison to demonstrate the impact of
LVSC. Figure 8 contrasts the results with and without LVSC implementation.
This comparison clearly shows the lack of continuity in the sampled frames
between batches when LVSC is not applied. In contrast, employing LVSC achieves
noticeable consistency, aligning the first frame of a subsequent batch with
the last frame of the previous batch.

Edit Prompt: Change the watermelon to hamburger.

Figure 8: In the process of video sampling, we utilize a batch size of 16. The
17th to 28th frames in the video are processed in the second batch, and they
reference the last four sampled frames from the preceding batch. By employing
the Long Video Sampling Correction (LVSC), we can ensure the content
consistency between sampled frames across different batches (green boxes in
the figure). In contrast, sampling two batches separately may lead to
inconsistencies at the boundaries where the batches change (rex boxes in the
figure).

##  Appendix D Effect of Motion Compensation in Long Video Sampling Correction

In our experiments, we observed that the presence of holistic camera motion
could degrade the transfer results of subsequent batches processed by the LVSC
model (see red boxes in Figure 9). This degradation arises because the same
regions across different frames require consistent score corrections. In other
words, score corrections should “travel” with the regions affected by camera
movement, ensuring that identical corrections are applied to the same areas
regardless of motion. To address this issue, we introduce a motion
compensation strategy. Specifically, we first employ the RAFT flow estimator
Teed & Deng (2020) to compute the optical flow between each reference frame
and the remaining frames in subsequent batches. The original equation for LVSC
(Equation 5) is then modified as follows:

$\displaystyle\tilde{\varepsilon_{\theta}}(z_{t})[:,m]$
$\displaystyle=\varepsilon_{\theta}(z_{t})[:,m]+(\frac{1}{N}\sum_{i=1}^{N}o(\varepsilon_{t}^{ref}[:,i]),i\rightarrow
m)-\varepsilon_{\theta}(z_{t})[:,m]$ (6)

Here, $m=[1,2,...,M]$ represents the indices of frames in the subsequent
batches, which contain a total of $M$ frames, $o(\cdot,i\rightarrow m)$ is a
warping function that uses optical flow to align the $i$-th reference frame
with the $m$-th frame in the subsequent batch. This modification ensures
better transfer quality by making the score corrections to be aware of camera
movement. The green boxes in Figure 9 reveal that applying motion compensation
significantly enhances content consistency and overall quality. This
improvement is particularly noticeable in the last few frames of the
subsequent batches.

Edit Prompt: Make it Van Gogh Starry Night style.

Figure 9: When the video exhibits extensive camera motion, the LVSC model
without motion compensation (LVSC without MC) suffers significant quality
degradation, particularly in frames from subsequent batches (red boxes in the
figure). Incorporating motion compensation (LVSC with MC) substantially
improves frame consistency with the reference and maintains high transfer
quality, even when the camera view changes dramatically in later batches
(green boxes in the figure).

##  Appendix E Effect of Sampling Hyperparameters and Picking Criteria

In our experiments, we observed that the video CFG and resolution have a
greater impact on the generated video than the text CFG. To streamline the
parameter search process, we thus focus solely on picking the video CFG and
resolution, effectively reducing the search space. Detailed visual effects of
the hyperparameter choices are provided in Figure 10. We opt for PickScore
Kirstain et al. (2023) as our automated selection criterion. Our choice is
motivated by the fact that PickScore aligns more closely with human perception
compared to the CLIP score, as indicated in Kirstain et al. (2023).

Edit Prompt: Make the rabbit robotic.

Figure 10: During sampling, the interplay between the scale of image
classifier-free guidance (CFG) and image size can significantly influence the
transfer result. In our evaluation, we sample videos by employing various
combinations of CFG scales and resolutions, utilizing the average PickScore
across all frames as the selection metric. The video with the highest
PickScore is designated as the final transfer result. As indicated by the
green box in the figure, the chosen video and its corresponding
hyperparameters bear the highest PickScore, thereby making it the final
selection.

##  Appendix F Qualitative Comparisons

We provide qualitative comparisons with baselines in Figures 11, 12, 13, 14,
15, 16, 17, 18 and 19

Figure 11: Swans gliding over a lake. $\rightarrow$ Pink flamengos gliding
over a lake. Figure 12: A cat and a dog playing on the street while a girl
walks around them.
$\rightarrow$ A cat and a dog playing on the beach while a girl walks around
them, golden hour lighting. Figure 13: A cat in the grass in the sun.
$\rightarrow$ A dog in the grass in the sun. Figure 14: Full view of the Earth
as it moves slowly toward the sun.
$\rightarrow$ Full view of the Earth as it moves slowly through a fireworks
display. Figure 15: Several goldfish swim in a tank.
$\rightarrow$ Several goldfish swim in a pond. Figure 16: A static shot of red
roses in sunlight, gently swaying in the breeze.
$\rightarrow$ A static shot of red roses in sunlight, gently swaying in the
breeze, origami style. Figure 17: A Delta Airlines aircraft descends onto the
runaway during a cloudless morning.
$\rightarrow$ A brightly colored cyberpunk aircraft descends onto the runway
during a cloudless morning, with a bustling cityscape in the background.
Figure 18: A ship sails on the sea.
$\rightarrow$ A ship sails on the lunar surface. Figure 19: A colourful
fireworks display in the night sky.
$\rightarrow$ A colourful fireworks display in the night sky, 8-bit pixelated.

##  Appendix G Failure Cases

Our model, InsV2V, occasionally encounters challenges in video transfer,
particularly when the object of interest is difficult to detect. Such
difficulties arise when the object is positioned close to the video’s edge, is
unusually small or large, or is partially obscured. These scenarios can lead
to the object either vanishing in the edited video or exhibiting inconsistent
appearances.

For instance, as depicted in Figure 20, when a person is situated near the
frame’s border and occupies a small area, InsV2V struggles to maintain the
person’s structural integrity, resulting in their disappearance from the
transferred video. However, when the person moves away from the border, InsV2V
can recognize them again, but with a notably altered appearance.

Figure 21 presents another scenario where the object, in this case, the front
part of a truck, is only partially visible in certain frames. While InsV2V can
accurately transfer the image when the truck is fully visible, it
misinterprets the partially visible truck as a car’s front, leading to
inconsistent results in the transferred video.

In summary, when object detection is hindered due to size, positioning, or
occlusion, InsV2V may not deliver satisfactory transfer results.

Prompt: Change background to tropical river.

Figure 20: Illustration of a failure case where InsV2V struggles to maintain
the structure of a person located near the video’s edge and occupying a small
area, resulting in the disappearance of the person from the transferred video.

Prompt: Make trucks made of wood (wooden style truck).

Figure 21: Demonstration of a failure scenario in InsV2V where a partially
visible truck is misidentified as a car’s front, leading to inconsistent
transfer results in the edited video.


\\{W_{12},\ldots,W_{15}\\}&=S_{4}\circ\left(s_{12}-p_{1}^{2}\right)\,,\\\
\\{W_{16},\ldots,W_{27}\\}&=S_{4}\circ\left(s_{15}-s_{34}\right)\,,\\\
\\{W_{28},\ldots,W_{33}\\}&=S_{4}\circ\left(s_{12}s_{15}-p_{1}^{2}s_{34}\right)\,,\\\
\\{W_{34},\ldots,W_{45}\\}&=S_{4}\circ\left(s_{12}s_{23}+p_{1}^{2}s_{45}-s_{12}s_{45}\right)\,,\\\
\\{W_{46},\ldots,W_{57}\\}&=S_{4}\circ\left(s_{15}s_{45}+p_{1}^{2}s_{23}-p_{1}^{2}s_{15}\right)\,,\\\
\\{W_{70},\ldots,W_{93}\\}&=S_{4}\circ\left(s_{12}s_{15}-s_{12}s_{23}-p_{1}^{2}s_{34}\right)\,,\end{aligned}$
(116)

where $S_{4}\circ x$ denotes the $S_{4}$-orbit of $x$, namely the set
$\\{\sigma\circ x\,|\,\sigma\in S_{4}\\}$.

The letters with non-trivial behaviour with respect to changing the sign of
one of the three-mass-triangle square roots but independent of $\tr_{5}$ are

$\displaystyle\begin{aligned}
\\{W_{118},\ldots,W_{123}\\}&=S_{4}\circ\left(\frac{p_{1}^{2}-s_{23}+s_{45}+\sqrt{\Delta_{3}^{(1)}}}{p_{1}^{2}-s_{23}+s_{45}-\sqrt{\Delta_{3}^{(1)}}}\right)\,,\\\
\\{W_{124},\ldots,W_{129}\\}&=S_{4}\circ\left(\frac{s_{13}-s_{12}-\sqrt{\Delta_{3}^{(1)}}}{s_{13}-s_{12}+\sqrt{\Delta_{3}^{(1)}}}\right)\,.\\\
\end{aligned}$ (117)

In particular, the letters $\\{W_{118},W_{123},W_{124},W_{129}\\}$,
$\\{W_{119},W_{122},W_{125},W_{128}\\}$ and
$\\{W_{120},W_{121},W_{126},W_{127}\\}$ are odd with respect to
$\sqrt{\Delta_{3}^{(1)}}$, $\sqrt{\Delta_{3}^{(2)}}$ and
$\sqrt{\Delta_{3}^{(3)}}$, respectively.

The parity-odd letters — namely those whose $\differential\log$ changes sign
when changing the sign of $\tr_{5}$101010In ref. Abreu:2021smk  the letters
are defined in terms of $\sqrt{\Delta_{5}}$, rather than $\tr_{5}$, and are
thus even under parity. We adopt their definition of the letters but replace
$\sqrt{\Delta_{5}}$ with $\tr_{5}$, to match the conventions used in the
computation of the planar integral families Abreu:2020jxa . Since
$\sqrt{\Delta_{5}}$ is invariant under any permutation of the massless legs
whereas $\tr_{5}$ changes sign with an odd-signature permutation, the sets of
permutations used here may differ from those of ref. Abreu:2021smk . — which
are free from the three-mass-triangle square roots are

$\displaystyle\\{W_{130},\ldots,W_{137}\\}=S_{4}\circ\left(\frac{a-\tr_{5}}{a+\tr_{5}}\right)\,,$
(118)

with

$\displaystyle
a=s_{12}s_{23}+s_{23}s_{34}-s_{34}s_{45}+s_{45}s_{15}-s_{12}s_{15}+p_{1}^{2}s_{34}\,.$
(119)

There are three parity-odd letters which also depend on one of the three-mass-
triangle square roots,

$\displaystyle\begin{aligned}
W_{186}&=\frac{\Omega^{--}\Omega^{++}}{\Omega^{-+}\Omega^{+-}}\,,\\\
W_{187}&=\sigma\circ W_{186}\quad\text{with}\ \sigma=\left(2435\right)\,,\\\
W_{188}&=\sigma\circ\left(\frac{1}{W_{186}}\right)\quad\text{with}\
\sigma=\left(2543\right)\,,\\\ \end{aligned}$ (120)

where

$\displaystyle\Omega^{\pm\pm}=s_{12}s_{15}-s_{12}s_{23}-s_{15}s_{45}\pm
s_{34}\sqrt{\Delta_{3}^{(1)}}\pm\tr_{5}\,.$ (121)

The letters $W_{186}$, $W_{187}$ and $W_{188}$ are all odd with respect to
$\tr_{5}$, and are odd with respect to $\sqrt{\Delta_{3}^{(1)}}$,
$\sqrt{\Delta_{3}^{(2)}}$ and $\sqrt{\Delta_{3}^{(3)}}$, respectively. The
definition of $W_{188}$ in eq. 120 requires an inverse because we are using
$\tr_{5}$ rather than $\sqrt{\Delta_{5}}\,$ (see also footnote 10).

The last four relevant letters are given by the four square roots of the
problem,

$\displaystyle\begin{aligned} W_{195}&=\sqrt{\Delta_{3}^{(1)}}\,,\\\
W_{196}&=\sqrt{\Delta_{3}^{(2)}}\,,\\\ W_{197}&=\sqrt{\Delta_{3}^{(3)}}\,,\\\
W_{198}&=\tr_{5}\,.\end{aligned}$ (122)

We stress that these letters are even with respect to changing the sign of any
square root.

Square root | Odd letters
---|---
$\sqrt{\Delta_{3}^{(1)}}$ | $W_{118}\,,W_{123}\,,W_{124}\,,W_{129}\,,W_{186}$
$\sqrt{\Delta_{3}^{(2)}}$ | $W_{119}\,,W_{122}\,,W_{125}\,,W_{128}\,,W_{187}$
$\sqrt{\Delta_{3}^{(3)}}$ | $W_{120}\,,W_{121}\,,W_{126}\,,W_{127}\,,W_{188}$
$\sqrt{\Delta_{3}^{(3)}}$ $\tr_{5}$ | $W_{130}\,,\ldots\,,W_{137}\,,W_{186}\,,W_{187}\,,W_{188}$
Table 7: Relevant letters which depend on the sign of the square roots of the
problem. On each row the letters in the right column are odd with respect to
changing the sign of the square root in the left column. We recall that
$\tr_{5}$ is related to $\sqrt{\Delta_{5}}$ through eq. 5.

##  Appendix D Positivity of the alphabet in the physical scattering region
$45\to 123$

The one-mass pentagon functions are expressed as iterated integrals with
integration kernels given by the logarithms of the alphabet letters (see
section 3.4). The representation in terms of iterated integrals highlights
that the pentagon functions may have singularities only on the loci where any
of the involved letters vanishes or diverges. Therefore we need to inquire
about the positivity of the alphabet letters in the region ${\cal P}_{45}$,
i.e. whether they have definite sign in the region ${\cal P}_{45}$. This has
important implications for the construction of the explicit representation of
the pentagon functions in section 4.2. In this appendix we study
systematically the positivity of the $108$ relevant letters of the alphabet
listed in eq. 35. We show that $20$ of them do not have fixed sign in ${\cal
P}_{45}$. We call the latter spurious letters. Out of them, $4$ are linear in
the Mandelstam invariants,

$\displaystyle{\cal S}_{\ell}=\\{$ $\displaystyle
W_{16},\,W_{17},\,W_{19},\,W_{20}\\}\,,$ (123)

and $16$ are quadratic,

$\displaystyle{\cal S}_{q}=\\{$ $\displaystyle
W_{35},\,W_{36},\,W_{38},\,W_{39},\,W_{40},\,W_{41},\,W_{43},\,W_{44},\,$
(124) $\displaystyle
W_{72},\,W_{74},\,W_{78},\,W_{80},\,W_{82},\,W_{84},\,W_{88},\,W_{90}\\}\,.$

We denote by ${\cal S}$ the set of all spurious letters,

$\displaystyle{\cal S}={\cal S}_{\ell}\cup{\cal
S}_{q}\subset\mathbb{A}_{S_{4}}^{{\rm rel}}\,.$ (125)

We prove this by rewriting the letters in such a way that their positivity
properties in ${\cal P}_{45}$ become obvious consequences of the inequalities
(18). For the sake of presentation we split the alphabet into several subsets
of letters having similar structure, to each of which we devote a subsection.
Finally, in subsection D.5 we show that the three-mass triangle functions
defined in eq. 55 have no branch cuts within ${\cal P}_{45}$.

###  D.1 Linear letters

The first $27$ letters, $\\{W_{i}\\}_{i=1}^{27}$, are linear in the Mandelstam
variables (3). By rewriting them as

$\displaystyle W_{1}=p_{1}^{2}\,,$ $\displaystyle W_{2}=s_{12}\,,$
$\displaystyle W_{3}=s_{13}\,,$ (126) $\displaystyle W_{4}=s_{14}\,,$
$\displaystyle W_{5}=s_{15}\,,$ $\displaystyle W_{6}=s_{23}\,,$ $\displaystyle
W_{7}=s_{24}\,,$ $\displaystyle W_{8}=s_{25}\,,$ $\displaystyle
W_{9}=s_{34}\,,$ $\displaystyle W_{10}=s_{35}\,,$ $\displaystyle
W_{11}=s_{45}\,,$ $\displaystyle W_{12}=s_{12}-p_{1}^{2}\,,$ $\displaystyle
W_{13}=s_{13}-p_{1}^{2}\,,$ $\displaystyle W_{14}=s_{14}-p_{1}^{2}\,,$
$\displaystyle W_{15}=s_{15}-p_{1}^{2}\,,$ $\displaystyle
W_{16}=s_{23}+s_{24}\,,$ $\displaystyle W_{17}=s_{23}+s_{25}\,,$
$\displaystyle W_{18}=s_{24}+s_{25}\,,$ $\displaystyle
W_{19}=-s_{23}-s_{34}\,,$ $\displaystyle W_{20}=-s_{23}-s_{25}\,,$
$\displaystyle W_{21}=(s_{13}-p_{1}^{2})+s_{23}\,,$ $\displaystyle
W_{22}=-s_{24}-s_{34}\,,\quad\quad$ $\displaystyle
W_{23}=(s_{14}-p_{1}^{2})+s_{34}\,,\quad\quad$ $\displaystyle
W_{24}=(s_{14}-p_{1}^{2})+s_{24}\,,$ $\displaystyle W_{25}=s_{25}+s_{35}\,,$
$\displaystyle W_{26}=(p_{1}^{2}-s_{15})-s_{35}\,,$ $\displaystyle
W_{27}=(p_{1}^{2}-s_{15})-s_{25}\,,$

we can immediately deduce their positivity properties in ${\cal P}_{45}$ from
the defining inequalities (18). Only the inequalities which are linear in the
Mandelstam invariants are required. The positivity properties of the linear
letters are summarized as follows:

$W_{1},\,W_{2},\,W_{3},\,W_{6},\,W_{11},\,W_{12},\,W_{13},\,W_{21},\,W_{22},\,W_{23},\,W_{26},\,W_{27}$ |  $>0$
---|---
$W_{4},\,W_{5},\,W_{7},\,W_{8},\,W_{9},\,W_{10},\,W_{14},\,W_{15},\,W_{18},\,W_{24},\,W_{25}$ |  $<0$
$W_{16},\,W_{17},\,W_{19},\,W_{20}$ |  $\gtrless 0$
(127)

###  D.2 Quadratic letters

$54$ letters, $\\{W_{i}\\}_{i=28}^{57}\cup\\{W_{i}\\}_{i=70}^{93}$, are
quadratic in the Mandelstam variables (3). For $11$ of them we find a
representation which makes their positivity a manifest consequence of the
linear inequalities (18) among Mandelstam invariants,

$\displaystyle\begin{aligned}
&W_{49}=s_{35}(s_{13}+p_{1}^{2})+p_{1}^{2}(s_{15}-p_{1}^{2})\,,\\\
&W_{51}=s_{34}(s_{13}+p_{1}^{2})+p_{1}^{2}(s_{14}-p_{1}^{2})\,,\\\
&W_{53}=s_{25}(s_{12}+p_{1}^{2})+p_{1}^{2}(s_{15}-p_{1}^{2})\,,\\\
&W_{55}=s_{24}(s_{12}+p_{1}^{2})+p_{1}^{2}(s_{14}-p_{1}^{2})\,,\\\
&W_{56}=s_{23}(s_{13}+p_{1}^{2})+p_{1}^{2}(s_{12}-p_{1}^{2})\,,\\\
&W_{57}=s_{23}(s_{12}+p_{1}^{2})+p_{1}^{2}(s_{13}-p_{1}^{2})\,,\\\
&W_{70}=s_{12}s_{24}+s_{34}(s_{12}-p_{1}^{2})\,,\\\
&W_{71}=s_{12}s_{25}+s_{35}(s_{12}-p_{1}^{2})\,,\\\
&W_{76}=s_{13}s_{34}+s_{24}(s_{13}-p_{1}^{2})\,,\\\
&W_{77}=s_{13}s_{35}+s_{25}(s_{13}-p_{1}^{2})\,,\\\
&W_{93}=s_{15}s_{35}+s_{34}(s_{15}-p_{1}^{2})\,,\end{aligned}$ (128)

i.e. both terms in each of the previous sums are simultaneously either
positive or negative.

In order to establish the positivity of the remaining quadratic letters, the
linear inequalities in eq. 18 are not sufficient. We have to appeal to the
Gram determinant inequalities. We thus express the letters in terms of
order-$3$ Gram determinants, $G_{1ij}$, and profit from their positivity in
${\cal P}_{45}$, namely $G_{1ij}>0$ for $i,j=2,\ldots,5$ and $i\neq j$. In
this way we can straightforwardly establish that the following quadratic
letters do not change sign in ${\cal P}_{45}$,

$\displaystyle\begin{aligned}
&W_{28}=\frac{4G_{125}}{s_{25}}\,,\qquad\qquad\qquad&&W_{29}=\frac{4G_{124}}{s_{24}}\,,\qquad\qquad\qquad&&W_{30}=\frac{4G_{123}}{s_{23}}\,,\\\
&W_{31}=\frac{4G_{135}}{s_{35}}\,,&&W_{32}=\frac{4G_{134}}{s_{34}}\,,&&W_{33}=\frac{4G_{145}}{s_{45}}\,,\\\
&W_{34}=\frac{4G_{124}}{s_{24}}+\frac{4G_{125}}{s_{25}}\,,\qquad\quad&&W_{37}=\frac{4G_{134}}{s_{34}}+\frac{4G_{135}}{s_{35}}\,,&&\\\
&W_{42}=\frac{4G_{124}}{s_{24}}+\frac{4G_{134}}{s_{34}}\,,&&W_{45}=\frac{4G_{125}}{s_{25}}+\frac{4G_{135}}{s_{35}}\,.&&\\\
\end{aligned}$ (129)

We recall that $G_{1ij}/s_{ij}$ is a quadratic polynomial in the Mandelstam
invariants, as can be seen explicitly in eq. 101. Similarly, we observe that
both terms in the following sums are of the same sign,

$\displaystyle\begin{aligned}
&W_{46}=-\frac{4G_{145}}{s_{45}}+s_{15}(s_{23}-s_{15})\,,&W_{47}=-\frac{4G_{145}}{s_{45}}+s_{14}(s_{23}-s_{14})\,,\\\
&W_{73}=-\frac{4G_{125}}{s_{25}}+s_{35}(p_{1}^{2}-s_{12})\,,&W_{75}=-\frac{4G_{124}}{s_{24}}+s_{34}(p_{1}^{2}-s_{12})\,,\\\
&W_{79}=-\frac{4G_{135}}{s_{35}}+s_{25}(p_{1}^{2}-s_{13})\,,&W_{81}=-\frac{4G_{134}}{s_{34}}+s_{24}(p_{1}^{2}-s_{13})\,,\\\
&W_{83}=-\frac{4G_{145}}{s_{45}}+s_{35}(p_{1}^{2}-s_{14})\,,&W_{85}=-\frac{4G_{145}}{s_{45}}+s_{25}(p_{1}^{2}-s_{14})\,,\\\
&W_{86}=-\frac{4G_{124}}{s_{24}}+s_{23}(p_{1}^{2}-s_{14})\,,&W_{87}=-\frac{4G_{134}}{s_{34}}+s_{23}(p_{1}^{2}-s_{14})\,,\\\
&W_{89}=-\frac{4G_{145}}{s_{45}}+s_{34}(p_{1}^{2}-s_{15})\,,&W_{91}=-\frac{4G_{145}}{s_{45}}+s_{24}(p_{1}^{2}-s_{15})\,,\\\
&W_{92}=-\frac{4G_{125}}{s_{25}}+s_{23}(p_{1}^{2}-s_{15})\,.\end{aligned}$
(130)

Finally, we have to employ more intricate identities to highlight the
positivity of the following quadratic letters,

$\displaystyle\begin{aligned}
&W_{48}=\frac{p_{1}^{2}(s_{23}+s_{34})^{2}+4G_{135}}{s_{13}-p_{1}^{2}}\,,\qquad&W_{50}=\frac{p_{1}^{2}(s_{23}+s_{35})^{2}+4G_{134}}{s_{13}-p_{1}^{2}}\,,\\\
&W_{52}=\frac{p_{1}^{2}(s_{15}-s_{34})^{2}+4G_{125}}{s_{12}-p_{1}^{2}}\,,\qquad&W_{54}=\frac{p_{1}^{2}(s_{23}+s_{25})^{2}+4G_{124}}{s_{12}-p_{1}^{2}}\,.\end{aligned}$
(131)

Let us recall that the letters in the previous equation are polynomial in the
Mandelstam variables: the denominators factor out once the numerators are
expressed in terms of Mandelstam variables.

We summarize the positivity properties of the quadratic letters given by eqs.
128, 129, 130 and 131 as follows:

$\begin{array}[]{l}W_{30},\,W_{33},\,W_{48},\,W_{50},\,W_{52},\,W_{54},\,W_{56},\,W_{57},\\\ W_{73},\,W_{75},\,W_{79},\,W_{81},\,W_{86},\,W_{87},\,W_{92},\,W_{93}\end{array}$ |  $>0$
---|---
$\begin{array}[]{l}W_{28},\,W_{29},\,W_{31},\,W_{32},\,W_{34},\,W_{37},\,W_{42},\,W_{45},\,W_{46},\,W_{47},\,W_{49},\\\ W_{51},\,W_{53},\,W_{55},\,W_{70},\,W_{71},\,W_{76},\,W_{77},\,W_{83},\,W_{85},\,W_{89},\,W_{91}\end{array}$ |  $<0$
$\begin{array}[]{l}W_{35},\,W_{36},\,W_{38},\,W_{39},\,W_{40},\,W_{41},\,W_{43},\,W_{44},\\\ W_{72},\,W_{74},\,W_{78},\,W_{80},\,W_{82},\,W_{84},\,W_{88},\,W_{90}\end{array}$ |  $\gtrless 0$
(132)

###  D.3 Three-mass square-root letters

The letters $W_{195},\,W_{196},\,W_{197}$ are the three-mass triangle square
roots, $\sqrt{\Delta^{(i)}_{3}}$ with $i=1,2,3$. They are given by order-$2$
Gram determinants (9), and thus they are positive in ${\cal P}_{45}$ due to
the proposition in eq. 94,

$\displaystyle W_{195},\,W_{196},\,W_{197}>0\,.$ (133)

Among the square-root letters, the letters $\\{W_{i}\\}_{i=118}^{129}$ contain
only the square roots of three-mass-triangle type. We observe that they are
neither singular nor vanish in the physical region ${\cal P}_{45}$, and their
signs are summarized as follows:

$W_{118},\,W_{123},\,W_{125},\,W_{126},\,W_{127},\,W_{128}$ |  $>0$
---|---
$W_{119},\,W_{120},\,W_{121},\,W_{122},\,W_{124},\,W_{129}$ |  $<0$
(134)

In order to show this, let us consider two prototypical examples.

The letter $W_{118}$,

$\displaystyle
W_{118}=\frac{s_{12}+s_{13}+\sqrt{\Delta^{(1)}_{3}}}{s_{12}+s_{13}-\sqrt{\Delta^{(1)}_{3}}}\,,$
(135)

could become singular or vanish only at
$(s_{12}+s_{13})^{2}=\Delta_{3}^{(1)}$. The latter constraint is equivalent to
$p_{1}^{2}\,s_{45}=0$, which is impossible inside the region ${\cal P}_{45}$.
Therefore, $W_{118}$ cannot change sign in ${\cal P}_{45}$.

The second example is given by the letter $W_{126}$,

$\displaystyle
W_{126}=\frac{s_{12}-s_{15}+\sqrt{\Delta^{(3)}_{3}}}{s_{12}-s_{15}-\sqrt{\Delta^{(3)}_{3}}}\,.$
(136)

The latter vanishes or becomes singular only at
$(s_{12}-s_{15})^{2}=\Delta_{3}^{(3)}$, which is equivalent to
$s_{12}s_{15}=p_{1}^{2}s_{34}$. Resolving the previous relation in one of the
Mandelstam variables, e.g. $p_{1}^{2}=s_{12}s_{15}/s_{34}$, and eliminating
this variable from $\Delta_{5}$, we observe that $\Delta_{5}$ turns into a
perfect square,

$\displaystyle\Delta_{5}\Bigl{|}_{p_{1}^{2}=\frac{s_{12}s_{15}}{s_{34}}}=(s_{12}s_{23}-s_{23}s_{34}-s_{15}s_{45}+s_{34}s_{45})^{2}\,,$
(137)

which cannot be negative, in contradiction with the quadratic constraint
$\Delta_{5}<0$ in eq. 18.

For the remaining algebraic letters involving only the three-mass triangle
square roots, one of the two arguments above is applicable.

###  D.4 Parity-odd letters and $\sqrt{\Delta_{5}}$

The remaining relevant letters that we need to consider,
$\\{W_{i}\\}_{i=130}^{137}\cup\\{W_{i}\\}_{i=186}^{188}\cup\\{W_{198}\\}$, all
contain $\sqrt{\Delta_{5}}$ in their expression. Since $\Delta_{5}<0$ in
${\cal P}_{45}$, $\sqrt{\Delta_{5}}$ is non-vanishing and purely imaginary in
the physical region. Thus, $W_{198}=\sqrt{\Delta_{5}}$ is purely imaginary,
while the other eleven letters take complex values on the unit circle. The
latter are parity-odd because the sign of the corresponding
$\differential\log$’s changes under the action of space-time parity $P$,

$\displaystyle P:\ \differential\log W_{i}\to-\differential\log
W_{i}\quad\text{for}\quad i=130,\ldots,137,186,187,188\,.$ (138)

Moreover, these letters cannot be equal to $1$, as that would imply
$\Delta_{5}=0$. In summary, for the parity-odd letters we have that

$\displaystyle|W_{i}|=1\quad\text{and}\quad W_{i}\neq 1\quad\text{for}\quad
i=130,\ldots,137,186,187,188\,.$ (139)

###  D.5 Three-mass triangles

In eq. 55 we defined three weight-two pentagon functions corresponding, up to
an overall algebraic factor, to the finite part of the three-mass triangle
Feynman integrals. Their explicit expressions (55) rely on the representation
(54) of the three-mass triangle function. Here we show that they are well
defined, namely real analytic, in the physical channel ${\cal P}_{45}$.

The following inequalities,

$\displaystyle s_{45}-s_{23}+p_{1}^{2}\pm\sqrt{\Delta_{3}^{(1)}}>0\,,\qquad
s_{45}+s_{23}-p_{1}^{2}\pm\sqrt{\Delta_{3}^{(1)}}>0\,,$ (140)

imply that the arguments of the logarithms and dilogarithms in the expression
for $f^{(2)}_{23}$ do not cross any branch cuts. The strict inequalities (140)
turn into equalities at the boundaries $p_{1}^{2}s_{45}=0$ and
$s_{23}s_{45}=0$ of ${\cal P}_{45}$, respectively. Similarly, the inequalities

$\displaystyle\begin{aligned}
&\pm(p_{1}^{2}+s_{34}-s_{25})+\sqrt{\Delta_{3}^{(3)}}>0\,,\\\
&\pm(p_{1}^{2}+s_{25}-s_{34})+\sqrt{\Delta_{3}^{(3)}}>0\,,\\\
&p_{1}^{2}-s_{25}-s_{34}\pm\sqrt{\Delta_{3}^{(3)}}>0\,,\end{aligned}$ (141)

guarantee absence of branch cuts in ${\cal P}_{45}$ for $f^{(2)}_{24}$. The
same holds true for $f^{(2)}_{25}$, thanks to

$\displaystyle\begin{aligned}
&\pm(p_{1}^{2}+s_{24}-s_{35})+\sqrt{\Delta_{3}^{(2)}}>0\,,\\\
&\pm(p_{1}^{2}-s_{24}+s_{35})+\sqrt{\Delta_{3}^{(2)}}>0\,,\\\
&p_{1}^{2}-s_{24}-s_{35}\pm\sqrt{\Delta_{3}^{(2)}}>0\,.\end{aligned}$ (142)

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