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11institutetext: ByteDance Inc.

# Superior and Pragmatic Talking Face Generation with Teacher-Student
Framework

Chao Liang 11 Jianwen Jiang 11 Tianyun Zhong 11 Gaojie Lin 11 Zhengkun Rong 11
Jiaqi Yang 11 Yongming Zhu 11

###### Abstract

Talking face generation technology creates talking videos from arbitrary
appearance and motion signal, with the "arbitrary" offering ease of use but
also introducing challenges in practical applications. Existing methods work
well with standard inputs but suffer serious performance degradation with
intricate real-world ones. Moreover, efficiency is also an important concern
in deployment. To comprehensively address these issues, we introduce
SuperFace, a teacher-student framework that balances  quality ,  robustness ,
cost and  editability . We first propose a simple but effective teacher model
capable of handling inputs of varying qualities to generate high-quality
results. Building on this, we devise an efficient distillation strategy to
acquire an identity-specific student model that maintains quality with
significantly reduced computational load. Our experiments validate that
SuperFace offers a more comprehensive solution than existing methods for the
four mentioned objectives, especially in reducing FLOPs by 99% with the
student model. SuperFace can be driven by both video and audio and allows for
localized facial attributes editing. Project Page.

###### Keywords:

Talking face generation High performance Practicability

Figure 1: Our proposed method outperforms state-of-the-art ones in the
following aspects: 1) Quality: producing higher-quality results (top left); 2)
Robustness: maintaining robustness even with poor-quality input (top right);
3) Editability: enabling users freely edit facial attributes (bottom left); 4)
Low cost: achieving comparable result by distillation with 99% reduction in
FLOPs (bottom right).

##  1 Introduction

Talking face synthesis has been applied in various scenarios, and related
technologies such as video translation and online conferences have
significantly impacted our daily lives. In recent years, the field has
witnessed marked advancements. However, prevailing attempts frequently
concentrate on boosting performance with regular inputs, overlooking the
complex factors encountered in practical applications. Specifically, users
might provide inputs with inconsistent quality, such as blurry appearance or
intense motion, which is extremely challenging to handle. Besides, current
methods have not adequately addressed the issues of cost-effectiveness and
editability, both of which are significant concerns for real-world deployment.
In general, an excellent method is expected to be:

  * $\bullet$

High-quality. Quality encompasses motion accuracy, visual clarity, as well as
the robustness to handle extreme inputs. Despite considerable progress,
previous works fail to fulfill both simultaneously. Many works [32, 37, 41,
55, 34] made substantial efforts on precise motion control, yet due to their
coarse motion representations, they still fall short in handling large
movements and subtle facial expressions. Recent works [16, 48, 54, 46] achieve
better visual quality under standard inputs through ID-aware techniques, but
stably generating clear results remains elusive for the diverse and variable
inputs in the real world.

  * $\bullet$

Pragmatic. The efficiency of algorithms shapes their practicality, especially
when high quality has already been achieved. Unfortunately, state-of-the-art
methods like MCNet [16] and Real-3d [46] always demand extensive computing
consumption for over 600G FLOPs to produce high-resolution $512\times 512$
outputs, which imposes great costs in real-world applications. Due to the
trade-off between computational expense and quality, directly reducing it
tends to degrade the results, so it is essential to develop a novel framework
that excels in both high quality and low cost. Moreover, processing cross-
modal inputs and editing local facial attributes are also common user-expected
capabilities, which existing methods often struggle with.

To comprehensively tackle the practical challenges mentioned above, We propose
SuperFace, a teacher-student based framework that delivers a high-quality,
robust, low-cost, and editable solution for talking head generation, as shown
in Fig. 1. We first designed a potent teacher model to ensure a high-quality
and robust generation. At its core stands: 1) a simulation for super-
resolution (SSR) strategy that trains the model’s generation and robustness
capabilities using constructed low-quality and high-quality pairs end-to-end,
enabling the teacher to create high-quality results from varied-quality
features. 2) a motion-enhancing mechanism (MEM) that leverages 3D
convolutional networks and efficiently integrates 3D facial priors via multi-
stage infusions, enabling the model to refine precise motion depiction.

Equipped with the powerful teacher model, we turned our attention to cost. To
break the trade-off between performance and efficiency, we introduce a
distillation strategy to train an identity-specific student model that
compresses the computational load by 99%. During distillation, the teacher’s
components and intermediate features can be delivered to the student without
any cost, serving either as specific modules or inputs. With just one image
for two hours’ training, the student model achieves performance comparable to
that of the teacher, with the ability of real-time inference on more low-end
devices. In our experiments, we verified that the student model requires two
orders of magnitude fewer FLOPs than current state-of-the-art methods.

To further enable flexible editing of facial attributes, we introduce a simple
mask training mechanism (MTM) that masks and replaces the global driving
signals generated by MEM with other local ones. Based on this, an auxiliary
audio-to-lip module is proposed to provide specific control for lip movements,
promoting speech-driven settings. These expanded functionalities offer the
benefits of crossmodal-driven settings and local editing, allowing for greater
versatility in practical synthesis.

In summary, our technical contributions are as follows:

  * $\bullet$

We propose SuperFace, a teacher-student framework, which avoids the zero-sum
trade-off between generation quality and computational cost and achieves
superior and pragmatic talking head generation.

  * $\bullet$

We propose SSR to learn image quality enhancement and MEM to facilitate
accurate motion both end-to-end, in the teacher model. Experiments demonstrate
that SuperFace not only significantly outperforms existing methods in quality
and robustness but also possesses crossmodal-driving and facial attribute
editing abilities.

  * $\bullet$

Based on teacher model, we propose a distillation scheme to train an identity-
specific student model that achieves performance similar to that of the
teacher while reducing computational costs by two orders of magnitude.

##  2 Related Work

Existing researches [37, 34, 41, 53, 48] in this field primarily concentrate
on enhancing visual quality and motion replication performance.

For visual quality, some works [29, 51, 54] extract ample appearance
information from a set of source images to minimize occluded regions.
Nonetheless, usability challenges are introduced due to the harsh condition of
multiple images. MCNet [16] learns a global facial representation space to
compensate more correlated information using the appearance pool, and
overcomes the ambiguous generation caused by the dramatic motions. Inspired by
the StyleGAN series [20, 21, 19], some works [47, 3, 38] borrow their ideas or
utilize their pre-trained models to regenerate clear talking head video.
However, they also exhibit poor controllability inherited from StyleGAN. Some
works [46, 48] incorporate pre-trained super-resolution networks as post-
processing modules to improve image quality, at the expense of disrupting the
end-to-end structure and increasing the inference cost. Another disadvantage
of these schemes is that they neglect time consistency, resulting in an
temporal incoherent generated video.

For motion accuracy, many studies [37, 49, 29, 41] symbolize motion as a
representation consisting of a set of learned keypoints along with their dense
fields. Such implicit semantics bring challenges for downstream tasks such as
local editing. To simplify the training process, some works [23, 22, 8, 56,
34, 48] utilize explicit 3D Morphable Models (3DMMs) [1, 31] or 2D facial
landmarks [25, 11] directly, which significantly increase the interpretability
of the model. However, this simplistic usage fails to explore the full
potential of these facial priors in describing motion, and may introduce
additional errors from the detection tools.

Besides the GAN-based approaches mentioned above, there are also some NeRF-
based [12, 28, 46] and Diffusion Model-based [36, 30, 14, 7] efforts. AD-NeRF
[12] is the first NeRF-based method, it generates a dynamic neural radiance
field using audio and renders it to a talking video. Real-3d [46] employs a
large image-to-plane model to distill 3D prior knowledge for better visual
quality. Despite these innovations, they hardly produce photo-realistic
results due to the inherent difficulty in modeling high-frequency details with
volume rendering. The Denoising Diffusion Probabilistic Models [15] perform
better than GANs in some image generation tasks, and thus emerges many studies
in this field. Some works [14, 7] utilize diffusion model to predict motion
latents and render the final video via VAE. EMO [39] successfully applied the
diffusion model to rendering, bypassing the need for intermediate 3D models or
facial landmarks. The biggest drawback of these methods is their substantial
computational cost, making deployment difficult.

To enable the application of deep networks with large parameter counts on
lower-end devices, researchers have not only designed lightweight structures
[17, 50] but have also proposed techniques such as parameter pruning [26, 44],
parameter quantization [5, 9], and knowledge distillation [2]. Unfortunately,
this line of research still remains largely unexplored in talking head
generation.

##  3 Methodology

The goal of this work is to achieve  high-quality ,  robust ,  low-cost , and
editable generation simultaneously. We address these issues progressively
using a teacher-student framework. For convenience, we define the teacher
model as $\Phi$ and the student model as $\phi$. The organizations of this
section are as follows: We introduce the superb teacher model in Sec. 3.1. In
Sec. 3.2, we describe the details of our distillation paradigm and student
model. In Sec. 3.3, we explain how to expand the functionalities including
crossmodal-driven settings and local facial attribute editing. Finally, in
Sec. 3.4, we introduce the training details as well as the loss functions for
both models.

###  3.1 Teacher Model

Figure 2: The pipeline of our proposed framework. SuperFace consists of a
teacher model and a student model. a) We first train an extremely powerful
teacher model using MEM (motion-enhancing mechanism) and SSR (simulation for
super-resolution). b) Then we distill its knowledge into an efficient student
through feature delivery.

Standard procedure. Talking Head Generation aims to animate a static portrait
using driving signals. We start with its standard procedure in a video-driven
setting. For easy understanding, readers may refer to the top of Fig. 2 while
disregarding the SSR and MEM modules. Given two images
$I_{s}\in\mathbb{R}^{3\times H\times W}$ refers to the source image and
$I_{d}\in\mathbb{R}^{3\times H\times W}$ refers to the driving one, the
teacher model first extracts a spatial appearance feature $\mathcal{F}$ from
$I_{s}$ using an appearance encoder. After the motion field $\mathcal{W}$ is
extracted from both $I_{s}$ and $I_{d}$ using a dense motion network (DMN), it
will be used to warp $\mathcal{F}$, and the warped appearance feature
$\mathcal{F}^{s\to d}$ will be fed into the generator (GEN) for decoding a
realistic facial image.

Motion-enhancing-mechanism. Motion field $\mathcal{W}$ aims to warp the
appearance features $\mathcal{F}$ of $I_{s}$ into $\mathcal{F}^{s\to d}$ to
align with the motion status of $I_{d}$. In the real world, head motion is
three-dimensional, thus $\mathcal{W}$ and $\mathcal{F}$ should contain depth
information to be 3D-aware for better 3D-consistent feature alignment.
Consequently, we process them into 3D spatial forms
$\mathcal{W}_{3D}\in\mathbb{R}^{C\times{H^{\prime}}\times{W^{\prime}}\times{D^{\prime}}}$
and
$\mathcal{F}_{3D}\in\mathbb{R}^{C\times{H^{\prime}}\times{W^{\prime}}\times{D^{\prime}}}$
with the help of 3D convolutional network. However, our experiments revealed
that capturing precise 3D motion solely from planar pixel inputs $I$ is
challenging for 3D networks, and additional 3D priors are necessary to provide
more reliable information.

With this in mind, we propose Motion-enhancing-mechanism (MEM) that leverages
3D face mesh to predict neural keypoints $p_{3D}\in^{k\times 3}$, which denote
the image’s dynamic state and serve as the inputs to the DMN. There are two
naive ways to infuse 3D face mesh: 1) treating it as a volumetric feature,
which necessitates expensive computational costs; 2) converting it into a
point cloud-like form, which sacrifices the inherent spatial relationships. To
avoid these two drawbacks, we convert the 3D mesh into a dual representation:
1) a 2D mesh image that captures the specific topological relationships of the
face mesh, and 2) a 3D landmarks that preserves the original facial depth
information. This approach maintains the spatial and connectivity
relationships of the face mesh with a lower computational cost. Furthermore,
MEM develops a simple yet effective multi-stage infusion. As shown in Fig. 3,
2D mesh image $m$ and 3D landmarks $c$ are successively infused (early and
late) to predict neutral keypoints $p_{3D}^{s/d}$, which serves as motion
signals fed into DMN. The 2D image $I$ also serves as an input for MEM to
model the motion of the background. Through multi-stage infusion of spatial
and structural information from 3D facial mesh priors, MEM can assist the
model in accurately capturing 3D-aware motion.

Simulation for super-resolution. Generally, the quality of images generated by
a model is consistent with its input, resulting in struggling when confronted
with inferior priors. Although post-processing techniques can improve image
clarity, there can be some side effects, as illustrated in Sec. 2. Inspired by
works [43, 42, 45, 10] in super-resolution, we introduce an end-to-end visual-
enhancing training strategy without any post-modules. To be specific, a high-
order degradation module is introduced to simulate real-world image
degradation, as shown in Fig. 5, which is performed on source image $I_{s}$ to
obtain the low-quality and high-quality image pairs
$(\overline{I}_{s},I_{s})$. During training, $\overline{I}_{s}$ is used to
extract the appearance features $\overline{\mathcal{F}}_{3D}$ for subsequent
generation, while $I_{s}$ serves as the label for supervised learning. The
process is similar to super-resolution tasks but can be implemented end-to-end
within our model, hence we refer to it as  SSR (Simulation for super-
resolution). Thus, by incorporating SSR, the generator learns to robustly
synthesize clear images stably from arbitrary features.

Figure 3: Our proposed MEM incorporates 2D inputs and 3D priors to accurate
depict of the motion. 3D priors are fully utilized through early/late infusion
and two distinct representation designs.

Figure 4: Pipeline of our SSR. We introduce a second-order degradation. Each
order consists of random blurring, resizing, adding noise, and JPEG
compression.

Figure 5: Pipeline of our MTM. We mask the vanilla driving signals and replace
them with other ones. It facilicates local editing and crossmodel-driving.

###  3.2 Distillation Paradigm

Utilizing MEM and SSR, we have developed a potent model equipped with the
ability to model 3D-aware motion and ensure robust synthesis. Employing this
as the teacher model, our goal is to lessen its computational demands,
enabling us to derive a student model for practical deployment. To this end,
we propose a distillation paradigm to distill teacher’s knowledge into an
identity-specific student model. Notably, the generalization of the student
model and the expansion of identities are discussed in Sec. 4.3. As depicted
in the bottom of Fig. 2, we design the student model with a structure similar
to that of the teacher, yet significantly more computationally economical,
replacing the 3D Convolutions with more lightweight 2D ones. During
distillation, teacher model first infers portrait animations using a single
image of a specific speaker and arbitrary driving videos. Both of the
generated videos and intermediate features are then serve as training data of
student model. Thanks to its effective network design, the teacher model is
capable of capturing precise motion details and modeling robust appearance
features. To inherit these merits, the distillation paradigm deliver teacher’s
prior knowledge into student, including the motion features $p_{3D}$ for
motion accuracy, and the appearance features $\overline{\mathcal{F}}_{3D}$ for
visual quality.

Specifically, we employ an appearance distributor $\phi_{d}$ to compresse
$\mathcal{W}_{3D}$ to $\mathcal{W}_{2D}$ and orthogonal projection to convert
keypoints $p_{3D}$ to $p_{2D}$. Additionally, the teacher’s discriminator is
also employed to serve an effective supervisory role during the distillation
process. Through such a design, the benefits of SSR and MEM proposed in Sec.
3.1 remain highly advantageous for the student model. In experiments, we also
verified that our proposed distillation strategy can significantly reduce the
data and time dependence of distillation training.

Since the appearance and keypoint features are transmitted from the teacher
model, the SSR and MEM proposed in our teacher model sitll work effectively
within the student. Additionally, the teacher’s discriminator is employed to
serve an effective supervisory role during the distillation process.

###  3.3 CrossModal and Local Editing

So far, we have addressed the issues of high quality, robustness, and low
cost, and now we turn our attention to the final challenge: editability.
Unfortunately, the ambiguous semantics of neural keypoints $p$ make it
difficult to control the generation of local attributes. To address this, a
semantically clearer and more easily learnable local driving signal is
introduced.

To eliminate the interaction between the different driving signals, we
introduce a mask training mechanism (  MTM ). For convenience, we use the
mouth landmarks $c^{loc}$ as an example of the driving signals that can be
edited. As depicted in Fig. 5, by predicting the global driving signals $p$
from the masked inputs $I$ and $m$ and employing the local ones $c^{loc}$ as
complement, MTM ensures that the semantics of two signals do not intertwine
and guarantees the decoupling of motion control across different regions.
During inferene, $c^{loc}$ and $p$ can originate from different sources to
realize localized facial attribute editing. Notably, this design is generic
and can be extended to arbitrary facial attributes.

Apart from extracting $c^{loc}$ directly from video, we also developed an
additional audio2lip module to offer crossmodal-driving capabilities. This
design provides the user with convenience that the control mechanism for
localized editing is greatly simplified.

###  3.4 Training strategy

To train SuperFace, we utilize loss functions similar to those used in
previous works [53, 41, 37]. It comprises:

Perceptual Loss $\mathcal{L}_{P}$.

The perceptual similarity loss [18] between $I_{d}$ and $y$ helps to improve
the perceptual quality. We minimize the pyramid perceptual loss between
features extracted from $I_{d}$ and $y$ through a pre-trained network.

GAN Loss $\mathcal{L}_{G}$.

. We employ the multi-scale PatchGAN [40] as our discriminator, which predicts
the probability whether the generated frame is comparable to the ground truth
at the patch-level.

Keypoint Loss $\mathcal{L}_{K}$.

To ensure that the extracted keypoints $p_{3D}$ evenly spread throughout the
face region rather than crowd around a confined neighborhood, we incorporate a
regularization as constraint. This involves measuring the distance between
each point pair of $p_{3D}$, with the model incurring a penalty if any
distance falls below a specified threshold.

Expression Loss $\mathcal{L}_{Exp}$.

Experiments results indicate that the motion can’t be accurately modeled if
the value of the expressions is too large. Therefore, we propose a
regularization term to constrain its range.

Head pose Loss $\mathcal{L}_{H}$.

A pre-trained network [35] is employed to detect the head poses of $y$ and
$I_{d}$. We minimize $\mathcal{L}_{H}$ so that they are consistent.

Equivariance Loss $\mathcal{L}_{Equ}$.

Due to $p_{3D}^{c}$ is learned unsupervised, we employ a equivariance loss to
ensure its validity. Specifically, we apply a random transformation $T(\cdot)$
to $I_{s}$ and obtain the transformed image $I_{Ts}$. The keypoints
$p_{3D}^{c}$ from $I_{s}$ and $p_{3D}^{c}T$ from $I_{Ts}$ should maintain the
same transformation relationship.

Reconstruction Loss $\mathcal{L}_{R}$.

This loss is to help the model converge quickly at the early stage of training
and accelerate the training process.

Local Loss $\mathcal{L}_{L}$.

To enhance the local high-frequency representation in the generated images, we
crop the eyes and mouth regions and compute localized perceptual loss as well
as landmarks loss.

For the teacher model, its total loss is:

$\displaystyle\mathcal{L}_{\Phi}=$ $\displaystyle\lambda_{\Phi
1}\mathcal{L}_{P}+\lambda_{\Phi 2}\mathcal{L}_{G}+\lambda_{\Phi
3}\mathcal{L}_{K}+\lambda_{\Phi 4}\mathcal{L}_{Exp}+\lambda_{\Phi
5}\mathcal{L}_{H}+$ (1) $\displaystyle\lambda_{\Phi
6}\mathcal{L}_{Equ}+\lambda_{\Phi 7}\mathcal{L}_{R}+\lambda_{\Phi
8}\mathcal{L}_{L}$

For the student model, its total loss is:

$\mathcal{L}_{\phi}=\lambda_{\phi 1}\mathcal{L}_{P}+\lambda_{\phi
2}\mathcal{L}_{G}+\lambda_{\phi 2}\mathcal{L}_{H}+\lambda_{\phi
4}\mathcal{L}_{R}+\lambda_{\phi 5}\mathcal{L}_{L}$ (2)

##  4 Experiments

In this section, we carry out comprehensive experiments to evaluate our
SuperFace qualitatively and quantitatively. SuperFace consists of a powerful
teacher model and a low-cost student model. We first introduce their specific
implementations in Sec. 4.1. The evaluation of SuperFace’s teacher model is
then conducted in Sec. 4.2, spanning video-driven and audio-driven settings,
as well as its robustness and editability. To verify the effectiveness of our
distillation paradigm, Sec. 4.3 presents the exploration results of the
student model in various dimensions. Ablation studies in Sec. 4.4 confirmed
that each proposed component is essential and contributes to the ultimate
performance.

###  4.1 Implementation Details

####  4.1.1 Datasets

Our teacher model was trained on the CelebV-HQ [57] dataset, a high-resolution
video dataset with 35k videos at a minimum resolution of $512\times 512$. A
simple data filtering process was performed, and eventually, around $31,000$
videos were retained for use. The student model was subsequently trained using
the teacher’s inference results on the HDTF dataset [52]. The test set for
video-driven settings derives from three sources: TK1KH [41], HDTF [52] and
CCV2 [13], from which 300 videos were sampled randomly. Additionally, 100 HDTF
videos entries with audio have been employed as our test set for audio-driven
evaluation.

####  4.1.2 Implementation Detail

We train our model on 8 NVIDIA A-100 (80G) machines. We used the ADAM
optimizer [24] with the learning rate $2\times 10^{-4}$ and
$(\beta_{1},\beta_{2})=(0.5,0.999)$. The numbers of keypoints $p$ fed into DMN
is set to. The batch size is set to 32 for all experiments. To provide 3D
priors and edit local facial attributes, we employ pretrained Mediapipe [27]
to extract 3D landmarks and render meshes. As for the trade-off loss weights,
we set $\lambda_{\Phi 1}=10,\lambda_{\Phi 2}=10,\lambda_{\Phi
3}=5,\lambda_{\Phi 4}=10,\lambda_{\Phi 5}=1,\lambda_{\Phi 6}=10,\lambda_{\Phi
7}=10,\lambda_{\Phi 8}=100,\lambda_{\phi 1}=10,\lambda_{\phi
2}=10,\lambda_{\phi 3}=10,\lambda_{\phi 4}=1,\lambda_{\phi 5}=100$.

####  4.1.3 Metrics

Several metrics are utilized to assess our SuperFace’s validity in terms of
both image quality and replication accuracy. For replication, we report the
cosine similarity (CSIM) of face embedding between $I_{s}$ and $y$ using a
recognition method [6]. We also evaluated the whole motion transfer quality
accessed by Average Keypoint Distance (AKD) [4], Average Expression Distance
(AED) [33] and Head Pose Distance (HPD). For visual quality, since our
SuperFace attempts to produce videos with higher clarity than the inputs,
comparison-based metrics are not applicable in this context thus no-reference
ones Entropy and Energy are employed. For audio-driven settings, we report
SyncNet Distance (Sync-D) which measures the synchronization between audio-
visual modalities.

###  4.2 Teacher Model Results

Figure 6: Qualitative video-driven comparisons with baselines. For better visualization, we highlight the artifacts using different anchor boxes. The blue bounding box concentrates on motion accuracy, whereas the red one represents visual quality. Table 1: Quantitative comparisons of our teacher with video-driven baselines. Methods | Visual | Same-id Motion | Cross-id Motion |  FLOPs (G) $\downarrow$
---|---|---|---|---
Ene$\uparrow$ | Ent/$10^{6}$ $\uparrow$ | CSIM$\uparrow$ | AKD$\downarrow$ | APD$\downarrow$ | AED$\downarrow$ | CSIM$\uparrow$ | APD$\downarrow$ | AED$\downarrow$
Face-v2v | 4.80 | 7.02 | 0.663 | 3.341 | 1.294 | 0.101 | 0.468 | 4.459 | 0.197 | 600
TPS MM | 4.81 | 6.34 | 0.668 | 1.897 | 1.195 | 0.083 | 0.416 | 4.435 | 0.152 |  120
MCNet | 4.76 | 5.07 | 0.668 | 1.570 | 1.322 | 0.086 | 0.378 | 4.413 | 0.154 | 200
ours 256 |  4.82 |  7.19 |  0.674 |  1.442 |  1.144 |  0.069 |  0.469 |  3.837 |  0.143 | 150
StyleHEAT | 4.80 | 3.95 | 0.337 | × | 2.417 | 0.148 | 0.277 | 4.778 | 0.200 | 1895
Hide-NeRF | 4.29 | 5.41 | × | × | 2.721 | 0.155 | × | 4.962 | 0.210 | 1065
Real3d | 4.85 | 7.22 | 0.649 | 1.638 |  0.822 | 0.083 | 0.478 | 3.744 | 0.176 | 610
ours 512 |  4.86 |  7.59 |  0.682 |  1.077 | 0.839 |  0.071 |  0.491 |  3.225 |  0.151 |  600

####  4.2.1 Video-Driven

We select several strong baselines using various technical routes. The
quantitative results are presented in Tab. 1. Obviously, our teacher model
outperforms state-of-the-art methods across all objective metrics, expection
of APD, where it is on the par with Real-3d. It is worth noting that our FLOPs
are comparable to those of baselines when generating videos at the same
resolution. We also present the qualitative comparisons in Fig. 5. It is
evident that our approach surpasses these baselines in terms of generation
quality and replication quality. Our technique is superior in capturing and
reconstructing tiny movements like blinking, lip motion, etc. In addition, our
SuperFace produces clearer portraits with rich texture details. High-frequency
information such as gaze, hair, and wrinkles is successfully reconstructed.

Moreover, our teacher model shows great robustness in producing high-quality
videos consistently, even when faced with large-posed or low-quality inputs.
The corresponding qualitative results are available in Fig. 7. Since talking
face synthesis is a video generation task, we highly recommend the reader
refer to our supplementary materials for more intuitive demos.

Figure 7: More visualization results of our teacher model. Top : it produces high-quality video enven when faced with poor-quality appearance and intense-motion inputs. Bottom : it supports the editing of localized facial attributes. Table 2: Quantitative comparisons of our teacher with audio-driven baselines.  Methods | AKD$\downarrow$ | APD$\downarrow$ | AED$\downarrow$ | CSIM$\uparrow$ | Sync-D$\downarrow$
---|---|---|---|---|---
MakeItTalk | 7.245 | 1.616 | 0.127 | 0.711 | 9.460
SadTalker | 7.299 | 1.003 | 0.115 | 0.678 | 7.867
Real3d | 6.574 | 0.995 | 0.104 | 0.674 |  7.774
ours |  0.907 |  0.945 |  0.068 |  0.758 | 9.354

####  4.2.2 Audio-Driven

Audio-driven talking head generation places a greater emphasis on lip
synchronization. Under this settings, we compare with MakeItTalk [56],
SadTalker [49] and Real3d-Protraits [46]. The qualitative results are shown in
Tab. 2. It can be observed that our AKD metric substantially surpasses those
of baselines, demonstrating SuperFace’s attention to global facial attributes
in audio-driven settings. In the realms of head motion and facial expressions,
our teacher model has also realized the most accurate reconstruction. We also
achieve better identity preservation performance. As our trainging didn’t
incorporate the SyncNet which is used to calculate Sync-D, our teacher shows
disadvantages compared to baselined that emolpyed it. Nevertheless, our
performance is on par with other models.

###  4.3 Student Model Results

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 8: The investigation results of the student model. From a row perspective, CSIM reflects the ability to preserve identity, Energy denotes visual quality, and AKD indicates the accuracy of the motion. Viewed column-wise, Steps represents an exploration into training duration, Data embodies an inquiry into the training datasize, and FLOPs pertains to the study of model complexity. Table 3: Quantitative comparisons of our student model with baselines.  Methods | Visual | Motion |  FLOPs (G) $\downarrow$
---|---|---|---
Ene$\uparrow$ | Ent/$10^{6}$$\uparrow$ | CSIM$\uparrow$ | APD$\downarrow$ | AKD$\downarrow$ | AED$\downarrow$
Face-v2v S | 4.41 | 2.26 | 0.522 | 1.528 | 6.643 | 0.248 | 6.0
MCNet S | 4.44 | 2.34 | 0.589 | 1.625 | 5.237 | 0.188 | 6.0
Face-v2v D | 4.51 | 2.83 | 0.563 | 1.373 | 5.678 | 0.174 | 6.0
MCNet D | 4.57 | 2.63 | 0.638 | 1.319 | 4.231 | 0.143 | 6.0
ours |  4.78 |  4.16 |  0.758 |  0.399 |  0.907 |  0.069 | 6.0

Given that early works in this field have overlooked computational costs,
there are no baselines with comparable complexity to our student model. Based
on teacher’s baselines [41, 16], we also established baselines for the student
by two principles. The first is simplifying them to 6G (consistent with our
student model) while retaining their structure designs, then retrained their
small models (S) using the same dataset and training strategy. The second is
similar to our distillation paradigm, which learns knowledge from their large
models to the distilled ones (D). The evaluation results are shown in Tab. 3.
It can be concluded that our student model has achieved a considerable lead in
all metrics. Baselines, regardless of whether they are S or D, suffer severe
performance degradation. Yet our student model manages to maintain consistency
with the teacher model, and its visualized results in Fig. 9(b) exhibit
natural attributes and clear visual quality. Another observation is that both
S versions fall short of their D versions, suggesting that distillation is
beneficial for reducing computational budgets.

As shown in Fig. 8, we vary the data size, training steps, and FLOPs of the
student model to explore its capabilities of identity-preserving (CSIM),
visual quality (Energy), and motion accuracy (AKD). It is worth noting that
our proposed model is not sensitive to dataset size or computational
complexity (FLOPs), which only have a subtle influence on our model. Hence, it
can be concluded that SuperFace’s student model can quickly converge to a
desirable state with minimal data (1 minute) and a short training duration
(2$\sim$5 hours). Besides, our student model is scalable and behaves better
when the training duration increases.

To explore the practical utility of the student model, we further investigated
its generalization capability by training it with several different
identities. Fig. 9(a) reveals that the identity-preserving ability (CSIM) of
our student model almost does not drop (0.76 $\rightarrow$ 0.74) when the
number of identities increases (1 $\rightarrow$ 20). At the same time, the
visual quality (Energy) slightly decreases but still keeps an acceptable level
(4.20 $\rightarrow$ 3.70). Visualized results in Fig. 9(b) also show that our
student model can generalize to multiple speakers while maintaining generation
quality, which greatly reduces the deployment cost in practical applications.

(a) The curve of speaker numbers and model performance.

(b) Several samples generated by a single student model.

Figure 9: The exploration to the generalization of our student model.

###  4.4 Ablation Studies

Ablation studies are performed to explore the contribution of each proposed
component. For the teacher model, we construct four variants including SSR,
early infuse and late infuse in MEM as well as our full model. The
quantitative results are shown in Tab. 4. Obvisouly, the version w/o SSR
exhibit degradation in visual quality metrics, while those w/o MEM show
decline in motion metrics. It demonstrates that the SSR helps the image
clarity and the MEM (infusions) facilitate motion accuracy.

For the student model, we examined the impact of the delivered features by
manipulating three variables: 1) w/o neural keypoints which learns $P_{2D}$
using a network instead of deriving from the teacher model; 2) w/o appearance
features which predicts $\mathcal{F}_{2D}$ from the image $I_{s}$ rather than
the teacher’s $\mathcal{F}_{3D}$; 3) w/o discriminator which fires the
teacher’s pretrained discriminator for supervision. The results presented in
Tab. 5 clearly reveal that optimal performance is only realized by the full
model, verifying the significance of each element we have proposed.

Table 4: Ablation studies of teacher model. Methods | Visual | Same-id Motion | Cross-id Motion
---|---|---|---
Ene$\uparrow$ | Ent/$10^{6}$$\uparrow$ | CSIM$\uparrow$ | AKD$\downarrow$ | APD$\downarrow$ | AED$\downarrow$ | CSIM$\uparrow$ | APD$\downarrow$ | AED$\downarrow$
w/o early fusion | 4.82 | 7.41 | 0.641 | 1.736 | 1.064 | 0.087 | 0.428 | 4.410 | 0.162
w/o late fusion |  4.89 |  7.75 | 0.658 | 2.477 | 1.183 | 0.094 | 0.445 | 4.443 | 0.181
w/o SSR | 4.79 | 6.65 |  0.685 | 1.299 | 0.881 |  0.071 | 0.486 | 4.380 | 0.163
full | 4.86 | 7.59 | 0.682 |  1.077 |  0.839 |  0.071 |  0.491 |  3.225 |  0.151
Table 5: Ablation studies of student model.  Methods | Visual | Motion |
---|---|---|---
Ene$\uparrow$ | Ent/$10^{6}$$\uparrow$ | CSIM$\uparrow$ | APD$\downarrow$ | AKD$\downarrow$ | AED$\downarrow$
w/o nk | 4.70 | 3.47 | 0.612 | 0.703 | 5.004 | 0.148
w/o app | 4.68 | 3.65 | 0.715 | 0.681 | 0.989 | 0.087
w/o disc | 4.69 | 3.80 | 0.720 | 0.614 | 0.969 | 0.074
full |  4.78 |  4.16 |  0.758 |  0.399 |  0.907 |  0.069

##  5 Conclusion

In this work, we propose SuperFace, a superior and pragmatic teacher-student
framework for talking head generation. We start by designing an powerful
teacher model using the super-resolution training and motion-enhancing
mechanism to achieve high quality and robustness. Then we propose a generic
distillation paradigm to distill its knowledge into a computationally
efficient architecture. A mask-training-mechanism resolves the issue of
semantic decoupling in driving signals, enabling local editing and cross-modal
control. Extensive comparisons demonstrate that our SuperFace outperforms
state-of-the-art approaches, and the student model inherits high performance
from the teacher with exhibits characteristics of low cost. Ablation studies
clearly show that our proposed components contribute to model’s performance.

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Cardy-Verlinde formula from boundary matrix model

Pavan Kumar Yerra<EMAIL_ADDRESS>Chandrasekhar
<EMAIL_ADDRESS>and Sudipta
<EMAIL_ADDRESS>

${}^{1},^{3}$ Institute of Physics, Sachivalaya Marg,

Bhubaneswar, Odisha, 751005, India

and

Homi Bhabha National Institute,

Training School Complex,

Anushakti Nagar, Mumbai, 400085, India

2 School of Basic Sciences

Indian Institute of Technology Bhubaneswar

Bhubaneswar, Odisha, 752050, India

Abstract

Cardy-Verlinde (CV) formula relates the entropy of a strongly coupled
conformal field theory (CFT) possessing AdS dual, at finite temperature, to
its total energy (with an appropriate insertion of additional internal energy
for charged systems) and Casimir energy. While the CV formula has been checked
for various CFTs, in the present work, we intend to verify it directly, by
exploiting a phenomenological matrix model, which is believed to qualitatively
capture features of strongly coupled ${\cal N}=4,SU(N)$ Super Yang-Mills
theory on $S^{3}$ at finite temperature and chemical potential at large $N$

##  1 Introduction

A couple of decades ago, Verlinde observed that the entropy of a strongly
coupled conformal field theory (CFT), possessing AdS dual, satisfies a Cardy-
like formula at finite temperature [1]. This later came to be known as the
Cardy-Verlinde (CV) formula. For CFTs on $R\times S^{n}$, it reads

$S=\frac{2\pi R}{n}{\sqrt{E_{c}(2E-E_{c})}}.$ (1.1)

Here, $S$ is the entropy associated with the CFT, $R$ is the radius of the
$S^{n}$, $E$ is the total energy and $E_{c}$ is the subextensive part of it.
This relation was shown to arise holographically from the thermodynamic
quantities associated with the AdS-Schwarzschild black hole. The CV formula
drew immediate attention to many as it satisfies a Bekenstein like entropy
bound in $n$ dimensions, namely,

$S\leq\frac{2\pi RE}{n}.$ (1.2)

Subsequently, the formula found its generalization to many other CFTs. One
example that will be important for our purpose is the one dual to an
electrically charged AdS black hole. The entropy was then found to obey [2]

$S=\frac{2\pi R}{n}{\sqrt{E_{c}[2(E-Eq)-E_{c}]}},$ (1.3)

where $E_{q}$ is the electrostatic potential energy.

While a direct computation [3] of the entropy of a weakly coupled CFT is shown
to violate the bound (1.2), for a strongly coupled CFT, a similar analysis is
missing. In this work we wish to take a step along this direction. Generally,
to describe a gauge theory on $S^{3}\times S^{1}$ at zero coupling, one writes
down an effective action involving the vev of the Wilson-Polyakov loop
variable on $S^{1}$ as the only light degree of freedom [4, 5, 6, 7, 8, 9, 10,
11]. Since it is hard to compute such an action at finite Yang-Mills coupling,
one may resort to phenomenological model belonging to the same universality
class of the boundary theory. Indeed, such a phenomenological model was
proposed in [6]. It contains two parameters which depend on the temperature
and $\lambda$, the ’t Hooft coupling. In [6, 12], these dependences were found
by exploiting the data arising from the dual supergravity description.
Subsequently, the model was generalized to incorporate the $R$-charge emerging
from the charged AdS black hole [7, 12]. This is done by allowing the model
parameters to depend on the chemical potential as well. Armed with the results
of these investigations, in the present work, we check if the model is
consistent with the CV formula. To this end, we compute the left and the right
hand sides of the equation (1.1) directly from the model and find qualitative
agreement. A similar agreement follows for the charged case as well when
compared against the equation (1.3). Due to the complexity of the model, at
places analytical handle becomes difficult and we had to resort to the
numerical computations.

The organization of this paper is as follows. In section-(2), we start with
the AdS-Schwarzschild black hole. In subsection-(2.1 ), we briefly review the
corresponding matrix model living on the boundary. Subsection-(2.2) contains a
direct check on the CV formula arising from the model. Subsequently, in
section-(3), working in the canonical ensemble, we generalize our computation
of the previous section to the electrically charged black hole case. In
subsection-(3.1), we review the extension of the matrix model to include non-
zero chemical potential. The computation leading to the agreement with the CV
formula is presented in the next subsection. Section-(4) contains some
remarks. We end the paper with two appendices where the thermodynamic
properties associated with the bulk black holes are summarized.

##  2 Cardy-Verlinde formula from boundary matrix model: zero chemical
potential

In this section, we start by considering the case of a conformal field theory
(CFT) represented by a matrix model dual to the Schwarzschild-AdS5 black hole
on the gravity side. We compute the entropy of the CFT from the effective
potential of the matrix model. Considering the CFT in a finite volume, the
Casimir energy can be obtained. Then, we numerically verify the Cardy-Verlinde
formula relating the entropy with its energy and Casimir energies.

###  2.1 Boundary matrix model: zero chemical potential

Our starting point is the phenomenological matrix theory, called the $(a,b)$
model, which has been shown to capture, among other aspects, the deconfinement
phase transition of $\mathcal{N}=4$, $SU(N)$ guage theory in the large $N$
limit. We only describe this model briefly, as details are available in [6,
5]. We also closely follow the set up in [8, 10, 11, 12], well suited for our
purposes. To this end, the partition function of the SYM theory is written as
a matrix integral over the effective action consisting of the operator
corresponding to the Wilson-Polyakov loop, i.e., $(\text{tr}U)/N$ as

$Z(\lambda,T)=\int dUe^{S_{\text{eff}}(U)}.$ (2.1)

Here, $U=P\,\text{exp}(i\int_{0}^{\beta}Ad\tau)$ is the $U(N)$ unitary matrix,
where $A(\tau)$ stands for the zero mode (of the time component) of the gauge
field on $S^{3}$. Generally, the action $S_{\text{eff}}(U)$ turns out to be a
polynomial in the traces of $U$ and its powers, which are allowed by imposing
the $Z_{N}$ symmetry. Now, by truncating the above action and retaining only a
couple of terms, one ends up with the aforementioned phenomenological $(a,b)$
model, which takes the form,

$Z(a,b)=\int
dU\,\text{exp}[a(\text{tr}U\,\text{tr}U^{{\dagger}})+\frac{b}{N^{2}}(\text{tr}U\,\text{tr}U^{\dagger})^{2}].$
(2.2)

Here, $a$ and $b$ are the key parameters of the model which contain nontrivial
dependence on the temperature $T$ and the ’t Hooft coupling $\lambda$.
Furthermore, the effecting potential ensuing from this set up can be expressed
in terms of the order parameter $\rho$ ( which is the expectation value of the
Polyakov loop $\frac{1}{N}\langle\text{tr}\,U\rangle$) that characterizes the
deconfined phase of the gauge theory as:

$\displaystyle V(\rho)$ $\displaystyle=$
$\displaystyle\frac{1-a}{2}\rho^{2}-\frac{b}{2}\rho^{4}\hskip
113.81102pt\text{for}\quad 0\leq\rho\leq\frac{1}{2}$ (2.3) $\displaystyle=$
$\displaystyle-\frac{a}{2}\rho^{2}-\frac{b}{2}\rho^{4}-\frac{1}{4}\text{log}[2(1-\rho)]+\frac{1}{8}\hskip
22.76228pt\text{for}\quad\frac{1}{2}\leq\rho\leq 1.$ (2.4)

Here, $\rho^{2}=(\frac{1}{N^{2}})\text{tr}\,U\text{tr}\,U^{\dagger}$ and the
saddle point equation is

$\displaystyle a\rho+2b\rho^{3}$ $\displaystyle=$ $\displaystyle\rho\hskip
56.9055pt\text{for}\quad 0\leq\rho\leq\frac{1}{2}$ (2.5) $\displaystyle=$
$\displaystyle\frac{1}{4(1-\rho)}\hskip
22.76228pt\text{for}\quad\frac{1}{2}\leq\rho\leq 1.$ (2.6)

The parameters satisfy the bounds $a<1$ and $b>0$. The dependence of the
parameters $a(T)$ and $b(T)$ on the temperature $T$ can be obtained
numerically making use of the bulk data, as in [6, 8, 12]. The result is
captured in Fig. 1.

(a)  (b)

Figure 1: Plots show the temperature dependence of the parameters $a(T)$ and
$b(T)$ for the phenomenological $(a,b)$ matrix model at zero chemical
potential. The red dots denote the data points, which are fitted by the blue
colored curves. The fitting formulas are
$a(T)=c_{1}\text{log}(T)+c_{2}T+c_{3}$, and $b(T)=c_{4}T+c_{5}$,
$(\text{where,}\,c_{1}=2.257,c_{2}=-4.579,c_{3}=4.171,c_{4}=11.239,c_{5}=-4.169.)$.

Now, utilizing the fitting curves for $a(T)$ and $b(T)$, the saddle point
equation (2.6) can be solved, yielding the following expression for
temperature:

$T(\rho)=\frac{c_{1}}{(c_{2}+2c_{4}\rho^{2})}\,\text{ProductLog}\Big{[}\frac{e^{\frac{1-4c_{3}(1-\rho)\rho-8c_{5}(1-\rho)\rho^{3}}{4c_{1}\rho(1-\rho)}}(c_{2}+2c_{4}\rho^{2})}{c_{1}}\Big{]}.$
(2.7)

As seen from Fig. 2a, $T(\rho)$ mimics the behaviour of the equation of state
of the theory in the bulk (see Fig. 7 in Appendix-A).

(a)

(b)

Figure 2: Case of $(a,b)$ matrix model at zero chemical potential: (a) Nature
of $T(\rho)$ qualitatively resembles the behaviour of the equation of state of
the bulk (cf. Fig. 7 in Appendix-A). (b) The effective potential $V(\rho)$ at
various temperatures $T$ is shown. Temperature of the curves increases from
top to the bottom. Blue curve is for $T_{\rm min}$, where as the dashed red
curve showing the HP transition is for $T_{\rm HP}$, with $a=0.3155$ and
$b=1.194$, $(\rho_{\rm min},\rho_{\rm HP})=(0.714,0.861)$.

Now, let us discuss the nature of the effective potential $V(\rho)$, which has
been plotted in Fig. 2b for various temperatures. In the $\rho>1/2$ region,
the phase structure of the bulk system emerges from the saddle points of
$V(\rho)$ [6]. At any temperature above a certain minimum temperature $T_{\rm
min}$, one gets two saddle points corresponding to large black holes (LBH) and
small black holes (SBH), with the former being stable and the later unstable.
$T=T_{\rm min}$ is the nucleation point for the creation of this black hole
pair. Minimum occurring at $\rho=0.861$ denotes the HP transition temperature
in the bulk, which corresponds to the deconfining temperature of the boundary
theory. The minima occuring at $\rho=0$ stands for the thermal
$\text{AdS}_{5}$ phase. It is useful to note that the supergravity description
is not valid in the $\rho\leq 1/2$ region and at $\rho=1/2$ the matrix model
exhibits a third order Grass-Witten transition [13], which has a natural
intepretation as a Horowitz-Polchinski point in string theory [14]. Since,
here the effective potential $V(\rho)$ plays the role of off-shell free energy
of the bulk theory (see equation-(A.8) in Appendix-A), in what follows, we use
it to derive other thermodynamic quantities, as is generally done in standard
treatment. The on-shell free energy $F$ can now be obtained, using the fitting
curves for $a(T)$, $b(T)$, and on substituting $T(\rho)$ in the effective
potential $V(\rho)$ in equation (2.4), as

$F=-\frac{\rho^{2}}{2}(c_{1}\text{log}T+c_{2}T+c_{3})-\frac{\rho^{4}}{2}(c_{4}T+c_{5})-\frac{1}{4}\text{log}[2(1-\rho)]+\frac{1}{8}.$
(2.8)

The computation of the entropy $S$ of the CFT (assuming a relation of the form
$dF=-SdT$ analogous to bulk free energy) turns out to be

$S=-\frac{\partial F}{\partial T}=-\frac{\partial F/\partial\rho}{\partial
T/\partial\rho}.$ (2.9)

Although, the above can be evaluated straightforwardly, the expression is
large and hence we do not present it here.

###  2.2 Cardy-Verlinde formula

Now, to derive the Cardy-Verlinde (CV) formula, we closely follow the set up
in [15]. We write the boundary metric, from the large $r$ behaviour of the
bulk metric (A.2), as

$ds^{2}=\frac{r^{2}}{l^{2}}(-dt^{2}+l^{2}d\Omega_{3}^{2}).$ (2.10)

After scaling the time parameter to $\tau=\frac{R}{l}t$, the above metric can
be recast as

$ds^{2}=\frac{r^{2}}{R^{2}}(-d\tau^{2}+R^{2}d\Omega_{3}^{2}),$ (2.11)

which is conformally equivalent to $\mathbb{R}\times S^{3}$, where $S^{3}$ now
has radius $R$. According to the holographic principle, the temperature
$\bar{T}$ and the free energy $\bar{F}$ of the CFT can be obtained by scaling
the respective expressions in (2.7) and (2.8) as [15]:

$\displaystyle\bar{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{\beta}}=\frac{l}{R}T,$ (2.12)
$\displaystyle\bar{F}$ $\displaystyle=$ $\displaystyle\frac{l}{R}F,$ (2.13)

with the CFT volume taken to be $\bar{V}=\omega_{3}R^{3}.$ Assuming the free
energy to be $\bar{F}=\bar{E}-\bar{T}\bar{S}$, we derive the energy $\bar{E}$
as a function of temperature $\bar{T}$ and volume $\bar{V}$ as

$\bar{E}=\frac{\partial(\bar{\beta}\bar{F})}{\partial\bar{\beta}}\Big{|}_{\rm\bar{V}}=\frac{\partial(\bar{\beta}\bar{F})/\partial\rho}{\partial\bar{\beta}/\partial{\rho}}\Big{|}_{\rm
R},$ (2.14)

while, the entropy $\bar{S}$ is

$\bar{S}=\bar{\beta}(\bar{E}-\bar{F})=S.$ (2.15)

This shows that the entropy of the CFT does not scale with $R$. Now, the
pressure $\bar{P}$ can be defined as

$\bar{P}=-\frac{\partial\bar{F}}{\partial\bar{V}}\Big{|}_{\rm\bar{\beta}}=-\frac{\partial\bar{F}(\bar{\beta},\rho)/\partial\rho}{\partial\bar{V}/\partial\rho}\Big{|}_{\rm\bar{\beta}},$
(2.16)

which satisfies the equation of state $\bar{E}=3\bar{P}\bar{V}.$ The Gibbs
free energy is $\bar{G}=\bar{F}+\bar{P}\bar{V}$, and thus the Casimir energy
can be defined by $\bar{E}_{c}=3\bar{G}$ [1]. The expressions for the
aforementioned thermodynamic quantities can be obtained, but are quite
lengthy. Hence, we directly show our result in Fig. 3, which confirms validity
of the following Cardy-Verlinde formula given as [1]:

$\bar{S}=\frac{2\pi R}{3}\sqrt{\bar{E}_{c}(2\bar{E}-\bar{E}_{c})}.$ (2.17)

There is however a mismatch of the curves around $\rho=1$ in Fig. 3, which we
attribute to some statistical errors and the approximations which were
considered in the current model.

Figure 3: For $(a,b)$ matrix model with zero chemical potential: Red curve is
for the LHS and the blue curve is for the RHS of the equation (2.17).

##  3 Cardy-Verlinde formula from boundary matrix model: non-zero chemical
potential

In this section, we consider a boundary matrix model with chemical potential
dual to Reissner-Nordstrom-AdS5 black holes in canonical (fixed charge)
ensemble. Using the effective potential of the matrix model, we perform a
computation similar to the one in the previous section. We also numerically
verify the Cardy-Verlinde formula, which now contains electrostatic potential
energy coming from the charge, in addition to the Casimir energy present
earlier.

###  3.1 Boundary matrix model at non-zero chemical potential

One can extend the $(a,b)$ matrix model shown in eqn. (2.2), with an
additional logarithmic term in the effective potential at a fixed nonzero
charge $q$. This will serve as a boundary dual for the Reissner-Nordstrom
$\text{AdS}_{5}$ black holes in the canonical ensemble. The effective action
of this model is [7]:

$S_{q}=S\big{(}a(T),\,b(T),\,\rho\big{)}+q\,\text{log}(\rho),$ (3.1)

which is supplemented by the following saddle point equations:

$\displaystyle\rho F+q$ $\displaystyle=$ $\displaystyle\rho^{2}\hskip
56.9055pt\text{for}\quad 0\leq\rho\leq\frac{1}{2}$ (3.2) $\displaystyle=$
$\displaystyle\frac{\rho}{4(1-\rho)}\hskip
22.76228pt\text{for}\quad\frac{1}{2}\leq\rho\leq 1\,,$ (3.3)

where $F(\rho)=a\rho+2b\rho^{3}$. The effective potential for this model is

$\displaystyle V(\rho)$ $\displaystyle=$
$\displaystyle\frac{1-a}{2}\rho^{2}-\frac{b}{2}\rho^{4}-q\,\text{log}(\rho)\hskip
113.81102pt\text{for}\quad 0\leq\rho\leq\frac{1}{2}$ (3.4) $\displaystyle=$
$\displaystyle-\frac{a}{2}\rho^{2}-\frac{b}{2}\rho^{4}-q\,\text{log}(\rho)-\frac{1}{4}\text{log}[2(1-\rho)]+\frac{1}{8}\hskip
22.76228pt\text{for}\quad\frac{1}{2}\leq\rho\leq 1.$ (3.5)

For the $\rho>1/2$ region, the behaviour of the parameters $a(T)$ and $b(T)$
can be computed for $T\geq T_{\rm cr}$ regime [7, 12]. The results are shown
in the Fig. 4. This of course requires the usage of bulk equation of state
(see the Appendix-B for details).

(a)  (b)

Figure 4: Case of $(a,b)$ phenomenological matrix model at non-zero chemical
potential (in the canonical ensemble): Plots show the temperature dependence
of the parameters $a(T)$ and $b(T)$ for $T\geq T_{\rm
cr}=\frac{4\sqrt{3}}{5\pi}$. Red dots denote the data points, whose fits are
represented by blue colored curves. The fitting curves are,
$a(T)=c_{1}T+c_{2}$, and$b(T)=c_{3}T+c_{4}$.
$(\text{Note:}\,c_{1}=-0.641573,c_{2}=0.611924,c_{3}=14.2782,c_{4}=-5.64198)$.

The fitting curves for $a(T)$ and $b(T)$ can now be used to solve the saddle
point equations 3.3 for temperature, giving:

$T(\rho,q)=\frac{4q(1-\rho)+8c_{4}(1-\rho)\rho^{4}+4c_{2}(1-\rho)\rho^{2}-\rho\,}{4(\rho-1)\rho^{2}(c_{1}+2c_{3}\rho^{2})}.$
(3.6)

The behaviour of the above expression extracted from curve fitting, matches
qualitatively with the known bulk formula, as is evident from fig. 5a. The
critical point can also be shown to lie at $(T_{\rm cr},\,q_{\rm
cr},\,\rho_{\rm
cr})=(\frac{4\sqrt{3}}{5\pi},\,\frac{1}{3\sqrt{15}},\,0.546293)$.

(a)

(b)

(c)

Figure 5: For the $(a,b)$ matrix model with finite chemical potential (in
canonical ensemble): (a) Plot of equation of state $T(\rho,q)$ for $q\leq
q_{\rm cr}$. From top to bottom, the charge increases. Dashed curve stands for
the case $q=q_{\rm cr}$ with the red dot denoting the critical point; Plots in
(b) and (c) are for the potential $V(\rho)$ for the cases $q=q_{\rm cr}$ (and
$T=T_{\rm cr}$) and for $q<q_{\rm cr}$, respectively.

Let us now summarize how the effective potential $V(\rho)$ at a fixed charge
$q$ characterizes the phases for various temperatures, particularly looking at
figures Fig. 5b, and Fig. 5c. The stable/unstable saddle points of $V(\rho)$
(for $\rho>1/2$) denote the corresponding stable/unstable black hole solutions
in the bulk (cf. Fig. 8 in Appendix-B). In the regime $q<q_{\rm cr}$ at
certain low temperatures, such as, $T<T_{01}$, one sees the presence of only
one saddle point, which can be interpreted as the small black hole in the
bulk. A raise in temperature to $T_{01}$ results in the nucleation of two
saddle points which in the bulk are the unstable intermediate and the stable
large black hole. When the temperature is raised further, such that,
$T_{01}<T<T_{02}$, there exist three saddle points (corresponding to small,
intermediate and large black holes in the bulk). As can be seen in the
aforementioned figures, for $T=T_{02}$, two of the saddle points merge
(namely, the merging of small and intermediate black hole branches). Finally,
for any temperature $T>T_{02}$, there is always a single saddle point,
representing the stable large black hole of the bulk. There is also an
interesting critical point at $q=q_{\rm cr}$, where all the three saddle
points coalesce and with this, the phase structure exhibited by the effective
potential $V(\rho)$ reproduces the known results from the bulk for $q\leq
q_{\rm cr}$ 111 One can refer [7], for further details of the phase
structure.. We should add that $\rho=0$ is not a solution in the non-zero
charge case. Since, close to the critical region (i.e., $q\leq q_{\rm cr}$),
the effective potential $V(\rho)$ reproduces the nature of the off-shell bulk
free energy, we utilize it to obtain other thermodynamic quantities as
follows. The on-shell free energy $F$, can be computed from the fitting curves
for $a(T)$, $b(T)$. This is practically done by substituting $T(\rho,q)$ in
the effective potential $V(\rho)$ in equation (3.5), to obtain

$F=-\frac{\rho^{2}}{2}(c_{1}T+c_{2})-\frac{\rho^{4}}{2}(c_{3}T+c_{4})-q\text{log}(\rho)-\frac{1}{4}\text{log}[2(1-\rho)]+\frac{1}{8}\equiv
E-TS,$ (3.7)

where similar to the situation in the bulk, we assume that the free energy
obeys $dF=\mu dQ-SdT$ 222Here, we have replaced $q$ with the physical charge
$Q$, using eqn. (B.5).. Now, the computation of entropy $S$ of the CFT results
in

$S=-\frac{\partial F}{\partial T}\Big{|}_{\rm Q}=-\frac{\partial
F/\partial\rho}{\partial T/\partial\rho}\Big{|}_{\rm
Q}=\frac{1}{2}\rho^{2}(c_{1}+c_{3}\rho^{2}).$ (3.8)

Also, the energy $E$ of the CFT is

$\displaystyle E$ $\displaystyle=$ $\displaystyle F+TS$ (3.9) $\displaystyle=$
$\displaystyle\frac{1}{24}\Big{(}3-12\rho^{2}(c_{2}+c_{4}\rho^{2})-6\text{log}[2(1-\rho)]-2\sqrt{3}Q\text{log}(\rho)\Big{)},$

where we assume that it satisfies the first law: $dE=TdS+\mu dQ$. The chemical
potential $\mu$ corresponding to the charge is

$\mu=\frac{\partial E}{\partial Q}\Big{|}_{\rm
S}=-\frac{\text{log}(\rho)}{4\sqrt{3}}.$ (3.10)

###  3.2 Cardy-Verlinde formula

It is now possible to derive the Cardy-Verlinde (CV) formula following the set
up in [2]. Taking the scaling of the boundary metric as in eqn. (2.11), the
energy $\bar{E}$, temperature $\bar{T}$, chemical potential $\bar{\mu}$,
charge $\bar{Q}$, and the entropy $\bar{S}$ of the CFT are scaled as:

$\displaystyle\bar{E}$ $\displaystyle=$ $\displaystyle\frac{l}{R}E,$ (3.11)
$\displaystyle\bar{T}$ $\displaystyle=$ $\displaystyle\frac{l}{R}T,$ (3.12)
$\displaystyle\bar{\mu}$ $\displaystyle=$ $\displaystyle\frac{l}{R}\mu,$
(3.13) $\displaystyle\bar{Q}$ $\displaystyle=$ $\displaystyle Q,$ (3.14)
$\displaystyle\bar{S}$ $\displaystyle=$ $\displaystyle S.$ (3.15)

These satisfy the first law:
$d\bar{E}=\bar{T}d\bar{S}+\bar{\mu}d\bar{Q}-\bar{P}d\bar{V}$. The pressure
$\bar{P}$, conjugate to the volume $\bar{V}=\omega_{3}R^{3}$, is given by

$\displaystyle\bar{P}$ $\displaystyle=$
$\displaystyle-\frac{\partial\bar{E}}{\partial\bar{V}}\Big{|}_{\rm\bar{S},\bar{Q}}=-\frac{\partial\bar{E}/\partial
R}{\partial\bar{V}/\partial R}\Big{|}_{\rm\bar{S},\bar{Q}}$ (3.16)
$\displaystyle=$
$\displaystyle\frac{1}{72R^{4}}\Big{(}3-12\rho^{2}(c_{2}+c_{4}\rho^{2})-6\text{log}[2(1-\rho)]-2\sqrt{3}Q\text{log}(\rho)\Big{)}.$

We now define the Casimir energy $\bar{E}_{c}$ (as the violation of the Euler
identity [1, 2]), and the electric potential energy $\bar{E}_{Q}$ [2], as:

$\displaystyle\bar{E}_{c}$ $\displaystyle\equiv$ $\displaystyle
3(\bar{E}+\bar{P}\bar{V}-\bar{T}\bar{S}-\bar{\mu}\bar{Q}),$ (3.17)
$\displaystyle\bar{E}_{Q}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\bar{\mu}\bar{Q}.$ (3.18)

Figure 6: For $(a,b)$ matrix model with non-zero chemical potential: Red curve
is for the LHS and the blue curve is for the RHS of the equation (3.19). Here,
we used $Q=0.1$.

Now, as shown in the Fig. 6, one can numerically verify the Cardy-Verlinde
formula [2]:

$\bar{S}=\frac{2\pi
R}{3}\sqrt{\bar{E}_{c}[2(\bar{E}-\bar{E}_{Q})-\bar{E}_{c}]}.$ (3.19)

##  4 Conclusion

To conclude, in this work, we made an attempt to verify the Cardy-Verlinde
(CV) formula directly working in the boundary gauge theory. Our computations
were facilitated by the phenomenological matrix model proposed to be an
effective model for the strongly coupled $\mathcal{N}=4$, $SU(N)$ gauge theory
at finite temperature and chemical potential at large $N$. Our results are
summarized in figures 3 and 6. While for small values of $\rho$, we find that
the CV formula is satisfied, there are discrepancies when $\rho$ approaches
one. Though the reasons are not immediately obvious to us, it could be because
of truncating the model to contain terms with only a few lower powers of
$\rho$ with possible statistical errors. With the inclusion of higher
curvature terms in the Einstein action and in the cases involving arbitrary
horizon topologies, interesting modifications of the CV formula have been
noted [2]. Using known methods to include such corrections in the boundary
matrix models [8, 9, 10], it should be a good exercise to check the validity
of CV formula in more general situations.

## Acknowledgements

One of us (C.B.) thanks the DST (SERB), Government of India, for financial
support through the Mathematical Research Impact Centric Support (MATRICS)
grant no. MTR/2020/000135. We thank IIT Bombay for warm hospitality during the
Indian Strings Meeting 2023, where this work drew to a close.

## Appendices

In the following two appendices, we summarize the procedure for obtaining the
off-shell free energy of Schwarzschild-AdS and the Reissner–Nordstrom-AdS
black holes, and also present their phase structure, which will be helpful to
refer to, when comparing with the phase structure shown by boundary matrix
model in the main text.

##  Appendix A Schwarzschild-AdS black holes

Staring point is the $5$-dimensional Schwarzschild-AdS black holes in the
bulk, with the action given below, followed by the line element [16]:

$I=-\frac{1}{16{\pi}G}\int d^{5}x\sqrt{-g}\left[R+\frac{12}{l^{2}}\right],$
(A.1) $ds^{2}=-V(r)dt^{2}+\frac{dr^{2}}{V(r)}+r^{2}d{\Omega}^{2}_{3},$ (A.2)

where $G$ denotes the Newton’s constant, $d{\Omega}^{2}_{3}$ is the metric on
the unit $3$-sphere $S^{3}$ with volume $\omega_{3}$. The function $V(r)$
takes the form

$V(r)=1-\frac{m}{r^{2}}+\frac{r^{2}}{l^{2}}\,$ (A.3)

with $l$ as the AdS length and $m$ related to the ADM mass $M$ of the hole.
The usual thermodynamic quantities, such as, energy $E$, temperature $T$,
entropy $S$, and finally the free energy $F$ of the hole can be written as a
function of the horizon radius $r_{+}$ as:

$\displaystyle E$ $\displaystyle=$ $\displaystyle M=\frac{3\omega_{3}}{16\pi
G}m$ (A.4) $\displaystyle=$ $\displaystyle\frac{3\omega_{3}}{16\pi
G}\big{(}r_{+}^{2}+\frac{r_{+}^{4}}{l^{2}}\big{)},$ $\displaystyle T$
$\displaystyle=$ $\displaystyle\frac{l^{2}+2r_{+}^{2}}{2\pi l^{2}r_{+}},$
(A.5) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\omega_{3}}{4G}r_{+}^{3},$ (A.6) $\displaystyle F$
$\displaystyle=$ $\displaystyle M-TS.$ (A.7)

The above quantities can be shown to satisfy the first law of thermodynamics:
$dE=TdS$, with the free energy obeying $dF=-SdT$. From the equation of state
i.e., the plot of temperature in Fig. 7a, one deduces the existence of a
minimum temperature $T_{\rm min}$, beyond which there can be two black holes,
known as, the small $(r_{+}<r_{\rm min})$ and large $(r_{+}>r_{\rm min})$. The
small black hole is locally unstable (with negative specific heat), and the
larger counterpart is stable. A key point in the phase structure is at a
different temperature $T_{\rm HP}>T_{\rm min}$, which corresponds to the
Hawking-Page (HP) transition, beyond which the large black hole branch is
preferred over othre phases.

(a)

(b)

Figure 7: Schwarzschild black holes in $\text{AdS}_{5}$: (a) Temperature $T$
plotted in terms of the horizon radius $r_{+}$, showing the existence of small
$(r_{+}<r_{\rm min})$ and large $(r_{+}>r_{\rm min})$ black hole branches for
$T>T_{\rm min}$. The Hawking-Page (HP) transition is seen at $T=T_{\rm HP}$.
(b) This plot shows the nature of Bragg-Williams free energy $f$ with respect
to the horizon radius $r_{+}$ at various temperatures $T$. Temperature of the
curves decreases from the bottom to the top. The blue curve is at $T_{\rm
min}$ and the dashed red curve corresponds to the temperature $T_{\rm HP}$.
Also, $(r_{\rm min},T_{\rm min})=(\frac{1}{\sqrt{2}},\frac{\sqrt{2}}{\pi})$
and $(r_{\rm HP},T_{\rm HP})=(1,\frac{3}{2\pi})$.

The HP phase transition can be understood straightforwardly with the
construction of Bragg-Williams (BW) off-shell free energy $f$ 333 Through out
the manuscript, we work with the units $\hbar=c=1$, and we set AdS length $l$,
volume of the three sphere $\omega_{3}$, and $16\pi G$ to 1. as [17]:

$f(r_{+},T)=M-TS=3r_{+}^{2}(1+r_{+}^{2})-4\pi r_{+}^{3}T.$ (A.8)

Here, the horizon radius $r_{+}$ is treated as an order parameter and the
temperature $T$ as an external parameter. The nature of the free energy $f$
for various temperatures can be inferred from Fig. 7b. The minima of the free
energy $f$, represent the locally stable large black hole phase, and the
maxima represent the locally unstable small black hole phase, while the AdS
space $(r_{+}=0)$ corresponds to the zero of free energy. The HP transition
happens when there is a degenerate minima that satisfies the two conditions
$(f=0\,\text{and}\,\frac{\partial f}{\partial r_{+}}=0)$. Further details of
the phase structure can be found in [18, 17].

##  Appendix B Reissner–Nordstrom-AdS black holes

Let us now consider charged black holes, namely, the Reissner–Nordstrom-AdS
black holes in $5$-dimensions in the canonical (i.e., fixed charge) ensemble.
The action and the line element are known to be [19, 20]:

$I=-\frac{1}{16{\pi}G}\int
d^{5}x\sqrt{-g}\left[R-F^{2}+\frac{12}{l^{2}}\right],$ (B.1)
$ds^{2}=-V(r)dt^{2}+\frac{dr^{2}}{V(r)}+r^{2}d{\Omega}^{2}_{3},$ (B.2)

with $V(r)$ and gauge potential noted to be:

$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle
1-\frac{m}{r^{2}}+\frac{q^{2}}{r^{4}}+\frac{r^{2}}{l^{2}},$ (B.3)
$\displaystyle A$ $\displaystyle=$ $\displaystyle\left(-{1\over c}{q\over
r^{2}}+\mu\right)dt.$ (B.4)

We get $c=2/\sqrt{3}$, and the parameter $q$ is related to the black hole’s
charge as

$Q=2\sqrt{3}\left({\omega_{3}\over 8\pi G}\right)q.$ (B.5)

The gauge potential is picked to vanish on the horizon at $r=r_{+}$, which of
course fixes the chemical potential

$\mu={1\over c}{q\over r_{+}^{2}}.$ (B.6)

This shows the electrostatic potential difference between the horizon and
infinity. The standard thermodynamic quantities such as, energy $E$,
temperature $T$, entropy $S$, and the free energy $F$ receive additional
corrections due to the presence of charge and read as:

$\displaystyle E$ $\displaystyle=$ $\displaystyle M=\frac{3\omega_{3}}{16\pi
G}m$ (B.7) $\displaystyle=$ $\displaystyle\frac{3\omega_{3}}{16\pi
G}\big{(}r_{+}^{2}+\frac{q^{2}}{r_{+}^{2}}+\frac{r_{+}^{4}}{l^{2}}\big{)},$
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{2r_{+}^{6}+l^{2}r_{+}^{4}-q^{2}l^{2}}{2\pi
l^{2}r_{+}^{5}},$ (B.8) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\omega_{3}}{4G}r_{+}^{3},$ (B.9) $\displaystyle F$
$\displaystyle=$ $\displaystyle M-TS.$ (B.10)

These once again follow the first law: $dE=TdS+\mu dQ$, and the free energy
satisfies $dF=\mu dQ-SdT$. Thermodynamics and phase transitions of charged
black holes in AdS have a rich history, whose phase structure closely follows
the van der Waals fluid model [19, 20, 21], which is captured by the equation
of state $T(r_{+},q)$, shown in Fig. 8a. There is a special critical charge
$q_{\rm cr}$ and temperature $T_{\rm cr}$, above which the black holes enter a
unique stable phase. Below this critical limit, the system can exist in any of
the three phases (namely, the small, intermediate, and the large black hole
branches). The small and large black hole branches are locally stable (due to
positive specific heat), but the intermediate branch is unstable. There is of
course a first order transition between the small black holes and the large
black holes, terminating in a second order critical point, which is a point of
inflection, obtained generally as

$\frac{\partial T}{\partial
r_{+}}=0,\quad\text{and},\quad\frac{\partial^{2}T}{\partial r_{+}^{2}}=0.$
(B.11)

This gives the critical point as $(T_{\rm cr},\,r_{\rm cr},\,q_{\rm
cr})=(\frac{4\sqrt{3}}{5\pi},\frac{1}{\sqrt{3}},\frac{1}{3\sqrt{15}})$.

(a)

(b)

(c)

Figure 8: The Reissner-Nordstrom black hole phases in $\text{AdS}_{5}$ (in the
canonical ensemble): (a) Plots of equation of state, i.e., $T(r_{+},q)$ for
various charges $q$, inferring the presence of three black hole branches
(i.e., the small, the intermediate, and the large) for $q<q_{\rm cr}$, with
one branch for $q>q_{\rm cr}$. Charge of the curves decreases from bottom to
the top. Dashed curve is plotted for $q=q_{\rm cr}$. Red dot corresponds to
the critical point; Off-shell free energy $f$ is shown for two cases, namely,
(b) $q=q_{\rm cr}$, and $T=T_{\rm cr}$. (c ) for $q<q_{\rm cr}$. The inset
plot shows the corresponding temperatures.

Now, we can construct the Bragg-Williams off-shell free energy $f$ as in
Appendix-A, where, in addition to the temperature, the charge is also an
external parameter. Horizon radius $r_{+}$ continues to be the order parameter
as before, with the free energy found to be [20, 22]:

$f=M-TS=\frac{3}{r_{+}^{2}}(r_{+}^{6}+r_{+}^{4}+q^{2})-4\pi r_{+}^{3}T.$
(B.12)

The phase structure of these charged black holes can now be obtained easily by
studying the free energy, whose extremal points precisely correspond to the
black hole solutions themselves. For instance, the minima (maxima) stand for
stable (unstable) phases. For $q>q_{\rm cr}$ and at any temperature $T$,
existence of a single extremal point indicates the presence of the stable
large black hole phase. In addition, for $q=q_{\rm cr}$ and $T=T_{\rm cr}$,
the three extremal points merge and signal the critical point, as shown in
Fig. 8b. Further, for $q<q_{\rm cr}$ and in the temperature range
$T_{01}<T<T_{02}$, free energy $f$ contains three extremal points representing
the small, intermediate, and large black holes branches (as also emphasized in
the Fig. 8c). One can refer to [19, 20, 22], for further details of the phase
structure, which our construction reproduces.

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